Journal of Geochemical Exploration 108 (2011) 62–72
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Journal of Geochemical Exploration j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j g e o ex p
Geostatistical methods to identify and map spatial variations of soil salinity P. Juan a, J. Mateu a, M.M. Jordan b,⁎, J. Mataix-Solera b, I. Meléndez-Pastor b, J. Navarro-Pedreño b a b
Department of Mathematics, Campus Riu Sec. University Jaume I. E-12071. Castellón, Spain Departamento de Agroquímica y Medio Ambiente, Universidad Miguel Hernández de Elche. E-03202. Elche (Alicante), Spain
a r t i c l e
i n f o
Article history: Received 29 March 2010 Accepted 11 October 2010 Available online 15 October 2010 Keywords: Bayesian methodology Electrical conductivity Spatial Gaussian linear mixed model Hierarchical modelling Sodium Soil salinity
a b s t r a c t The problem of estimating and predicting spatial distribution of a spatial stochastic process, observed at irregular locations in space, is considered in this paper. Environmental variables usually show spatial dependencies among observations, with lead one to use geostatistical methods to model the spatial distributions of those observations. This is particularly important in the study of soil properties and their spatial variability. In this study geostatistical techniques were used to describe the spatial dependence and to quantify the scale and intensity of spatial variations of soil properties, which provide the essential spatial information for local estimation. In this contribution, we propose a spatial Gaussian linear mixed model that involves (a) a non-parametric term for accounting deterministic trend due to exogenous variables and (b) a parametric component for defining the purely spatial random variation due possibly to latent spatial processes. We focus here on the analysis of the relationship between soil electrical conductivity and Na content to identify spatial variations of soil salinity. This analysis can be useful for agricultural and environmental land management. © 2010 Elsevier B.V. All rights reserved.
1. Introduction There are two main types of soils salinity: dryland salinity, which occur on land not subject to irrigation; and irrigated land salinity. Both of them describe areas where soils contain high levels of salts that can affect plant productivity and soil organisms (Navarro-Pedreño et al., 1997). Under arid or semi-arid conditions and in regions of poor natural drainage, there is a real hazard of salts accumulation in soils. The processes by which soluble salts cause salinity and sodicity in soils include: (i) the application of waters containing salts; (ii) weathering of primary and secondary minerals in soils; (iii) organic matter decay; (iv) watertable instability. The importance of each of those causes depends on soil type, climate, and agricultural management. Accumulation of dispersive cations, such as Na, in soil solution and the exchange phase (K, Mg, Ca) affect the physical properties of soil, such as structural stability, hydraulic conductivity, infiltration rate and erosivity. Historically, the physical behaviour of salt-affected soils has been described in terms of the combined effects of soil salinity, as measured by the EC of a saturation extract, and exchangeable sodium percentage (ESP) on flocculation and soil dispersion. Based on this, the U.S. Salinity Laboratory Staff (1954) described a saline soil (ECN 4 dS/m; ESPb 15) and a saline-alkali soil (EC N 4 dS/m; ESPN 15). By 1979, the term “alkali” was listed as obsolete by the Soil Science Society of America (although it is still used by farm advisors and others), and the word “sodic” was used in its place with the definition: “a soil having an ESPN 15”. Outside the United
⁎ Corresponding author. Fax: +34 966658340. E-mail address:
[email protected] (M.M. Jordan). 0375-6742/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.gexplo.2010.10.003
States, the word “alkali” is generally used in a narrower context referring only to soils with: (i) high sodicity and high pH (ESP N 15, pH N 8.3) and (ii) containing soluble bicarbonate and carbonate (Na/[Cl + SO4] N 1) (Gupta and Abrol, 1990). The use of the term “alkali” thus allows practical distinctions between saline, sodic, and alkali soils in terms of soil management (Bhumba and Abrol, 1979; Gupta and Abrol, 1990). In this paper, “saline” and “sodic” are used as defined by the Soil Science Society of America, while “alkali” is used as defined above. There is a number of factors that influence the vulnerability of sites to salinization (Oster et al., 1996). These factors include: (i) the position of a site within a landscape (Manning et al., 2001), and (ii) soil type and rainfall. It has been postulated that the combination of information on these and other factors allows prediction of sites that are vulnerable to salinity (Navarro et al., 2001). Data sets consisting of rainfall, topography, soil type, and other relevant spatial information can be compiled into a GIS to determine spatial patterns of salinization, and to predict areas that may be at risk. Spatial variations and interrelations among variables related to salinity, such as Na content or EC, are complex. Thus, full understanding, estimating and mapping of the spatial characteristics of soil salinity facilitates accurate risk assessment (Florinsky et al., 2000) and remedying of environmental problems. Most soil properties of scientific interest vary continuously in space and time, and cannot be practically measured or recorded everywhere. To represent their spatial variations, values of individual variables or class types at unsampled locations must be estimated from data of those variables recorded at sample sites. The need to precisely continuous spatial variations is clear, and geostatistics is largely the relevant theory for addressing that need. It embraces a set of stochastic techniques that take into account both the random and
P. Juan et al. / Journal of Geochemical Exploration 108 (2011) 62–72
structured nature of spatial variables, the spatial distribution of sampling sites and the uniqueness of any spatial observation (Journel and Huijbregts, 1978; Goovaerts, 1997). Spatial statistics is one of the major methodologies of environmental statistics, and geostatistical methods are an important part of spatial statistics with wide applications in environmental surveys (e.g., soil science). In most cases, soil data can be considered as partial realizations of a random function (stochastic process) over a region, i.e., a spatially continuous process, as characterized by Cressie (1993). Typically, samples are taken at a finite set of locations in a region and are used to estimate quantities of interest such as the values of the property of interest at unvisited locations. Data of this kind are often called geostatistical data. An important tool in geostatistics is kriging, which refers to a least square linear predictor that, under certain stationarity assumptions, requires at least the knowledge of covariance parameters and the functional form for the mean of the underlying random function. Kriging does not take uncertainty into account in the prediction, but uses plug-in estimates as if they were true. Bayesian inference, in contrast, provides a way to incorporate parameter uncertainty in the prediction by treating the parameters as random variables and integrating over the parameter space to obtain the predictive distribution of any quantity of interest (Feyen et al., 2002). In this paper, we attempt to combine soil properties using spatial statistical techniques under a modelling framework that we refer to as a Gaussian Spatial Linear Mixed Model (GSLMM). This is a general and flexible class of models for handing spatial variation shown by individual variables in a particular environment. The use of a Bayesian approach is also compatible with the GSLMM and, thus, is shown in the paper. Various geostatistical techniques have been discussed in the statistical literature. For example, the relationship between universal kriging and cokriging with regression modelling has been discussed by Stein and Corsten (1991). A complete derivation of a Bayesian approach to kriging has been discussed by Kitanidis (1986), among others. In addition, Curreriro and Lele, 1999 have developed a hierarchical approach to spatial modelling, and they have shown how that approach
63
coincides with both hierarchical Bayesian modelling and mixed linear models. Here, we combine all these techniques into the GSLMM framework. Thus, the focus of this paper is to model the spatial relationship between soil properties: extractable Na content and EC. Both of these properties are clearly associated with soil salinity and their spatial estimation and prediction are of prime scientific interests for further agricultural or environmental applications (Knudsen et al., 1982). The aim of this research was to provide quantitative assessments of the complex (spatial) distributions of Na content and electrical conductivity and interactions between them, in order to determine soil salinity.
2. Study area, data and methods Alicante is a Mediterranean province of southeast Spain with a high variety of soil types. It is a transition area from arid and semi-arid to temperate-humid. These characteristics of that province have a great influence in the variety of cultivated plants (horticulture, delicious trees, citrus, cereals). Although most of the province is under the influence of the Mediterranean sea, its orography and lithology produce an unusual variety of environments (also different kinds of soils) (Antolin, 1998). It is known that this province has saline soils due to important factors: (i) original lithology with presence of Keuper, Tertiary-age clays (Pollifrone and Ravaglioli, 1973); (ii) the situation of saline lagoons which affected soil characteristics (Kout and Dudas, 1995; Navarro-Pedreño et al., 1997); and (iii) long term irrigated agriculture. Water movement into soils is the key factor in management of salt-affected soils. In contrast, the variability of soil salinity could be high in regions with a great variety of microclimate (Harrach and Nemeth, 1982), as is the case of Alicante. Note that Alicante is a region where (a) the climate changes from the south (semi-arid) to the north (template climate) (Fig. 1) and, thus (b) rainfall varies from 250 mm to 600 mm per year (Perez, 1994; Andreu, 1997). It is clear, therefore, that analysis of soil salinity in
Fig. 1. Spatial distribution of rainfall (mm) corresponding to data from 30 years (Navarro-Pedreño et al., 2007).
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Alicante is essential for understanding of environmental degradation processes in the province. 2.1. Soil sampling and analysis The Environmental Protection Administration of Alicante province initiated a collaborative research program to establish the presence of certain kind of salts in the region. Over 400 hundred soil samples (arable layer) from 46 different agricultural sites (located with GPS) were taken during a period of two years (Fig. 2). Samples were dried at room temperature and EC (1:5 w/v water extraction) and extractable Na content (ammonium acetate 1 M extraction) were determined. Soil solution extracts were analysed using atomic absorption spectrometry to determine the Na concentration. 2.2. Geostatistical theoretical framework The basic format for univariate geostatistical data is {(ui, z(ui)), i = 1,...,n}, where ui identifies a spatial location and z(ui) is a scalar measurement taken at location ui. It follows that the basic form of a geostatistical model is a real-valued stochastic process {Z(u): u ∈ A}, which, in turn, is typically considered to be a partial realization of a stochastic process {Z(u): u ∈ ℜ2}. The measurement process Zi can be regarded as a noisy version of an underlying random variable S(ui), the value at location ui of a {S(u): u ∈ ℜ2} that is of prime scientific interest. We call S(u) the signal. The basic model is then extended to one with two ingredients: a stochastic process S(u); and a statistical model for the measurements, Z = (Z1, ..., Zn) conditional on {Z(u): u ∈ ℜ2}. The objectives of geostatistical analysis are broadly of two kinds: estimation and prediction. Estimation refers to inference about the parameters of a stochastic model for the data. These may include parameters of direct scientific interest (e.g., defining a regression relationship between a response variable and an exploratory variable, and parameters of indirect interest (e.g., the covariance structure of a model for S(u)). Prediction refers to inference about the realization of
the unobserved signal S(u). Specific prediction objectives might include prediction of the realized value of S(u) at arbitrary location u within a region of interest A, or prediction of some property of the complete realization of S(u) for all u in A. Before interpolation and prediction, we need to know the structure of the spatial variation, and this is done through variogram (or co-variogram) estimation. Several authors have proposed methods for the improvement and robustness of variogram estimation (Cressie and Hawkins, 1980; Cressie, 1993; Curriero and Lele, 1999). To map the spatial variation in soil salinity or to highlight soil degradation by certain salts that are present in the soil, we need to perform spatial predictions. For prediction, we applied our GSLMM methodology, firstly with matrix X = 1 to obtain ordinary kriging (OK) predictions for each variable, and, secondly with X considered in a trend matrix to obtain cokriging and external trend kriging for the relationship between Na content and EC. After having learned about the co-regionalization model, based on the experimental cross-variogram and cross-variogram model, we can use that knowledge of the spatial relations between two or more variables to predict their values by co-kriging. Typically, the aim is to estimate just one variable, which we may regard as the principal or target variable, from data of that variables plus data of one or more other variables, which we regard as auxiliary variables. In this sense, the cokriging is a natural extension of kriging when a multivariate variogram or covariance model and multivariate data are available. Co-kriging and external trend kriging models are designed for exploiting over-sampled variables to better estimate under-sampled variables. The classical methodology of kriging does not take into account uncertainty in parameter estimation, and this could affect prediction results. Therefore, we used the Bayesian framework to analyze the form of the predictive distributions for the two soil properties considered in this study. 2.2.1. Gaussian spatial linear mixed models (GSLMM) This section describes a full interactive model that gives an explicit expression for function f by means of GSLMM. In the GSLMM, it is
Fig. 2. The locations of sampling sites (Navarro-Pedreño et al., 2007).
P. Juan et al. / Journal of Geochemical Exploration 108 (2011) 62–72
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considered that a variable Z is a noisy version of a latent spatial process, the signal S(u). The distribution of the noise is assumed to be Gaussian and conditionally independent on S(u). The model is specified by: 1. Covariates: the mean part of the model is described by the term X (ui)β, where X(ui) is a vector of spatially referenced non-random variables at location ui and β is the mean parameter vector. 2. The underlying spatial process {S(u): u ∈ ℜd} is a stationary Gaussian process with zero mean, variance σ2 and correlation function ρ (h; ϕ), where ϕ is the correlation function parameter and h is the vector distance between two locations. 3. Conditional independence: the variables Z(ui) are assumed to be Gaussian and conditionally independent on the signal, 0 2 ð1Þ Zðui ÞjS∼N Xðui Þ β + Sðui Þ; τ In some applications we may want to consider a decomposition of the signal S(u) into a sum of latent processes Tk(u) scaled by σ2k. Which this consideration, the model can be rewritten, in a hierarchical way, as:
Fig. 4. QQ plots of log-transformed values of EC and Na content. Red line: trendline – dotted red lines: envelopes.
Level 1: K
ZðuÞ = XðuÞβ + SðuÞ + εðuÞ = XðuÞβ + ∑ σ k Tk ðuÞ + εðuÞ k=1
ð2Þ
Level 2: Tk(u) ~ N(0, Rk(ϕk)), T1,…,Tk mutually independent and ε(u) ~ N(0, τ2I) Level 3: (β, σ2, ϕ, τ2) ~ pr(·), where pr(·) defines a prior probability distribution. The model components are described by: is a random vector with components Z(u1),…, Z(un), related to the measurements at locations u. X(u)β = μ(u) is the expectation of Z(u). X(u) is a matrix of fixed covariates measured at locations u. β is a vector parameter. If there
0.0010
Density
0.0
0.0000
0.0005
0.2 0.1
Density
0.3
0.4
0.0015
Z(u)
are no covariates, X = 1 and the mean reduces to a single constant value at all locations. Tk(u) is a random vector at sample locations, of a standardised latent stationary spatial process Tk at locations u. It has zero mean, variance of one and correlation matrix Rk(ϕk). The elements of Rk(ϕk) are given by a correlation ρk(h; ϕk) with parameter ϕk. If the process is isotropic, this parameter and the distance h are scalar parameters. The processes T1,…,Tk are mutually independent. The signal S is defined by the sum of scaled latent processes SðuÞ = ∑Kk = 1 σ k Tk ðuÞ. σk is a scale parameter. ε(u) denotes the error (noise) vector at the locations u, i.e. a spatially independent process (spatial white noise) with zero mean and variance τ2.
4
5
6
7
8
0
500
1500
2000
6
7
0.6 0.4
Density
0.0
0.2
0.003 0.002 0.001 0.000
Density
1000
EC
log(EC)
0
200
600
Na
1000
3
4
5
log(Na)
Fig. 3. Histograms raw and log-transformed values of Na content and EC. Red line: empirical line frequency polygon – blue dotted line: theoretical line frequency polygon.
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P. Juan et al. / Journal of Geochemical Exploration 108 (2011) 62–72
Table 1 Summary Statistics for the original and log-transformed variable.
Min. 1st Qu. Median Mean 3rd Qu. Max. Skewness Kurtosis
EC
Na
log(EC)
log(Na)
74.200 174.081 302.638 408.088 536.969 1536.167 1.545 2.741
30.600 84.850 192.926 230.319 326.488 618.400 1.370 2.278
4.307 5.160 5.710 5.730 6.286 7.337 0.044 -0.855
3.421 4.441 5.262 5.082 5.788 6.427 -0.290 -0.845 Fig. 5. Empirical and fitted spherical variograms for the log-transformed EC.
This model-based specification can be related to conventional geostatistics terminology as follows: 1. The term trend refers to the mean part of the model, Xβ; 2. A latent process corresponds to a structure in the variogram; 3. A value of σ2k corresponds to a partial sill. The sill is the value of ∑ K1 σ 2k ; 4. The nugget effect is quantified as τ2. In the geostatistics literature, this term refers to variation at small distances plus measurements error. 5. The total sill is given by the sum of the sill and the nugget effects. 2.2.2. Bayesian inference Let us focus now on parameter estimation and prediction results from a Bayesian analysis of geostatistical data. Consider a model that is simpler than Eq. (2) and defined as: Z(u) = X(u)β + σT(u), with Tu ~ N(0, Rz(ϕ)) and in a third level defining a prior probability for pr (β, σ2, ϕ). One of the most important issues within this context is the analysis of the uncertainty associated with the mean parameter. 2.2.2.1. Uncertainty in the mean parameter. In this case only the mean parameter β is unknown. The covariance parameters are known and the covariance matrix is written as V(σ 2,φ ) = σ 2R(φ ), and denoted * * * * by σ 2R (The subsymbol * denotes known parameter). The model * considered here corresponds to the common situation in geostatistics where the mean is filtered and the covariance parameters are estimated by some method and plugged-in for predictions. The joint probability distribution for (Z, Z0) without the nugget effect and with only one latent process, is 2 ðZ;Z0 j β; σ ; ϕ Þ∼N
X 2 Rz β; σ r0 X0
r R0
ð3Þ
2
2
ðβj Z; σ ; ϕ Þ∼Nðmβ ; σ Vβ Þ
ð6Þ
the posterior gives −1
2
0 −1
ðβ j Z; σ ; ϕ Þ∼NððVβ + X Rz XÞ
−1
−1
0 −1
−1
2
0 −1
ðVβ mβ + X Rz zÞσ ðVβ + X Rz XÞ
ˆ N ; σ 2 V ˆ Þ ∼Nð β βN
2
−1 0 −1 −1 −1 Vβ mβ E½Z0 j Z = ðX0 −r0 R−1 z XÞðVβ + X Rz XÞ h i 0 −1 0 −1 −1 0 −1 −1 0 −1 + r Rz + ðX0 −r Rz XÞðVβ + X Rz XÞ X Rz z 0 −1
2
0 −1
−1
0 −1
−1
Var½Z0 j Z = σ ½R0 −r Rz r + ðX0 −r Rz XÞðVβ + X Rz XÞ
ð8Þ
0 −1
0
ðX0 −r Rz XÞ
ð9Þ
2.2.2.3. Posterior for model parameters: flat prior. Assuming a flat (noninformative) prior for the mean parameter, we have p(θ) ∝ 1, and the posterior distribution gives 0 −1
2
−1
ðβj Z; σ ; ϕ Þ∼NððX Rz XÞ
0 −1
2
0 −1
−1
X Rz zÞ; σ ðX Rz XÞ
ˆ N ; σ V βÞ ˆ Þ∼Nð β 2
ð10Þ Now, the mean and variance of the predictive distribution can be 1 calculated from Eqs. (8) and (9) with V− β ≡ 0, 0
−1
2
0
−1
ð11Þ 0
−1
0 −1
−1
Var½Z0 j Z = σ ½R0 −r Rz r + ðX0 −r Rz XÞðX Rz XÞ ð4Þ
Þ
Now, the mean and variance of the predictive distribution is
0 −1
ðZ j β; σ ; ϕ Þ∼NðXβ; σ Rz Þ
−1
ð7Þ
ˆ + r Rz z E½Z0 j Z = ðX0 −r Rz XÞ β
and the associated marginal and conditional distributions are 2
2.2.2.2. Posterior for model parameters: conjugate prior. Assuming a Normal or Gaussian prior for the mean parameter
0
−1
0
ðX0 −r Rz XÞ ð12Þ
Finally, the posterior for known mean parameter β can also be obtained from Eqs. (8) and (9) considering Vβ− 1 N N X ' Ry− 1X or Vβ ≡ 0.
and 2
0
−1
2
0
−1
ðZ0 j Z; β; σ ; ϕ Þ∼NðX0 β + r Rz ðz−XβÞ; σ ðR0 −r Rz rÞÞ
ð5Þ
Table 2 Variogram fitting parameters for ordinary kriging estimation.
Direction of anisotropy Variogram model Nugget Sill Minor range (m) Major range (m)
log(Na)
log(EC)
79.4 Spherical 0.34776 0.7362 83,871 92,640
285.5 Spherical 0.14517 0.67051 79,879 92,407
Fig. 6. Empirical and fitted spherical variograms for the log-transformed Na content.
P. Juan et al. / Journal of Geochemical Exploration 108 (2011) 62–72
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Let us assume that we have sample observations z1,..., zn that are realizations of Z(u1),..., Z(un), where {Z(u): u ∈ A}, A p R2}, is the analyzed random field. For λ ≥ 0, we have ( ZðuÞ =
Fig. 7. Empirical and fitted spherical cross-variance for the log-transformed values of Na content and EC derived by co-kriging.
gλ−1 ðSðuÞÞ
if SðuÞ∈gλ ðj0; ∞ jÞ;
0
otherwise
ð14Þ
where {S(u): u ∈ A} is a Gaussian random field of mean μ(u) = d(u)Tβ and covariance Cov(S(u), S(u')) = C(u, u'). To carry out the geostatistical work, we used the ArcGIS 8.2 software. Maps were generated using the ArcMap module of this software and geostatistical analyses were performed with the Geostatistical Analyst extension. The Bayesian methodology was performed using the free R software and the corresponding built-in geoR library (http://www.r-project.org/). 3. Results and discussion
2.2.2.4. Relationships with conventional geostatistical methods. Some of these considerations can be related to conventional geostatistical methods (Journel and Huijbregts, 1978; Goovaerts, 1997). Under the Bayesian perspective, conventional geostatistical methods can be interpreted as prediction procedures that only take into account the uncertainty in the mean parameters. 1. If X ≡ 1 and X0 ≡ 1 (constant mean), the mean and variance in Eqs. (11) and (12), respectively, coincide with the OK predictor and 2 the OK variance (σOK ). 2. If X and X0 are trend matrices with rows given by data coordinates or a function of them, the mean and variance in Eqs. (11) and (12) coincide with the universal or trend kriging (UK or KT, respective2 ly) predictor and the universal or trend kriging variance (σKT ). 3. If X and X0 are trend matrices with covariates measured at data and prediction locations, respectively, the mean and variance in (Eq. (11)) and (Eq. (12)) coincide with the kriging with external trend (KTE) predictor and the kriging with external trend variance 2 (σKTE ). 2.2.3. Data transformation To guarantee the Gaussian assumption of the geostatistical data (Christensen et al., 2001), we use a family of Box–Cox transformations indexed by a parameter λ that has to be estimated from data (Box and Cox, 1964): λ gλ ¼ ðz 1Þ=λ if λ≠0; logðzÞ if λ ¼ 0:
ð13Þ
We worked with logarithms as both variables needed a Box–Cox transformation, which yielded a value for parameter λ = 0. Box–Cox transformation is needed to normalize the data and to remove the skewness from the original variables. By using the logarithm transformed variables, we make ensure that the variables approximately have Gaussian distributions. It is known that histograms (Fig. 3) and QQ-plots (Fig. 4) provide for visual examination of fitting to normality, but here we complement these graphical procedures with some statistics to objectively confirm the normality assumption (Table 1).The spherical model was initially chosen as the parametric family that best fitted the empirical variograms of both variables (Table 2). Thus, the best-fit variogram models were specified as the sum of two structures, by a nugget-effect term, and a spherical model. The parameters of each variogram were obtained using the weighted least squares procedure. Anisotropy was encountered for both variables and in all cases, 0 is the north direction. Thus, an analysis of the predominant influence direction was carried out for each variable. Note that the spatial dependencies are defined up to 92 km for both Na content and EC. The range is important in terms of controlling upper limits of the spatial dependencies in prediction processes. Note also that the nugget effect values are relatively small, indicating that measurement errors or small-scale variations are well-controlled and, thus, variations among locations are mainly due to spatial dependencies exhibited by the soil variables. The cross-covariance models show small nugget effect values (Figs. 5–7). Before we selected these models, we derived several tentative models fitted to the data and we selected the best model in terms of lowest prediction errors (Tables 3–5). In this case,
Table 3 Alternative models fitted to the logarithm of EC. In bold the one selected. 0 – North direction. Variable
Spherical
Model type Major range Minor range Direction Sill Nugget Lag size Number lags Prediction errors Mean Root-mean-square Averaged standard error Mean Standardized Root-mean-square Standardized
92,407 79,879 295.5 0.6705 0.1451 8134.4 12 0.01035 0.5846 0.5257 0.01214 1.103
log(EC) Spherical
Exponential
Gaussian
Spherical
Spherical
Spherical
26228 20000 120 0.425 0.06 8134.4 12
96419 93084 281.6 0.7205 0.08655 8134.4 12
92337 75835 299.7 0.64822 0.24004 8134.4 12
117680 75417 302.4 0.6275 0.1662 8134.4 17
71120 67896 281.3 0.6084 0.1447 6000 12
73078 73078 – 0.62386 0.14142 8134.4 12
0.0004035 0.6055 0.5547 −0.009817 1.132
0.01007 0.5839 0.5283 0.00995 1.119
0.004954 0.6025 0.5422 0.00797 1.099
0.002605 0.5899 0.5368 −0.00001 1.088
0.00788 0.5816 0.5367 0.00689 1.08
0.01273 0.578 0.5301 0.01583 1.086
68
P. Juan et al. / Journal of Geochemical Exploration 108 (2011) 62–72
Table 4 Alternative models fitted to the logarithms of Na. In bold the one selected. 0 – North direction. Variable
log(Na)
log(Na)
log(Na)
log(Na)
log(Na)
Model type
Spherical
Spherical
Exponential
Gaussian
Spherical
Major range Minor range Direction Sill Nugget Lag size Number lags Prediction errors Mean Root-mean-square Average standard error Mean Standardized Root-mean-square Standardized
92,640 83,871 79.4 0.7362 0.3477 8134.4 12
37,600 30,000 22.5 0.696 0.032 8134.4 12
96,419 91,813 266.2 0.843 0.2326 8134.4 12
92,407 75,889 76.9 0.674 0.468 8134.4 12
67,904 67,904 – 0.68475 0.31036 8134.4 12
−0.00237 0.8049 0.7221 −0.00152 1.099
0.0024 0.9105 0.5336 −0.0000681 1.693
0.007202 0.8311 0.7003 0.008857 1.161
−0.00573 0.7981 0.7426 −0.00371 1.056
0.001907 0.8175 0.7094 0.004226 1.134
the mean, root-mean-square and averaged standard error close to zero, and the root-mean-square standardized were close to one. The results of comparison show experimental cross-variance and cross-variance models fitted with an experimental model with sill of 0.30844 and range of 89,745 to 96,419 m represent the best models (Figs. 8–10). In these figures, red dots represent experimental variograms (crosscovariances) and yellow curves are the fitted variograms (crosscovariances). Anisotropy was detected by analyzing the major and minor ranges (Tables 3–5) together with Fig. 11. The spatial predictions of Na content and EC in the study area are shown in Fig. 12 to 15. For both soil variables, the maximum standard errors of the predictions (Figs. 12 and 13) are less than 8% of the maximum data values in logarithm scale, indicating that the resulting spatial predictions obtained by co-kriging techniques can be trusted. The spatial prediction maps show that salinity in the southern, arid to semi-arid part of the study area, is higher than in the northern, temperate–humid part of the study area. Although only the relationship between EC and extractable Na content was analyzed and rainfall distribution was not taken into account, rainfall is apparently associated with these results because in the northern part of study area (Fig. 1) higher amounts of meteoric water can leach soils such that in the upper soils horizons are present in lower concentrations. However, the northwestern part of the province has saline soils due to important factors of soil formation; there are, the presence of Tertiary (Keuper) marls and clays with high content of calcium sulphates (gypsum or CaSO4·2H2O; anhydrite or CaSO4), halite (or NaCl) and existence of saline lagoons like the “Laguna de Salinas”. In general,
soils in the northern part of study area show less degradation (in terms of salinity) compared to those in the southern part where the values of extractable Na and EC are the highest.
Table 5 Alternative models fitted to the cross relationship between logarithm of CE and logarithm of Na. In bold the one selected. 0 – North direction.
Fig. 9. Experimental and fitted exponential variogram for the log-transformed Na content.
Fig. 8. Experimental and fitted exponential variogram for the log-transformed of EC.
Cokriging log(EC) − log(Na) Model type
Spherical
Exponential
Major range Minor range Direction Sill Nugget Lag size Number lags Prediction errors Mean Root-mean-square Average standard error Mean standardized Root-mean-square Standardized
92,342 79,743 286.4 0.2898
96,419 89,745 280.3 0.30844
8134.4 12 0.009591 0.5346 0.5132 0.01075 1.043
8134.4 12 0.009608 0.5057 0.5154 0.01032 1.008
Fig. 10. Empirical and fitted exponential model for log-transformed values of Na content and EC derived by co-kriging.
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Fig. 11. Graph showing the main direction.
In the case of applying the relationships of the two variables in prediction, we show this by predicting Na content as target variable using EC as the subsidiary variable. For this analysis, we took the parameters for co-kriging from the co-regionalization model given
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above. The resulting map (Figs. 12 and 13) basically highlight the main features obtained by OK but the results of co-kriging show smaller standard errors (Figs. 14 and 15). As these predictions take into account the spatial relationship of EC, this technique may be considered as more appropriate one to predict spatial variation of soil salinity in terms of Na content. In this sense, suppose we have sampled EC over a fine grid covering the whole region of interest. Suppose further that data on EC were originally collected at the sites that are different from sites where Na content had been measured. Thus, we first interpolate EC using OK (Fig. 13) and then used the interpolated data as auxiliary variable in the regression model to act as an external drift factor for predicting the spatial distribution of Na contents in the whole region. This result (Fig. 16) shows that the main features of soil salinity in terms of Na content can be highlighted using EC as auxiliary variable in co-kriging such that the standard errors are slightly lower, compared to prediction of Na content using OK. The reason maybe is that we have used interpolated data of EC in external drift kriging instead of using data of EC at the same sites where Na contents were be measured.
Fig. 12. The spatial distribution of Na content obtained by OK (left) and corresponding distribution of standard errors (right).
Fig. 13. The spatial distribution of EC obtained by OK (left) and corresponding distribution of standard errors (right).
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Fig. 14. The spatial distribution of Na content obtained by cokriging (left) and corresponding distribution of standard errors (right).
Fig. 15. The spatial distribution of EC obtained by cokriging (left) and corresponding distribution of standard errors (right).
Thus, we have shown here the usefulness of Bayesian framework to analyse the form of the predictive spatial distributions of both soil properties considered in this study. A Gaussian prior for the mean parameter was used and this gave the mean and variance of the predictive distribution. Maps show the distributions of Na content and EC for different sites within the province of Alicante obtained by Gaussian spatial linear mixed model of spatial prediction. The quality of the predictions varies depending on locations, which could not be shown using classical methodology. However, it is more appropriate to carry out this study using more data. The soil EC can be measured using, for example, a contact sensor to obtain great quantity of data, which could be used to accurately map some soil variables and, especially, those related to soil salinity. 4. Conclusions Soil salinity can change abruptly due to local characteristics. The methodology proposed here, which could be ameliorated with more
soil data, was shown to be adequate in predicting the spatial distribution of soil characteristics in the study region. With the proposed methodology, it is possible to display relationships between soil salinity parameters in a medium scale map. This is a starting point to apply the proposed methodology in order to derive information that is useful in planning to mitigate soil salinity risk. We conclude that the proposed methodology does not result in over-estimation in predicting the spatial distribution properties to model land degradation due to salinization. The proposed method can also be useful in revealing a sline areas linked with discharges of saline waters or saline aquifers. This study has elucidated the spatial correlations and variations in soil measures related to salinity. The results of the geostatistical analyses can be applied in making decisions regarding environmental monitoring, remediation, land management and planning. The spatial predictions, from an economical point of view, have special and particular importance before agricultural transformation of the land,
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Fig. 16. Predicted values at three selected sampling sites (see Fig. 1) using Gaussian distribution. Graphs on the left are for log-transformed Na content, those on right and are for logtransformed EC. Dot lines represent the mean of predictive distribution, and solid lines represent the real values.
or environmental restoration, or selection of the most appropriate species adapted to soil salinity. The methodology proposed here can also be used for environmental quality assessment and planning in large and medium scales. Acknowledgments Authors would like to thank CAM (Caja de Ahorros del Mediterráneo), and Generalitat Valenciana for their support. We are specially grateful to Dr. John Carranza for the very constructive and thorough review of this manuscript. References Andreu, J., 1997. Contribución de la sobreexplotación al conocimiento de los acuíferos kársticos de Crevillente, Cid Cabeco d'Or (provincia de Alicante). Ph.D. thesis. Universidad de Alicante. 447 pp.
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