General and Local Estimation of Local Average of Geotechnical Parameters in Reliability Analysis

Abstract eng:
A method to evaluate uncertainty involved in spatial variability and resulting statistical estimation error of geotechnical parameters are proposed in this study. The soil profile is model by a stationary random field (RF). The local average at an arbitrary point in the RF is estimated from limited samples. The two estimation methods are proposed, namely the general estimation and the local estimation. Relative positions of samplings and of the construction are not considered in the former, whereas they are taken into account in the latter. If a local average, ZV(x), of a stationary RF is to be estimate by a weighted average of the observed values, Z(xi)i = 1,... n) as ŻV(x) = ∑ni=1viZ(xi), the estimation variance of ŻV(x), i.e. σ2V, is defined as below: σ2V=E[{ŻV(x)-ZV(x)}2]= E[{∑ni=1viZ(xi-ZV(x)}2] =σ2Z{Γ2(V/θ)+∧2(n,LN,v)-2∑ni=1viγV(ZV(x),Z(xi ))} Each term of the equation above can be interpreted as follows: 1. The first term describes the variance reduction by taking local average. This is the effect pointed out by Vanmarcke(1977), which can be evaluated by Γ2(V/θ). 2. The second term exhibits the statistical estimation error associated with the population mean μ>Z. 3. The third term is quantifying the contribution of observed values on the estimation of the local average. Thus, it is a function of relative location of the observations and that of local averaging. As understood from these interpretation, the first term is indicating spatial variability of the soil parameter. The sum of the second and third term is indicating the statistical estimation error of the mean of the local average, and this term can be arranged to evaluate the estimation variance for the general and the local estimation of ZV(x).

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