A case study on optimizing the geotechnical site investigation using Kriging method

One of the major tasks in geotechnical investigation is the stratigraphy distribution and the physico-mechanical properties of strata encountered in the investigation area. In order to reduce the project risks associated with uncertainties in predicting the distribution of strata (in area and in depth), a geotechnical investigation plan is usually designed with as many as possible of the boreholes. And this, in turn, increases the investigation costs. On the contrary, the owner of the project is expected to gather as much information about the subsurface soils as possible at the lowest cost. To deal with this contradiction, geotechnical engineers not only mobilize their knowledge on the investigation area and their experiences in the field of geotechnical investigation but also should be supported by geostatistical tools, especially the interpolation method of Kriging. Based on the real data taken from a geotechnical investigation project in Saigon (Vietnam), this paper will introduce two geotechnical investigation plans: a) an actual investigation plan; b) an optimized investigation plan designed with the support of Kriging method. From these two plans, the ability of Kriging in optimization of geotechnical investigation will be evaluated.


INTRODUCTION
To understand the behavior and characteristics of the ground, geotechnical engineers build up a geotechnical investigation plan in which several methods such as excavation, drilling, penetration… are applied to gather as much geological data as possible.
However, the contradiction arises during that planning.The more amount of data is collected, the higher reliability of data is and therefore it is very costly and time-consuming.On the contrary, if less amount of data is gathered then this sampling strategy will be costeffective and time-saving but the data reliability is lower.
To overcome the contradiction "less data -more information", geotechnical engineers should be supported by geostatistical tools, especially Kriging method to unveil the spatial characteristics and make use of intrinsic information in available geological data.
Kriging method investigates the spatial relationship of geological data by building up a semi-variogram (see  This paper is a pilot study to demonstrate the optimization of geotechnical investigation plan by using Kriging method for the top layer in the study area. The study area is located in the south of Saigon, Vietnam (see Fig. 2).In an actual geotechnical investigation plan, the project owner had conducted 41 boreholes and 20 CPTu tests in order to understand the geological conditions of the study area.
According to boring logs of 41 boreholes, the top layer is a soft soil layer whose bottom elevation values vary from -32.2m to -16.5m.Meanwhile, the elevation of borehole collars vary from +0.3m to +1.6m.

METHODOLOGY
To optimize the sampling network, the geotechnical investigation is undertaken in multi stages.The location of new boreholes in the current stage is decided from the error map of the previous stage (see Fig. 4).
The multistage investigation process will be terminated when the error variance at the current stage is not higher than the expected error value or the difference between two prediction maps from two consecutive stages is not statistically significant.A statistical method TOST (Two One-Sided t-Test) is applied to test the equivalence of two prediction maps from two stages.The null hypothesis in TOST equivalence test is that the two prediction maps are totally different.And vice versa, the alternative hypothesis is that the two prediction maps are similar.More information about the TOST equivalence method should be referred in [4], [5].
The TOST method compares two group means and their two one-sided α-level confidence intervals by comparing them to a predefined equivalence limit (  ).
An indifference region of  = ±25% of the standard deviation is commonly used.If the indifference region completely encompasses the confidence interval then the two populations are deemed significantly similar (the null hypothesis is rejected).If not, then the null hypothesis is not rejected.In this case, there is a lack of sufficient evidence to conclude that the two prediction maps are similar.At each interpolation stage, the semivariogram models will be validated with the independent data set of 20 CPTu holes.

Interpolation of stage 1
At the initial stage, 10 boreholes are selected randomly to cover the whole area (see Fig. 5).Their coordinates and the corresponding bottom elevations of the top layer are presented in Table 4.
The semi-variogram of stage 1 and its parameters are displayed in Fig. 6.Apparently, the quality of the semivariogram is not good enough and results in the high error variance in the error map (see Fig. 9).

Interpolation of stage 2
Based on the error map of stage 1, 10 new boreholes which locate at the positions of high error on the error map are added for the interpolation of stage 2 (see Fig. 5).The semi-variogram of 20 boreholes used in stage 2 is presented in Fig. 6.The prediction map and error map of stage 2 are presented in Fig. 8 and Fig. 9, respectively.
The variation of estimated error values of stage 2 is narrower than that of stage 1.TOST test for the difference between two prediction maps of stage 1 and stage 2 proves that there is no strong evidence to conclude that two prediction maps are similar (the null hypothesis could not be rejected).

Interpolation of stage 3
Twelve new boreholes added in stage 3 are based on the result of the error map of stage 2 (see Fig. 5).The result of the interpolation and error variance are displayed in Fig. 8 and Fig. 9, respectively.
The semi-variogram and its parameters of 32 boreholes are in Fig. 7.The theoretical semi-variogram is fitted well with the experimental semi-variogram up to the distance of 1200m.Beyond this range, the difference between theoretical and experimental semivariograms is gradually increased.This behavior could be caused by a spatial trend imposed on the bottom elevation data of the top layer.
TOST test for the difference between stage 2 and stage 3 proves that there is no strong evidence to conclude that two prediction maps are similar.

Interpolation of stage 4
All boreholes are used in stage 4 which is similar to the actual investigation plan carried out by the project owner of this project (see Fig. 5).The result of interpolation and error variance are displayed in Fig. 8 and Fig. 9, respectively.The semi-variogram and its parameters are presented in Fig. 7. TOST test for the difference between stage 3 and stage 4 confirms that the difference between two prediction maps is not statistically significant.Therefore, it is reasonable to stop the investigation at stage 3 with 32 boreholes instead of at stage 4 with 41 boreholes.

VALIDATION OF THE SEMI-VARIOGRAM MODELS
The reliability of the semi-variogram models will be validated by using CPTu data as the independent data set.The validation results presented in Table 5 show that the quality of the interpolation gradually increase from stage 1 to stage 3.The interpolated values at CPTu holes using the semi-variogram models of stage 3 and stage 4 are not significantly different.

CONCLUSIONS
The approach of multistage investigation plan together with TOST equivalence test shows that the geotechnical investigation plan with 32 boreholes could reveal the same information as the actual investigation plan with 41 boreholes conducted by the project owner of the project.
Geostatistics tools, especially Kriging method are really helpful to unveil the spatial characteristics of geological data and to optimize the sampling network.
Using geostatistics tools or Kriging method will reduce the uncertainty and increase the reliability of information derived from available geological data.

TÀI LIỆU THAM KHẢO
Fig 1) which presents the relationship between semivariances of data pairs   h  with their distances lag(h).From this chart, some important parameters should be taken into account: theoretical semi-variogram model, nugget, sill and range.Readers who are interested in Kriging method could get more information from [1]-[3].

Figure 1 .
Figure 1.A typical semi-variogramThe value of data at one unsampling point is estimated by the formula:n p i i i 1 ẑ w z = = 

Figure 2 .Figure 3 .
Figure 2.Location of the study area

Figure 4 .
Figure 4. Flowchart of the multistage geotechnical investigation

Table 1 .
Equivalence test for maps of stage 1 and 2

Table 2 .
Equivalence test for maps of stage 2 and 3

Table 3 .
Equivalence test for maps of stage 3 and 4

Table 4 .
Bottom elevation of the top layer from 41 boreholes in the study area

Table 5 .
Results of the validation with CPT data (independent data set)