Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol23.Issue8.2000:

Dynamic modeling of long-term sedimentation in the Yadkin River

basin

Jagdish Krishnaswamy

a,*

, Michael Lavine

a

, Daniel D. Richter

b

, Karl Korfmacher

c a

Nicholas School of the Environment, Duke University, Durham, NC 27708, USA

b

Institute of Statistics and Decision Sciences, Duke University, Durham, NC 27708, USA

c

Environmental Studies Program, Denison University, Granville, OH 43023, USA

Abstract

Modeling of sediment transport in relation to changing land-surface conditions against a background of considerable natural variability is a challenging area in hydrology. Bayesian dynamic linear models (DLMs) however, o€er opportunities to account for non-stationarity in relationships between hydrologic input and basin response variables. Hydrologic data are from a 40 years long

record (1951±1990) from the 5905 km2_{Yadkin River basin in North Carolina, USA. DLM regressions were estimated between}

log-transformed volume-weighted sediment concentration as a response and log-log-transformed rainfall erosivity and river ¯ow, respec-tively, as input variables. A similar regression between log-transformed river ¯ow and log-transformed basin averaged rainfall was also analyzed. The dynamic regression coecient which re¯ects the erodibility of the basin decreased signi®cantly between 1951 and 1970, followed by a slowly rising trend. These trends are consistent with observed land-use shifts in the basin. Bayesian DLMs represent a substantial improvement over traditional monotonic trend analysis. Extensions to incorporate multiple regression and

seasonality are recommended for future applications in hydrology. Ó _{2000 Elsevier Science Ltd. All rights reserved.}

Keywords:Dynamic linear models; Bayesian time-series methods; Hydrology; Sedimentation; Land-use change; Yadkin River basin

1. Introduction

River systems are highly dynamic and are controlled by a complex of ecologic, climatic, and geomorphic processes. The movement of sediment through a river system is dicult to predict and control [38], at least in part because sediment results from soil and channel erosion that may have occurred hours to centuries in the past. Sedimentation is one of the world_Õs primary water pollution problems, mainly because it is so readily ac-celerated by human activities such as forest conversion, farming, surface mining, road construction, and the growth of suburban and urban communities. Bruijnzeel [5] in a review of land-use e€ects on tropical humid basin hydrology noted the non-availability of adequately gauged catchments, the inability of most studies to ac-count for large-scale weather patterns, and infrequent

examples of rigorous statistical techniques. This is also the general case even for temperate systems. The hyd-rologic and sedimentation response of large river sys-tems to shifts in land use are neither well documented nor easily tested experimentally. The task of separating changing human in¯uences from natural variability is made therefore complex and requires sophisticated modeling of frequently sampled long-term data of high quality.

The Yadkin River basin in North Carolina o€ers an opportunity to test our ability to detect changes in sedimentation due to changing land-use. The availability of over 40 years of high quality time-series data during a period of major land-use change makes this basin im-portant for testing new methods that can address changing sedimentation response.

Previous attempts at long-term trend analyses of sedimentation processes have largely relied on graphical techniques or detection of monotonic trends in sediment discharge [10,14,31]. These methods are limited to linear trends over time and are unable to e€ectively separate land-use changes from climatic variability. In addition they cannot be used to interpret the changes in a mechanistic framework.

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*

Corresponding author.Present address: Wildlife Institute of India, P.O. Box 18, Chandrabani, Dehradun 248001, India.

E-mail addresses: jug@duke.edu (J. Krishnaswamy), michael @stat.duke.edu (M. Lavine), drichter@duke.edu (D.D. Richter), korfk@denison.edu (K. Korfmacher).

We present a simple dynamic Bayesian regression model that overcomes these limitations and is a rea-sonable compromise between mechanistic complexity and statistical validity. The use of Bayesian time-series methods in this study is motivated by its potential for capturing the complex and changing hydrologic and ¯uvial dynamics of large basins.

1.1. Yadkin River basin

The river basin above Yadkin College is 5905 km2_in

area, mainly draining the western Piedmont of North Carolina, but also the Blue Ridge escarpment of the Appalachian Mountains in both western North Carol-ina and Virginia (Fig. 1). Detailed characteristics of the basin are described elsewhere [14].

In the highly erodible Piedmont of south eastern North America, conversion of forests for agriculture greatly accelerated soil erosion and river sedimentation since the 18th century [19]. As a result, many stream and river channels, runo€ source areas, and ¯ood plains have large sediment deposits that are periodically transported downstream [20]. Piedmont river systems are transporting 10-fold or more sediment than they were prior to European colonization [21].

During the 20th century, soil erosion from cultivated lands has been decreasing in the Piedmont, mainly as a result of row cropped land being abandoned or con-verted by farmers to less erosive uses [35]. Soil man-agement has also improved on Piedmont farms that remain under cultivation [34]. These agricultural changes are widely suggested to have caused rapid decreases in river sediment transport [1,9,18,26,35]. Other sediment

studies of the Piedmont and other regions suggest a more complex and long-term river basin response to land-use [8,19,21,33]. The Piedmont, for example, is now entering a post-agricultural era in which urban and suburban development and highway construction have become increasingly common across the region.

The speci®c objective of this study was to model the sediment response of the Yadkin basin using Bayesian dynamic linear regression models, to determine their ability to detect long term trends in basin sedimentation response to changes in land-surface properties.

2. Methods

2.1. Modeling basin sedimentation processes

Land use and vegetation determine the land surface and soil conditions that control the erodibility of the soils in the basin. Rainfall supplies the energy to dis-lodge available exposed soil. The transport capability as expressed in stream discharge controls the ultimate ex-port of sediment from the basin. The dynamic sediment discharge from a basin can be modeled as a simple function of changing input hydrologic energy (rainfall erosivity), transportation capability (stream ¯ow), and supply of erodible and transportable sediment. The change in supply can be associated with deforestation, subsequent land use, intensi®cation of agriculture, as well as mass movement events such as earthquakes and landslides. All these contributory factors can vary in time leading to a non-stationary supply function for sediment. That is,

Stˆf1…Et;Tt;SUt†; …1†

where St; Tt; EtandSUt are sediment output, ¯uvial transport capability, hydrologic energy (rainfall erosiv-ity), and sediment supply from the basin, respectively and o_St=o_t;o_Et=o_t;o_Tt=o_{t; and}o_SUt=o_{t are their} re-spective rates of change.

Empirical data based approaches are used to model some measure of sedimentation (i.e. sediment concen-tration or ¯ux) as a non-linear power function of stream ¯ow or discharge

CtˆaQbt; …3†

where Ct is usually ¯ux averaged mean monthly sus-pended sediment concentration (mg lÿ1_{) and Qt is}

monthly average of corresponding stream ¯ow (m3 _sÿ1₎

at time t. In practice, log transformed sediment con-centration and corresponding ¯ow at a gauging station have been used to ®t static linear regression models

log10CtˆA‡Blog10…Qt† ‡e: …4†

The parameters a and b (or A and B) may reveal watershed and channel characteristics, including area, geomorphology, vegetation, and hydro-climatic factors [2,4,11,22±24,27,37]. In general, the coecient A will be higher (all else being equal) for watersheds with higher rates of sedimentation. Thus the coecientAfunctions as a base line supply parameter.Bmay be considered a measure of rate at which hydrologic energy is converted to geomorphic work [22,29]. Changes inBover time at a gauging station could be an indication of increasing sensitivity of the upstream watershed to hydrologic forcing. Many researchers [1,2,20,25,41] have related parameterBto the availability of sediment in relation to available hydrologic energy. However, the parametersA

andB are inversely correlated in the watershed and as statistical coecients as well [29].

Researchers have used various methods to detect changes in water-quality variables such as sedimenta-tion. The most frequently used method is to ®t a global static linear regression between sediment concentration and some function of river ¯ow, as described above. This is followed by graphical and non-parametric trend analyses of the time-ordered residuals [10]. Other methods of detecting change include input±output type models in which the watershed is treated as a lumped linear system with ¯ow as input and sediment yield rate as output [32], transfer function models of suspended sediment that are based on autocorrelation structure of the time-series [17] and Kalman ®lter applications [3].

In this paper we ®t a parsimonious regression model with parameters that are mechanistically realistic [12,13,36] and that evolve over time. The parameters and their uncertainty at any timetare speci®ed as prior

probability distributions. The parameters evolve with time by incorporating new data to update distributions. Information loss through time is speci®ed by discount-ing older data. This method uses the Kalman ®lterdiscount-ing equations and is known as a dynamic linear model (DLM). It was applied to assess changes in sediment-¯ow relationships in the Terraba basin of Costa Rica [15].

Details of the DLM regression and applications in hydrology are discussed elsewhere [16,28,39].

Many researchers have used ¯ow as the independent variable in regressions to predict sediment concentra-tion. A more appropriate variable to predict sediment is available hydrologic energy or rainfall [7,14,32]. This is particularly applicable where rainfall erosivity varies seasonally and inter-annually. The use of this variable in sedimentation studies has been hampered by non-availability of basin-wide rainfall intensity data. In this study we will use basin averaged rainfall erosivity in addition to stream ¯ow separately as regression vari-ables. In this paper we follow the basic procedure and consider DLMs with observation equations using a monthly time-step:

log₁₀CtˆAt‡Btlog₁₀…EROSIVITYt† ‡et; …5†

log₁₀…FLOWt† ˆAt‡Btlog10…RAINFALLt† ‡et; …6†

log10CtˆAt‡Btlog10…FLOWt† ‡et: …7†

Ct is monthly volume-weighted sediment concentration, FLOWt is monthly stream ¯ow, RAINFALLt is

monthly basin rainfall, EROSIVITYt is a function of

rainfall intensity, and et is a stochastic noise variable.

Since there is no term for supply, changes in sediment availability related to land-surface conditions will be re¯ected in the temporal dynamics of coecientB. We will de®ne coecient B in Eq. (5) as the basin-wide erodibility coecient at any timetand interpret changes as re¯ecting changes in land-surface conditions. Changes in the parameters Bt over time due to land-surface processes such as forest conversion, reforest-ation and urbanizreforest-ation will be the main focus of the analyses.

Details of the DLM are available elsewhere [28,39]. Operational details are given in Appendix A. Here a brief description is given. In the DLM representation, consider the sediment system of Eq. (5):

Observation equation:

Ytˆlog10CtˆFt0ht‡mt; mtN‰0;VtŠ; …8†

where

Ftˆ …1;log10…EROSIVITY†† and htˆ …At;Bt†:

The so-called ``dynamic'' aspect is modeled through the system equation. This equation is the deterministic relationship between parameter values at any time tto their values at time tÿ1. The regression coecient vector htˆ …At;Bt† performs a random walk about its

previous level and its probability distribution is updated at each time-step using Bayes theorem.

htˆhtÿ1‡xt; xtTn…tÿ1†‰0;WtŠ; …9†

p…htjDt† / p…htjDtÿ1†p…Ytjht†; …10†

whereTis studenttdistribution on appropriate degrees of freedom corresponding to progress in processing the time-series step by step and Dt is information available at time stept.

The observation and evolution equation together comprise the DLM. The covariance matrix Wt models the rate of change in regression parameters. We choose

Wt through time by the method of discount factor [39]. The discount factordis a number in [0, 1]. LetRtbe the covariance matrix ofhgiven data up to timetÿ1, from

p…htjDtÿ1† ˆTn…tÿ1†‰at;RtŠ, that is the prior for time stept andCtÿ1be the posterior covariance matrix at time-step tÿ1. Then,

RtˆCtÿ1‡Wt; …11†

Wtˆ …dÿ1_ÿ

1†Ctÿ1; …12†

RtˆCtÿ1dÿ1: …13†

E€ective use of a discount factor adds some extra vari-ance to the posterior at timetÿ1 to generate the prior for time t. Large values of d model a process that changes slowly through time. In fact, dˆ1 yields a process that is static through time and reduces to the usual linear regression model. Smaller values ofdlead to greater rates of decay of past information in relation to more recent data. Small values ofdmodel a process that changes rapidly. Determining appropriate discount factors for regression variables and implications for hydrologic modeling are discussed elsewhere [16,28]. In the current study the discount factors for the levelAtand regression coecients Bt were kept the same based on statistical correlation between similar variables in tra-ditional sediment rating curves [29]. In this application it is interpreted as a dynamic regression intercept. In ap-plications where the level coecient At is of interest as well, it is desirable to make At orthogonal and inde-pendent of the regression coecient Bt [28,39]. This is achieved by centering the regression variable time-series by subtracting the mean and dividing by the standard deviation. However, this does not a€ect the general shape of the time-trend ofBt, only its magnitude. Cen-tering is also recommended in multiple regression models to make the regression variables comparable. In the current application, the regression slope Bt is the

parameter of interest, and the original scale of the ob-served time-series was used.

In order to initiate the DLM, prior estimates of model parameters are required at timet0. In the absence

of informative priors from previous data, mechanistic intuition or expert opinion, two practical methods are available. One can ®t a static model…dˆ1†, equivalent to ordinary regression that assumes no change in re-gression coecients over time. The means and variances for regression parameters can be used to specify priors, along with a point estimate of the observation variance. The other alternative is to use reference analyses [28,39]. The reference initial prior speci®cation for an unknown but constant observation varianceVcase is de®ned via

p…h1;VjD0† / Vÿ1: …14†

The joint prior and posterior distributions of the state vector and the observation variance for tˆ1;2;. . .are

derived and described elsewhere [39]. A small part of the time-series data is used and after n‡1 observations (nˆnumber of unknown parameters including the un-known observation variance) proper priors with large uncertainty are obtained and the regular DLM updating can proceed. E€ectively posterior covariances do not exist for t<n‡1. So setting Wtˆ0 fortˆ

1;2;. . .;n‡1 is recommended. This is because until

timetˆn‡1, we have essentially only one observation worthy of information for each unknown parameter. Thus, it is not possible to detect or estimate any changes in parameters until tˆn‡1. At this time, the full dy-namic model with suitable, non-zero evolution covari-ance matrices becomes operational. In this study, reference analyses were used to specify priors. To check sensitivity to priors we also used the static regression method and found very little di€erence in the posteriors and the shape of the time-trend of the parameters of interest.

In order to evaluate alternative models in discount space, a model performance diagnostic is required that can be plotted as a likelihood surface for discount combinations. Observed predictive density is model likelihood, a measure of goodness of predictive per-formance is de®ned as

Pt_P…YtjDt

ÿ1† …15†

in the one-step ahead forecast mode, where Yt is the actual observed response and Dt is information:

p…htÿ1jDtÿ1†;Ytÿ1, etc. at timet. Thus the error sequence etˆYtÿft where ft is the one step ahead predicted forecast based on Dtÿ1 is used to calculate the model

Once the one-step ahead estimates of the regression coecients and their distributions, p…htÿkjDtÿkÿ1† are

obtained, the retrospective probability distributions, conditional on all the data p…htÿkjDt† for all k, can be

obtained through a backward recursive technique [28,39]. These estimates are particularly recommended when the objective is to assess historical changes in processes, rather than forecasting ability. In this study, only the ``smoothed'' retrospective estimates of the re-gression coecient and their 90% probability intervals are presented.

2.2. Data

The daily suspended sediment record of the Yadkin River collected at the gauging station at Yadkin College, North Carolina is one of the longest-duration daily suspended sediment records in the world. At Yadkin College, suspended sediment has been sampled by the US Geological Survey as part of the NASQAN network at least daily since January 1951 to September 1990.

Concentrations of suspended sediment are estimated by depth-integrated stream collections. Corresponding stream¯ow at Yadkin College has been continuously recorded. Details of sampling are described elsewhere [14,31]. The daily records were aggregated to generate monthly records of average and volume weighted-sedi-ment concentration and corresponding stream ¯ow.

Daily precipitation at eight 15-min recording stations (Fig. 2) were aggregated on a monthly time-step and Thiessen polygons were utilized to generate an area weighted, spatially averaged, basin rainfall. Daily

ero-sion index (EI) variable [30] was generated for each of the eight stations using the EI equation:

EIˆaPb_{‡e; …16†}

where EI is the calculated EI variable (MJ mm haÿ1

hÿ1_{),Pis the daily (storm) rainfall,ethe random}

com-ponent, and a and b are model parameters that vary seasonally. The higher value for parameter a in the summer season re¯ects the rainfall intensity di€erences between summer convective thunderstorms and winter frontal systems. Daily values were aggregated on a monthly time-step and area weighted by Theissen polygons to obtain a basin wide monthly rainfall ero-sivity. The processed time-series used in this study are shown in Fig. 3.

3. Results and discussion

3.1. Choice of discount factor

The log-likelihood results for Eqs. (5) and (7) favor a discount factor of 0.95, and for Eq. (6) a value closer to 0.90 (Table 1). In this study Eqs. (5) and (7) are the chief focus, and to be consistent, 0.95 was chosen for all three equations. In addition, the principles of conservative trend detection favor a value close to 1 [15,28]. However sensitivity analyses for discount factor was performed. This is discussed later. Comparisons of the time-trend of regression coecients using the static prior and refer-ence analyses priors gave almost identical results.

However, the graphical displays of the results presented for discussion are based on the reference analyses priors.

3.2. Dynamic regression coecients

A regression coecient that changes over time is in-dicative of changes in the process that links the inde-pendent variable to the response variable. Analysis of the DLM yields time-series of estimates of Bt, the re-gression slope at time t. Figs. 4±6 show how the slope (with 90% probability intervals) changes over time for regressions (5)±(7). The annual cycle is evident in Figs. 5 and 6, and is not expressed in the erodibility coecient (Fig. 4). This is attributed to the incorporation of seasonal changes in rainfall energy explicitly in the re-gression variable itself. All three ®gures indicate a change occurring sometime in the late 1960s or early 1970s. Fig. 4 shows that following a period of decrease between 1951 and the change point, there was an in-crease in sediment per unit basin hydrologic energy …o_St=o_Et†, suggesting an increase in basin erodibility.

Fig. 3. Time-series of monthly hydrologic data for Yadkin basin.

Table 1

The observed predictive density expressed as log-likelihood for di€er-ent values of discount factors

Discount factor

Log-likelihood

Eq. (5) Eq. (6) Eq. (7) 0.85 )116 193 )226

0.9 )93 231 )307

0.95 )77 201 )128

1.00 )101 123 )146

Fig. 5 shows an increased ¯ow per unit rainfall …o_Ft=o_Rt†, in the latter period. Fig. 6 indicates an in-crease in sediment per unit ¯ow …o_St=o_{Tt†, indicating a} more accessible source of sediment, especially in the late 1960s to 1980s period.

3.3. Model ®t and sensitivity analyses

In order to assess the model-®t performance of the selected models, the log-likelihood based on one-step ahead forecasts (Table 1) and ®tted vs observation plots (Fig. 7) based on ®ltered, retrospective smoothed pre-dictions were evaluated. The superiority of the non-stationary models (higher log-likelihood and better ®t) as compared to the static or non-stationary models is clearly evident. This indicates that the observed changes in the regression coecients over time are a strong in-dicator of changes in the system in which the time-series data were sampled. The shape of the time-trend is gen-erally robust towards choice of discount factor espe-cially for Eqs. (5) and (6) (Fig. 8). Thus the observed changes in Bt cannot be attributed as an artifact of choice of discount factor. Analyses of residuals from the

models also indicated no serious violation of assump-tions.

3.4. Dynamic intercept At

Although the parameter of interest is clearly the re-gression slope coecient,Bt, results for the interceptAt

are presented as well (Fig. 9). The time-trend of At in cases where it is modeled as an intercept (rather than a level) is expected to be non-orthogonal to Bt. The neg-ative correlation between At and Bt corresponds to similar correlation between the intercept and regression slope observed in traditional static sediment rating curves and other hydrologic regressions [29].

3.5. Sedimentation in the Yadkin basin

All the observed changes as re¯ected in the dynamic regression coecients above are consistent with, and possibly explained by, changes in land use throughout the basin starting in the late 1960s that reversed the declining trends in basin erodibility and run-o€.

The basin is mainly forested in the mountains and is largely agricultural across much of the Piedmont with

Fig. 5. Time-series of dynamic ¯ow coecient with 90% probability intervals: slope of rainfall in regression of log ¯ow with log rainfall as independent variable.

the exception of the Winston-Salem area, which has undergone urbanization and suburbanization, especially since the 1970s. The land use of the larger southern Piedmont is in general rapidly changing from a pri-marily rural, agricultural landscape to one that is dominated by a mix of uses (Fig. 10). A remotely sensed land-use land-cover database [14,31] estimates that the area under row crops decreased from 679 (11.52%) to 330 km2_{(5.6%) during the period of record, reverting}

to forest and pasture. On the other hand residential and urban areas have grown from 391 to 704 km2

(5±18%). Urban development is particularly concen-trated close to the main stem of the river near the mouth of the basin.

Previous analyses using the traditional trend tech-niques suggest that ¯ow adjusted sediment transport is decreasing at the rate of 1:150:3790% conf mg kmÿ2

yrÿ1_{. According to the non-parametric time trend}

ana-lyses [9] the Yadkin was transporting 30% less sediment in the 1990s than in the 1950s [30]. The e€ects of the shift from rural changes to urbanization could not be ana-lyzed using these techniques, since they assume a single monotonic trend through time. However only the existence of a monotonic trend can be demonstrated and

linkages to changes in land-surface conditions are not possible. The current application of the DLM in this study demonstrates that a graph of the back-®ltered estimates of the regression coecient or slope of rainfall erosivity provides a much stronger and convincing evi-dence for a real change in basin erodibility.

The regrowth of forests and pastures on abandoned agricultural areas would explain the decrease in the ability of rainfall to erode soil in the basin between 1951 and 1970. The more recent rising trend in the erodibility coecient and changes in the rainfall-¯ow processes may be related to the increase in urban areas and road construction. These generate impervious surfaces close to the main stem of the river leading to quick run-o€ and higher sediment delivery ratio. This recent growth of more impervious surfaces prone to run-o€ and conse-quent erosion accelerated in the late 1960s to early 1970s. The increase in the availability of sediment be-tween mid-1960s and early 1980s as re¯ected in the ris-ing trend in the dynamic sediment coecient (regression 7) in Fig. 6 is attributed to the recent urban develop-ments closer to the main stem of the river near the basin mouth.

The agricultural changes have substantially decreased gross soil erosion on extensive rural areas of agricultural land throughout the Piedmont region [14,31]. It is pos-tulated, however, that the sources of sediment in the Yadkin River are not simply decreasing but are rather shifting from being largely a result of agricultural ac-tivities to being a result of a variety of human acac-tivities, increasingly associated with urban and suburban de-velopment [40]. The Yadkin basin, like much of the Piedmont as a whole is increasingly a€ected by urban and suburban developments and highway construction. In the Yadkin, such developments have been particu-larly pronounced in rapidly growing communities sur-rounding Winston-Salem, NC. The continued e€ects of urbanization in stabilizing the decline in overall basin surface erodibility and perhaps increasing sedimentation will perhaps be revealed by DLMs estimated in the near future.

4. Conclusion and future directions

The DLM regression approach is successful in mod-eling non-stationary hydrologic processes as illustrated for the Yadkin basin. The changes in the regression coecients were consistent with predicted e€ects of changes in land-use. The dynamic erodibility coecient as de®ned in this paper can be used to analyse the e€ects of complex shifts in land-surface conditions on basin sedimentation separate from natural climatic variability. In general the dynamic regression approach would be most suitable for assessing changes in relationships be-tween di€erent hydrologic and ecological ¯uxes over

time. These could then be related to changes in land-use or shifts in hydro-climatology. Impact of infrequent but high intensity events such as earthquakes and landslides, which may have a lingering in¯uence on hydrology and sedimentation, are more easily detectable using this method. The change points as well the duration of the in¯uence of such events on sediment or rainfall-¯ow processes could be analysed using the dynamic re-gression approach. This method lends itself to a more mechanistic interpretation (of the changes in the re-gression coecients) compared to time-ordered residual analysis after ®tting a static regression model or non-parametric tests for monotonic trend such as the Seasonal Kendall and Mann±Kendall with Sen slope estimator. This is primarily because changes in regres-sion coecients over time are more likely to be related to real changes in the process that links an independent variable such as rainfall erosivity to a response variable such as sediment concentration.

In this study only one regression variable was used to model a response variable. Extension to multiple re-gression applications with single or di€erent discount factors for level and the di€erent regression variables is

recommended [16]. In hydrologic systems, the contri-bution of di€erent input or forcing variables to the re-sponse may display varying degrees of hydrologic ``memory'' or dynamism as the system evolves in time. In the DLM this can be approximated by setting dif-ferent discount factors for the individual regression variables. One example in hydrology, where di€erent discount factors could be applied is in a multiple re-gression of stream ¯ow modeled as a function of basin storage and rainfall. The rainfall coecient is likely to be more dynamic compared to basin storage which may have a longer ``hydrologic'' memory.

Seasonality, driven primarily by the annual solar cy-cle, is an integral feature of many ecological and geo-physical systems. In many hydrologic systems the seasonality of the annual cycle is expressed in ¯uxes of hydrologic variables such as ¯ow, rainfall, rainfall ero-sivity, run-o€ coecients, erosion and sedimentation. In sedimentation for example, concentrations at a given discharge mostly decrease as the run-o€ season pro-gresses and sediment is ¯ushed out of a system leading to exhaustion. This leads to a typical clockwise loop between ¯ow and sediment concentration. However, the

Yadkin basin is unique in displaying an anticlockwise seasonal hysteretic loop [14].

Incorporation of a seasonal component in intercept or regression coecients can be done by a modi®cation of the system evolution matrix, G and the regression variable vector, Ft [16]. These models can potentially capture the seasonality in real hydrologic ¯uxes and regression or intercept coecients over time can be

de-composed into a long-term trend and a seasonal oscil-lation. These could be used to determine changes in the amplitude of annual seasonality over time. This type of analyses can be used to determine changes in climate and phenomenon such as ENSO, and their impact on hydrologic processes.

These approaches need to be applied to other hy-drologic and geophysical time-series data where there is an identi®ed response variable and one or more ex-planatory variables. This will lead to an enhanced un-derstanding of the dynamism of hydrologic systems and their sensitivity to climate change as well as anthropo-genic in¯uences.

Acknowledgements

Michael Hofmockel of the Duke University Forest Soils laboratory helped with ®gures and formatting. We thank two anonymous reviewers for useful comments and suggestions. The Ashoka Trust for Research in Ecology and the Environment supported and facilitated this study.

Appendix A

A brief outline of the DLM from [39] is provided below to enable implementation.

Consider the case of when observation variance

V ˆ/ÿ1 is unknown and constant and the system evolution is a random walk about previous values.

Ytˆobservation for response variable, Ftˆ indepen-dent variables regression matrix, htˆregression

pa-rameters state vector, Wtˆsystem evolution variance. d‰0;1Š is the discount factor.Dtˆinformation available at time t, Stˆ point estimate of observation variance,

ntˆdegrees of freedom, N, T and G are Normal, Stu-dentTand Gamma distributions, respectively.etis one step ahead forecast error and ft is the forecast. Initial priors are speci®ed: m0;C0;S0;n0ˆ1 or obtained after

reference analyses using a small part of the initial data. Then:

Observation:

YtˆFt0ht‡mt; mtN‰0;VŠ:

System:

htˆhtÿ1‡xt; xtTn…tÿ1†‰0;WtŠ:

Information:

…htÿ1jDtÿ1† Tn…tÿ1†‰mtÿ1;Ctÿ1Š;

…/jDtÿ1† G‰ntÿ1=2;ntÿ1Stÿ1=2Š:

Forecast:

…YtjDtÿ1† Tn…tÿ1†‰ft;QtŠ;

…htjDtÿ1† Tn…tÿ1†‰at;RtŠ:

Fig. 9. Posterior estimates of interceptAtwith 90% probability

inter-vals and relationship with regression slopeBt. AllR2 values are for

pˆ0.

RtˆCt‡Wt; Wtˆ …dÿ1ÿ1†Ctÿ1;

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