The penetration depth for frequencies larger than 1.4 GHz (L-band) is less than 10 cm. This is valid for loamy soils with a volumetric water content larger than 0.2 g/cm3.
3.2 Kriging
In the best of all worlds relevant data at every desired point would be available.
However, that is not practically possible and interpolation is therefore used to predict unknown values from data at known locations. There exist numerous techniques for interpolating irregularly spaced data. Some of the commonly used methods are e.g. nearest neighbour interpolation, spline interpolation and weighted moving average methods.
Kriging is a weighted average method used in geostatistics and first introduced by the South African mining engineer E. Krige. This method uses a semi-variogram to express the spatial variation and it minimizes the error of the predicted values. In this section the most important elements of kriging are presented. This includes a description of ordinary kriging, which is a technique well suited for modelling the spatial irregularities we are facing in this project.
A detailed introduction to most geostatistical methods is provided in Isaaks and Srivastava (1989) [36] and Huijbregts (1978) [41]. Analysis of irregularly distributed points is given in Hartelius (1996) [33]. An explanatory introduction to geostatistics and kriging with applications is found in Nielsen (1999) [55] and for a thorough analysis of regularly and irregularly sampled spatial, multivariate, and multi-temporal data refer to Nielsen (1994) [54].
3.2.1 Geostatistics
The fundamental concept in geostatistics isregionalized variables. These vari-ables have been introduced because the spatial variation of any continuous sur-face often is too irregular to be modeled by a simple mathematical function.
Instead the variation can be described by a stochastic surface and the measur-able quantity is then called a regionalized varimeasur-able.
The regionalized variable theory states, that for each position x in a domain D there exists a measurable quantity z(x). The quantity z(x) is called a re-gionalized variable and D is typically a subset of R2 or R3. z(x) is consid-ered a particular realization of a random variable Z(x). The set of random
variables {Z(x)|x ∈ D} constitutes a random function and Z(x) is often as-sumed to follow a normal or log-normal distribution. Z(x) has expectation value E{Z(x))} = µ and covariance Cov{Z(x), Z(x+h)} = C(x,h). Here z(x) andz(x+h) are quantities measured at two points in spacexandx+h and separated byh. Ifµis constant overD,Zis said to be first order stationary.
If the covariance is constant overDalso, i.e.C(x,h) =C(h),Zis second order stationary.
3.2.2 Semivariogram
The autocovariance function betweenZ(x) andZ(x+h) is
C(x,h) =E{[Z(x)−µ][Z(x+h)−µ]}.
The variability in space is also described by the semivariance, which is defined
γ(x,h) = 1
2E{[Z(x)−Z(x+h)]2}.
Theintrinsic hypothesisstates that the semivariogram is a function of the dis-placement vectorhand not of the positionx, that is
γ(x,h) =γ(h).
The intrinsic hypothesis is less restrictive than second order stationarity. For the intrinsic hypothesis second order stationarity is not assumed for Z(x), but rather the first order differences Z(x+h)−Z(x). Second order stationarity for Z(x) implies the intrinsic hypothesis but not vice versa. If second order stationarity is assumed or imposed the relation between the semivariogram and the autocovariance is
γ(h) =C(0)−C(h),
whereC(0) =σ2.
3.2 Kriging 29
Theexperimental semivariogramcan be estimated using
ˆ
whereN(h) is the number of point pairs separated byh.
In order to characterize the experimental semivariogram a number of models can be fitted. A frequently used model, which is also applied in this project, is the spherical model γ∗(h) with nugget effect. A reason for this is the easy interpretability of the parameters. By setting |h|=hwe assume isotropy and get
The constantC0 is the nugget effect, which is a discontinuity ath= 0 due to measurements errors and short range spatial variations. The quantityC0/(C0+ C1) is therelative nugget effectwhereC0+C1is the maximum level of semivari-ance also called the sill (= σ2). The range of influence is R corresponding to the maximum semivariance. Other models used are e.g. the cubic, exponential, Gaussian and the linear model.
3.2.3 Ordinary kriging
At the unsampled locationx0we consider a value ˆz0, which we wish to estimate as a weighted average of the values zi sampled at locations around it. The unbiased linear estimator is given by
ˆ
where wi are the weights, w0 is a constant and N refers to the number of neighbours to ˆz0. The expected error of estimation is
E(Z0−Zˆ0) = E(Z0−w0−wTZ)
= µ0−w0−wTµ, (3.4)
and this unbiasedness gives
µ0−w0−wTµ= 0. (3.5)
Thekriging variance, or the minimum mean square errorσE2, is
σ2E = E(Z0−Zˆ0)2
= σ2+wT(Cw−2Cov(Z0,Z)), (3.6)
whereC is the dispersion matrix ofz.
Inordinary kriging(OK) we suppose thatE(Zi) =µ0for theN neighbours and by combining (3.4) and (3.5) it follows that
E(Z0−Zˆ0) =µ0(1−wT1)−w0= 0.
This implies thatw0= 0 andwT1= 1. In the case ofsimple kriging (SK) the meanµ0 is known and the constraintPN
i=1wi= 1 is ignored.
The minimum variance of error is obtained by minimizing (3.6) under the con-straintPN
i=1wi= 1. This can be done using the Langrangian multiplierλand solving the equations∂[σE2+2λ(wT−1)]/∂wi = 0 and∂[σE2 + 2λ(wT −1)]/∂λ= 0.
This leads to the ordinary kriging system
Cw+λ1 = Cov(Z0,Z) (3.7)
1Tw = 1.
3.2 Kriging 31 from the semivariogram. Using (3.6) and (3.7) the estimated kriging variance is
σOK2 =σ2−wTCov(Z0,Z)−λ.
A variety of other kriging techniques has been developed. Of these are e.g. sim-ple kriging, universal kriging, co-kriging. In simsim-ple kriging the mean is known and second order stationarity is assumed. Universal krigingis a technique where a drift polynomial is used to model a non-stationary trend surface. In case the original variables have been undersampled, the covariation between different variables can be taken into account in the reducing of the estimation variance.
This is referred to asco-kriging. Ordinary kriging is well suited for interpolating surfaces whereµ0 is not constant i.e. the lack of first order stationarity. Hence the ordinary kriging technique described above is applied in this project.
Some very important advantages of kriging:
• Kriging is exact and it is the Best Linear Unbiased Estimator (BLUE). If a value at a location that has been sampled is estimated, the kriging system will return the sample value as the estimator and a kriging variance zero.
• The kriging system has a unique solution if and only if the covariance matrixCis positive definite, this also ensures a non-negative kriging vari-ance.
• The kriging system and the kriging variance depend only on the covari-ance function (semivariogram) and on the spatial lay-out of the sampled supports and not on the actual data values. If a covariance function is known or assumed this has important potential for minimizing the esti-mation variance in experimental design (i.e. in the planning phase of the spatial lay-out of the sampling scheme).
The programs used to compute semi-variograms and to do the ordinary kriging (OK) is a part of GSLIB which is a library of geostatistical programs [23].