HYDROLOGICAL PROCESSESHydrol. Process. 21, 3529–3531 (2007)Published online 1 October 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/hyp.6889

Non-stationarity in long time series: some curious reversals inthe ‘memory’ effect

Tim Burt1* andFred Worrall2

1 Department of Geography, DurhamUniversity, Science Laboratories,South Road, Durham DH1 3ER, UK2 Department of Geological Sciences,Science Laboratories, South Road,Durham DH1 3LE, UK

*Correspondence to:Tim Burt, Department of Geography,Durham University, ScienceLaboratories, South Road, DurhamDH1 3ER, UK.E-mail: t.p.burt@durham.ac.uk

Received 13 August 2007Accepted 16 August 2007

Two decades ago in Hydrological Processes, one of us (Burt et al., 1988)reported stream nitrate levels in a small catchment of mixed land use(the Slapton Wood catchment) over a period of 15 years (1970–1985).Multivariate analysis showed that variations in annual mean nitrateconcentration were controlled by antecedent hydrological conditions,rather than by runoff conditions in the given year itself. This suggestedthat there was a strong lag or ‘memory’ effect in the hydrologicalsystem, with dry years being followed by higher concentrations thanexpected in subsequent years, while wet years exhausted the system sothat subsequent years had lower than expected concentrations.

Two decades on, the Slapton nitrate record is still being maintainedand it seems trite to note that the record is now, of course, much longerthan the 15-year series we originally examined. The important point isthat the much longer record includes unexpected changes in response,in addition to changes that we might have forecast 20 years ago. Herewe look briefly at the 35-year record (October 1970–September 2005)and compare it to another long time series, dissolved organic carbon(DOC) records (using water colour as a surrogate) for the River Teesin northeast England (Worrall and Burt, 2004). We present correlationsbetween annual mean concentration and annual rainfall totals, with thelatter lagged by 1 (rf-1) and 2 years (rf-2) in addition to the currentyear (rf). Correlations for a 15-year period are repeated using a ‘movingwindow’ (Worrall et al., 2003) with the results being plotted against thefinal year of the series (Figure 1).

What is immediately apparent in both cases is that three out offour lagged variables show a complete reversal of sign over the periodof analysis. For the Slapton nitrate record, both lagged variableshave moved from negative to positive correlations with the dependentvariable. For the Tees, the response is exactly the opposite for therf-2 variable: positive correlations at the start of the series change tostrongly negative by 1999, although with some move back towards zerocorrelation after that. These reversals were a complete surprise to us,but are they more common than we have realized? And what are thepossible reasons behind the reversals–why might the ‘memory’ effect ofa system change over time? One important word of caution: for n = 15,the 5% significant level is a correlation of ±0·51 and ±0·44 at 10%.This means that the trends plotted may have little meaning, althoughsignificant results are achieved in most cases at both ends of the analysisperiod.

Table I reminds us in general terms how sequences of wet and dryyears might work out in terms of concentration, depending on whetherthe lag effects are positive or negative. The two catchments includedhere have very different characteristics so we might expect the processmechanisms to differ, although as we argue below, the general effectmay have a similar basis. The Slapton Wood catchment (1 km2) isan intensively farmed, lowland area; the Tees, a very much largerarea (818 km2), is dominated in its upland headwaters by blanket peatmoorland, wetter and colder than the conditions at Slapton. We can,therefore, expect the soil nutrient cycles to be very different.

Copyright 2007 John Wiley & Sons, Ltd. 3529

T. BURT AND F. WORRALL

(a)

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1985 1990 1995 2000 2005

Pea

rson

cor

rela

tion

(r)

r Nrfr Nrf-1r Nr-2

(b)

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1985 1990 1995 2000 2005

Pea

rson

cor

rela

tion

(r)

r Crfr Crf-1r Cr-2

Figure 1. Correlations between annual rainfall (mm) and (a) nitrate(mg l−1 NO3-N) and (b) water colour (Hazen units) for the SlaptonWood and River Tees catchments respectively. Correlations arebetween nitrate (N) or water colour (C) and this year’s rainfall (rf,dashed line), last year’s rainfall (rf-1, black line) or rainfall 2 yearsago (rf-2, grey line). Each correlation is for a 15-year period withresults plotted for the last year in the series. For n = 15, the 5%

significance level is a correlation of ±0·51 and ±0·44 at 10%

Table I. Interpretations of the ‘memory effect’. The impactof a particular sequence of wet and dry years depends onwhether the lag correlation (‘memory’) is positive or negative

(a) Negative memory

This year

Dry Wet

Last year Dry Not as low asexpected

Higher thanexpected

Wet Lower thanexpected

Not as high asexpected

(b) Positive memory

This year

Dry Wet

Last year Dry Lower thanexpected

Not as high asexpected

Wet Not as low asexpected

Higher thanexpected

In the original Slapton paper, we speculated thatdry years represented transport-limited conditions(i.e. no leaching losses) so that the next year, whether

Table II. Calculations of annual concentration of nitrate andwater colour using regression equations. Wet and dry yearsare defined ±1 standard deviation from the mean. Nitrateconcentrations are in mg NO3-N l−1; colour is measured in

Hazen units

(a) Slapton Wood nitrate, 1971–1985

Thisyear dry

Thisyear wet

Last year dry 6·3 6·4Last year wet 4·9 5·0(Average conditions predict a concentration of 5·6)

(b) Slapton Wood nitrate, 1987–2001

Thisyear dry

Thisyear wet

Last year dry 8·1 9·2Last year wet 9·2 9·3(Average conditions predict a concentration of 8·7)

(c) Tees colour, 1972–1986

Thisyear dry

Thisyear wet

Last year dry 72 91Last year wet 94 113(Average conditions predict a concentration of 93)

(d) Tees colour, 1987–2001

Thisyear dry

Thisyear wet

Last year dry 104 123Last year wet 104 123(Average conditions predict a concentration of 113)

wet or dry, will have higher than expected nitrateconcentrations. In contrast, wet years would mean asupply-limited situation; because nitrate stores havebeen depleted, concentrations will be lower thanexpected the following year. However, other explana-tions are possible, and for nitrate the impact of a dryyear may be as much to do with lack of mineraliza-tion (i.e. lack of supply) as with lack of flow. Thus,it could be argued that the early situation at Slapton(‘negative’ memory) is generally supply-limited, whilethe end of the period is transport-limited (i.e. no limi-tation of nitrate). This explanation has the advantageof being applicable in all cases, rather than needingseparate explanations depending on whether the pre-vious year was wet or dry. Table II shows how thelag effect works in reality, using regression equationsto calculate sequences of annual mean concentration.It shows that the negative memory effect dominatesthe earlier period at Slapton; nitrate concentrationsdepend more on past than present conditions. Thisdependence on previous conditions is exactly the samein the later period, except that the memory effect isnow positive. Could this mean that the soils at Slaptonare now nitrogen saturated?

Copyright 2007 John Wiley & Sons, Ltd. 3530 Hydrol. Process. 21, 3529–3531 (2007)DOI: 10.1002/hyp

INVITED COMMENTARY

A consistent explanation would require the Tees tohave moved from a condition of transport limitationto supply limitation. At first sight, this would seem atodds with the steady increase in DOC concentrationsobserved for the Tees (Worrall and Burt, 2004), butit may be that, despite the upward trend, year-to-year variations are now more sensitive to rainfallthan before, perhaps reflecting the warmer conditionstoday and the tendency for stronger seasonality withwetter winters and drier summers. From Table II itseems clear that the Tees is always transport-limited,given that the current year’s rainfall is always muchmore of an influence compared to Slapton. There wassome dependence on the previous year’s conditions inthe earlier period, but this has now disappeared. Itseems that the reversal in the memory effect betweenthe two periods has not so much moved the Tees into asupply-limited state but rather has simply eliminatedthe positive memory effect; nevertheless, this meansthat the system does appear less transport-limitedthan before.

We would welcome comments both in relation tothe results presented and to know whether otherresearchers have noted reversals similar to thosedescribed here. We recognize, of course, that statisticalresults will require a process-based analysis if reliableexplanations are to be generated. In both these cases,we hope to do just that, in due course.

References

Burt TP, Arkell BP, Trudgill ST, Walling DE. 1988. Stream nitratelevels in a small catchment in south west England over a period of15 years (1970–1985). Hydrological Processes 2: 267–284.

Worrall F, Burt TP. 2004. Time series analysis of long term riverDOC records. Hydrological Processes 18: 893–911.

Worrall F, Swank WT, Burt TP. 2003. Changes in nitrate export dueto ecological succession, land management and climate: developing asystems approach to integrated catchment response. Water ResourcesResearch 39(7): 1–14.

Copyright 2007 John Wiley & Sons, Ltd. 3531 Hydrol. Process. 21, 3529–3531 (2007)DOI: 10.1002/hyp