Inadmissible evidence: knowledge and prediction in land and riverscapes

Inadmissible evidence: knowledge and prediction

in land and riverscapes

Graham Harrisa,*, A.L. Heathwaiteb

aUniversity of Tasmania, Sandy Bay, Hobart, Tasmania 7005, AustraliabCentre for Sustainable Water Management, The Lancaster Environment Centre, Lancaster University, Lancaster LA1 4YQ, UK

Received 30 November 2003; revised 1 May 2004; accepted 1 July 2004

Abstract

Analyses of daily water quality data from two streams draining a pair of small coastal catchments in South Devon, England

reveal that what conventionally would be thought to show random noise, has a discernable structure and is clear evidence of

process. Catchment and aquatic systems are non-equilibrium systems and climate drivers cause fluctuations in water quality

both in terms of the concentrations of individual parameters and in the correlations between parameters. The data reveal non-

linear coupling at small scales and show evidence of fractal properties both of which may be evidence of self-organised

phenomena at small scales in catchments and streams. These data show that: (a) water quality and catchment nutrient export

data may be strongly aliased, and (b) there is a fundamental degree of indeterminacy underlying the data we can collect and the

knowledge we can generate from the data. New techniques of data based modelling that use the data itself to define more

parsimonious predictive models are needed because such an approach recognises the partial nature of our knowledge and

requires adequate monitoring and adaptive management programs.

Crown Copyright q 2004 Published by Elsevier B.V. All rights reserved.

Keywords: Fractal; Catchments; Streams; Nutrients

1. Introduction

In a world of overwhelming environmental degra-

dation it is important that we get our basic assump-

tions and tools correct if we are to make progress in

reversing or at least steadying the rate of degradation

and addressing its consequences. Whether or not we

are able to perceive the environmental problems being

0022-1694/$ - see front matter Crown Copyright q 2004 Published by El

doi:10.1016/j.jhydrol.2004.07.020

* Corresponding author.

E-mail addresses: [email protected] (G. Harris), louise.

[email protected] (A.L. Heathwaite).

created, and whether or not we are able to respond

effectively will depend on what evidence we can

muster for environmental degradation, how we

explain the phenomena observed and how we predict

and plan for the outcomes of our actions and set

policy. Traditionally, prediction in landscape and

aquatic science has placed considerable reliance on

various forms of dynamical simulation models. Too

often this reliance does not question the (often built-

in) assumption that the dominant processes are

understood and that catchments, rivers and estuaries

are equilibrium systems with prediction based around

Journal of Hydrology 304 (2005) 3–19

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G. Harris, A.L. Heathwaite / Journal of Hydrology 304 (2005) 3–194

central tendencies at timescales from hours to weeks.

Nonetheless, there are some who question this

reliance (Beven, 1989, 1993, 2000; Neal, 1997,

2002, 2004).

Here it is illustrated that catchments, and the water

bodies draining them, contain much contingent small-

scale information, and as a consequence, many widely

accepted models have limited predictive ability: new

analyses of observational data are provided to back up

this proposition. For this, our work focuses on

understanding the outcomes of land use change on

the quality of receiving waters in catchments. Here we

are dealing with the interface between terrestrial and

aquatic systems. This interface is critical but complex

because as the medium changes from land to water,

different rules, rates and dynamics apply. Viewed as a

space-time matrix of possible outcomes, researchers

have assumed that space factors predominate in

terrestrial systems, and time factors are more critical

in aquatic systems. Thus attaining reliable evidence to

demonstrate change and to study pattern and process

across this interface is a challenge.

2. Landscapes and ‘riverscapes’

Catchments and basins are the natural biophysical

units in which human actions have made massive

changes in land use for agriculture and urban

development. Hydrology has been altered through

the building of dams and weirs, extracted water for

irrigation and urban use, and fragmented habitats.

Because biodiversity controls ecosystem and land-

scape function, such intervention in landscapes

changes the spatial and temporal scales of variability

and reduces biodiversity, which in turn leads to

changes in hydrology and the cycling of energy and

nutrients in catchments. It is possible to use water

quality data as indicators of catchment function and

biogeochemistry, and there is growing evidence of

close linkages between changes in land use, hydrol-

ogy, climate, water quality and the condition of our

rivers and estuaries both on contemporary (Harris,

2001; Heathwaite, 1993; Heathwaite et al., 1996;

Heathwaite and Sharpley, 1999; Williams et al., 2002)

and historical (Neal et al., 2004) timeframes. Here, the

question of scale is critical because while biogeo-

chemistry ‘sums’ processes across spatial and

temporal scales in catchments, quite clearly small

scale microbial processes dominate many large scale

patterns (Harris, 1999, 2001).

‘Riverscapes’—aquatic ecosystems within catch-

ments—are particularly useful as model ecological

systems because of the wide range of spatial and

temporal scales encountered: from the large and slow

(kilometres to decades) in catchments, to the small

and fast (micrometres to seconds) in water. Wiens

(2002) describes six central tenets of landscape

ecology which we suggest can be made in the aquatic

context for ‘riverscapes’: (1) patches differ in quality,

(2) patch boundaries affect flows, (3) patch context

matters, (4) connectivity is critical, (5) organisms are

important, and (6) scale is important. In this paper, we

explore what we might learn from these spatial and

temporal controls in the context of water quality

dynamics. For land- and riverscape restoration to be

successful there is an urgent need to better understand

and explain the linkages and pathways between land

use, soil nutrient status and water quality.

Much of what we see in catchments and their water

quality signal seems to emerge from small scale

pattern and process. For example, catchment soils and

their condition are closely linked to water quality

(Engstrom et al., 2000; Markewitz et al., 2001),

especially through small scale interactions with pore

structure and microbial processes (e.g. Harris, 1999).

Kirchner et al. (2000) noted that catchments show

fractal properties arising from the multiplicity of flow

pathways at a range of scales and that white noise

inputs in rainfall are thus turned into fractal 1/f

distributions in catchment exports. Kirchner (2003)

subsequently termed this the ‘double paradox’ of

catchment function. Bishop et al. (2004) provided one

explanation for the highly variable chemistry in

catchment runoff by invoking both vertical and

horizontal heterogeneity in flow paths and responses

to rainfall.

3. Riverscapes from a non-equilibrium viewpoint

Catchments and their aquatic signatures are not

equilibrium systems. There is much contingency and

chance in ecosystem outcomes, particularly in

response to the vagaries of previous floods, droughts

and other extreme events. Catchment systems show

G. Harris, A.L. Heathwaite / Journal of Hydrology 304 (2005) 3–19 5

variability at a wide range of scales; the choice of scale

selected for observation is usually arbitrary and often

made on the basis of either human convenience or on

the availability of research funding. Because many of

the species and processes in aquatic ecosystems are

small and things happen quickly, there are aspects of

their ecology which can be studied within reasonable

time frames from a human point of view. Thus

observations are usually carried out for short periods.

Long time series of data from catchments spanning

decades are rare and often poorly sampled (but see e.g.

Likens and Bormann, 1995). Added to observational

constraints, land and water quality data are notoriously

noisy and flout the standard statistical assumptions of

Gaussian distributions and constant means and var-

iances; this has caused problems with their measure-

ment and interpretation.

The little we do know about catchments and their

receiving waters shows that there are significant

periodicities in the climatological drivers of these

systems at inter-annual, seasonal, 40–50 days, 5–10

day, 1–2 days and diurnal scales (see Burt and

Heathwaite, 1996a; Harris, 1980, 1987; Harris et al.,

1988; Harris and Baxter, 1996; Trudgill et al., 1991).

Much of the high frequency end of the spectrum is very

poorly understood because the data we collect is

weekly at best, and often more infrequently than that.

This makes it difficult to judge the significance of small

scale disequilibrium, and we may be missing important

information here. Water quality time series in

particular show great variability (e.g. Heathwaite

et al., 1996; Johnes and Butterfield, 2002; Wood

et al., in press; Haygarth et al., in press) This variability

has usually been described as being caused by ‘noise’,

and assumed to be the consequence of sampling and

analytical errors. Harris (2003) showed that there is

much information carried in the correlation structure

within water quality data and that what was thought to

be noise was, in fact, evidence of much small scale

information.

Harris (1987) examined the water quality dynamics

of Hamilton Harbour, Ontario, generating water

quality time series ranging from scales of seconds to

years in order to compute the temporal and spatial

variance spectra for the Harbour. The data were

statistically problematical: means and variances were

not constant over the time series, and neither means nor

variances were normally distributed. Detailed analysis

of the relationships between the water quality par-

ameters (e.g. nitrate, phosphate, total N and total P,

etc.) revealed what looked like more noise; all that

appeared to be extractable from the data were variance

spectra, (geometric) means and broad trends. Problems

with the analysis and interpretation of water quality

time series means researchers often resort to using long

term means for sampling stations in catchments (e.g.

Harris, 1999, 2001). Others (e.g. Petersen et al., 2001)

have tried alternative statistical techniques such as

principal component analysis, to decompose time

series of water quality data but, other than broad

distinctions between ‘discharge modes’ and ‘biologi-

biological modes’ little progress appears to have been

made. More recently, a growing number of reports in

the literature describe important small-scale variability

in water quality data from estuaries (Hatje et al., 2001)

and coastal waters (Seuront et al., 2002). For coastal

waters, Seuront et al. (2002) suggested that ‘the

observed small-scale nutrient patches could be the

result of complex interactions between hydrodynamic

conditions, biological processes related to phytoplank-

ton populations, and the productive efficiency of

bacterial populations’.

The time series of water quality measured in rivers,

lakes and coastal waters is therefore, likely to be

influenced by a number of non-equilibrium processes

operating over a wide range of scales and arising both

in interactions between rainfall and soils in the

catchment and processes that operate in situ. These

interactions will vary depending on the relative

strengths of the catchment and in situ influences.

4. Evidence and complexity in earth surface

systems

Reliable evidence for the relationship between

catchment processes, landscape management and

water quality response plays a fundamental role in

our understanding of the way that human activity

modifies the functioning of ecological systems.

Empirical evidence is needed to inform process

understanding. Because our knowledge is partial and

our predictive powers are limited there is a real need

for rapid feedback from knowledge to response.

A downward spiral of misinformation can develop if

our assumptions regarding what we should be


measuring or how we should interpret the evidence

generated are flawed. It is critical to decide whether

we are dealing with equilibrium systems or not and to

learn whether this matters and at what scale.

When we work at the interface of land and water

systems, non-equilibrium and non-linear processes are

especially important owing to their impact on

dynamics, spatial and temporal heterogeneity (Wu

and Loucks, 1995), and discontinuities and multiple

equilibria at a range of scales (O’Neill, 1999). Many

properties of landscapes show fractal-like variability

across a wide range of temporal and spatial scales

(Lohle and Li, 1996; Sole et al., 1999; Li, 2000; Brown

et al., 2002). It also appears that the concentrations of

elements in stream flow show 1/f scaling properties and

memories of events at all scales (Kirchner et al., 2000;

Feng et al., 2004). While fractal properties may be used

as evidence of self-organised criticality (Bak et al.,

1988) care should be exercised in the interpretation of

apparently fractal data in ecology because pattern does

not necessarily imply process and many data sets can

show apparently fractal properties for a number of

reasons (Halley et al., 2004). Strictly speaking, data

about process should be multi-fractal rather than

fractal (Halsey and Jensen, 2004) and some of the

best data from planktonic systems which span many

scales seems to show this (Seuront et al., 1999).

However, in many cases, the amount of information

available is limited and detailed analysis to the degree

required for such an assessment is often missing:

Kirchner et al. (2004) shows using continuous

monitoring of conductivity and pH that there is

considerable information potentially available with

very fine resolution river water quality data over long

time periods, but the frequency of observation required

may be of the order of minutes to hours. Whatever the

generating mechanism, because variability in many

data can show apparently self similar properties across

a wide range of scales, prediction, particularly for

things like community composition and catchment

dynamics is a matter of some difficulty (Lawton, 1999).

As we begin to deal with and understand complexity

in all its forms, and as we develop the necessary

conceptual tools, the philosophy of scientific enquiry is

changing. We are shifting from attempts to articulate

‘theories of everything’ to more partial explanations that

recognise the intrinsic complexity and logical depth of

natural phenomena, and the necessarily partial nature

of evidence and explanation (Cartwright, 1999).

Fox Keller (2002) noted that there is a strong distinction

between the philosophy and practice—between the

status of evidence and explanation—in physics and

biology. In physics, theory has precedence; evidence is

collected to confirm or refute the fundamental theoreti-

cal basis of the science. In biology on the other hand,

evidence has precedence over theory - so that biological

explanations are contingent and may take the form of

explanations or descriptions of natural history (Gould,

1991). Our understanding of catchment dynamics is

beginning to shift away from theory laden ‘physics

envy’ to a more sensitive reliance on what the data

actually tells us (e.g. Young, 1998). Beven (2002) has

argued for a kind of philosophical realism in catchment

modelling; a recognition of the partial and incomplete

nature of our knowledge because of differences between

places and of complexity in pathways and process. Neal

(1997) argues that there is a need towards assessing

structural rather than parameter uncertainty within

hydrochemical models within the theme of emergent

properties: by exploring different structural models,

some feel for the range of potential environmental

impacts can be gained. Neal (1997) also argues that an

assessment of parameter uncertainty provides no

information of any use for deciding the overall

predictive error given the uncertainty in the model

structure for what are highly heterogeneous catchment

systems. Data gathering networks provide a baseline of

information but ‘missing variables, low resolution,

inadequate duration, temporal and spatial gaps, and

declining coverage are pervasive limitations’ (Clark et

al., 2001). In recognising that our knowledge is partial,

and that any evidence upon which catchment policy and

management action is based is partial, there is an urgent

need to fully exploit the available evidence using

appropriate tools, so we can move forward, take action,

monitor progress and learn from our experiences.

5. Water quality data from two small catchments

in England

5.1. Time series analyses of the Gara and Start

catchment data

Here, time series of water quality data from two

adjacent sub-catchments of the 46 km2 Slapton


catchment, southwest Devon, UK, are examined. The

data analysed were from the Gara and Start catch-

ments which comprise the majority of the flow into

Slapton Ley. Details of catchment hydrology, stream

water quality and nutrient exports along with long

term studies at this site may be found in Burt and

Heathwaite (1996a,b) Burt et al. (1996) and

Heathwaite, Johnes (1996) and . Daily time series of

water quality data from two adjacent catchments were

reconstructed in part from the data presented in

Heathwaite and Burt (1991) and Johnes and

Heathwaite (1997). Data presented by Burt and

Heathwaite (1996a,b) and Burt et al. (1996) showed

that the sampling period was not exceptional in terms

of temperature or rainfall in these catchments.

Fig. 1. Time series of discharge (m3 sK1), NO3–N (mg lK1) and PO4–P (m

Oct 1987–1 Oct 1988.

Furthermore, 1987–88 does not stand out in any of

the long term weather records for this period being of

average rainfall and evaporation (Burt and Horton,

2001).

An initial exploration of the daily water quality

data at Higher North Mill in the Gara catchment and at

Deer Bridge in the Start catchment from 1 October

1987 to 1 October 1988 was carried out using classical

time series techniques (Box and Jenkins, 1976). The

data revealed both climate driven variability through

periodic rainfall events and long term trends (Figs. 1

and 2). The data clearly showed all the usual problems

which frequentist statistical approaches must tackle

when dealing with hydrological and water quality

time series; non-Gaussian distributions, non-

g lK1) for the Start sub-catchment at Deer Bridge, Slapton, Devon, 1

Fig. 2. Time series of discharge (m3 sK1), NO3–N (mg lK1), and PO4–P (mg lK1) for the Gara sub-catchment at Higher North Mill, Slapton,

Devon 1 Oct 1987–1 Oct 1988.

Fig. 3. Frequency of statistically significant autocorrelations in both

raw and log transformed data series for the Start catchment at Deer

Bridge and the Gara catchment at Higher North Mill, Slapton,

Devon.


stationarity of means and variances and temporal

trends (Harris, 1987). Acknowledging these problems,

univariate partial autocorrelation functions were

computed for the data series using Minitab. Log

transformation and detrending of the data were used

where appropriate. Despite the expectation that there

would still be problems with this analysis because of

non-stationarity and differing epochs in the data there

were, nonetheless, statistically significant autocorre-

lations in almost all the series (discharge, dissolved

(!0.45 mm) inorganic NO3–N, NH4–N, PO4–P and

suspended sediment) with the significant periodicities

clustering around scales of 10–15 and 45–50 days

(Fig. 3). Inspection of the original data showed that

these scales were due to the climatological drivers,

especially periodic rainfall events in the first 6 months

(winter) of the sampling period (Figs. 1 and 2). The

time series analysis of the daily rainfall record from 1

Oct 1987 to 1 Oct 1988 showed significant auto-


correlation at a scale of 39 days and a peak in spectral

power at 42 days.

5.2. A new approach using moving correlations

Given the character of the data series, a new

approach to the analysis of water quality data using

moving correlation windows was used to display

the inter-relationships and correlations between

bivariate time series of water quality parameters

as they change over time. The concept is similar to

moving averages, with correlation (R2) values

calculated for moving windows of defined widths,

moving stepwise along the length of time series

data. Traditionally, time series analysis assumes

that the data and the generating functions of cross-

correlations are stationary over the entire length of

the series (Box and Jenkins, 1976), so a moving

correlation approach has been used in a number of

applications where it was expected that the

magnitude or sign of the correlation between a

pair of variables would change with time. This

approach has been used in climatology (Kumar and

Dash, 2001) and in medical science (Balestreri

et al., 2004), and a similar technique involving

moving spatial windows rather than moving

temporal windows has also been published (Bolvi-

ken et al., 1997). The application of this technique

to water quality data was justified based on work

by Harris (1987) that discussed the existence of

epochs within such data where the sign of

correlations between parameters changed during

sampling periods.

Moving correlation analysis can easily be carried

out using spreadsheets. The statistical significance of

the correlation coefficients obtained by this analysis

can be found from standard tables (e.g. Bishop, 1966)

by assuming that the number of degrees of freedom is

one less than the width of the correlation window

used. To an approximation, the variance of the

correlation estimator is inversely proportional to the

bandwidth of the window used, so that the product of

bandwidth and variance is constant (Jenkins and Watt

1968). This means that moving correlation windows

with narrow bandwidths reveal short epochs of

changing signs in the data but these are noisy,

whereas windows with larger bandwidths can only

reveal longer epochs but do so with greater

confidence.

The correlations between the water quality data

over 4–6 day periods (chosen to be shorter than

the short term 10–15 day periodicity in the date

series) showed strong switches between positive

and negative correlations over the length of the

time series. (Harris (2003) examined similar trends in

water quality data from rivers, lakes and coastal

waters and came to similar conclusions). Whilst the

correlation between PO4–P and NH4–N was often

positive there were significant periods when the

correlation was strongly negative. The probability

density functions of the correlations (pdfs) were

highly non-Gaussian and decidedly non-random. By

varying the width of the moving window it was

possible to show that most time series had statisti-

cally significant patterns of variability in the band

width (moving correlation window) from 4 to 10 data

points. Below 4 data points the sample sizes were so

small as to require unrealistically high R2 values for

significance (for 3 data points R2O0.95). Above 10

data points (where the required R2 for pZ0.05 O0.602) the moving correlation averaged out signifi-

cant (and large) switches in the correlation coefficient

to the extent that statistical significance was rapidly

lost.

Water quality data appear to show decidedly non-

random correlated fluctuations between parameters

but these correlations change sign during short epochs

in the data. Here lies an explanation for the fact that

cross-correlation patterns between variables in water

quality time series that might reasonably be expected

to be statistically significant are frequently not so and

hence are dismissed as noise. Without a moving

window approach, the band width of the entire series

is usually too long to reveal the high frequency

correlation patterns in the data. Also weekly (or less

frequent) sampling of these catchments would lead to

severe aliasing problems and would also lead to the

data being dismissed as noise.

Because random time series can also show these

apparent switches in the sign of the correlation

between series when treated with moving correlation

techniques it was necessary to test for the significance

of the patterns in the cross-correlations. Kolmogorov–

Smirnoff (K–S) tests were use here to assess for

significant differences in the probability density

Fig. 4. (a) Four-day moving correlation (MC) between daily PO4–P (!0.45 mm) and NH4–N data for samples collected at Deer Bridge in the

Start river catchment, Slapton Ley, Devon. For full sampling and analysis details see Heathwaite et al. (1989); (b) Four- day moving correlation

between the daily PO4–P (!0.45 mm) and NO3–N data for samples collected at Higher North Mill in the Gara river catchment Slapton Ley,

Devon. For full sampling and analysis details see Heathwaite et al. (1989).


functions (pdfs) of the distributions of the correlation

coefficients before and after randomisation of one of

the series in pair-wise comparisons. Providing the

time series are long and the sampling density is high,

the K-S test is a sensitive discriminator of non-random

patterns in the high frequency switches in the sign

of the correlation coefficient shown in Fig. 4.

The significance of changes in the cumulative pdf

before and after randomisation may be estimated from

the criterion: at pO0.05, DO1.36O((n1Cn2)/n1n2),

where n1 and n2 are the lengths of the series (Siegel,

1956). For 365 data points a deviation between the

cumulative pdfs of more than 15% is highly

significant. The results of the K–S tests for PO4–P

and NH4–N are shown in Fig. 5 and suggest that the

patterns of variations in the sign of the correlation

shown in Fig. 4 were certainly not random.

The moving correlation approach yields small

scale correlation patterns that have physical interpret-

ations. In the Start and Gara data the PO4–P and

NH4–N (!0.45 mm) data showed statistically signifi-

cant 4- and 6-day moving correlation patterns (i.e.

K–S tests involving randomisation of the data sets

showed significant effects, Fig. 5) as did the NH4–N

and suspended sediment (!0.45 mm) data. Broadly,

the correlations between these parameters were

positive. Evaluation of the correlations between the

NO3–N and PO4–P data sets (Fig. 4b) showed

apparently random correlation patterns (no effect of

series randomisation). Nitrate–nitrogen and sus-

pended sediment data were broadly negatively

correlated over these time scales. This result is

consistent with a close inspection of the original

data where it was clear that the pathways of move-

ment of the PO4–P and NH4–N were different from

NO3–N. Orthophosphate and NH4–N concentrations

peaked at peak flows indicating rapid responses to

rainfall events, whereas NO3–N showed dilution at

peak flows and a maximum concentration 24–48

hours after rainfall (Heathwaite et al., 1989). Clearly,

the sources and pathways of the different water quality

parameters differed and this was revealed by the

correlation structures (or lack thereof) in the data.

5.3. Coherence between adjacent catchments

Throughout the entire time series the overall

pattern of fluctuations in dissolved inorganic N:P

ratios in the two catchments (NO3–NCNH4–N/PO4–P

as a molar ratio) were similar and well correlated

(Fig. 6a). Both data series showed high NO3–N at the

beginning of the sampling period with a declining

trend thereafter (Figs. 1 and 2). Catchment exports

were driven by large winter rainfall events and the

inter-annual pattern of NO3–N export from

the catchments. Burt and Heathwaite (1996b); Burt

Fig. 5. (a–c) Probability density function plots for the 4-day moving correlations (MCs) between PO4–P and NH4–N before and after

randomisation of one of the series and the corresponding cumulative probability plots. By the Kolmogorov–Smirnoff test for data series 365

points long any difference in cumulative probability greater than 10% is likely to be significant at the 5% level.


et al. (1996) discussed the long term, inter-annual,

pattern of NO3–N dynamics in the Slapton catchments

and showed that they were largely determined by

inter-annual variability in temperature, rainfall and

evaporation with larger NO3–N export after relatively

dry years. Thus at annual scales, the two catchments

showed similar dynamics, but in detail the responses

over shorter time scales were very different.

Analysis of the relationship between the dissolved

inorganic N:P ratios in the two catchments using 6

day moving correlations showed significant period-

icities at scales of 42 days and decidedly non-random

pdfs (Figs. 6b–d) so that comparison of the original

moving correlation data series with its randomised

counterpart showed very significant changes

(Fig. 6e).

A comparison of the correlation patterns between

NH4–N and PO4–P at 4–6 day scales in these small

adjacent catchments indicated poor coherence (Fig. 7).

The combination of steep slopes, permeable soil over

impermeable bedrock, and high rainfall, encourages

the production of large volumes of subsurface runoff

(Burt and Heathwaite, 1996b). Quickflow (i.e. that

fraction of precipitation which rapidly reaches the

stream channel and is usually associated with high

discharge) accounts for a small (c. 1%) of the annual

Fig. 6. (a) Correlation over the year of the dissolved inorganic N:P ratios at Deer Bridge and Higher North Mill; (b) Six-day moving correlation

between the two series; (c) and (d) Probability density functions (pdfs) of the 6-day moving correlation (in b) before and after randomisation of

one of the series; (e) Corresponding cumulative probability plots of the two pdfs. Any difference in cumulative probability between the two plots

greater than 10% is likely to be significant at the 5% level.


Fig. 7. Temporal coherence between the correlation patterns for the

two adjacent sub-catchments of Slapton Ley: the Start (DB) and

Gara (HNM) using a 4-day moving correlation and PO4–P vs. NH4–

N. Replotted from data presented in Fig. 4a and b. (MCZmoving

correlation).

Fig. 8. Double logarithmic power law plot of the frequency and

magnitude of changes in the differenced time series of daily NH4–N

data from Higher North Mill, Gara river catchment, Slapton Ley.


runoff total, with much of the baseflow in the

catchment occurring in winter in the form of delayed

flood hydrographs which peak several days after the

quickflow response. Orthophosphate, NH4–N and

suspended sediment delivery to surface waters in the

catchment are associated with rapid flowpaths,

whereas NO3–N delivery is delayed (Burt et al.,

1996). While the Gara and Start catchments are to a

degree ‘paired’ being side by side in a similar climate,

there are in fact important differences between them in

terms of their underlying geology and land use. The

Gara catchment is steeper and largely wooded on the

steeper slopes whereas the Start catchment has gentler

slopes and a greater area of agricultural land. These

differences are reflected in the detailed differences in

their correlation structures and the general lack of short

term coherence between them.

So, while large scale correlations in the data (scales

of seasons and years) reflected annual and inter-

annual climatological drivers (Fig. 6), the smaller

scale non-stationary behaviour of the chemical

parameters around these long term trends - the

behaviour of the manifold of residuals if you like—

is not noise but strongly reflects drivers, pathways and

processes in the catchments.

All the water quality data examined showed

evidence of self organised criticality and fractal

properties in that the data showed ‘fat tailed’ log-

normal or power law distributions. Double logarith-

mic plots of the frequency versus magnitude of events

in the data all showed significant power law

relationships with slopes varying between K1.5

and K2. Correlation coefficients of all the power

law plots were between 0.87 and 0.95. A sample plot

is shown in Fig. 8. These plots were noisy at the low

frequency, high magnitude ends of the plots because

of the relatively short data sets examined. Evaluation

of the results of these analyses for PO4–P and NH4–N

seemed to show consistent differences in slope

between the two nutrients: the PO4–P plots had slopes

close to K2 whereas the NH4–N plots had slopes

closer to K1.5.

6. Discussion—a new look at what is inadmissibleevidence

There is known to be a temporal and spatial

spectrum of fluctuations in water quality data

collated from waterbodies. This variability is driven

partly by external fluctuations in climate and

hydrodynamics, and partly by fluctuations in

internal biological and physiological processes

(Harris, 1980, 1987; Heathwaite, 1994; Neal,

1997, 2002; Robson et al., 1992). For lakes and

coastal waters in particular, there appears to be a

set of diffusion-reaction processes operating that are


driven by stirring and turbulence in the water

column (Seuront et al., 2002; Harris, 2003). Such

processes, together with strong non-linear coupling,

can give complex emergent spatial and temporal

patterns such as spiral waves and patchiness in

many ecosystems (e.g. Bjornstad et al., 2002).

The results presented here suggest there is similar

evidence for strong non-linear coupling at small

scales in soils and aquatic environments that also

generate complex spatio-temporal patchiness. This

may be evidence for fractal properties and self-

organised criticality in these data (Kirchner et al.,

2000; Feng et al., 2004). If the sampling interval is

such (days) that processes are resolved then the

variability in water quality data is not just noise. If

the sampling interval is too long to resolve

important small scale pattern and process (weeks

or longer) then the data sets are severely aliased

and are just noise (Harris, 1987). Much water

quality data has been collected at too infrequent

intervals and is of little real value other than to

compute long term averages and display inter-

annual patterns. Weekly or less frequent water

quality sampling in these catchments cannot revel

the true scales of pattern and process and the

results frequently cannot be distinguished from

noise. This is shown both in this study for

agricultural and elsewhere for upland forested

(Kirchner et al., 2004) catchments.

Viewing the changing sign of the moving corre-

lation between water quality parameters shown in

Figs. 4 and 6 as the slope of the relationship between,

in this case PO4–P and NH4–N or suspended

sediment and NH4–N, at any point in time; then we

are seeing, using the moving correlation window, the

unfolding of a possibly self-organised critical mani-

fold arising from small scale reaction diffusion

relationships (Seuront et al., 2002). Such manifolds

are very characteristic of aquatic ecosystems. For

lakes at least, their properties can be revealed by quite

simple models (e.g. Gragnani et al., 1999; Lima et al.,

2002), and can have very counterintuitive outcomes

(Huppert et al., 2002). The fluctuations in water

quality data for the small rivers shown in Fig. 4 are

surprising: they may reflect both in-stream diffusion-

reaction processes and also self-organised criticality

driven by catchment inputs at a range of scales

(Kirchner et al., 2000). Given the scale of interactions

that have been revealed by these data, we face a

fundamental problem of whether we can ever know

the initial conditions and the time history of the

contingent interactions for each small patch of water,

its source and the particular flow pathway in the

catchment. Both spatial and temporal patchiness is

important in terrestrial and aquatic systems and can

lead to fractal data patterns (Kirchner et al., 2000;

Kirchner, 2003; Feng et al., 2004). There is, therefore,

a fundamental problem of indeterminacy, lack of

adequate non-aliased data and incomplete parameter-

isation. For example, the lack of temporal coherence

between the two catchments of Slapton Ley (Fig. 6)

and the evidence of fractal properties in the data

(Fig. 7) both strongly suggest that modelling these

systems is going to be difficult because of indetermi-

nacy and small scale dynamics. Clearly there are

small scale processes and events which appear to

dominate the observed patterns in the data and the

usual approach to documenting and predicting change

based around means and central tendencies is, we

argue, inadmissible.

Data availability and parameterisation at the

correct scale frequently limits our ability to interface

catchment processes with aquatic ecosystem

response. Model parameterisation has been the

subject of much discussion the hydrological literature

(Loague and Freeze, 1985; Beven, 1989, 1993, 2002).

The problems of scale, missing data and the lack of

inclusion of key processes were discussed by Haus

(1990). Clark et al. (2001) raised the issue of partial

and incomplete data and this, together with the

problems of climate prediction means that there are

some fundamental problems with calibration, vali-

dation and prediction (Orekes et al., 1994). The usual

approach for parameter estimation in data sparse

situations is to define a ‘lumped’ parameter at a large

scale, removing the bias in model predictions by

defining an ‘effective parameter’. This approach has

been criticised on the basis that the data are often

insufficient to constrain the chosen parameter values

(e.g. Beven, 1993). The problem of fundamental

indeterminacy described above which, together with

self-organised criticality, non-linear coupling and

shifting stoichiometries over a wide range of scales,

may shake the foundations of dynamical simulation

modelling.


7. The implications for modelling and prediction

Research linking catchment processes and water

quality impact has traditionally been based on either

simple empirical relationships or on simulation

modelling. Empirical models have shown considerable

success - most notably the relationship between total

phosphorus export from catchments and algal biomass

response (Vollenweider, 1968). This relationship still

forms the basis of much lake management worldwide

providing probabilistic predictions of algal blooms.

Dynamical simulation models of catchments and

receiving waters replace probabilistic distributions

with deterministic equations; the resulting models are

strongly scale dependent. In catchment modelling,

this approach relies on large sets of differential

equations to describe ecological processes, major

functional groups and their interactions. The models

deterministically represent noisy, non-equilibrium

systems and average out the natural variability at a

wide range of temporal and spatial sales. No attempt

is made to represent all interactions, relying instead

on a ‘lumped’ representation of the ecosystem. These

models can either be used heuristically to synthesise

knowledge and to guide further data collection or

they can be used to predict the outcomes of

management action and other forms of human

intervention. Either way they form the basis of

explanations of catchment processes and use aver-

aged and simplified space-time distributions and

interactions.

Of course, an ability to fit the data does not, in

itself, make any definitive statement about the

adequacy of the model structure or its parameterisa-

tion (Klemes, 1986; Rykiel, 1996). So doubts remain,

and include concerns regarding the philosophical

basis of dynamical simulation modelling, first voiced

by Berlinski, 1976, and the ability to calibrate and

validate such models (Orekes et al., 1994). The latter

concern is critical given that calibration data is also

partial and noisy because any set of catchment and/or

water quality data is a sample from the normal spatial

and temporal variability encountered in natural

systems. Consequently, such data are aliased and are

sparse samples that are unrepresentative of the full

spectrum of natural variability (Clark et al., 2001). To

get round this, analysis of water quality data often

relies on mean values on the assumption that observed

variability is noise and the parameters are uncorre-

lated. Simulation models can only model the central

tendencies in such data - not the actual small scale

variability (Harris, 1999). Although dynamical simu-

lation models of water bodies can show large scale

hysteresis effects (e.g. Harris et al., 1996; Murray

and Parslow, 1999a,b) which are quite realistic (e.g.

Scheffer et al., 2001), the existence of small scale non-

linearity and cross-correlations between parameters is

of concern, and unlikely to be just ‘noise’ in most

cases. This is because our evidence is based on partial

information and our explanations rely on unrepresen-

tative, deterministic dynamical models. Observations

bear out this assertion. Magnusson et al. (1990), for

example, showed that the temporal unfolding of a

suite of properties from a group of ‘similar’ lakes was

unpredictable, and Reynolds (1986) had the same

problems with attempts at ecosystem scale ‘controls’

in large enclosures.

What the presence of small scale dynamics shows

is that hysteresis and non-linearity at small scales may

be sufficient to render the predictions of simulation

models and management interventions unsafe at

larger scales or higher levels. In the data reported

here, the molar export ratios of dissolved inorganic

N:P change over the year by more than two orders of

magnitude (O600 to ! 5, Fig. 6) and the 6 day

moving correlation series showed significant 42 day

periodicities so the Higher and Lower Ley at Slapton

is being loaded by drastically changing stoichio-

metries both at inter-annual scales and at scales of 40–

50 days - long enough to stimulate the growth of

different algal species in the lake from month to

month and from year to year. Herein must lay some

explanation for the initiation of algal blooms of

different species in receiving waters.

Systems which show evidence of self-organised

criticality and fractal properties also frequently show

emergent properties and surprising behaviours.

Clearly there are fundamental limitations to our

ability to measure and predict the variability in

catchments, rivers and estuaries. Stochastic events

can lead to hysteresis effects and state shifts in natural

ecosystems (Scheffer et al., 2001). Any predictions

had, therefore, better be probabilistic and hedged

about with uncertainty because there may be more

than one set of parameterisations and more than one

set of parameter values that can give equally


acceptable predictions of the observational data

available. This is the problem of non-uniqueness, or

equifinality described by Beven (2000). As Neal

(1997) pointed out, there are issues of structural rather

than parameter uncertainty that needs to be addressed:

the point is that with parameter uncertainty, there is

the assumption that the model is in essence structu-

rally correct and this is inconsistent with the complex-

ity that is observed for the highly heterogeneous

studies being modelled. We suggest therefore, that

there is a problem, not only with the evidence

collected, but also with attempts at synthesis and

explanation.

Raupach et al. (2002) described a method for

evaluating the prediction bias in simulation models.

They suggest that the reliability of the aggregated

model depends critically on the accurate estimation of

the means, variances and co-variances of model

variables and parameters. The authors showed that

considerable bias may be introduced if the higher

order terms in models are large (i.e. if there are

significant non-Gaussian distributions in the pdfs of

the values and the co-variances are significant - as

demonstrated in the water quality data presented

here). Clearly, if estimates of the means, variances

and co-variances are not available, then attempts to

improve the performance of an aggregated model by

further measurement of only the mean variables and

parameters are: “futile, because measurements of

mean quantities cannot eliminate bias arising from the

interaction between model non-linearity and small-

scale variability” (Raupach et al., 2002). This problem

is exemplified in landscapes and riverscapes where,

for example, changes to catchment land use not only

changes the annual averaged nutrient loads to rivers

and estuaries but also changes flood frequencies and

the frequency and magnitude of events. Webster and

Harris (2004) have used simple models to show that

estuaries are surprisingly sensitive to changes in the

frequency and magnitude of nutrient loads. Annually

averaged loads of single nutrient parameters are not

sufficient to describe the response of these receiving

waters to catchment exports. Changing stoichi-

ometries over ecologically significant time scales

will only complicate the picture. A further compli-

cation arises because the underlying stochastic drivers

of catchment systems, as well as the (self-organised)

internal dynamics lead to the observed biodiversity

(Gragnani et al., 1999; Huisman and Weissing, 1999);

and we know that the biodiversity controls the overall

function, often in idiosyncratic ways (Emmerson

et al., 2001). While there are some overall patterns

of response, dynamical simulation models do not

predict the dynamics, emergence, contingency and

biodiversity of real world systems.

This analysis leads us to conclude that the

fundamental question that needs to be answered is

how often do the small scale contingency, indetermi-

nacy, non-linearity and the cross-correlations high-

lighted above generate instability at higher levels so

as to render predictions of central tendencies unsafe?

8. New technologies for data collection—the way

forward?

New technologies are now available, particularly

in the area of hydrology and water quality, which

make it possible to collect more complete and

representative data sets. Automatic sampling equip-

ment, in situ chemical analytical facilities and new

automatic probes, coupled with telemetry and data

storage for remote data acquisition make it possible

to collect high frequency data from catchments and

rivers which are more representative of the full

range of spatial and temporal scales of variability.

These technological advances mean it is viable to

shift away from arbitrarily chosen sampling

schemes to a more complete analysis of the most

significant time and space scales - and then to

optimise the sampling accordingly. This, in itself,

will produce more concrete evidence of the links

between land use and river water quality and

ecology degradation and provide a firmer baseline

for ‘evidence based’ policy development and

management interventions. Such data may be used

to derive parsimonious data-driven models upon

which to base management action and policy

development (Kokkonen et al., 2003; Young,

1998, 1999, 2000). Models may be developed

which use as much information as is available in

the data.

In a world of climate and land use change we need

to know what is changing over what time scales and

what evidence of change is required.


Acknowledgements

This paper could not have been written without key

intellectual inputs from Richard Vollenweider in

Canada whose insights and letter started it all. As

far as we know Richard Vollenweider was the first to

suggest that moving correlation approaches should be

applied to the analysis of water quality data. In

addition we want to thank Peter Young (Lancaster

University) and Phil Haygarth (IGER) who have

helped with the development of these ideas. David

Fox helped us with statistical advice.

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Documents

Inadmissible evidence: knowledge and prediction in land and riverscapes