Thickening and-Or Thinning Upward Patterns in Sequences of Strata - Tests of Significance - Sedimentology, Harper, 1998

Thickening and/or thinning upward patterns in sequencesof strata: tests of signi®cance

CHARLES W. HARPER JRSchool of Geology and Geophysics, University of Oklahoma, Norman, OK 73019, USA

INTRODUCTION

Geologists often characterize a sequence of strataas exhibiting one or more thickening and/orthinning upward sequences (e.g. Pickering et al.,1989; Figs 1±5). They may postulate that thesequence as a whole exhibits a single thickening(or thinning) upward pattern. More commonly,

they recognize two or more thickening (or thin-ning) upward subsequences within the sequence,or a combination of thickening and thinningupward patterns.

Alleged thickening and thinning upward pat-terns are especially important in turbidite studies,where models in vogue interpret thickeningupward sandstone sequences (in a sequence of

ABSTRACT

Geologists commonly purport that successions of strata show one or more thickening

and/or thinning upward trends, often prompting colleagues to argue that the `trends'

are subjectively identi®ed, unproven or nonexistent. Parametric and randomization

tests are proposed to evaluate the null hypothesis of random succession against a

variety of alternative postulates of trend. In place of test statistics in vogue that merely

compare each bed thickness with that of the beds immediately above and below it, test

statistics based on Kendall's S and Tau that make sequence-wide (or subsequence-

wide) comparisons of bed thicknesses are advocated. The test statistic used and the

exact form of the test depends on the alternative model considered: against the

alternative of a single thickening (and/or thinning) upward trend, Kendall's S or

equivalently Kendall's Tau are recommended. These statistics make pair-wise

comparisons of beds, comparing bed thicknesses with their positions in the vertical

sequence. Against the alternative of trends in g subsequences recognized a-priori, e.g.

those separated by breaks such as thick sequences of hemipelagic shale, test statistics

proposed include: the weighted sum of the g Tau coef®cients calculated for the

individual subsequences (if subsequences are alleged to be all thickening or all

thinning upward), and the weighted sum of the absolute value, or square, of the Tau

coef®cients (if subsequences are alleged to include both thickening and thinning

upward patterns). Tests can indicate that a sequence has one or more subsequences

which are nonrandom, but it will not indicate which. To test each subsequence for

signi®cance, test each of g subsequences at a level of signi®cance � a/g, thus

achieving an overall, sequence-wide, level of signi®cance � a. Against the alternative

g subsequences recognized post-hoc, i.e. purely on the basis of observed thickness

patterns, a family of test statistics are proposed, each equal to the maximum value of

the appropriate test statistic (de®ned for subsequences recognized a-priori) that is

attainable by partitioning the total sequence of beds into 1, 2,¼. up to g subsequences.

Both same-type and mixed subsequences alternatives arise. Each test proposed is

applied to several different sequences, mostly turbidites.

Sedimentology (1998) 45, 657±696

Ó 1998 International Association of Sedimentologists 657

658 C. W. Harper

Ó 1998 International Association of Sedimentologists, Sedimentology, 45, 657±696

alternating sands and shales) as formed by pro-grading delta lobes and thinning upward se-quences as formed by gradual channel ®llingand abandonment (Hiscott, 1981; Mutti, 1984;Walker, 1984; Shanmugam & Moiola, 1988; Picke-ring et al., 1989). Aggradational lobes are charac-terized by a lack of well de®ned trends in bedthickness (Shanmugam & Moiola, 1988). Picke-ring et al. (1989; p. 7) referred to the associationof thinning-upward and thickening-upward se-quences in submarine-fan-successions with orig-inal depositional setting (e.g. fan channels vs. fanlobes) as à powerful concept that developedduring the 1970s'. However, they added that theassociation has been questioned because `thesequences have not been rigorously de®ned usingstatistical techniques, and are subjectively identi-®ed to the extent that there is little agreementamong researchers as to their validity.'

The problem with attributing signi®cance to`patterns' of thickening (and/or thinning) upwardsequences or subsequences is that apparent pat-terns can easily appear even if the succession ofbed thicknesses is random (Hiscott, 1981). Forexample, while Moiola & Shanmugam (1984)initially interpreted ¯ysch deposits in the Jack-fork Group (Mississippian), Oklahoma and Ar-kansas as turbidites, recognizing thickening andthinning upward patterns (Fig. 2l,m,n); they latershow the sand beds to be predominantly debris¯ow and slump deposits, and so plausibly attrib-ute the thickness `trends' to `chance occurrences

of random debris ¯ows and slumps' (Shanmugam& Moiola, 1995).

This paper discusses ways of testing the nullhypothesis of random succession against variousalternatives. For testing against the alternativeof one or more thickening (thinning) upwardpatterns, this paper argues against using nearest-neighbour methods in favour of tests based onKendall's S and Tau. First, tests against thealternative of a single thickening (and/or thin-ning) upward trend within a sequence are sug-gested. Then, the more dif®cult problem of testingagainst the alternative that one or more thicken-ing and/or thinning upward subsequences areembedded in a sequence is investigated. Thelatter problem is less intractable if the subse-quences are recognizable a-priori as separated byrecognizable breaks (e.g. erosion surfaces, thicksequences of hemipelagic shale). This case isconsidered, ®rst against an alternative of same-type subsequences (one or more thickening, orone or more thinning upward subsequences), andthen against the alternative of mixed subsequenc-es (i.e. a combination of thickening and thinningupward subsequences, including symmetric se-quences as a special case). Finally, the case whereone or more thickening and/or thinning upwardsubsequences are purported to be recognizedposthoc, that is to say purely on the basis ofobserved thickness patterns, is considered. Herealso the alternative of same-type subsequences isconsidered ®rst followed by that of mixed subse-quences.

Approaches designed to evaluate for thickeningor thinning upwards could be used to assessalleged coarsening or ®ning upward patterns (e.g.those shown by Steel & Gloppen, 1980; Steel &Thompson, 1983, their Figs 8 & 14; see also refs.cited in Graham, 1988; p. 62). In turbidites,thickening upward sequences appear to coarsenupward, and thinning upward sequences to ®neupward (Potter & Scheidegger, 1966; Sadler,1982). The approaches discussed here might beused also in other contexts. For example, Duller &Floyd (1995) plotted chemical composition (10major elements, 19 trace elements) for 861 suc-cessive Ordovician-Silurian greywacke samplesfrom Scotland; Brant & Elias (1989) argued forglobal increasing and decreasing trends in max-imum tempestite thickness for latest Precambrianto the Recent; Steel & Thompson (1983) plottedmatrix percentage estimated at outcrop of braidedstream conglomerates; Dec (1992) plotted maxi-mum particle size for Devonian alluvial fandeposits (see also Fig. 2o, p, 3e, 5a, b).

Fig. 1. Bar charts showing bed thicknesses in strati-graphic sequences. On each chart, beds are plotted ashorizontal bars with oldest at bottom to youngest at top.Bar widths indicate bed thickness (as percentage ofmaximum thickness). Tables to right of each chart showvalues of Rij (see de®nition Table 2). Rightmost columngives row totals. For further discussion see text. Se-quence descriptions and test results are in Table 3A,3B, 7A (Table numbers are given in brackets below). (a)Turbidite Sandstones, San Salvatore Sandstone, BobbioFormation (Miocene), Apennines, Italy. Mutti, 1974,Fig. 5, Section 13 (Table 3A). (b) Hypothetical se-quence (Table 7A). (c) Hypothetical sequence. Ran-domized succession of bed thicknesses shown in (b)(Table 3A). (d) Hypothetical sequence (Tables 3A, 3Band 7A). (e) Turbidite Sandstones, Tourelle Formation(Ordovician), QueÂbec (Hiscott, 1980; Fig. 15F) (Ta-bles 3A and 3B). (f) Same sequence as that shown in(b). Two subsequences. For further discussion see text.(g) Same sequence as that shown in (c). Two subse-quences. For further discussion see text. (h) Same se-quence as that shown in (d). Three subsequences. Forfurther discussion see text.

Thickening/thinning: Tests of signi®cance 659


Fig. 2. Bar charts showing bed thicknesses (fan ratio or maximum particle size where indicated) in stratigraphic sequences. Oneach chart, beds are plotted as horizontal bars with oldest at bottom to youngest at top. Bar widths indicate bed thickness (fanratio or maximum particle size) as percentage of maximum observed. Number of beds in a sequence is shown at top of plot.Sequence descriptions and test results are given in Table 3A. (a) Turbidite sandstones, Tourelle Formation (Ordovician),Quebec (Hiscott, 1980; Fig. 15j). (b) Turbidite sandstones, Tourelle Formation (Ordovician), Quebec (Hiscott, 1980; Fig. 13,col. 2). (c) Turbidite sandstones, Tourelle Formation (Ordovician), Quebec (Hiscott, 1980; Fig. 15e). (d) Turbidite sandstones,Tourelle Formation (Ordovician), QueÂbec (Hiscott, 1980; Fig. 8, col. 3). (e) Turbidite sandstones, Tourelle Formation (Ordo-vician), Quebec (Hiscott, 1980; Fig. 15L). (f) Turbidite sandstones, Carboniferous Flysch, Southern Morocco (Graham, 1982Fig. 3). (g) Turbidite sandstones, Carboniferous Flysch, Southern Morocco (Graham, 1982, Fig. 4; 1988, Fig. 2á53; beds 194±243;selected out of 298 beds sequence shown in Fig. 4(a) herein as an example of an obvious thickening upward sequence). (h)Turbidite sandstones, Shimanto Group (Jurassic-Cretaceous), Oigawa District, Japan (Kimura, 1966; Fig. 19, Loc. 119). (i)Turbidite sandstones, Beds 96±143 from sequence shown in Fig. 4(a) herein with sequence of beds arranged to show obviousthinning upward sequence. (j) Turbidite sandstones, beds shown in (g) herein with sequence of beds arranged to show obviousthickening upward sequence. (k) Turbidite sandstones, Antola Formation (Upper Cretaceous), Northern Apinnines, Italy(Martini, Sagri & Doveton, 1978; Fig. 8, S. Donato Section). (l) Turbidite sandstones (or possibly debris ¯ow and slumpdeposits). Jackfork Formation (Upper Mississippian), DeGray Dam, Arkansas (Moiola & Shanmugam, 1984; Fig. 5. (m) Turbiditesandstones (or possibly debris ¯ow and slump deposits). Jackfork Formation (Upper Mississippian), DeGray Dam, Arkansas(Moiola & Shanmugam, 1984; Fig. 6). (n) Turbidite sandstones (or possibly debris ¯ow and slump deposits). Jackfork Formation(Upper Mississippian), DeGray Dam, Arkansas (Moiola & Shanmugam, 1984; Fig. 7). (o) Fan Ratio (� sand-mud ratio ´ degreeof contrast ´ maximum clast size); Fluvial and Lacustrine Sandstones, Bechsrieth Formation (Upper Carboniferous), SouthernGermany (Dill, 1992; Fig. 4). (p) Maximum Particle size (mean value of 10 largest clasts), Devonian alluvial fan conglomerates,Hornelen Basin, Norway (Steel et al., 1977; Fig. 3, Fan Cycle A).

660 C. W. Harper


Fig. 3. Bar charts showing bed thicknesses (or Wt.% CaCO3 where indicated) in stratigraphic sequences. On eachchart, beds are plotted as horizontal bars with oldest at bottom to youngest at top. Bar widths indicate bed thickness(or Wt.% CaCO3) as percentage of maximum observed. Number of beds in a sequence is shown at top of plot. Wheresequence is partitioned into subsequences by breaks, partitions are marked by wide gaps with dashed lines. Sub-sequences are numbered in square brackets. Sequence descriptions and test results are given mainly in Table 5A. Thetable(s) where each sequence is analysed are given in brackets below. (a) Alternating Channellized and Non-Chan-nellized cycles, Turbidite sandstones, Bullfrog fan, Mineral King roof pendant, Sierra Nevada, California (Busby-Spera, 1985; Fig. 7, Cycles 18±26). [Table 5A]. (b) Hemipelagite beds in alternating mud turbidite/hemipelagite bedsequence (turbidite thicknesses shown in Fig. 3C). Quaternary, South Shetland Trench, West Antarctica (Porebskiet al., 1991, Fig. 23). [Table 3A, 5A]. (c) Mud turbidite beds in alternating mud turbidite/hemipelagite bed sequence.Quaternary, South Shetland Trench, West Antarctica (Porebski et al., 1991, Fig. 23). [Table 3A, 5A]. (d) Turbiditesandstones, Macigno Fm. (Oligocene), Northern Apennines, Italy (Gibaudo, 1980, Fig. 5; Hiscott, 1981; Fig. 1).[Tables 3A, 3B, 5A, 5B and 7A]. (e) Weight Percent Calcite, Pelagic limestones and marlstones. Tropic Shale (UpperCretaceous), Sit Down Bench, Utah (Sethi & Leithold, 1994; Fig. 4). [Table 3B, 5A and 7A].



TESTS OF SIGNIFICANCE BASED ONNEAREST-NEIGHBOUR COMPARISONS

Nearest-neighbour comparison tests

Many of the tests used by geologists to evaluate anull hypothesis of random succession use a teststatistic that merely compares each bed thicknesswith that of the beds immediately above andbelow it. Tests statistics include: numbers ofturning points; the number of runs up or down;numbers of runs up or down of length N betweenturning points; length of the longest run; numberof positive vs. negative differences or ties betweenadjacent beds; the sum of squares of the differen-ces in thickness of adjacent beds normalized by

the sum of squares of deviations of thicknessesfrom the mean (Durbin-Watson test); and thecorrelation coef®cient between pairs of consecu-tive bed thicknesses. Tests based on these teststatistics are described in Appendix 3.

Nearest neighbour tests are useful in Markovchain analysis of vertical successions of facies(Hiscott, 1980; Harper, 1984a, b). Markov analysismay be applied to bed thickness data if discretelithofacies can be recognized, and if each facieshappens to have a characteristic range of bedthickness (Hiscott, 1980; Schweller et al., 1987;Pickering et al., 1989).

Sadler et al. (1993) and Drummond & Wilkin-son (1993) effectively use nearest-neighbour teststo analyse cycle-thickness in successions of

Fig. 4. Bar charts showing bedthicknesses in stratigraphic se-quences. On each chart, beds areplotted as horizontal bars with old-est at bottom to youngest at top. Barwidths indicate thickness (as per-centage of maximum thickness).Number of beds in a sequence isshown at top of plot. Where se-quence is partitioned into subse-quences by breaks, partitions aremarked by wide gaps with dashedlines. Subsequences are numberedin square brackets. Sequence de-scriptions and test results are givenmainly in Tables 5B and 5C. Thetable(s) where each sequence is an-alysed are given in brackets below.(a) Turbidite sandstones, Carbonif-erous Flysch, Southern Morocco(Graham, 1982; Fig. 4; 1988,Fig. 2á53. [Tables 3A, 5B, 5C and7B]. (b) Turbidite sandstones,Dezeadeash Fm. (Jurassic-Creta-ceous), Yukon, Canada (Lowey,1980; Table pp. 116±149; Lowey,1992) [Table 5B].

662 C. W. Harper


Fig. 5. Bar charts showing bed thicknesses (or other variables) in stratigraphic sequences. On each chart, beds are plotted ashorizontal bars with oldest at bottom to youngest at top. Bar widths indicate bed thickness (dO18 (%%) calcium carbonate ormaximum particle size) as percentage of maximum variable size. Number of beds in a sequence is shown at top of plot.Sequence descriptions and test results are given mainly in Table 7A or 7B. The table(s) where each sequence is analysed aregiven in brackets below. (a) dO18 (%%) Calcium Carbonate, Annual layers of Glacial Ice, Eemian (135 000±115 000 y.b.p.)portion of the Greenland Ice-core Project Summit Ice core, Central Greenland (Greenland Ice-core Project (GRIP) members 1993)[Table 7A]. (b) Maximum Particle size (mean value of 10 largest clasts), Devonian alluvial fan conglomerates, Hornelen Basin,Norway (Steel et al. (1977; Fig. 3, Fan Cycle C) [Table 7A]. (c) Turbidite sandstones, San Salvatore Sandstone (Miocene),Northern Apinnines, Italy. (Mutti, 1974; Fig. 5 ± Section 2) [Tables 3B and 7A]. (d) Turbidite sandstones, San SalvatoreSandstone (Miocene), Northern Apinnines, Italy. (Mutti, 1974 Fig. 5 ± Section 12) [Tables 3B and 7A]. (e) Turbidite sandstones,Carbonate fan deposits, Loma del Toril Fm. (Upper Jurassic), Spain (Ruiz-Ortiz, 1983; Fig. 5, megasequences IV±VIII). [Ta-bles 3B and 7A]. (f) Turbidite sandstones, Murca Formation (Lower Cretaceous) submarine fan deposits, Bogata Trough.Pimpirev & Sarmiento, 1993, Fig. 6(c). [Tables 3B and 7A]. (g) Turbidite sandstones, Macigno Fm. (Oligocene), NorthernApennines, Italy (Gibaudo, 1980, Fig. 7, Level 10; Hiscott, 1981; Fig. 2) [Tables 3B and 7A]. (h) Turbidite sandstones, MacignoFm. (Oligocene), Northern Apennines, Italy (Gibaudo, 1980, Fig. 9, Level 6) [Tables 3B and 7A]. (i) Turbidite sandstones,Macigno Fm. (Oligocene), Northern Apennines, Italy (Gibaudo, 1980, Fig. 9, Level 4) [Tables 3B and 7A]. (j) Turbidite sand-stones, Macigno Fm. (Oligocene), Northern Apennines, Italy (Gibaudo, 1980, Fig. 9 Level 15) [Tables 3B and 7A]. (k) Turbiditesandstones, Macigno Fm. (Oligocene), Northern Apennines, Italy (Gibaudo, 1980, Fig. 7 Level 45) [Tables 3B and 7B]. (l)Turbidite sandstones, Macigno Fm. (Oligocene), Northern Apennines, Italy (Gibaudo, 1980. Fig. 11, Levels 17±18; Hiscott,1981; Fig. 3) [Table 7B]. (m) Turbidite sandstones, Las Tortulas Formation (Devonian-Carboniferous), Chilean Andes (Bahlburg& Breitkreuz, 1993; Fig. 5b). [Table 7B].



shallowing upward cycles in shallow marinecarbonate platform environments using `Fischerplots'. (To avoid confusion, note that the thick-nesses of each cycle are measured not thethickness of beds within cycles). First Sadleret al. and Drummond & Wilkinson divide cyclesin a succession into two categories: those with athickness (or log (thickness)) greater than themean cycle thickness for the succession and thoseless than it. Then Sadler et al. and Drummond &Wilkinson test for runs (groupings) of thicker thanaverage and thinner than average cycles due tothe interaction of eustatic sea-level ¯uctuationand basin subsidence on rates of accommodationspace creation (Drummond & Wilkinson, 1993). Atendency for both thick and thin cycles to beclustered in runs would typify cycles generatedduring long-term change in sea-level or subsi-dence. An excessive alternation of thick and thincycles might result from cycle thickness datawhich have a strongly hierarchical structure(Drummond & Wilkinson, 1993; See Appendix 3).

Problems with using nearest-neighbourcomparisons as a test against thickening orthinning upward trends

While `nearest-neighbour' tests are useful in theabove geologic contexts, none are particularlysuitable to evaluate the null hypothesis of randomsuccession of bed thicknesses against an alterna-tive model that a particular directional trendexists, e.g. a single thickening upward trend.

As a test of the null hypothesis against thealternative of one or more trends, nearest-neigh-bour tests are ¯awed in two ways: (1) they havelow power to detect a trend; and (2) they havemultifaceted alternatives (so a rejection of thenull hypothesis does not corroborate a particulartrend).

Tests that merely involve `nearest-neighbour'comparisons have low power against a speci®calternative, e.g. that of one or more thickening orthinning upward trends. In other words, applica-tion of the tests often results in failure to reject thenull hypothesis when it is false. Kendall & Ord(1990; p. 21) note that à test based on turningpoints will perform poorly as a test for trend,because random ¯uctuations imposed on a mildtrend will have much the same set of turningpoints as if the trend were absent.' Waldron(1987; p. 140) makes the same point: a runs testèxamines only transitions between immediatelyadjacent beds, thus it may fail to identify cycleswhen noise is superimposed on the pattern.'

Milenkovic (1989) provides an actual numericalexample. The same argument for low powerapplies to other tests listed in Appendix 3.

For example, consider sequences which showobvious thickening upward (Fig. 2g, 2j, the sub-sequences labelled [1] in Fig. 4a) and thinningupward (Fig. 2i and the subsequence labelled [2]in Fig. 4a) trends. All are highly signi®cant usingtests based on Kendall S (or Tau). Nearest-neighbour runs tests and length of longest runstests were applied to both thickness data and tolog (thickness) data for these ®ve sequences.Twenty tests in all were carried out as outlinedin Drummond & Wilkinson (1993) (see Appendix3), yet, only one test shows the observed sequenceas signi®cant at the 0á05 level. (The exception is aruns test of log (thickness) applied to Fig. 2j.Similarly, applying the Durbin-Watson test, onlythree of the ®ve sequences are signi®cant at the0á05 level.

As noted above, nearest-neighbour tests pose aneven more serious problem: they evaluate the nullhypothesis against a catch-all alternative. Bhatta-charyya (1984) referred to catch-all alternativehypotheses as òmnibus alternatives' to random-ness.

Any one of an omnibus of trends, including amultitude of thickening and/or thinning upwardtrends, could lead us to correctly reject the nullhypothesis. Alternatively there might be morechanges from thickening upward to thinning thanwould be expected by chance (Drummond &Wilkinson, 1993). Or perhaps neighbouring bedstend to have similar thicknesses. Or beds of likethicknesses may tend to occur together in packets,an alternative brie¯y explored by Pickering et al.(1989). Thus, even if the null hypothesis isrejected, we cannot view this as corroborating aparticular alternative hypothesis, e.g. that there isa single thickening (or thinning) upward trend(Miller & Kahn, 1962).

Waldron (1987) suggests smoothing by moving-averages as a way to increase the power of testsinvolving the signs of differences between con-secutive beds. He shows that the expected num-ber of positive differences in n-point movingaverages between adjacent beds in a randomsequence of N beds is (N ) n)/2 with a standarddeviation of Ö[(N + n)/12]. Ties could be handledas in method (3), in Appendix 3. This is a step inthe right direction for this reason: it gets us awayfrom the shackles of comparing each bed thick-ness merely with that of its nearest neighbours.However, Waldron (1987) shows that using two-fold moving averages in effect merely compares

664 C. W. Harper


each bed thickness with that of beds two bedsaway; n-fold moving averages with that of beds nbeds away. A better procedure still would be touse a test statistic that compares the thickness ofeach bed with that of every other bed in thesequence (or subsequence) studied. Fortunately,Kendall & Gibbons (1990) have de®ned such astatistic: Kendall's S coef®cient and the closelyrelated Kendall's Tau.

In sections that follow, ways to test the nullhypothesis of random succession of bed thick-nesses against various trend alternatives arediscussed. The use of test statistics based onKendall's S (and Tau) that make `sequence-wide'(or subsequence-wide) comparisons of bed thick-nesses are advocated (Harper, quoted in Pickeringet al., 1989; p. 84). The test statistic used and theexact form of the test will depend on thealternative model considered.

PARAMETRIC VS. RANDOMIZATIONTESTS

Suppose a geologist suspects that beds show atendency to thicken upward (Fig. 1a) or thin

upward (Fig. 1e) through a sequence. What isneeded is a test statistic that quanti®es thistendency. We can then provide tests to determinewhether the value of the test statistic obtainedfrom a particular sequence is statistically signi®-cant. Two types of tests ± parametric and ran-domization ± are available.

For each test statistic proposed, both a para-metric test, i.e. a test having an analytic solutionfor the distribution of the test statistic under thenull hypothesis, if one is available, as well as arandomization test (Edgington, 1987) are provid-ed. Table 1 details the two types of tests. Para-metric tests provide approximations to the prob-ability of attaining the observed results even if p-values are very small (Mielke, 1991). Where botha parametric test and a corresponding random-ization test are available, the latter may be used toassess the validity of the analytic solution in theformer. If an analytic expression is not known,only a randomization test is provided.

Tests were implemented on an IBM mainframeusing programs written in SAS that are availablefrom the author. Randomization tests of longsequences took up to several minutes to run.

Table 1. Procedure for testing a null hypothesis against an alternative, e.g. that there are two thickening upwardsubsequences.

Using a Parametric test: Using a Randomization test:

Choose a level of signi®cance a (optional). Choose a level of signi®cance a (optional).

De®ne a test statistic T with a probability distributionthat can be expressed analytically. The statisticshould measure how well a given sequence exhibitsthe pattern characterized in the alternative hypothesis.

De®ne a test statistic T. The statistic should measurehow well a given sequence exhibits the patterncharacterized in the alternative hypothesis.

Calculate the T for the observed sequence. Calculate the T for the observed sequence.

Compute Prob (T), the probability, under thenull hypothesis, of obtaining a value of the teststatistic as extreme or more extreme than thatactually observed.

Generate a random sequence, i.e. a random shuf¯eof all bed thicknesses.

Calculate the T for the random sequence.If test statistic is approximately distributed as normalwith a mean = 0 & a given standard deviation, thenZ � (T/Standard deviation) will be distributed asunit normal, so that Prob (T) = the probability ofobtaining a value of Z as extreme or more extremethan that actually observed. The latter can be gottenfrom a table for the unit normal distribution.

Repeat last two steps many times (several thousandtimes say). Determine Probr (T) = the proportion ofactual plus randomized datasets that give a teststatistic as high or higher than the oneactually observed.

Report the observed data as signi®cant at thelevel Prob (T). If a level of signi®cance µ waschosen in step (1), and Prob (T) is greater than orequal to µ, then reject the null hypothesis that thesequence is random; otherwise do not rejectthe null hypothesis.

Report the observed data as signi®cant at the levelProbr (T). If a level of signi®cance µ was chosen instep (1), and Probr (T) is greater than or equal to µ,then reject the null hypothesis that the sequence israndom; otherwise do not reject thenull hypothesis.



Table 2. Test against alternative of a single thickening or thinning upward trend.

Given: A stratigraphic sequence of beds numbered 1, 2, . . . N from lowest to highest, each bed i with a thickness(or other measurable property) ti, and an alternative hypothesis of a single trend.

Pairwise comparison of beds rankings jRijj:Beds may be rank ordered by position in the sequence and by thickness (or other variate) t. For each of the

[N (N)1)/2] pairs of beds i and j, i above j, the Rank Order Matrix jRijj compares beds i and j:Rij � + 1 if ti > tj (i.e. the upper bed is thicker than the lower bed i).Rij � 0 if ti � tj (beds i and j are tied (i.e. of equal thickness)).Rij � )1 if ti < tj (i.e. the upper bed is thinner than the lower bed i).

Sample space:As set of possible outcomes, consider all possible permutations of the beds amongst the N stratigraphic positions.

Test statistics:Test null hypothesis against a single trend using test statistic S � Kendall's (or equivalently, Tau � Kendall's Tau)de®ned as follows:

S � Kendall's S � Pi � n

i � 1

Pj � n

j � i�1

Rij �1�

The range of S is )[N (N)1)/2]-Ties £ S £ [N (N)1)/2]-Ties, where Ties � the number of bed pairs with tied values.

Tau � S/[N(Nÿ 1�=2� �2�The range of Tau is )(1 )Ties/( N( N)1)/2)) £ Tau £ + (1 )Ties/( N( N)1)/2)) where Ties is the number of bed pairswithtied values. (See Kendall & Gibbons 1990, pp. 1±8, 40±43).

Parametric test: Randomization test:

For N ³ 10, the distribution of S can be closelyapproximated as normal with a mean of zero anda standard deviation of:

Results using above parametric test above test can bechecked against those using a randomization testtabulating:

Ö [(1/18) N(N)1)(2 N+5) )Tsum] (3)

Where Tsum � Sum over all runs r of consecutivebeds with the same thickness (in a listing of bedsordered by bed thickness) of (1/18)ur(ur)1)(2ur+5),and where ur � number of beds in run r (Kendall& Gibbons, 1990, p. 60±66).

Itera � Number of datasets analysed in arandomization test (including the dataset actuallyobserved):

Probr(S) � Proportion of actual plus randomizeddatasets which produce a value of S as great orgreater than the value calculated for the datasetactually observed. (7)

Test S using

Z � (S)1)/(Standard deviation of S) (4)

which is distributed as unit normal. Use numeratorequal to S)1 rather than S to obtain correction forcontinuity (Kendall & Gibbons, 1990, p. 65±69).

For test against thinning upward alternative, useProbr(S) � Proportion of actual plus randomizeddatasets which produce a value of S equal to or lessthan the value calculated for the dataset actuallyobserved. (7¢)

Prob(S) � Probability of obtaining a value of S asextreme or more extreme than that observed � Prob(Z), (5)where Prob(Z) � Probability of obtaining a valueof Z as extreme or more extreme than that observed.

Result of randomization test: Report observed datasigni®cant at level a.a � Probr(S).

A test using S is equivalent to a test using Kendall's Tau,except that the correction for continuity is harder to applyusing Tau. For N ³ 10, Kendall's Tau is distributed asnormal with a mean of zero and a standard deviation of:

(Standard deviation of S)/(N(N)1)/2). (6)

See Eq. (3) above for standard deviation of S (Kendall &Gibbons, 1990). If N < 10 then use tables in Kendall & Gibbons(1990, Appendix 1) and Sillito (1947, Table 2) to obtainProb(S) and Prob(Tau).

Result of parametric test: Report observed data signi®cantat level a.a � Prob(S).

666 C. W. Harper


TESTING AGAINST A SINGLE THICKEN-ING/THINNING UPWARD TREND

To test the null hypothesis against the alternativethat there is a single thickening (or thinning)upward sequence, using a parametric test basedon Kendall's S (or equivalently Kendall's Tau) issuggested. Optionally the test may be corroborat-ed with a randomization test (Table 1). Suggestedtest procedures are detailed on Table 2.

The test statistic

One should choose a test statistic that would leadto rejection of the null hypothesis if there is astatistically signi®cant tendency to thicken (orthin) upward through a sequence. Table 2 detailsthe use of the coef®cients recommended: Ken-dall's S, or equivalently Kendall's Tau. Thesestatistics compare beds pair-wise, comparing therank order of bed thicknesses with their order inthe vertical sequence.

Figure 1a±e shows examples of computation ofKendall's S and Tau. The contribution of eachdistinct bed pair Rij to S (the numerator of Tau) isshown in a table on each ®gure.

Another coef®cient, Spearman's Rank Correla-tion Coef®cient (Kendall & Gibbons, 1990) ishighly correlated with Kendall's S (or Kendall'sTau) and gives virtually identical results (Harper,1984c). Assuming the null hypothesis, the prod-uct moment correlation between the two coef®-cients calculated for N beds � 2 (N + 1)/Ö[2N(2N + 5)] which rapidly tends to 1 for increasingN. Thus, for N � 20 it is 0á99 (Kendall & Gibbons,1990; p. 102).

Lowey (1992) used the product moment corre-lation coef®cient and slope of least square regres-sion line. These test statistics are especiallyunsuitable as they assume a linear relationbetween expected values of bed thickness andstratigraphic position. Thus, they have low powerto detect a nonlinear trend.

The probability under the null hypothesis ofobtaining a value of the test statistic as extremeor more extreme than that actually observed

As a test of the null hypothesis, a test of S isequivalent to a test of Tau. It is easier to apply acorrection for continuity to the normal approxi-mation using S. Recommended test procedure isdetailed in Table 2.

If the succession is random, then any permu-tation of the observed bed thicknesses is as likely

to occur as any other. Prob(S) in Table 2 is theprobability that a given permutation of bedthicknesses will have a Kendall S (or Tau)coef®cient as extreme or more extreme than thatactually observed. Of course, in a test for a singlethinning upward sequence, Prob(S) would be theprobability of obtaining a value of the test statisticas small or smaller than the value actuallyobserved. If there is no geologic reason to expecta thickening or a thinning upward sequence, thena two sided test (against the alternative of eitherthickening or thinning upward), is called for. Fora two-sided test use twice the value of Prob(S)calculated for the single-sided case.

Results of application of test

Several sequences, mostly turbidites, were testedagainst the alternative of a single thickening orthinning upward trend using both parametric andrandomization tests. Results are shown on Ta-bles 3A and 3B. The reader may wish to ®rstvisually inspect each sequence plot and make asubjective guess as to whether it shows a pattern;then check results in Tables 3A and 3B.

TESTING AGAINST ONE OR MORESUBSEQUENCES HAVING THICKENING/THINNING UPWARD TRENDS

Randomization tests involving subsequences

Given a sequence having (or purported to have) gsubsequences, randomization can be carried out:(1) shuf¯ing each of the g subsequences individ-ually each time; or (2) shuf¯ing the entiresequence of beds as a whole each time. If no tiesexist, the two ways of randomization produceidentical results, as both result in a randompermutation of the rank order of bed thicknesseswithin each subsequence. With ties the two wayscan have slightly different results, as the numberof ties assigned to a given subsequence may varyfrom one randomization to the next. As there isno geologic basis for holding the number of ties ineach subsequence constant, the second proceduremakes more sense geologically and is used here.Randomization tests carried out on the sequenceshown in Fig. 3d using both procedures yieldedidentical results.

Randomization tests must be used if parametrictests are not available. All the analytic solutionsfor parametric test statistics involving subse-quences are at best only approximations if oneor more subsequences have less than 10 beds.



Table 3A. Results ± parametric tests against alternative of a single thickening or thinning upward trend.

(Figure) Stratigraphic unit

Alleged thickeningor thinningupward trend

No. ofbeds

Results S/[N(N)1)/2] � Tau[Eqs. (1), (2)Table 2]

ResultsParametric testProb(Tau) � Prob(S)[Eq. (5), Table 2]

(1a) Turbidite sandstones,San Salvatore Sandstone(Miocene), Northern Apinnines,Italya

Thickening 9 13/36 � + 0á361 0á10

(1c) Hypothetical sequence.Randomized succession of bedthicknesses shown in Fig. (1b).

Tested for onethickening here

10 9/45 � + 0á200 0á237

(1d) Hypothetical sequence. Tested for onethickening here

10 28/45 � + 0á622 0á0077

(1e) Turbidite sandstones,Tourelle Formation (Ordovician),Quebec.b

Thinning 9 )6/36 � )0á167 0á306

(2a) Turbidite sandstones,Tourelle Formation (Ordovician),Quebec.b

Thickening 23 32/253 � + 0á126 0á206

(2b) Turbidite sandstones,Tourelle Formation (Ordovician),Quebec.b

Thinning 13 )42/78 � )0á538 0á006

(2c) Turbidite sandstones,Tourelle Formation (Ordovician),Quebec.b

Thinning 20 )120/190 � )0á632 0á00005

(2d) Turbidite sandstones,Tourelle Formation (Ordovician),Quebec.b

Thinning 19 +23/171 � + 0á135k 0á28 + 0á5 � 0á78!

(2e) Turbidite sandstones,Tourelle Formation (Ordovician),Quebec.b

Thickening 27 2/351 � + 0á006 0á492

(2f) Turbidite sandstones,Carboniferous Flysch,Southern Morocco.b

Alleged randomsequence

63 )146/1953 � )0á075 0á1948

(2g) Turbidite sandstones,Carboniferous Flysch, SouthernMorocco (beds 194±243; selectedout of 298 beds sequence shownin Fig. 4a as an example of anobvious thickening upwardsequence).

Thickening 50 388/1225 � + 0á317 0á0006

(2h) Turbidite sandstones,Shimanto Group(Jurassic-Cretaceous),Oigawa District, Japan.d

Thickening 148 255/10878 � +0á023 0á337

(2i) Turbidite sandstones,Beds 96±143 extracted fromsequence shown in Fig. 4awith sequence modi®ed toshow obvious thinning upwardsequence.

Thinning 48 )701/1128 � )0á621 2á21 E±10

668 C. W. Harper


Table 3A. (Contd.)



No. ofbeds

Results S /[N(N-1)/2] � Tau[Eqs. (1), (2)Table 2]

ResultsParametric testProb(Tau) � Prob(S)[Eq. (5) Table 2]

(2j) Turbidite sandstones,Beds 194±243 (� Beds shown inFig. 2g) with sequence modi®edto show obvious thickeningupward sequence.

Thickening 50 530/1225 � + 0á433 4á76 E±6

(2k) Turbidite sandstones,Antola Formation (UpperCretaceous), Northern Apinnines,Italy. Succession at ®rst glancesuccession appears random, butif four thickest beds are ignoredthen a thickening upwardpattern is obvious.e

Several thickening;just tested here forone thickening

48 352/1128 � + 0á312 0á0009

(2l) Turbidites sandstones(or possibly debris ¯ow andslump deposits), JackforkFormation (Upper Mississippian),DeGray Dam, Arkansasf

Thinning 8 )22/28 � )0á785 0á0028

(2m) Turbidites sandstones(or possibly debris ¯ow andslump deposits), JackforkFormation (Upper Mississippian),DeGray Dam, Arkansasf

Random 29 )70/406 � )0á172 0á096

(2n) Turbidites sandstones(or possibly debris ¯ow andslump deposits), JackforkFormation (Upper Mississippian),DeGray Dam, Arkansasf

Thickening.However, ifignore thickestbed, appearsrandom

13 1/78 � + 0á012! 0á500!

(2o) Fan ratio (� sand-mud ratioX degree of contrast X maximumclast size); Fluvial and LacrustineSandstones. Bechsrieth Formation(Upper Carboniferous), SouthernGermanyj

Decreasing upward 41 )439/820 � )535 4á194 E±7

(2p) Maximum particle size(mean value of ten largest clasts),Devonian alluvial fan conglomer-ates, Hornelen Basin, Norwayi

Increasing upward 105 3652/5460 � + 0á669 0á000000

(3b) Hemipelagite beds in alternatingmud turbidite/hemipelagite bedsequence. Quaternary, SouthShetland Trench, West Antarcticag

Thickening 17 35/136 � + 0á257 0á079

(3c) Mud turbidite beds inalternating mud turbidite/hemipelagite bed sequence.Quaternary, South ShetlandTrench, West Antarcticag

Thinning 17 )28/136 � )0á206 0á129

(3d) Turbidite sandstones, Lower 21beds � Subsequence [1] only.Macigno Fm. (Oligocene),Northern Apennines, Italyh

Thickening 21 75/210 � + 0á357 0á009



Two different geologic contexts

A test against one or more thickening and/orthinning upward subsequences is needed in twofundamentally different geologic contexts:1 A geologist may recognize two or more subse-quences separated by recognizable and geological-ly signi®cant breaks (erosion surfaces and/orintervening strata of different origin). These are a-priori groupings of strata recognized independentof thickness data. He/she may then test against thehypothesis of one or more within-subsequencetrends. For example the two turbidites subse-quences shown in Fig. 3d are separated by thickshale intervals. Also, Yose & Heller (1989) recog-nize a sequence of turbidite beds partitioned intodiscrete subsequences by megabreccia debrissheets.2 A geologist may subdivide a sequence intosubsequences including one or more thickeningand/or thinning upward subsequences merely byinspection of bed thicknesses (�post-hoc group-ings of strata).

Many (e.g. Meddis (1984; p. 288 291), Faith(1991)) make the fundamental distinction be-tween a-priori hypotheses (made before datais analysed) vs. post-hoc ones (constructed afteran analysis of data). Meddis notes `when ahypothesis is chosen after the data have beenscrutinized, the researcher is in the position ofsomeone who decided in advance to considerevery possible hypothesis and report only themost signi®cant'.

Testing two or more subsequences recognizeda-priori (separated by recognizable breaks) are®rst considered. Testing against two or moresubsequences postulated post-hoc (purely on thebasis of observed thickness patterns), one or morewith thickening and/or thinning upward trends,are then considered. In both cases, testing againstsame-type (all thickening or all thinning upward)subsequences, and against mixed (thickening arethinning upward) subsequences are considered.

TESTING SUBSEQUENCES RECOGNIZEDA-PRIORI (SEPARATED BY RECOGNIZ-ABLE BREAKS) AGAINST ONE OR MORESUBSEQUENCES HAVING THICKENING/THINNING UPWARD TRENDS

General form of the test statistic

Consider a sequence of N beds divided into gsubsequences (e.g. Figure 1b, 1f showing twosubsequences, Fig. 1d, 1h showing three). A testusing Kendall's Tau calculated for the sequenceas a whole is not suitable as it would have lowpower against the alternative of one or moresubsequence trends. For example the sequenceshown in Fig. 1b; it has two thickening upwardsubsequences, yet Kendall's Tau equals ±0á111.

Mielke (1984, 1991) and Orlowski et al. (1993)address the closely analogous problem of evalu-ating multivariate data observations on a parti-tioning of N objects into g mutually exclusive

Table 3A. (Contd. )



No. ofbeds

Results S /[N(N)1)/2] � Tau[Eqs. (1), (2)Table 2]

ResultsParametric testProb(Tau) � Prob(S)[Eq. (5) Table 2]

(3d) Turbidite sandstones,Upper 14 beds � Subsequence [2]only. Macigno Fm. (Oligocene),Northern Apennines, Italyh

Thickening 14 )15/91 � )0á165k 0á316 + 0á5� 0á816!

(4a) Beds 96±143 � Subsequence [2]only. Carboniferous Flysch,Southern Moroccoc

Thinning 48 )264/1128 � )0á234 0á0095

(4a) Lower 95 beds � Subsequence[1] only. Carboniferous Flysch,Southern Moroccoc

Thickening 95 1563/4465 � +0á350 2á50 E±7

a Mutti, 1974, Fig. 5-Section 13, b Hiscott, 1980, Figs 8, 13, 15, c Graham, 1982 Figs 3, 4; 1988, Fig. 2á53; d Kimura,1966, Fig. 19, Loc. 119; e Martini, Sagri & Doveton, 1978, Fig. 8, S. Donato Section; f Moiola & Shanmugam, 1984, Figs5±7; g Porebski, 1991, Fig. 23; h Gibaudo, 1980, Fig. 5; Hiscott, 1981, Fig. 1, i Steel et al., 1977, Fig. 3, Fan Cycle A;j Dill 1992, Fig. 4; k Note sign.

670 C. W. Harper


Table 3B. Results ± parametric tests against alternative of a single thickening or thinning upward trend. Exampleswhere both parametric and randomization tests were done. (Except for the ®rst two, the sequences listed wereactually tested primarily for multiple subsequences using a randomization test, but test statistics against single trendwere also computed, and are listed here to enable comparison of parametric test and randomization test results).

(Figure) stratigraphic unit

Allegedthickening orthinningupward trend

No. ofbeds

ResultsTau [Eq. (2)Table 2]

ResultsParametric testProb(Tau) �Prob(S)[Eq. (5), Table 2]

Results Randomizationtest Proportion ofsuccesses out ofItera trials] �Probr(Tau) � Probr(S)

(1d) Hypothetical sequence Thickening 10 +0á622 0á0077 5/1000 � 0á005

(1e) Turbidite sandstones,Tourelle Formation(Ordovician), Quebeca,j

Thinning 9 )0á167 0á306 305/1000 � 0á305

(3d) Turbidite sandstones,Macigno Fm.(Oligocene), NorthernApennines, Italyf,j

Two thickeningk 35 0á008 0á476 918/2000 � 0á4590

(3e) Weight Percent Calcite,Pelagic Limestones andMarlstones, Tropic Shale(Upper Cretaceous),Sit Down Bench, Utahg,j

Three decreasingupwardk

25 )0á423 0á0015 1/2000 � 0á0005

(5c) Turbidite sandstones,San Salvatore Sandstone(Miocene), NorthernApinnines, Italyb,j

Three thickeningk 24 +0á529 0á0001 1/1898 � 0á0005

(5d) Turbidite sandstones,San Salvatore Sandstone(Miocene), NorthernApinnines, Italyb,j

Two thickeningk 18 +0á327 0á031 21/901 � 0á023

(5e) Turbidite sandstones,Carbonate fan deposits,Loma del Toril Fm.,(Upper Jurassic),Spainc,j

Five thinningk 46 )0á043 0á6686 641/964 � 0á665

(5f) Turbidite sandstones,Murca Formation(Lower Cretaceous)submarine fan deposits,Bogata Troughh,j

Seven thickeningk 45 )0á059 0á728 735/1000 � 0á735

(5g) Turbidite sandstones,Macigno Fm.(Oligocene), NorthernApennines, Italyd,j

Five thickeningk 64 +0á104 0á113 114/1000 � 0á114

(5h) Turbidite sandstones,Macigno Fm. (Oligo-cene), Northern Apen-nines, Italye,j

Four thickeningk 31 +0á174 0á085 112/1101 � 0á102

(5i) Turbidite sandstones,Macigno Fm.(Oligocene), NorthernApennines, Italye,j

Four thickeningk 45 +0á223 0á015 11/916 � 0á012



Table 3B (Contd. )


Allegedthickening orthinningupward trend

No. ofbeds

ResultsTau [Eq. (2)Table 2]

ResultsParametric testProb(Tau) �Prob(S)[Eq(5) Table 2]

Results Randomizationtest Proportion ofsuccesses out ofltera trials] �Probr(Tau) � Probr(S)

(5j) Turbidite sandstones,Macigno Fm. (Oligo-cene), Northern Apen-nines, Italye,j

Three thickeningk 30 -0á064 0á6862 624/907 � 0á688

(5k) Turbidite sandstones,Macigno Fm. (Oligo-cene), Northern Apen-nines, Italyi,j

One thickening &one thinningk

16 )0á083 0á674 2004/3000 � 0á668

a Hiscott, 1980, Fig. 15F; b Mutti 1974, Fig. 5-Sections 2, 12; c Ruiz-Ortiz, 1983, Fig. 5, megasequences IV-VIII; d Gi-baudo, 1980, Fig. 7, Level 10; Hiscott, 1981, Fig. 2; e Gibaudo, 1980, Fig. 9, Level 6, 4, 15; f Gibaudo, 1980, Fig 5;Hiscott, 1981, Fig. 1; g Sethi & Leithold, 1994, Fig. 4; h Pimpirev & Sarmiento, 1993, Fig. 6c; i Gibaudo, 1980, Fig. 7Level 45; j See Table 7 for main discussion; k but here tested against alternative of one thinning upward trend;l Itera � No. of datasets analysed � denominator of ratio shown. Eqs. 7 & 7¢, Table 2.

Table 4A. Test of subsequences of ®xed size recognized a-priori (recognized at outset) against alternative of one ormore within-subsequence trends.

Given: A stratigraphic sequence of N beds separated by breaks that subdivide the sequence a-priori intog subsequences of ®xed size, and an alternative hypothesis that postulates one or more of the subsequences exhibita thickening and/or thinning upward trend.

Sample space:For parametric tests: As set of possible outcomes, consider all permutations of beds within each subsequence.

For randomization tests: As set of possible outcomes, consider all possible permutations of the N beds amongst the Nstratigraphic positions in the g ®xed subsequences.

General form of test statistics:

Testw;f �Xi � g

i � 1

Ciw Function�Taui�: w � 1; 2; 3 �8�whereg � Total number of subsequences in sequence.Taui � Kendall's Tau coef®cient calculated for subsequence i.

w � weighting coef®cient index. Selects one of threeweighting coef®cients. See Ciw.

f � function index. Selects one of 5 functions of Taui

calculated for a subsequence i:Ciw � weighting coef®cient of type w for the i-thsubsequence

Function f(Taui) � simple function of Taui oftype f.

[Three weighting coef®cients are proposed: (w � 1,2,3): (f � 1,2,3,4,5):Function1(Taui) � Taui,Function2(Taui) � Absolute Value Taui � jTauij,

Ci1 � �Ni�Ni ÿ 1��=2 Pj � g

j � 1

NjNj ÿ 1�=2�,

Function3(Taui) � (Taui)2,

Function4(Taui) � (Taui/Standard deviation of (Taui))2,

Function5(Taui) � (()1)i+1) Taui

Ci2 � Ni/N andN � Total number of beds in sequence.

Ci3 � 1 (for a Chi-square test)]. Ni � no. of beds in subsequence i.

672 C. W. Harper


groups. They ®rst de®ne an average between-object distance function (� ei) for all distinctpairs of objects in each group Si (i � 1, . . . ,g), andthen de®ne a test statistic d as

d �Xi � g

i � 1

Cini

where Ci � weighting coef®cient for i-th group.I suggest combining Kendall's Tau coef®cients

for the g subsequences into a single test statisticusing weighted averages of functions of thecoef®cients.

The general form of the test statistic proposed,Testwf, is given in Table 4A, Eq. (8):

Testwf �Xi � g

i � 1

CiwFf�Taui�

whereTaui � Tau coef®cient for subsequence i,

Ciw � weighting coef®cient of type w for the i-thsubsequence, andFf (Taui) is a simple function of Taui.

Three weighting coef®cients are proposed:

Ci1 � �Ni�Ni ÿ 1�=2�.Xj � g

I � 1

Nj �Nj ÿ 1�=2ÿ �Ci2 � Ni=N

Ci3 � 1 (for a Chi-square test)

where Ni � no. of beds in subsequence i,Nj � no. of beds in subsequence j, and N � totalnumber of beds in sequence.

Mielke (1984, 1991) and Orlowski et al. (1993),in a different context, recommend a weightingcoef®cient Ni/N, like Ci2 listed above, as anef®cient choice of weighting coef®cient, but notetheir coef®cient equal to Ci1 listed above, isinef®cient. However, in all examples consideredbelow, the two choices of Ci1 and Ci2 give test

Table 4B. Test of subsequences of ®xed size recognized a-priori (recognized at outset) against alternative of one ormore within-subsequence trends.

Test of subsequences recognized a-priori against alternative of one or more thickening or thinning upward sub-sequences of same type:

Test null hypothesis using test statistic Wtau_onew � Testw,1, w � 1,2:

Wtau_onew � Weighted sum of within-subsequence Kendall's Tau coef®cients calculated for g subsequences usingweighting coef®cients of type w (w � 1,2).

Wtau onew �Pi � g

i � 1

CiwTaui (9)

where Ciw � weighting coef®cient of type w for the i-th subsequence, and Taui � Tau coef®cient for subsequence i.

Parametric tests: Randomization tests:

If the g subsequences each have 10 or more beds, thenWtau_onew is distributed as normal with a mean ofzero and a standard deviation of:

Xi� g

i� 1

�Ciw

uuut �StandarddeviationofTaui��2 �10�

where Ciw are weighting coef®cients of type w(Wadsworth & Bryan, 1960, pp. 159±160).Test null hypothesis usingZ � Wtau_onew/(standard deviation of Wtau_onew) (11)Z is distributed as unit normal.Prob(Watu_onew) � Probability of obtaining avalue of Watu_onew as extreme or more extreme thanthat observed � Prob(Z). (12)Prob(Z) � Probability of obtaining a value of Z asextreme or more extreme than that observed.Result of parametric test:Report observed data signi®cant at level a.a=Prob (Wtau onew�:

ltera � Number of datasets analysed in a randomiza-tiontest (including the dataset actually observed). (Wholesuccession of N beds is randomized each iteration).

To test against thickening upward subsequencescalculate Probr(Wtau_onew) � Proportion of actualplusrandomized datasets which produce a value ofWtau_onew as great or greater than the value calculatedfor the dataset actually observed. (13)

To test against thickening upward subsequences countProbr(Wtau_onew) � Proportion of actual plusrandomized datasets which produce a value ofWtau_onew equal to or less than the value calculatedfor the dataset actually observed. (13¢)

Result of randomization test:Report observed data signi®cant at level a.

a=Probr (Wtau onew�:



Table 4C. Test of subsequences of ®xed size recognized a-priori (recognized at outset) against alternative of one ormore within-subsequence trends.

Test of subsequences recognized a-priori against alternative of one or more thickening or thinning upward sub-sequences of mixed type.

Test null hypothesis using test statistics having general form given in Table 4A, Eq. (8), viz.:

Testw;f �Xi� g

i� 1

Ciw Functionf(Taui�: w = 1,2,3

Speci®cally use:(a) Wtau_absw � Testw,2 based on Function2(Taui) � Absolute Value Taui, w � 1,2,(b) Wtau_sqrw � Testw,3 based on Function3(Taui) � Tau2

i , w � 1,2,(c) Wtau_chisqr � v2 � Test3,4, based on Function4(Taui) � (Taui/Standard deviation of Taui)

2, w � 3, and(d) Wtau_symw � Testw,5 based on Function5(Taui) � (()1)i+1) Taui. w � 1,2.

Parametric tests:

(a) Wtau_absw:

Wtau_absw � Weighted sum of absolute values ofwithin-subsequence Kendall's Tau coef®cients calcu-lated for g subsequences using weighting coef®cients oftype w (w � 1,2).

Wtau absw �Xi� g

i�1

CiwjTauij �14�

where Ci are weighting coef®cients of type w andjTau(i)j � Absolute value of Tau calculated for sub-sequence i.

If the g subsequences each have 10 or more beds, then

Wtau_absw (Wtau_absw ³ 0) is distributed as twice thenormal distribution with a mode of zero and a standarddeviation equal to the standard deviation of

Wtau_onew (15)

(see Eq. (6) above for latter)]. (Wadsworth & Bryan,1960, pp. 153, 159±160).

Test null hypothesis using

Z2 � Wtau_absw/(Standard deviation of Wtau_absw).(16)Z2 is distributed as twice the unit normal distribution.

Prob(Wtau_absw) � Probability of obtaining a value of

Wtau_absw as extreme or more extreme than that ob-served � Prob(Z2)

Prob(Z2) � Probability of obtaining a value of Z2 asextreme or more extreme than that observed �2[Unitnormal (Wtau_absw/standard deviation ofWtau_absw)], where unit normal (z) � Probability ofobtaining a value taken from the unit normal distribu-tion equal to or greater than z. (17)

(b) Wtau_sqrw:

Watu_sqrw � Weighted sum of squares of within-sub-sequence Kendall's Tau coef®cients calculated for gsubsequences using weighting coef®cients of type w(w � 1,2).

Wtau sqrw �Xi� g

i�1

CiwTau2i �18�

The distribution of Wtau_sqrw is not known. Use arandomization test.

(c) Wtau_chisqrWtau chisqr �Unweighted sum of squares of Within-Subsequence Kendall's Tau coef®cients divided bytheir standard deviation:

Wtau chisqr �Xi� g

i� 1

Taui=(standard deviation of Taui��2

�19�If the g subsequences each have 10 or more beds, thenWtau_chisqr is distributed as v2 with g degrees offreedom (Wadsworth & Bryan, 1960, pp 161±162).Prob(Wtau_chisqr) � Probability of obtaining a value ofWtau_chisqr as or more extreme than that observed.(20)

(d) Wtau_symw

To test against the alternative of g/2 symmetric sub-sequences (thickening followed by thinning upwardsubsequence pairs) I suggest using this modi®ed ver-sion of Wtau_onew [Table 4B, Eq. (9)]:

Wtau symw �Xi� g

i� 1

�ÿ1�i�1CiwTaui �21�

Wtau_symw sums + Ciw Taui for purportedly thicken-ing upward subsequences and )Ciw Taui for purport-edly thinning upward subsequences.Wtau_symw is distributed as Wtau_onew [see Eqs. (9)±(12) above].Prob(x) � Probability of obtaining a value of x as ex-treme or more extreme than that observed. (22)Result of parametric tests:Report observed data signi®cant at level aa � Prob(Wtau_absw) using test statistic Wtau_absw.a � Prob(Wtau_chisqr) using test statistic Wtau_chisqr.a � Prob(Wtau_symw) using test statistic Wtau_symw.

674 C. W. Harper


statistics with virtually the same variance andstandard deviation, i.e. with the same ef®ciency(Dixon & Massey, 1957; p. 74). See table inAppendix 4 that compares estimates of thestandard deviation of test statistics using Ci1 withthat of those using Ci2.

Testing against same-type (all thickening or allthinning upward) subsequences

Testing sequence as a whole using weightedaverage of Kendall Tau coef®cients as teststatistic

To test sequences as a whole against the alterna-tive of subsequences of same type, tests areproposed using test statistic Wtau_onew as out-lined in Table 4B, Eq. (9):

Wtau onew �Xi � g

i � 1

Ciw Taui

whereTaui � Tau coef®cient for subsequence i, andCiw � weighting coef®cient of type w for the i-

th subsequence.Either of the ®rst two choices of weighting

coef®cients listed above and in Table 4A (Ci1 orCi2) can be used. However, the ®rst coef®cient,Ci1, is intuitively meaningful in the context ofcombining Kendall's Tau coef®cients. To see this,consider a given partition of N beds into gsubsequences. Each subsequence of Ni beds willhave Kendall's S � Si and a Tau coef®cientTaui � (Si)/[Ni(Ni ) 1)/2] calculated for [Ni

(Ni ) 1)/2] distinct bed thickness pairs.One simple intuitive measure of trend

equals the sum of the Si values for all g

subsequences divided by the total number ofbed thickness pairs compared in all g subse-quences. Yet, such a test statistic is equivalentto Wtau_one1 above using a weighting coef®-cient Ci1.

Thus, for the ®rst choice of Ci1, the test statisticWtau_one1 is the weighted average of the Taui

values for the g subsequences. It equals the sum ofthe S values for the g subsequences divided by thesum of the total number of bed thickness pairscompared in all subsequences. As an example,assume the sequence shown in Fig. 1b, f consistsof turbidite sand beds separated into two subse-quences by a thick shale break. Figure 1f illus-trates how Wtau_one1 � (10/20) (10/10) + (10/20) (10/10) � 1 is calculated. Similarly assumethe sequence shown in Fig. 1c, g consists of sandbeds separated into two subsequences by a shalebreak. Figure 1g illustrates how Wtau_one1 � (3/24)(1/3) + 21/24)(13/21) � 0á58 is calculated. Fi-nally, assume the sequence shown in Fig. 1d, hconsists of sand beds separated into three subse-quences by thick shale breaks. Figure 1h illus-trates how Wtau_one1 � (6/12)(1) + (3/12)(1)+ (3/12)(1) � 1 is calculated.

If each of the g subsequences has more than 10beds, then under the null hypothesis, the withingroup Tau coef®cients, Taui, may be treated asindependent, normally distributed variates with amean of zero and a standard devia-tion � Standard deviation Taui (Table 4B). Thus,their weighted sum Wtau_onew, will also benormally distributed with mean of zero andstandard deviation equal to the weighted sumgiven in Eq. (10), Table 4B. Prob (Wtau_onew),the probability of obtaining a value of Wtau_onew

as extreme or more extreme than that observed,

Table 4C. (Contd.)

Randomization tests:

Itera � Number of datasets analysed in a randomization test (including the dataset actually observed).(Whole succession of N beds is randomized each iteration).

Probr(Wtau_absw) � Proportion of actual plus rando-mized datasets which produce a value of Wtau_absw asgreat or greater than the value calculated for the datasetactually observed. (23)Probr(Wtau_sqrw) � Proportion of actual plus rando-mized datasets which produce a value of Wtau_sqrw asgreat or greater than the value calculated for the datasetactually observed. (24)Probr(Wtau_chisqr) � Proportion of randomized data-sets which produce a value of Wtau_chisqr as great orgreater than the value calculated for the dataset actuallyobserved. (25)

Probr(Wtau_symw) � Proportion of randomized data-sets which produce a value of Wtau_symw as great orgreater than the value calculated for the dataset actuallyobserved. (26)

Result of randomization tests:Report observed data signi®cant at level a.

a � Probr(Wtau_absw) using test statistic Wtau_absw.a � Probr(Wtau_sqrw) using test statistic Wtau_sqrw.a � Probr(Wtau_chisqr) using test statisticWtau_chisqr.a � Probr(Wtau_symw) using test statistic Wtau_symw.



Table 5A. Results ± tests of subsequences of ®xed size recognized a-priori against alternative of one or more thickening or thinning upward subsequences ofsame type.

(Figure) stratigraphic unitWay beds are partitionedinto subsequences

Alleged thickeningor thinningupward trends

No. ofbeds

Teststatistics

ResultsParametric test

ResultsRandomization test

ResultsTests of signi®canceof individualsubsequences

Wtau_one1

Wtau_one2

Wtau_chisqre

Wtau_abse1

Wtau_abse2

Prob(Wtau_one1)Prob(Wtau_one2)Prob(Wtau_chisqr)Prob(Wtau_abs1)Prob(Wtau_abs2)[Table 4,Eqs. (8)±(26)].

Probr(Wtau_one1)Probr(Wtau_one2)Probr(Wtau_chisqr)Probr(Wtau_abs1)Probr(Wtau_abs2)Itera � No. ofdatasetsanalysed �denominator ofratio shown.

Prob(Taui) for eachsubsequence [i] of the gsubsequences labelled [1]to [g] on ®gure andSubsequences (if any) thatare signi®cant at acomparison-wise error ratea/g (corresponding to anexperiment-wiseerror rate a).g

(3a) Alternating channelizedand non-channelizedcycles,Turbidite sand-stones, Bullfrog fan,Mineral King roof pendant,Sierra Nevada, Californiaa

Nine non-channelizedthickening alternatingwith eight channelizedthinning (+ one 2 bedchannelized subsequence)

462 + 242� 704

Wtau_one1

� 0á375Wtau_one2

� 0á365Wtau_chisqr� 260

Wtau_abs1

� 0á380Wtau_abs2

� 0á379

Prob(Wtau_one1)� 0á000000


Prob(Wtau_chisqr)� 0á000000

Prob(Wtau_abs1)� 0á000000


Probr (Wtau_one1)� 1/11 � 0á091

Probr (Wtau_one2)� 1/11 � 0á091

Probr (Wtau_chisqr)� 1/11 � 0á091

Probr (Wtau_abs1)� 1/11 � 0á091

Probr (Wtau_abs1)� 1/11 � 0á091

Prob(Taui) for sub-sequences labelled [1] to[18] with values for ninechannelized subsequencesunderlined: 5E-11*, 0�2397,1E-9*, 2 bed subsequence,0á00031*, 0�085, 0á000023*,0�00245*, 0á1995, 0�024,8E-10*, 0�014, 7E-8*,0�00035*, 0á0000265*,0�000272*, 0á24, 6�5E-8*.

Channelized intervalspartitioned by interveningnon-channelized intervals,and vice-versa

Disregarding 2 bedsubsequence [4], all buttwo non-channelized andfour channelizedsubsequences aresigni®cant at the0á05/17 � 0á0029 level. Ofthose signi®cant at thatlevel, all but one are sig-ni®cant at the 0á01/17 � 0á00059 level). Sig-ni®cant values marked byasterisks(*)

676

C.

W.

Harp

er

Ó1998

Inte

rnatio

nal

Asso

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of

Sed

imen

tolo

gists,

Sed

imen

tolo

gy,

45,

657±696

(3b) & (3c) Alternatingmud turbidites andhemipelagite beds.Quaternary, SouthShetland Trench,West Antarctica.

(3B) shows hemipelagitesequence, (3C) mudturbidite sequence.b

Intermeshed mudturbidite andhemipelagite sequences

Hemipelagitesthickening upward.Turbidites thinningupward.

Two setsof 17

Wtau_one10

� 0á232Wtau_one2

� 0á232Wtau_chisqr� 3.49

Wtau_one2

� 0á232

Wtau_abs2

� 0á232






Probr(Wtau_one1)� 71/2000� 0á0355

Probr(Wtau_one2)� 71/2000 � 0á0355

Probr(Wtau_chisqr)� 360/2000 � 0á180Probr(Wtau_abs1)� 271/2000 � 0á1355

Probr(Wtau_abs1)� 271/2000 � 0á1355

Turbidites:Prob(Taui) � 0á129Hemipelagites:Prob(Taui) � 0á079Neither is signi®cantat the 0á05/2 � 0á025 level

Hemipelagite but not tur-bidite sequence is signi-®cant at 0á20/2 � 0á10 level.

(3d) Turbidite sandstones,Macigno Fm. (Oligocene),Northern Apennines, Italyc

Two thickening 35 Wtau_one1

� 0á199Prob(Wtau_one1)� 0á043

Probr(Wtau_one1)� 254/5001 � 0á051

Prob(Taui) for subsequence[1] � 0á00925 which is sig-ni®cant even at 0á02/2 �0á01 level. However, Prob(-Taui) for subsequence [2] �0á8328 which is not sig-ni®cant.

Divided into partitions byan intervening thickshale interval.

Wtau_one2


Probr(Wtau_one2)� 542/5001 � 0á108

Wtau_chisqr� 6.63


Probr(Wtau_chisqr)� 121/5001 � 0á024

Wtau_abs1

� 0á299Prob(Wtau_abs1)� 0á010

Probr(Wtau_abs1)� 76/5001 � 0á015

Wtau_abs2

� 0á280Prob(Wtau_abs2)� 0á013

Probr(Wtau_abs2)� 112/5001 � 0á022

(3e) Weight PercentCalcite, Pelagic Limestonesand Marlstones. TropicShale (Upper Cretaceous),Sit Down Bench, Utahd f

Three decreasingupward

25 Wtau_one1


Probr(Wtau_one1)� 1/2999 � 0á0003

Prob(Taui) for sub-sequences [1], [2], &[3] �0á00134, 0á047* (from Sil-leto, 1947, Table), and0á0083 (from Kendall &Ord, 1990 Table) respec-tively. All but one markedby asterisk (*) are sig-ni®cant even at0á03/3 � 0á01 level.

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imen

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45,

657±696

can be obtained from a table for the unit normaldistribution (Table 4B, Eqs (10)±(12).

Table 4B also outlines a randomization testbased on Wtau_onew.

Results: testing sequence as a whole

Table 5A gives results of parametric and randomiza-tion tests against the alternative of one or moresubsequences of same type. The Table gives results,for three turbidite sequences 3a, 3b/c, and 3d and oneLimestone sequence 3e. The relevant test results arethose for Wtau_one1 and Wtau_one2. (Table 5A alsoshows results for three other test statistics applicableto subsequences of mixed type; these are discussedbelow). As suggested earlier, the reader may wish to®rst visually inspect each sequence plot and make asubjective guess as to whether it shows a pattern;then check results in Table 5A.

Table 5B shows test results for Wtau_one1 andWtau_one2 for ®ve turbidite sequences.

In all examples given, the proportion of valuesof each test statistic as extreme or more extremethan that observed obtained from the randomiza-tion test very closely matches the correspondingprobability obtained using a parametric test.

Testing each of g subsequences at a levelof signi®cance a/g

The above test can indicate that a sequence has oneor more subsequences which are nonrandom, but itwill not indicate which of the subsequences con-tribute(s) the nonrandom character. One cannotsimply test two or more subsequences individuallyat a level of signi®cance of 0á05 for example. Forevenif the null hypothesis is true, theodds areoftenhigh that one or more of the subsequences testedwill appear signi®cant (SAS Institute, 1990; p. 942±44). For example, Lowey (1980; 1992) performedtests of signi®cance on 40 separate subsequences ofturbidites in the Dezadeash Formation of the Yukonusing the Kendall Tau coef®cient. He found threesubsequences showing signi®cant thinning up-ward trends at the 0á05 level. However, under thenull hypothesis the odds of three or more successesin 40 trials is 0á32 (Simpson et al., 1960; p. 137;Lindgren et al., 1978; p. 68). He found ®ve subse-quences showing signi®cant thinning or thickeningupward trends at the two-sided 0á10 level; the oddsof at least ®ve successes at 0á10 level under the nullhypothesis is 0á37.

To test each subsequence for signi®cance, testeach of g subsequences at a level of signi®-cance � a/g, thus achieving an overall, allT

able

5A

.(C

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.

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�0á6

60

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0007

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aB

usb

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,1985,

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.7,

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s18±26;

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.23;

cG

ibau

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1980,

Fig

5;

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Fig

.1;

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,1994,

Fig

.4;

eL

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s(W

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_ch

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Wta

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,&

Wta

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)are

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are

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have

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029.

678 C. W. Harper


Table 5B. Results ± tests of subsequences of ®xed size recognized a-priori against alternative of one or more thickening or thinning upward subsequences ofMixed type.

(Figure) stratigraphic unitWay beds are is partitionedinto subsequences

Allegedthickeningor thinningupward trends

No. ofbeds Test statistics



Results Tests of signi®cance ofindividual subsequences

Wtau_oned1

Wtau_oned2

Wtau_chisqrWtau_abs1

Wtau_abs2

Wtau_sqr1

Wtau_sqr2

Prob(Wtau_one1)Prob(Wtau_one2)Prob(Wtau_chisqr)Prob(Wtau_abs1)Prob(Wtau_abs2)Prob(Wtau_sqr1)Prob(Wtau_sqr2){Table 4, Eqs.(8)±(26)}.

Probr(Wtau_one1)Probr(Wtau_one2)Probr(Wtau_chisqr)Probr(Wtau_abs1)Probr(Wtau_abs2)Itera � No. ofdatasets analysed� denominator of

ratio shown.

Column lists: Prob (jTauij) fortwo-sided test for each subsequence[i] of the g subsequences labelled [1]to [g] on ®gure and Subsequences(if any) that are signi®cantat a comparison-wise errorrate a/g (corresponding to anexperiment-wise error rate a).e

(3d) Turbidite sandstones,Macigno Fm.(Oligocene),Northern Apennines,Italya Divided intopartitions by anintervening thickshale interval.

Two thickening.Tested for onethickening and onethinning upwardsubsequence here

35 Wtau_one1

� 0á199Wtau_one2

� 0á148Wtau_chisqr� 6á63

Wtau_abs1

� 0á299Wtau_abs2

� 0á280






Probr(Wtau_one1)� 254/5001 � 0á051

Probr(Wtau_one2)� 542/5001 � 0á108


Probr(Wtau_abs1)� 76/5001 � 0á015

Prob(Wtau_abs2)� 112/5001 � 0á022

Use jTauij as test statistic for twosided test Lower subsequence [1]Prob(jTauij) � 0á00925 times2 � á0185 signi®cantat 0á05/2 � 0á025 level.Upper subsequence [2]Prob(|Taui|) � 0á3328 times2 � 0á666 which is not signi®cant.

(4a) Turbidite sandstones,Carboniferous Flysch,Southern Moroccob

Three thick shaleintervals dividesuccession into fourwell de®nedsubsequences:Beds 1±95, 96±143,144±258, 259±298.

Three symmetric,post-hoc, but heretested for four sub-sequences (Threethickening, onethinning) re-cognized a-priori.

All 298beds

Wtau_one1

� 0á351Wtau_one2

� 0á275Wtau_chisqr� 89á9

Wtau_abs1

� 0á392Wtau_abs2

� 0á351






Probr(Wtau_one1)� 1/10 � 0á1



Probr(Wtau_abs1)1/10 � 0á1

Probr(Wtau_abs2)1/10 � 0á1

Use jTauij as test statistic for twosided test: Prob (jTauij) for sub-sequences [1] to [4] � 5á0E-7,0á019*, 2á8E-14, 0á278*.Subsequences [1] & [3] marked byasterisks (*) are signi®cant at the0á05/4 � 0á0125 level.).Subsequence [2] is signi®cantat the 0á10/4 � 0á025 level.

Th

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657±696

Table 5B. (Contd.).

(4b) Turbidite sandstones,Dezeadeash Fm.(Jurassic-Cretaceous),Yukon, Canadac Loweydivided sccessioninto partitions ofrelatively thick beds,separated by intervalsof 6 or more thin(<30 cm) beds.

40 sub-sequences(only 17 shown),three thickening,three thinning,rest random.

729(only 300shown inFig. (5E)

Wtau_one1

= 0á035Wtau_one2

� 0á024Wtau_chisqr= 53á3

Wtau_abs1

� 0á133Wtau_abs2

� 0á157



Prob(Wtau_chisqr)� 0.077






Probr(Wtau_abs1)� 7/55 � 0á127


Use jTauij as test statistic fortwo-sided test. Out of fortysubsequences, only one, sub-sequence [12] shown on Fig. 4b,is ``signi®cant'' even at the0á33/40 � 0á00825 level. It hasProb(Taui) � 0á008.

(4b) Turbiditesandstones,Dezeadeash Fm.(Jurassic-Cretaceous),Yukon, Canadac

Lowey dividedsuccession into partitionsof relatively thick beds,separated by intervals of6 or more thin (<30 cm)beds.

Subsequencesshown in Fig. (5E)[Actually only 17out of 40 total] aretested alone here.

300 Wtau_one1

� )0á034Wtau_one2

� )0á031Wtau_chisqr� 38á2

Wtau_abs1

� 0á200Wtau_abs2

� 0á220Watu_sqr1

� 0á057Watu_sqr2

� 0á072






(no analyticsolution forProbr(Wtau_sqrw)

Probr(Wtau_one1)� 1088/1300 � 0á837

Probr(Wtau_one2)� 1046/1300 � 0á8046




Probr(Watu_sqr1)� 2/300 � 0á007

Probr(Watu_sqr2)� 1/300 � 0á003

Use |Taui| as test statistic for two-sided test. Prob(|Taui|) for sub-sequences labeled [1] to [17] on Fig.4b: 0á012, 0á720 (using Kendall &Ord, 1990, table), 0á628, 0á078, 1á0(using Sillito, 1947, table), 0á238(using Kendall & Ord, 1990, table),0á042, 0á400, 0á082, 1á0, 0á510,0á008*, 0á594, 0á494, 0á084, 0á214,0á470.Out of 17 values of Prob( |Taui|)listed above, only that for sub-sequence [12], marked by an aster-isk(*), is signi®cant even at the 0á20/17 � 0á012 level.

aGibaudo, 1980, Fig. 5; Hiscott, 1981, Fig. 1). bGraham 1982, Fig. 4; 1988, Fig. 2á53. cLowey, 1980, table pp. 116±149; Lowey, 1992. dFirst two statistics, Wtau-one1 & Wtau_one2, are designed to test against same type rather than mixed subsequences, but values are shown for comparison. Statistics appropriate fortesting against alleged pattern are shown in boldface. eUsing an experiment-wise error rate a, declare a subsequence i signi®cant at level a if Prob(jTauij) <= a/g.For example for 17 subsequences, declare a given subsequence i signi®cant at level 0á05 if Prob(jTauij) < � 0á05/17 � 0á0029. Prob(jTauij) � Twice Prob(Taui).

680

C.

W.

Harp

er

Ó1998

Inte

rnatio

nal

Asso

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tion

of

Sed

imen

tolo

gists,

Sed

imen

tolo

gy,

45,

657±696

Table 5C. Results ± tests of subsequences of ®xed size recognized a-priori against alternative of one or more symmetric (thickening followed by thinning)subsequences.

(Figure) stratigraphic unitWay beds arepartitioned into subsequences

Allegedthickening orthinningupwardtrends

No. ofbeds Test statistics



Results Tests of signi®cance ofindividual subsequences

Wtau_symb1

Wtau_symb2

Wtau_chisqrWtau_abs1

Wtau_abs2

Prob (Wtau_sym1)Prob(Wtau_sym2)Prob (Wtau_chisqr)Prob(Wtau_abs1)Prob(Wtau_abs2)[Table 4, Eqs. (8)-(26)].

Probr (Wtau_sym1)Probr(Wtau_sym2)Probr (Wtau_chisqr)Probr(Wtau_abs1)Probr(Wtau_abs2)Itera � No. ofdatasets analysed� denominator of

ratio shown.

Column lists: Prob(Taui) forone-sided test for each sub-sequence [i] of the g sub-sequences labelled [1] to [g] on®gure and Subsequences (ifany) that are signi®cant at acomparison-wise error rate a/g(corresponding to an experi-ment-wise error rate a). Usingan experiment-wise error rate a,declare a subsequence i sig-ni®cant at level a if Prob(Taui) <= a/g.

(4a) Turbidite sandstones,Carboniferous Flysch,Southern Moroccoa

Assume the successionis divided into sixpartitions by 5intervening thick shaleintervals. In the lowerpacket, a Shale breakdoes occur at the transi-tion from thickening toa thinningupward subsequence.Similar breaks do notoccur in the top twopackets, but, for sakeof illustration, assumethat breaks did occurat middle of thesesymmetric appearingsubsequences (at beds up236 and 285 beds frombase (see Fig).

Grahamrecognizedthree symmetricsubsequencespost-hoc, buthere tested forthree symmetricsubsequencesrecognizeda-priori:subsequences[1] & [2] groupedtogether, [3] &[4] on Fig. 4a.

All 298beds

Wtau_sym1

� 0á292Wtau_sym2

� 0á186(Wtau_chisqr� 91á8

Wtau_abs1

� 0á373Wtau_abs2

� 0á376

Prob(Wtau_sym1)� 0á00000

Prob(Wtau_sym2)� 0á00000



Prob(Wtau_abs2

� 0á00000

Probr(Wtau_sym1)� 1/36 � 0á028

Probr(Wtau_sym2)� 1/36 � 0á028




Use Taui as test statistic fora one sided test. Prob(Taui)for the 6 subsequences thatmake up the three symmetricsubsequences listed bottomto top: 2á50E-7, 0á009547, 1á37E-9, 0á00076, 0á00063, 0á0162*.Out of the six subsequences,®ve (marked by asterisk(*))are signi®cant at the 0á05/6 � 0á0083 level; All aresigni®cant at the 0á10/6 � 0á0170 level. A one-sidedrejection region should be usedhere in test against the alter-native of three symmetric(thickening followed bythinning upward) sub-sequences.

aGraham 1982, Fig. 4; 1988, Fig. 2.53. bStatistics appropriate for testing against alleged symmetric pattern are shown in boldface.

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imen

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657±696

sequence-wide, level of signi®cance � a. Underthe null hypothesis the odds of one or moresubsequence testing as signi®cant at the a/g levelis less than or equal to a (Miller, 1981, p. 67±70;Harper, 1984c; SAS Institute, 1990; p. 942±944).This procedure is especially useful if g is not toolarge and P-values encountered are all small(Miller, 1981; p. 8).

As an example, consider the two subsequencesshown in Fig. 3d. Hiscott (1981) calls the lowersubsequence `quite convincing' but the upper onenot. The probability of obtaining a value ofKendall's Tau as extreme or more extreme thanthat of the lower subsequence is 0á009 which issigni®cant at the a/g � 0á02/2 level; the corre-sponding probability for the upper subsequenceis 0á828 which is not signi®cant.

Results: testing each of g subsequences

Table 5A gives results of tests of signi®cance ofindividual subsequences of same type. As sug-gested earlier, the reader may wish to ®rstvisually inspect each sequence plot and make asubjective guess as to which subsequences show apattern; then check results in Table 5A.

Testing against mixed (both thickeningand thinning upward) subsequences

Testing sequence as a whole using weightedaverage of Kendall Tau coef®cients as teststatistic

Suppose N beds are divided into g subsequences oneormoreofwhichappears to thickenupward,andoneor more to thin upward. For such a case, I proposeparametric and randomization tests based on fourtest statistics Wtau_absw, Wtau_sqrw, Wtau_chisqrand Wtau_symw as outlined in Table 4C.

Consider ®rst Wtau_absw (Table 4C, Eq. (14)):where

Wtauÿÿabsw �Xi� g

i�1

CiwjTauij

jTauij � absolute value of Taui, and Ciw �weighting coef®cient for i-th subsequence.

As a rationale for using Wtau_absw consider thefollowing: Any one of the g subsequences, forexample the i-th, couldbe tested individuallyagainstan alternative hypothesis that there is either athickening or a thinning upward pattern usingKendall's Tau and a two-sided rejection region. Thetest is equivalent to testing jTauij, the absolute valueof Taui, using a one-sided upper rejection region.

If Taui [)1 £ Taui £ 1] is distributed as anormal distribution with a mean and mode ofzero and a standard deviation � Standard devi-ation Taui, then its absolute value, jTauij,[0 £ jTauij £ 1], is distributed as half-normal, ortwice the normal distribution, i.e. as:

function (jTauij) � 2 (Normal (0, Standard Devi-ation of Taui)),

[0 £ jTauij £ 1], which has a mode at zero and astandard deviation � Standard deviation of Taui.(Wadsworth & Bryan, 1960; p. 153).

Further, if each of the g subsequences has morethan 10 beds, then under the null hypothesis, thewithin group Tau coef®cients, jTauij, may betreated as independent, normally distributedvariates with mode of zero and a standarddeviation � Standard deviation Taui. Theirweighted sum is distributed as given in Table 4C,Eq. (15)±(17).

Consider next the test statistic Wtau_sqrw

de®ned as:

Wtauÿÿsqrw �Xi� g

i� 1

CiwTau2i

(Table 4C, Eq. (18)). No parametric test using it isavailable as its distribution is not known. Ran-domization tests based on it should give resultscomparable to randomization tests based onWtau_absw.

As a third alternative we can use Wtau_chisqr:

Wtauÿÿchisqr �Xi� g

i� 1

Ci3

�Taui=(Standard Deviation of Taui��2

where Ci3 � 1. (Table 4C, Eq. (19)).If each of subsequences have 10 or more beds,

each Taui/(Standard Deviation of Taui) is distrib-uted as unit normal, and their equally weightedsum will be distributed as Chi-Square with gdegrees of freedom (Wadsworth & Bryan, 1960; p.161±162). The Chi-square test weights each par-tition segment equally. If partitions differ appre-ciably in size, Wtau_absw or Wtau_sqrw are bettertest statistics as they weight subsequences inproportion to their size.

Geologists commonly recognize symmetric sub-sequences (thickening followed by thinning up-ward sandstone subsequence pairs, or vice-versa).Higgs (1991) interprets coarsening followed by®ning upward subsequences as regressive-trans-gressive cycles with central sandstone beds

682 C. W. Harper


representing regressive maximum. Pimpirev &Sarmiento, 1993, Figs. 7 & 9) interpret thickening-thinning turbidite subsequences as progradation-al-retrogradational and thinning-thickening sub-sequences as the reverse. See also Figs 4a and 5k.

To test against the alternative of g/2 symmetricsubsequences (thickening followed by thinningupward subsequence pairs) I suggest usingWtau_symw (Table 4C, Eq. (21), a modi®ed ver-sion of Wtau_onew (Table 4B, Eq. (9)):

Wtau symw �Xi� g

i�1

�ÿ1�i�1Ciw Taui

Given a succession of symmetric subsequences,the test statistic sums +Ciw Taui for lower seg-ments of symmetric sequences and )Ciw Taui forupper segments. Thus, both the lower and upperportion of any symmetric pair will make apositive contribution to the statistic. Againstthinning followed by thickening subsequencepairs use ()1)i in place of ()1)i + 1. Wtau_symw

is distributed as Wtau_onew (see above).

Results: testing sequence as a whole

Table 5B shows results of tests against the alter-native of subsequences of mixed type for ®veturbidite sequences: Fig. 3d; Fig. 4a, assumingfour subsequences; Fig. 4b, all 729 beds; andFig. 4b, 300 beds only. The relevant test resultsare those for Wtau_chisqr, Wtau_abs1, andWtau_abs2 (and Wtau_sqr1, and Wtau_sqr2 calcu-lated for one sequence only). (Table 5B alsoshows results for two other test statistics(Wtau_one1, Wtau_one2) applicable to subse-quences of same type; these are discussed above).As suggested earlier, the reader may wish to ®rstvisually inspect each sequence plot and make asubjective guess as to whether it shows a pattern;then check results in Table 5B.

As noted earlier, Table 5A gives results ofparametric and randomization tests against thealternative of one or more subsequences of sametype. While test results for Wtau_chisqr,Wtau_abs1, and Wtau_abs2 are not relevant to testsagainst same type, they were calculated for pur-poses of illustration and are listed in the Table.

Table 6A. Test of subsequences recognized post-hoc (after examining thicknesses) against alternative of one or morewithin-subsequence trends.

Given: A stratigraphic sequence of N beds and an alternative hypothesis that postulates post-hoc that the sequence isdivided into g subsequences one or more of which exhibit a thickening and/or thinning upward trend.

Sample space:As set of possible outcomes, consider all possible partitionings P of the sequence into 1, 2, . . . up to g subsequences,and all possible permutations of the N beds amongst the N stratigraphic positions in the g subsequences.

General form of test statistics:Max(Testw,f) � Maximum value of Testw,f attainable by partitioning the N beds into 1,2. g subsequences.(27)Min(Testw,f) � Minimum value of Testw,f attainable by partitioning the N beds into 1,2. g subsequences. (28)Test statistic Testw,f de®ned above in Table 4A, Eq. (8).

For alleged thickening upward or mixed subsequences use Max (Testw,f), for thinning upward subsequences use Min(Testw,f).

If the number of beds were small the test statistic could be calculated by examining all possible partitions of theobserved sequence into 1, 2,. . . g subsequences and calculating Max(Testw,f) for each partition. However, for a largenumber of beds the number of possible partitions becomes astronomic making it impractical to examine all possiblepartitions. For example, the number of distinct partitions of 48 beds into 4 subsequences of one or more beds each is47!/(4!(43!)) � 178 365. For 100 beds in the sequence, it is 3á8 million.

In such cases, I suggest we estimate the test statistic Max(Testw,f) as follows:(1) calculate Kendall's Tau for the sequence as a whole, and tentatively record this value as the initial estimate of Max(Testw,f).(2) examine all possible partitionings of the sequence into two subsequences. Determine that partitioning whichmaximizes Testw,f; if the maximum attainable Testw,f is greater than the current estimate of Max(Testw,f) (it invariablyis) then record it as the current estimate. If g � 2 then stop, else(3) examine all possible further partitions of the two subsequences into three subsequences. Find an additional thirdpartition which maximizes the Testw,f; if the maximum attainable Testw,f is greater than the current estimate of Max(Testw,f) (it invariably is), then record this value as the current estimate. If g � 3 stop, else(4) repeat step 3 for four, ®ve. . . g subsequences.



Table 6B. Test of subsequences recognized post-hoc (after examining thicknesses) against alternative of one or morewithin-subsequence trends.

Randomization test against alternative ofsubsequences recognized post-hoc including two ormore thickening or thinning upward subsequencesof same type:

Randomization test against alternative of subsequencesrecognized post-hoc including two or more thickening orthinning upward subsequences of mixed type:

Test null hypothesis using Max(Wtau_onew),w � 1,2 (de®ned below) for thickening upward,Min(Wtau_onew), w � 1,2 (de®ned below) for thinningupward alternative and Max(Wtau_symw), w � 1,2(de®ned below) for symmetric alternative.

Test null hypothesis usingMax(Wtau_absw), w � 1,2 (de®ned in Eq. (32) below),and/or Max(Wtau_sqrw), w � 1,2 (de®ned in Eq. (33)below), and/or Max(Wtau_chisqr) (de®ned in Eq. (34)below).

Max(Wtau_onew) � Max(Testw,1)Max(Wtau_onew) � Maximum weighted sum ofwithin-subsequence Kendall's Tau coef®cientsattainable for g subsequences using weightingcoef®cients of type w (w � 1,2). (29)

Max(Wtau_symw) (for symmetric sequences,Eq. (35) below).

Similarly, Min(Wtau_onew) � Min(Testw,1) �Minimum weighted sum. (29¢)

Max(Wtau_absw) � Max(Testw,2) � Maximum weightedsum of absolute values of within-subsequence Kendall'sTau coef®cients attainable for g subsequence usingweighting coef®cients of type w (w � 1,2). (32)

Itera � Number of datasets analysed in arandomization test (including the dataset actuallyobserved):

Max(Wtau_sqrw) � Max(Testw,3) � Maximum weightedsum of squares of within-subsequence Kendall's Taucoef®cients attainable for g subsequences usingweighting coef®cients of type (w � 1,2). (33)

Probr(Max(Wtau_onew)) � Proportion of actual plusrandomized datasets which produce a value ofMax(Wtau_onew) as great or greater than the valuecalculated for the dataset actually observed. (30)

Max(Wtau_chisqr) � Max(Test3,4) � Maximumunweighted sum of squares of [within-subsequenceKendall's Tau coef®cients each divided by its standarddeviation] attainable for g subsequences. (34)

Probr(Min(Wtau_onew)) � Proportion of actual plusrandomized datasets which produce a value ofMin(Wtau_onew) equal to or less than the valuecalculated for the dataset actually observed. (31)

Max(Wtau_symw) � Max(Testw,5) � Maximum weightedsum using Wtau_symw. (35)

Result of randomization test: Itera � Number of datasets analysed in a randomizationtest (including the dataset actually observed):

Report observed data signi®cant at level a Probr(Max(Wtau_absw)) � Proportion of actual plusrandomized datasets which produce a value ofMax(Wtau_absw) as great or greater than the valuecalculated for the dataset actually observed. (36)

a � Probr(Max(Wtau_onew)) using test statisticProb(Max(Wtau_onew))a � Probr(Min(Wtau_onew)) using test statisticProb(Min(Wtau_onew))

Probr(Max(Wtau_sqrw)) � Proportion of actual plusrandomized datasets which produce a value ofMax(Wtau_sqrw) as great or greater than the valuecalculated for the dataset actually observed. (37)

Probr(Max(Wtau_chisqr)) � Proportion of actual plusrandomized datasets which produce a value ofMax(Wtau_chisqr) as great or greater than the valuecalculated for the dataset actually observed. (38)

Probr(Max(Wtau_symw)) � Proportion of actual plusrandomized datasets which produce a value ofMax(Wtau_symw) as great or greater than the valuecalculated for the dataset actually observed. (39)

684 C. W. Harper


A test using Wtau_symw was applied to Fig. 4a,assuming six subsequences, and results are givenin Table 5C (the Table also shows results forWtau_chisqr, Wtau_abs1 and Wtau_abs2).

In all examples given, the proportion of valuesof each test statistic as extreme or more extremethan that observed obtained from the randomiza-tion test very closely matches the correspondingprobability obtained using a parametric test.

Testing each of N subsequencesat a level of signi®cance a/g

For subsequences of mixed type, use proceduresas outlined above for the case of subsequences ofsame type.

Results: testing each of g subsequences

Tables 5B and 5C give results of tests of signi®-cance of individual subsequences of mixed type.As suggested earlier, the reader may wish to ®rstvisually inspect each sequence plot and make asubjective guess as to which subsequences show apattern; then check results in Tables 5B±C.

TESTING AGAINST TWO OR MORESUBSEQUENCES POSTULATEDPOST-HOC (PURELY ON THE BASISOF OBSERVED THICKNESS PATTERNS)INCLUDING ONE OR MOREWITHIN-SUBSEQUENCE THICKENING/THINNING UPWARD TRENDS

Introduction

Tests described in the previous section aredesigned for ®xed subsequences separated by

breaks. They should not be applied to subse-quences recognized by scanning a sequence post-hoc. Long random successions typically containimbedded patterns of thickening/thinning up-ward subsequences that could be singled outpost-hoc.

If a sequence appears post-hoc to be dividedinto, for example, three thickening upward sub-sequences of say 15, 22 and 32 beds each, it is notenough to ask what is the probability of obtainingthree such subsequences a-priori. A better ques-tion is: what is the probability that a randomsequence of the beds could be partitioned intothree subsequences (or into one, or into twosubsequences) of any length that show a within-subsequence thickening upward pattern as ex-treme or more extreme than that actually ob-served.

General form of the test statistic

The form of the test statistics suggested are givenin Table 6A. These are based on test statisticsde®ned for the ®xed partition case in Table 4A±C.

They are the maximum value of test statisticsde®ned in Table 4A, Eq. (8), that are attainable bypartitioning the total sequence of N beds into 1, 2,¼.up to g subsequences. (As it turns out, in allcases studied, this maximum value is the valueobtained by a partitioning into g subsequences).

Testing against same-type (all thickening or allthinning upward) subsequences

Testing sequence as a whole using most ex-treme value of weighted average of Kendall Taucoef®cients attainable by partitioning as teststatistic

Table 6B. (Contd.)

Randomization test against alternative ofsubsequences recognized post-hoc including two ormore thickening or thinning upward subsequencesof same type:

Randomization test against alternative of subsequencesrecognized post-hoc including two or more thickening orthinning upward subsequences of mixed type:

Report observed data signi®cant at level a.a � Probr(Max(Wtau_absw)) using statisticMax(Wtau_absw).a = Probr(Max(Wtau_sqrw)) using statisticMax(Wtau_sqrw).a = Probr(Max(Wtau_chisqr)) using test statistic(Max(Wtau_chisqr).a = Probr(Max(Wtau_symw) usingProbr(Max(Wtau_symw)).



Table 7A. Results ± randomization test against alternative of subsequences recognized post-hoc including one or more thickening or thinning upwardsubsequences. Same type.


Alleged thickening orthinning upward trends(recognized post hoc) No. of beds

Test statistic Max(Wtau_one1)[Thickening upward]Min(Wtau_one1) [Thinningupward]. [Eqs. (29) & (29'),Table 6].

ResultsProbr (Max(Wtau_one1))Probr(Min(Wtau_one1)).Itera � No. of datasetsanalysed � denominator ofratio shown.]. [Eqs. (30)±(31),Table 6].

(1b) Hypothetical sequence Two thickening 10 Max(Wtau_one1) � 1 2/1000 � 0á002(1d) Hypothetical sequence Three thickening 10 Max(Wtau_one1) � 1 14/1000 � 0á014(3d) Turbidite sandstones, Macigno Fm.

(Oligocene), Northern Apennines, ItalyfTwo thickening 35 Max(Wtau_one1) � 0á266 242/2000 � 0á121

(3e) Weight Percent Calcite, PelagicLimestones and Marlstones, TropicShale (Upper Cretaceous), Sit DownBench, Utahg

Three decreasing upward 25 Max(Wtau_one1) � 0á661 3/1000 � 0á003

(5a) d018(%%) Calcium Carbonate,Annual layers of Glacial lce,Eemian (135,000±115,000 y.b.p.)portion of the GRIP Summit Ice core,Central Greenlandi

Three increasing upward 166 Max(Wtau_one1) � 0á683 1/100 � 0á01

(5b) Maximum Particle size (mean valueof ten largest clasts), Devonian alluvialfan conglomerates, Hornelen Basin,Borwayh

Six increasing upward 62 Max(Wtau_one1) � 0á539 1/300 � 0á0033

(5c) Turbidite sandstone, San SalvatoreSandstone(Miocene), NorthernApinnines, Italyb

Three thickening 24 Max(Wtau_one1) � 0á529 56/2000 � 0á028

(5d) Turbidite sandstones, San SalvatoreSandstone (Miocene), NorthernApinnines, Italyb

Two thickening 18 Max(Wtau_one1) � 0á457 49/1000 � 0á049

(5e) Turbidite sandsstones, Carbonatefan deposits, Loma del Toril Fm.(Upper Jurassic), Spainc

Five thinning 46 Max(Wtau_one1) � 0á484 48/1000 � 0á048

686

C.

W.

Harp

er

Ó1998

Inte

rnatio

nal

Asso

cia

tion

of

Sed

imen

tolo

gists,

Sed

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tolo

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45,

657±696

The test statistic

To test against the alternative of g thickening (orthinning) upward subsequences, I suggest arandomization test based on Max(Wtau_onew),Eq (29), Table 6B (or Min(Wtau_onew), Eq. (29)¢)as a test statistic. The proposed test is outlined inTable 6B. Max(Wtau_onew) is equal to the max-imum value of Wtau_onew attainable by parti-tioning the sequence into 1,2. . .g subsequences.

To see how the test statistic Max(Wtau_onew) iscomputed consider the sequence shown inFig. 1b. Assume the succession is divided post-hoc into two subsequences giving a value ofWtau_one1 � 1. The sequence is randomized foritera (� 1000 for example) times. For each ran-domized sequence Wtau_one1 is computed for thetwo-subsequence partition which maximizesWtau_one1. For example one randomization ofthe sequence shown in Fig. 1b is shown in Fig. 1c;the partitioning of the randomized sequencewhich maximizes Wtau_one1 � 0á58 is shown inFig. 1g; thus Max(Wtau_one1) � 0á58. The ran-domized dataset would be counted as one whichproduces a value of Max(Wtau_one1) less than thevalue calculated for the dataset actually observed.In a test of the sequence shown in Fig. 1d and hagainst the alternative of three thickening upwardsubsequences, Max (Wtau_one1) � 1 for the se-quence actually observed.

Results

Results for 15 different sequences, mostly turbid-ites, are shown on Table 7A. Again, the readermay wish to ®rst visually inspect each sequenceplot and make a subjective guess as to whether itshows a pattern; then check results in Table 7A.

Testing against mixed (both thickeningand thinning upward) subsequences

Testing sequence as a whole using most ex-treme value of weighted average of Kendall Taucoef®cients attainable by partitioning as teststatistic

The test statistic

Given a sequence of N beds that appears ad-hoc tobe divided into g subsequences, some thickeningupward and some thinning upward, using arandomization test identical to the one justdiscussed with one exception is suggested. Thetest should use a test statistics Max(Wtau_absw),(5

f)T

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Max(W

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0á5

51

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.(O

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64

Max(W

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0á3

94

77/1

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0á0

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.(O

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31

Max(W

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0á4

41

381/1

200�

0á3

175

(5i)

Tu

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.(O

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es,

Italy

eF

ou

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icken

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45

Max(W

tau

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0á4

80

11/1

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0á0

11

(5j)

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san

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.(O

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30

Max(W

tau

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0á4

57

59/1

000�

0á0

59

aH

iscott

,1980,

Fig

.15F

;b

Mu

tti

1974,

Fig

.5-S

ecti

on

s2,

12;

cR

uiz

-Ort

iz,

1983,

Fig

.5,

(megase

qu

en

ces

IV-V

II);

dG

ibau

do,

1980,

Fig

.7,

Level

10;

His

cott

,1981,

Fig

.2;

eG

ibau

do,

1980,

Fig

.9,

Levels

6,4

,15;

fG

ibau

do,

1980,

Fig

.5;

His

cott

,1981,

Fig

.1;

gS

eti

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eit

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,1994,

®g.

4;

hS

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1977,

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.3,

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Table 7B. Results ± randomization test against alternative of subsequences recognized post-hoc including one or more thickening or thinning upwardsubsequences. Mixed, including symmetric type.

(Figure) Stratigraphic unit Alleged thickening orthinning upwardtrends recognizedpost-hoc

No. of beds Test statisticsMax(Wtau_sqr1)Max(Wtau_abs1)Max(Wtau_sym1)e

[latter used to testagainst symmetricsubsequences][Max(Wtau_chisqr))not used]

ResultsProbr(Max(Wtau_sqr)Probr(Max(Wtau_abs1)Probr(Max(Wtau_sym1)Itera � No. of datasetsanalysed � denominator ofratio shown. ]{Eqs. (29)±(39), Table 6]

(4a) Turbidite sandstones, Beds 144±258�Subsequence [3]. Carboniferous Flysch,Southern Moroccoc

Beds 144±258 Middlesymmetric (thickeningfollowed by thinning)subsequence [3]

115 Max(Wtau_sqr1)�0á267Max(Wtau_abs1)�0á516Max(Wtau_sym1)�0á523

1/106�0á0091/43�á0231/78�0á013

(4a) Turbidite sandstones, Beds 259±298�Subsequence [4]. Carboniferous Flysch,

Southern Moroccoc

Beds 259±298 Uppersymmetricsubsequence [4]


1/200�0á0051/100�0á0101/80�0á0125

(5k) Turbidite sandstones, Macigno Fm.(Oligocene), Northern Apennines, Italyd

One thickening & onethinning


229/1000�0á229187/1000�0á187136/3000�0á0453

(5l) Turbidite sandstones, Upper 41 beds[beds above lowest dashed line] Macigno Fm.(Oligocene), Northern Apennines, Italya

Subsequences [2]±[10].Two thickening andseven thinning

(upper 41of 60)

Max(Wtau_sqr1)�0á412Max(Wtau_abs1)�0á574

410/1000�0á41064/106�0á604

(5l) Turbidite sandstones, Macigno Fm.(Oligocene), Northern Appennines, Italya

Subsequences [1]±[10].Three thickening andseven thinning

60 Max(Wtau_sqr1)�0á501Max(Wtau_abs1)�0á696

1/99�0á011/99�0á01

(5m) Turbidite sandstones, Las TortulasFormation (Devonian-Carboniferous),Chilean Andesb

Five thinning andthree thickening

147 Max(Wtau_sqr1)�0á201Max(Wtau_abs1)�0á412

1/19�0á0531/13�0á076

a Gibaudo, 1980, Fig. 11, Levels 17-18.; Hiscott, 1981, Fig. 3; b Bahlburg & Breitkreuz, 1993, Fig 5b;c Graham, 1982, Fig. 4; 1988, Fig. 2.53; d Gibaudo, 1980, Fig. 7 Level 45;e Statistics appropriate for testing against alleged pattern are shown in boldface.

688

C.

W.

Harp

er

Ó1998

Inte

rnatio

nal

Asso

cia

tion

of

Sed

imen

tolo

gists,

Sed

imen

tolo

gy,

45,

657±696

Max(Wtau_sqrw) and/or Max(Wtau_chisqr) de-®ned in Table 6B as test statistics. Wtau_chisqrweights subsequences equally, so Max(Wtau_chi-sqr) should be used only if equal weighting isappropriate.

To test for a single coarsening followed by®ning upward sequence use Max(Wtau_symw)(Eq. (39), Table 6C).

Results

Results for ®ve different turbidite sequences areshown on Table 7B. The ®rst three examples,Fig. 4a, subsequences [3], [4], Fig. 5k, are testedagainst the alternative on a single symmetric(thickening followed by thinning) upward se-quence. The last three, Fig. 5l, upper ninesubsequences [2]±[10], Fig. 5l, all subsequences[1]±[10], and Fig. 5m are tested against thealternative of mixed type. Again, the readermay wish to ®rst visually inspect each sequenceplot and make a subjective guess as to whetherit shows a pattern; then check results inTable 7B.

SUMMARY OF KEY POINTS

1 Tests for trend using test statistics based onKendall's S and Tau are superior to those basedon nearest-neighbour test statistics. They makesequence-wide (or subsequence-wide) compari-sons of bed thicknesses rather than merelycompare each bed thickness with that of the bedsimmediately above and below it.2 Both parametric tests (where available) andrandomization tests are suitable. Procedures areoutlined in Table 1.3 To test against the alternative of a singlethickening (and/or thinning) upward trend, useKendall's S or equivalently Kendall's Tau.Table 2 gives details of test procedures. Ta-bles 3A and 3B give test results for sequencesanalysed.4 Multiple subsequence trends arise in twocontexts: those where subsequences are recog-nized a-priori (separated by breaks such as thicksequences of hemipelagic shale) and those pos-tulated post-hoc (purely on the basis of observedthickness patterns). In both cases, purportedsubsequences may be same-type or mixed (thelatter including both thickening and thinningupward subsequences, with symmetric subse-quence pairs as a special case).

5 As a test against the alternative of subsequencesrecognized a-priori, use as a test statistic theweighted sum of functions of the Tau coef®cientcalculated for the individual subsequences:

(a) Against the alternative of one or morethinning upward subsequences of same type(either all thickening or all thinning) use theweighted sum of Tau coef®cient. Tables 4A and4B give test details. Table 5A gives test results forsequences analysed.

(b) Against the alternative of mixed (boththickening and thinning upward) subsequencesuse the absolute value of Tau coef®cient, [Taucoef®cient]2, or [Tau coef®cient/(Standard Devia-tion of Tau coef®cient)]2; against the alternative ofsymmetric subsequence pairs use ((± 1)i+1) (Taucoef®cient). Tables 4A and 4C give test details.Tables 5B and 5C give test results for sequencesanalysed.

6 The above tests can indicate that a sequence hasone or more subsequences which are nonrandom,but it will not indicate which. To test eachsubsequence for signi®cance, test each of gsubsequences at a level of signi®cance � a/g,thus achieving an overall, sequence-wide, level ofsigni®cance � a. Tables 5A±C give test resultsfor sequences analysed.7 As a test against the alternative of subsequencesrecognized post-hoc, use as a test statistic themaximum value of the appropriate test statistic(cited in item (5) above) that is attainable bypartitioning the total sequence of beds into 1,2,. . .up to g subsequences. Table 6A gives detailsof tests. In particular:

(a) Against the alternative of one or morethinning upward subsequences of same type(either all thickening or all thinning) use themaximum value of the weighted sum of functionsof Tau coef®cient (i.e. of the test statistic cited insection 5(a) above). Table 6B gives details of test.Table 7B gives test results for sequences ana-lysed.

(b) Against the alternative of mixed (boththickening and thinning upward) subsequencesuse the maximum value of the absolute value ofTau coef®cient, [Tau coef®cient]2, or [Tau coef-®cient/(Standard Deviation of Tau coef®cient)]2;against the alternative of symmetric subse-quence pairs use the maximum value of ((±1)i+1) (Tau coef®cient). Table 6B gives details oftest. Table 7B gives test results for sequencesanalysed.

Notation and test statistics are summarized inAppendices 1 and 2.



A POSSIBLE DIRECTION FOR FUTURERESEARCH

Suppose we are not able to reject a null hypoth-esis of random succession. It could be that anonrandom pattern does exist but we lack suf®-cient data to show it. Or it could be that thesequence is for all practical purposes trulyrandom. Can we rule out the possibility that thelatter holds, and yet the observed sequence ofbeds is generated by a deterministic but chaoticprocess? Successive values of quantitative vari-ables from a very simple purely deterministicnonlinear process can look like random noise(May, 1989; Middleton, 1990).Consider the logistic equation:

xt�1 � vxt�1ÿ xt�

where x1,. . .xn represent the value of a variate atsuccessive times and v is a constant. For example,xt may represent sedimentation rate at time t.Prokoph & Barthelmes (1996) discuss its applica-tion to nonlinear sedimentary sequences.

Vivaldi (1989) notes that this is the formula fora parabola. Casti (1989; p. 228) discusses thepiecewise-linear function:

xt�1 � 2 xt if 0 � � � 1=2

xt�1 � 2ÿ 2xt if 1=2 � xt � 1;

topologically equivalent to the above with v � 4.Suppose that successive bed thicknesses (Di) in

a regime are so determined, with bed thicknessesscaled so that they will vary up to a maximumthickness of 4 metres:

Di�1 � v[Di�1ÿ Di�� Ei

where Di,¼,Dn represent bed thicknesses mea-sured in fractions of maximum thickness and Ei isan èrror' term that models random ¯uctuations inthickness of bed deposited from one stratigraphiclevel to the next. Note that Di will vary between 0and 1, the latter corresponding to a bed 4 metres.Clearly the random term could be important in areal world situation. However, to see the effects ofthe deterministic component, let us set Ei nearzero. Suppose the latter holds. What pattern ofsequence of bed thicknesses, if any, would result?For v > 3á57, the output of such a process wouldlook `for all the world like samples from somerandom process' (May, 1989; Vivaldi, 1989; Mid-dleton, 1990; 1991).

One may test whether or not the logisticequation holds by plotting the thickness ofadjacent beds. A few sequences of bed thickness-es have been studied and shown obviously not tofollow the logistic equation. For example, Archeret al. (1995; Fig. 5a±c) plotted scattergrams ofsuccessive-lamina thicknesses for Carboniferoussilty tidal rhythmites in the Eastern Interior Coalbasin; these show no pattern at all, let alone aparabolic function. Nor do any of the sequencesshown in Figs 1±5 herein ®t. On the other hand,imagine a sequence of strata made up of in-placeaccumulations of skeletons of a benthonic organ-ism. Hastings & Higgins (1994) use the logisticequation to model population size of a livingcoastal marine species distributed along a one-dimensional habitat such as a coast with pelagiclarva that are widely redistributed each genera-tion. They ®nd that population size can varychaotically for periods of over eight thousandyears (Hastings & Higgins, 1994; Figs 1 & 2). Soperhaps output of skeletal debris derived fromsuch a population, and hence thicknesses ofskeletal beds would vary chaotically also. Thus,a purely deterministic process of bed thicknessmay effectively mimic sequences due to randomprocesses.

ACKNOWLEDGMENTS

Thanks to Dr R. N. Hiscott for suggesting the topic tome and Dr J. W. F. Waldson for helpful suggestions thatimproved the manuscript.

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Manuscript received 12 February 1996;revision accepted 26 August 1997.

692 C. W. Harper


Appendix 1. Notation used.

Indices:

i � subsequence index. Ranges over g subsequences.

f � function index. Selects one of 5 functions of Taui calculated for a subsequence i:Functionf (Taui) � simple function of Taui of type f. (f � 1,2,3,4,5):Function1 (Taui) � Taui,Function2 (Taui) � Absolute Value Taui � jTauij,Function3 (Taui) � Taui

2,Function4 (Taui) � (Taui/Standard deviation of Taui)

2,Function5 (Taui) � ((± 1)i+1) Taui.

w � weighting coef®cient index. Selects one of three weighting coef®cients Ciw:Ciw � weighting coef®cient of type w for the i-th subsequence (w � 1,2,3):

Ci1 � �Ni (Niÿ 1�=2�.Xj� g

j� 1

Nj�Nj ÿ 1�=2�

Ci2 � Ni/NCi3 � 1 (for a Chi-square test).

Variables:g � total number of subsequences in a sequence.N � Total number of beds in sequence.Ni � No. of beds in subsequence i.Rij: for two beds i & j, i above j, having thicknesses (or other variate values) ti & tj:

Rij � + 1 if ti > tj (i.e the upper bed is thicker than the lower bed i.Rij � 0 if ti � tj (beds i and j are tied (i.e. of equal thickness).Rij � ± 1 if ti < tj (i.e. the upper bed is thinner than the lower bed i.

Si � Kendall's S calculated for subsequence i.Taui � Kendall's Tau coef®cient calculated for subsequence i.

Prob (test statistic) � Probability of obtaining a value of test statistic as extreme or more extreme than that actuallyobserved.

Probr (test statistic) � Proportion of actual plus randomized datasets which produce a value of test statistic asextreme or more extreme than the value calculated for the dataset actually observed.



Appendix 2. Test statistics used. Equation numbers same as in Tables 2±6. See Appendix 1 for notation.

Test statistics de®ned for test against single trend alter-native.

S � Kendall's S �Xi�n

i� 1

Xj�n

j� i�1

Rij �1�Tau � Kendall's Tau = S/[N (Nÿ 1�=2� �2�

Test statistics de®ned for case where subsequences are recognized a-priori.

Testw;f = the general form of several proposed test statis-tics:

Wtau sqrw � Test�w; 3� :

Testw;f �Xi�g

i�1

CiwFunctionf�Taui�: w � 1; 2; 3 �8� Wtau sqrw �Xi� g

i� 1

CiwTaui2 w � 1; 2: �18�

Wtau onew � Test�w; 1� : Wtau chisqr � v2 � Test�3; 4�Wtau onew �

Xi�g

i�1

CiwTaui w � 1; 2: �9� Wtau chisqr �Xi� g

i� 1

�Taui=�Standard

deviation of Taui��2 �19�Wtau absw � Test�w; 2� : Wtau symw � Test�w; 5�

Wtau absw �Xi�g

i�1

CiwjTauij w � 1; 2: �14� Wtau symw �Xi� g

i� 1

�ÿ1�i�1CiwTaui w � 1; 2: �21�Against thinning followed by thickeningsubsequence pairs use �ÿ1�i in place of �ÿ1�i�1 in Eq. (21).

Test statistics de®ned for case where subsequences are recognized post-hoc.

Max (Testw,f) � the general form of several proposed test statistics � the maximum (or minimum) value of Testw,f

de®ned above that is attainable by partitioning the total sequence of N beds into 1, 2, ¼¼up to g subsequences.Max (Wtau_onew) � the maximum value of Wtau_onew attainable by partitioning the sequence into 1,2. . . . . .gsubsequences.Min (Wtau_onew) � Max (±Wtau_onew) � the minimum value of Wtau_onew attainable by partitioning the sequenceinto 1,2¼..g subsequences.Max (Wtau_absw) � the maximum value of Wtau_absw attainable by partitioning the sequence into 1,2. . . . . .gsubsequences.Max (Wtau_sqrw) � the maximum value of Wtau_sqrw attainable by partitioning the sequence into 1,2. . . . . .gsubsequences.Max (Wtau_chisqr) � the maximum value of Wtau_chisqr attainable by partitioning the sequence into 1,2. . . . . .gsubsequences.

Max (Wtau_symw) � the maximum value of Wtau_symw attainable by partitioning the sequence into 1,2. . . . . .gsubsequences.

694 C. W. Harper


Appendix 3. Nearest-neighbour comparison tests.

Tests commonly used by geologists include:(1) A test of number of turning points in the sequence studied, or, equivalently, the number of runs up or down

(Miller & Kahn, 1962, p. 331±332; Heller & Dickinson, 1985; Waldron, 1986, 1987; Milenkovic, 1989, p 109±110 112; Kendall & Ord, 1990, p. 18±20). The expected number of turning points in a random sequence of Nbeds is (2/3) (N-2), and the expect number of runs up or down is one less; both have a standard deviation ofÖ[(16N-29)/90]. For expected number of runs and standard deviation for the case involving ties see Fama (1965;p. 74±76). Sadler et al. (1993, p. 363±364) discuss a runs test which simply divides all thicknesses into twocategories: those thicker than and those thinner than the mean (see also Bradley, 1968, p. 262±3; Drummond &Wilkinson, 1993, p. 688±691).

(2) A test of numbers of runs up or down of length d between turning points, or a test of the length of the longestrun. Kendall & Ord, 1990 (p. 20) show that the expected number of runs up or down of beds of length d(counting upper turning point bed) in a random sequence of N beds is [[2 (N)d)2) (d2 + 3d + 1)/(d + 3)!]. Theydescribe a chi-square test based on this test statistic comparing observed and expected run lengths of 1,2, and³ 3. Bradley (1968, p. 264±269) provides a formula for the expected length of longest run. Drummond &Wilkinson (1993, p. 690±692) divide bed thicknesses into two categories: those thicker and those thinner thanthe mean; they use the length of the longest run as a test statistic.

(3) A test of the signs of the difference between consecutive beds (Waldron,1987, p. 140±142; Kendall & Ord, 1990,p. 21). The expected number of positive differences between adjacent beds in a random sequence of N beds is(N±1)/2 with a standard deviation ofÖ[(N + 1)/12]. For each tie of adjacent bed thicknesses that occurs in the sequence, Waldron (1987) suggestsreducing the effective number of beds N by 1. For another way to handle ties see Fama (1965, p. 77±78). Famaprovides expressions for the expected number of positive differences, negative differences and ties.

(4) A test using the sum of squares of the differences in thickness of adjacent beds normalized by the sum of squaresof deviations of thicknesses from the mean (Durbin-Watson test). Drummond & Wilkinson (1993, p. 691±693)provide test statistics and chart showing 95% con®dence interval.

(5) A test involving the information correlation coef®cient of the ®rst order advocated by Milenkovic (1989, p. 110±116). Pairs of consecutive bed thickness (x1, x2), (x2, x3), (x3, x4),¼,(xN)1, xN), may be tested for correlationusing the ordinary correlation coef®cient calculated from these (N±1) pairs. The ordinary correlation coef®cientcan be used to test against an alternative hypothesis that two variates are linearly related. However, there is noreason to suppose a linear relation between successive bed thicknesses. For such cases, Milenkovic re-commends the use of the Linfoot information coef®cient of correlation. It may be applied to a series irrespectiveof the shape of its distribution, and is amenable to a one-sided or two-sided test (Milenkovic, 1989 for details oftest).

(6) Subjective subdivision of the sequence into `states' (= `facies'), i.e. packets of strata characterized by range ofbed thickness, and then testing for preferred sequences of states using markov-chain analysis (Harper, 1984a,b).Like the previous tests that merely compare thicknesses of adjacent beds, this one just analyses differencesbetween adjacent bed packets, and introduces an additional subjective grouping. It often makes good geologicsense to analyse strata in this way, but it is at best a indirect way of searching for thickening or thinning upwardtrends.



Appendix 4. Estimates of standard deviation for weighting coef®cients 1 vs. 2 (Table 4B, Eq. (9); Table 4C Eq. (14)) inparametric tests of subsequences of ®xed size.

(Figure) Stratigraphic unitNo. ofbeds

Coef®cient, standarddeviation w � 1

Coef®cient, standarddeviation w � 2

(3a) Channellized/Non-ChannellizedTurbidite subsequences, Mesozoic, California

704 Wtau_one1, 0á029a Wtau_one2, 0á025a

(3b) & (3c) Intermeshed mud turbidite &hemipelagite subsequences, Quaternary,West Antarctica

34 Wtau_one1, 0á124 Wtau_one2, 0á125

(3d) Turbidite sandstones, Macigno Fm.,Oligocene, Italy


(3e) Weight % Calcite, Pelagic Limestones andMarlstones, Cretaceous, Utah


(4a) Turbidite sandstones, Carboniferous FlyschSouthern Morocco

298 Wtau_abs1, 0á041a Wtau_abs2, 0á040a

(4a) Turbidite sandstones, Carboniferous Flysch,Southern Morocco

298 Wtau_abs1, 0á043 Wtau_abs2, 0á040

(4b) Turbidite sandstones, Mesozoic, Yukon,Canada


(4b) Turbidite sandstones, Mesozoic, Yukon,Canada


aSee Table 4B, Eq. (9), Table 4C, Eq. (14).

696 C. W. Harper


Documents

Thickening and-Or Thinning Upward Patterns in Sequences of Strata - Tests of Significance - Sedimentology, Harper, 1998