spatial patterns for watersheds erosion

Spatial Patterns for Watersheds Erosion

Fotis Maris & Kyriaki Kitikidou

Received: 5 June 2009 /Accepted: 22 June 2011 /Published online: 9 July 2011# Springer Science+Business Media B.V. 2011

Abstract This article describes a simultaneously autore-gressive model applied to the erosion data collected at 17natural lake watersheds in Greece. The methodologyconsiders spatially correlated random area effects takinginto account the information provided by neighbouringtorrents/streams. The article discusses the gain obtainedfrom modelling the spatial correlation among small arearandom effects useful in representing the unexplainedvariation of the small area target quantities.

Keywords Erosion . Small area models . Spatial models .

Watersheds

1 Introduction

In 1997, a team of researchers from the AristotleUniversity of Thessaloniki started an environmentalstudy for 17 lake watersheds in Greece. Greek lakewatersheds cover about 3,885 km2. One of the majorobjectives of the project was the estimation of the surfacewater pollution by watershed. Because erosion on agriculturalland is known to be a major component of water pollution aswell as the major source for lake sedimentation, a survey wasdesigned to estimate the amount of erosion delivered tostreams in the watershed.

The total erosion from agricultural land representedthe variable of primary interest to the researchers. In thepresent study, the estimator for average erosion wascalculated for each watershed [7]. Since only 51 siteswere drawn, which is not sufficient to achieve reliableestimates at the watershed level, it was necessary to use a

small area estimation technique to improve the precisionof this estimator.

Τhe aim of this article is to apply an estimator and ameasure of its variation (root mean squared error(RMSE)) for the survey data collected as a part ofthe study taking into account the spatial dimension ofthe data, adapting a model with spatially correlatedrandom effects. An empirical best linear unbiasedpredictor (EBLUP; [4]) combined with a simultaneouslyautoregressive model (SAR), to consider the first- andsecond-order level of spatial dependence of the data, hasbeen implemented.

Opsomer, Botts and Kim [8] and Petrucci and Salvati[10] discussed a small area estimation procedure (EBLUPestimator), which used explicit linking models based onrandom area effect. These effects were considered uncorre-lated but, in most of the environmental applications, itshould be more reasonable to assume that the random areaeffects between the neighbourhood areas are correlated andthe correlation decreases to zero as distance increases.Following this reasoning, Cressie [3] also used a condi-tional autoregressive spatial model (CAR) for small areaprediction although an estimator of the MSE of the EBLUPestimator has not been given in detail.

2 Materials and Methods

2.1 Methodology

Spatial models are a special case of the general linear mixedmodels, with an error variance–covariance matrix of theform [3]:

E ui; uj� � ¼ 4 tð Þ ð1Þ

where τ is the coefficient in a SAR or CAR error process,and ui, uj are the random area effects.

F. Maris :K. Kitikidou (*)Dimokritos University of Thrace, Greece,Pandazidou 193,68200 Orestiada, Greecee-mail: [email protected]

Environ Model Assess (2012) 17:267–273DOI 10.1007/s10666-011-9284-0

The basic area model includes random area effects andthe m×p matrix of area specific auxiliary covariates X isrelated to the m×1 vector of the parameters of inferentialinterest θ (total y, mean y):

q ¼ Xb þ Zv ð2Þwhere b is the regression parameters vector of dimensionp×1, Z is a matrix of known positive constants ofdimensions m×m, with m the number of small areas (avector (0,0…0,1,…0) with value 1 referred to the itharea, i=1, 2, …, 17 in our study), v=ρWv+u is the m×1vector of dependent random variables according to a SARerror process with mean 0 and variance–covariance matrixs2u I� rWð Þ I� rWT

� �� 1. In this matrix, ρ is the spatial

autoregressive coefficient, W is the spatial weight matrixfor θ, I is an m×m identity matrix and u � N 0; s2

uI� �

. If Wis defined in row standardized form, it is row stochastic,ρ∈(−1, 1), and it is called the spatial autocorrelationparameter [1]. Moreover, it assumes that the m×1 vectorof direct estimators bq is unbiased:

bq ¼ q þ e ð3Þwhere e represents the sampling errors with mean 0 andknown m×m variance matrix diag(Ψi). Combining Eqs. 1and 2, with e independent of v, the resulting model is:

bq ¼ Xb þ Z I� rWð Þ�1uþ e ð4Þ

The error terms v and e have m×m covariance matrices,respectively:

G ¼ s2u I � rWð Þ I � rWð ÞTh i�1

ð5Þ

and

R ¼ diag y ið Þ ð6ÞThe covariance matrix of the studied variable is

V ¼ Rþ ZGZT ¼ diagðy iÞ þ Zs2u I� rWð Þ I� rWT

� �� 1ZT

ð7Þ

According to model (4), the spatial best linear unbiasedpredictor (spatial BLUP) estimator of θi is:

eqSi s2u; r

� � ¼ xibb þ bTi s2u I� rWð Þ I� rWT

� �� 1n o

ZT

� diag y ið Þ þ Zs2u I� rWð Þ I� rWT

� �� 1ZT

n o�1 bq � Xbb� �ð8Þ

where bb ¼ XTV�1X� ��1

XTV�1bq and bTi is the 1×m vector(0,0,…,0,1,0,…,0) with 1 in the ith position.

The MSE of the spatial BLUP estimator eqSi s2u; r

� �,

depending on two variance components s2u; r

� �, can be

expressed as [11]:

MSE eqSi s2u; r

� �h i¼ g1i s

2u; r

� �þ g2i s2u; r

� � ð9Þ

with

g1i ðs2u; rÞ ¼ bTi s

2u I� rWð Þ I� rWT

� �� 1

�s2u I� rWð Þ I� rWT

� �� 1gZT � diag y ið ÞþZs2

u I� rWð Þ I� rWT� �� 1

ZTg�1

Zs2u I� rWð Þ I� rWT

� �� 1bi

and

g2i s2u; r

� � ¼ xi � bTi s2u I� rWð Þ I� rWT

� �� 1ZT


� �� 1ZT

n o�1X

0BB@1CCA

� XT diag y ið Þ þ Zs2u I� rWð Þ I� rWT

� �� 1ZT

n o�1X

� �1

�xi � bTi s

2u I� rWð Þ I� rWT

� �� 1ZT


� �� 1ZT

n o�1

0B@1CAXT

The term g1i s2u; r

� �is due to the estimation of random

effects and is of order O(1), while the term g2i s2u; r

� �is due

to the estimation of β and is of order O(m−1) for large m.In practical applications, s2

u and ρ are unknown andreplaced by estimators bs2

u and br. Hence, a two-stageestimator eqSi bs2

u;br� �is obtained and called spatial empirical

BLUP (spatial EBLUP):

eqSi bs2u;br� � ¼ xibb þ bTi bs2

u I� brWð Þ I� brWT� �� 1

n oZT

� diag y ið Þ þ Zbs2u I� brWð Þ I� brWT

� �� 1ZT

n o�1

� bq � Xbb� �ð10Þ

with bTi =(0,0,…,0,1,0,…,0) and 1 referred to the ith area.The spatial EBLUP has the following properties, [6]:

& It is unbiased for θi;

& E eqS bs2u;br� �h i

is finite;

& bs2u and br are any invariant estimators of s2

u and ρ(assuming their independence).

The variance components s2u and ρ can be estimated

either by maximum likelihood (ML) or restricted maximumlikelihood (REML) methods. The ML and REML estima-tors are robust and they work well even under nonormaldistributions [5].

268 F. Maris, K. Kitikidou

The MSE of spatial EBLUP eqSi bs2u;br� �

is:

MSE eqSi bs2u;br� �h i

� g1i s2u; r



� �ð11Þ

where g3i is obtained by [6]:

g3i s2u; r

� � ¼ tr

bTi C�1ZTV�1 þ s2uC

�1ZT �V�1ZC�1ZTV�1� ��

bTi AZTV�1 þ s2uC

�1ZT �V�1ZAZTV�1� �� 24 35V

�bTi C�1ZTV�1 þ s2

uC�1ZT �V�1ZC�1ZTV�1

� �� bTi AZTV�1 þ s2

uC�1ZT �V�1ZAZTV�1

� �� 24 35T

V bs2u;br� �

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;ð12Þ

with C ¼ I� rWð Þ I� rWð ÞTh i

, A ¼ s2u �C�1 2rWWT��

2WÞC�1Þ and V bs2u;br� �

is the asymptotic covariance matrixof bs2

u and br. The neglected terms in the approximation are oflower order than O(1). An estimator of MSE eqSi bs2

u;br� �h ican

be expressed as:

MbSE eqSi bs2u;br� �h i

� g1i bs2u;br� �þ g2i bs2

u;br� �þ g3i bs2u;br� �

ð13Þ

if bs2u and br are REML estimators. If the ML method is used,

the MSE eqSi bs2u;br� �h i

can be expressed as:

MSE eqSi bs2u;br� �h i

� g1i bs2u;br� �� bTML bs2

u;br� �rg1i bs2u;br� �

þg2i bs2u;br� �þ g3i bs2

u;br� � ð14Þ

If the term bTML bs2u;br� �rg1i bs2

u;br� �is ignored, the use of

ML estimators could lead to underestimation of the MSEapproximation. This term is given by:

rg1i bs2u;br� � ¼ bTi

C�1 � C�1ZTV�1Zs2uC

�1�

A�ð AZTV�1Zs2uC

�1�

(þs2

uC�1ZT �V�1ZC�1ZTV�1

� �Zs2

uC�1 þ s2

uC�1ZTV�1ZC�1

��þs2

uC�1ZT ð�V�1ZAZTV�1

�Zs2

uC�1 þ s2

uC�1ZTV�1ZA

��)bi

and

bTML s2u; r

� � ¼ 1

2mI�1 s2

u; r� � tr XTV�1X

� ��1XT �V�1ZC�1ZTV�1

� �X

h itr XTV�1X

� ��1XT �V�1ZAZTV�1

� �X

h i264375

8><>:9>=>;

where I�1 s2u; r

� �is the matrix of the expected second

derivatives of b; s2u; r

� �with respect to s2

u and ρ and isgiven by:

I s2u; r

� � ¼ 1

2tr V�1ZC�1ZTV�1ZC�1ZT � 1

2tr V�1ZC�1ZTV�1ZAZT �

1

2tr V�1ZAZTV�1ZC�1ZT � 1

2tr V�1ZAZTV�1ZAZT �

264375

w i t h C ¼ I� rWð Þ I� rWð ÞTh i

a n d A ¼ s2u �C�1�

2rWWT � 2W� �

C�1Þ.

Spatial Patterns for Watersheds Erosion 269

2.2 Application

According to the sampling design and data collectionmethodology described by Opsomer, Botts and Kim [8]and Petrucci and Salvati [10], each small area (domain-lake) was divided in plots (streams; total 265), each plotwas sequentially labelled and a systematic sampling ofplots was selected. The fractional interval [13] was fixed inorder to select 4 units from each small area (domain). Notall these 4×17 units were included in the sample. Fromeach domain, a simple random sample of 3 units wasdrawn. Then, within each watershed, three plots wereselected and a sample of 51 units was obtained. The finalsample can be reasonably assimilated to a simple randomsample from the domains and the sampling variance Ψi atthe domain level can be estimated by

1� niNi

� bs2i

ni

�

where ni=3 and Ni is the number of plots in the ith area[5, 7].

Auxiliary data at the watershed level are required inorder to apply an area-level random effect model. For thisapplication the land use and the topography data, consid-ered as major determinants of the erosion, were available.Data related to these factors were available for the studyregion in the form of digital elevation and land useclassification coverages. A set of variables for a predictionmodel was constructed combining slope class and rocktypes [7].

In order to detect the spatial pattern (spatial associa-tion and spatial autocorrelation) of the average erosion,some standard global spatial statistics have been calcu-lated: Moran’s I, Geary’s C [2], G statistics [9]. Thestandardized Moran’s I is analogous to the correlationcoefficient and its values range from 1 (strong positivespatial autocorrelation) to −1 (strong negative spatialautocorrelation). Geary’s C ranges between 0 and 2.Positive spatial autocorrelation is found with valuesranging from 0 to 1 and negative spatial autocorrelationis found between 1 and 2. Finally, G statistic is an index ofspatial clustering of a set of observations over a definedneighbourhood. The above indexes have been calculatedon the sample and they show a positive spatial correlation

Index Value

Moran’s I 0.298

Geary’s C 0.566

G statistic 0.148

Table 1 Values of Moran’s I,Geary’s C and G statistics

Table 2 Comparison of small area estimators: direct estimates for the mean of y, RMSEs and biases B

Smallarea

Post-stratified regression EBLUP Spatial EBLUP

Direct estimatesfor the mean of y

RMSEsof theestimates

Biases B Direct estimatesfor the mean of y


Biases B Direct estimatesfor the mean of y


Biases B

1 71.470 3.239 0.345 67.848 2.276 1.613 72.738 1.675 0.374

2 63.769 3.856 −0.248 62.896 2.574 −0.128 63.889 0.519 −0.1503 117.478 5.121 −0.435 117.044 2.146 0.687 118.599 0.761 −0.0474 23.225 2.546 0.247 25.131 2.526 2.392 25.369 2.509 −0.1425 102.061 4.634 0.638 99.760 2.651 0.600 102.023 0.687 0.375

6 101.555 5.304 −0.161 100.443 3.017 1.711 103.428 1.743 0.095

7 169.693 5.403 0.118 167.885 2.375 −4.071 165.503 4.105 0.382

8 82.609 3.219 −0.540 83.541 2.166 −0.553 82.596 0.684 −0.3809 117.967 4.295 −0.109 117.203 1.952 −2.558 115.518 2.579 0.236

10 47.693 2.794 −0.562 44.866 2.180 −2.687 45.567 2.757 −0.44611 125.163 4.634 −0.823 128.917 2.033 1.112 127.097 1.163 −0.40412 169.785 6.369 −1.290 181.759 1.989 1.696 172.771 1.789 −0.17413 63.292 2.673 −0.011 65.247 1.832 1.860 65.163 1.925 0.098

14 84.804 3.082 0.244 84.405 1.872 0.543 85.103 0.669 0.122

15 151.789 4.764 −0.425 150.334 1.696 −2.611 149.603 2.647 0.063

16 129.755 4.473 0.083 129.872 1.940 −0.079 129.593 0.358 −0.10417 48.887 2.498 0.016 50.077 2.091 1.286 50.157 1.415 0.149

Average 98.294 4.053 −0.171 98.660 2.195 0.048 98.513 1.646 0.003


of the average erosion and indicate the possibility ofclusters of high values of the study variable (see Table 1).

To implement an EBLUP estimator combined with aSAR model a neighbourhood structure W has to bedefined for the 17 small areas. The spatial weight matrixrepresents the potential interaction between locations. Ageneral spatial weight matrix can be defined by asymmetric binary contiguity matrix, which can be gener-ated from the topological information based on theadjacency criterion: the spatial weight wij is set equal toone if area i shares an edge with area j and 0 otherwise.For an easier interpretation, the general spatial weightmatrix is defined in row standardized form, in which therow elements sum to one. Spatial EBLUP methodology isdescribed in Petrucci and Salvati [10].

In order to assess the use of the spatial EBLUPmethodology, this section presents results that focus onestimating the small area means for a survey variable Y. Westudied the relative performances of the post-stratifiedestimator (direct estimator), EBLUP and spatial EBLUPestimators. To this end, the population of y values wasgenerated from a linear mixed model with random areaeffects of the neighbouring areas correlated according to the

SAR dispersion matrix with established spatial autoregres-sive coefficient, defined byyij ¼ 0:35xij þ ui þ eij þ x1 2=

ij , i=1,2,…,m, j=1,2,…Niwhere xij is the value of the auxiliaryvariable x for unit j in the small area i, ui is the random area

Fig. 2 Normal q–q plot to check the normality of the standardizedresiduals r

Fig. 1 Map of the 17 Greek lakes with spatial EBLUP estimates for average erosion


specific effect and eij is the individual error [12]. Wegenerated independent random variables u ¼ u1; u2; :::; um½ �Tand e ¼ e11; e12; :::; eij; :::emNm

� �Trespectively, from a

MVN 0; s2u I� rWð Þ I� rWT

� �� 1n o

and N(0,σ2) with

σ2=5.44; xij values were generated from a uniform distribu-tion between 0 and 1.000. The empirical sampling distribu-tion of the estimates is obtained under four different patternsof spatial correlation with ρ equal to ±0.20, ±0.80 andneighbourhood structure (W) based on the lakes. The Wmatrix was kept fixed for all runs.

To make comparisons between estimators undercustomary repeated sampling, (T=250) samples of aver-age size n ¼ 30were selected from each of the fourpopulations by simple random sampling. For each sampledrawn, the mean of each small area was estimated by thepost-stratified estimator, EBLUP and spatial EBLUPestimators.

For each estimator, we computed the average bias (B)and the root mean squared error (RMSE), defined asfollows:

B ¼ 1

m

Xmi

1

T

XTt¼1

bY it=Yi � 1� ��

��

RMSE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

m

Xmi¼1

1

T

XTt¼1

bY it � Yi� �2

vuut

3 Results and Discussion

The results for B and RMSE are reported in Table 2. It isclear from Table 2 that the EBLUP and the spatial EBLUPestimators perform significantly better than the directestimator in terms of RMSE and B, leading to smallervalues. Overall, the spatial EBLUP estimator is better thanthe traditional EBLUP: the introduction of spatial informa-tion leads to an improvement in terms of RMSE (1.646compared to 2.195) and to a reduction in the bias (0.003compared to 0.048).

Looking at the RMSE estimator and its performancewith respect to a coverage of 95% of confidence intervals,the estimated RMSE is very sensitive to the spatialcorrelation value. The RMSE estimators are lower andconsequently the width of the confidence intervals issmaller, given the nominal level of coverage (95%). Thishappens by maintaining or improving the level of empiricalcoverage of the confidence intervals obtained with theEBLUP estimator.

Figure 1 displays the map of the lake watersheds withSpatial EBLUP estimates for average erosion (m3/year).The value of the estimated spatial autocorrelation coeffi-cient br is 0.889 (s.e.=0.0085) using the ML procedure and0.887 (s.e.=0.0087) with the REML method, whichsuggests a strong spatial relationship. The territorialdistribution of spatial EBLUP estimates is more variablethan that obtained with traditional EBLUP and makesevident the specific characteristics of sub-watersheds. Theadditional spatial information inserted in our estimatorreduces the smoothing effect resulting from the applicationof traditional EBLUP, without paying through precision ofthe estimates.

An evaluation of our spatial model is performed by treatingthe standard residuals r ¼ eqS s2

u;br� �� Xb= diag Vð Þð Þ1 2= asN(0,1). In particular, to check the normality of thestandardized residuals r and to detect outliers, a normal q–q plot is examined (Fig. 2). One outlier r is shown incorresponding to neighbouring areas; it can be attributed to aparticular microclimate which characterizes that region(Vistonida lake). No other significant deviations from theassumed model are observed. The Shapiro–Wilk W statisticgives the value of 0.901 for small area effects, showing a pvalue of 0.071 that suggests no discrepancy with thehypothesis of normality.

In conclusion, considering the case study, the use of thespatial EBLUP methodology, which takes into account theSAR model in the small area estimation, reduces theconfidence interval, allowing one to obtain an appreciableimprovement of the small area estimates.

References

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