Geo Spatial Parameter Estimation Essay

Estimation of GeoSpatial Parameter (Lugeon values) using Variogram technique.

Abstract: This article gives brief overview on developing Geo-Spatial variational map of parcel of land where few tests has already been carried out. The calculation procedure is demonstrated with dummy Lugeon test values. Such maps may be helpful for practicing engineers who are working in groundwater movement to visualize and estimate the ground water movement.

Introduction

In Geo-statistics we encounter the situation when there is need to auto-correlate of one or more variables in the space or sometimes in space-time for following purposes: to make estimate at unobserved locations to give information about the accuracy of prediction to reproduce spatial

The variogram estimator is calculated as: γ_E (h)=1/(2*N(h)) (z_(x+h)-z_x )^2

A typical variogram is shown in Figure 2.

Figure 1 Variogram is calculated by searching the combination of two points of which the distance is between h-0.5Δh and h+00.5Δh Figure 2 General shape of variogram

Variogram models

The variogram is modeled by approximating a theoretical model parametric curve. Such fitted curve is called variogram estimator (or models). Some of the popular models are:

Linear model γ(h)=a*h

Spherical model γ(h)=c[3/2 (h/a)-1/2 (h/a)^3 ] when h<a γ(h)=c when h<a

Exponential model γ(h)=c (1-exp⁡(-h/a) )

Kriging

Apart from developing variogram map, it is essential to develop estimating function. This is done by method called Kriging. Kriging is method to interpolate the observations on a space (xyz) coordinates at unknown points. It uses weighted average of the neighboring points to estimate the value of an unobserved point.

The value of function at any point is given

As a rule of thumb, the half maximum distance is

% suitable range for variogram analysis. hmd = max(D(:))/2; % the maximum number of lags max_lags = floor(hmd/lag); % Now the separation distances are classified and the classical

% variogram estimatoris calculated

%

% here SEL is the selection matrix defined by the lag classes in LAG, DE is

% the mean lag, PN is the number of pairs and GE is the variogram estimator. LAGS = ceil(D/lag); for i = 1 : max_lags

SEL = (LAGS == i);

DE(i) = mean(mean(D(SEL)));

%PN(i) = sum(sum(SEL == 1))/2;

GE(i) = mean(mean(G(SEL))); end %plot the variogram plot(DE,GE,'o' ) var_z = var(z); b = [0 max(DE)]; c = [var_z var_z]; hold on plot(b,c, '--r') yl = 1.1 * max(GE); ylim([0 yl]) xlabel('Averaged distance between