Simulation experiments

4.2.2.1 Random field generation by frequency-domain technique

In order to investigate the performance of the proposed approach using the simulated data, random values of the model parameters are generated based on frequency-domain technique and the observed settlement data is then calculated by Eq. (4-14), using the generated parameters and the assumed standard deviation of the observation error, σε. The detail formulations for this technique can be found in Section 4.1.2.1.

4.2.2.2 Improvement of the estimation by considering spatial correlation structure

A series of simulation experiments was performed based on the procedure previously described in Section 4.1.2.1. It is assumed that total number of the observation points is 36 (n = 36) and these points are arranged in a square grid pattern with even spacing of s and total width of L, as shown in Figure 4-6. It was also decided to limit the number of simulations for each experiment to 50 (Nsim = 50). For the purpose of recognizing the performance of the approach, these selections seem sufficient.

For simulation of the model parameters, it is assumed that the mean and standard deviation of the random field of m1 are 100 (cm) and 10 (cm), and those of m0 are -100 (cm) and 10 (cm), respectively. These imply that coefficient of variation (COV) = 0.1. It is assumed that the settlement is observed for 21 times (i.e. total observation time step, K = 21) from the day 10^th to 1000^th. Note that, by substituting m1 = 100 (cm) and m0 = -100 (cm) into Eq. (4-14), the estimated settlement at the day 1000^th is 200 (cm). For the current study, the observation error, σε, is assumed to be 10 (cm).

By assigning the desired values of auto-correlation distance (η), random values of the model parameters together with the observed settlement at each observation point can be generated, as described in Section 4.1.2.1. It should be emphasized that the generated data actually represents a set

of the settlement data observed from an area which the true values of settlement model parameters and underlying spatial correlation structure are known.

Based on the generated observation data, the procedure proposed in Section 3.2 is performed to back-estimate the model parameters. The prior means of the model parameters (see Eq. (4-18)) are assumed to be equal to those used for the data generation, i.e. mˆ ( )_1,0 x₁ = mˆ ( )_1,0 x₂ = … = mˆ ( )_1,0 x_m = 100 (cm), and mˆ ( )_0,0 x₁ = mˆ ( )_0,0 x₂ = … = mˆ ( )_0,0 x_m = -100 (cm). As for σ^{m1,0 and} σ^m0,0 (see Eq. (4-19)), COV of 0.4 is assumed, i.e. σ^{m1,0 =} σ^m0,0 = 40 (cm). This relatively large value of COV is assumed in order to limit the influence of prior information, which commonly does not know in practice. The auto-correlation distance and the observation error are also assigned the same values as those used for generating the simulated data, namely the true values, in order to focus only on the effect of considering spatial correlation. In other words, the model selection process presented in Section 3.3 is not included in the calculations in this section.

In order to examine the advantages of considering spatial correlation structure, the Bayesian estimation, using the observed settlement of each observation point to estimate the model parameters of that point itself, i.e. ignoring spatial correlation structure, is also performed based on the same conditions with the considering one. This is actually equivalent to the case which η = 0 is assumed.

The estimations based on these two different conditions are compared and presented in this section.

The different model parameters are randomly generated for 50 times (Nsim = 50) and the estimation errors are calculated by the terms of mean and standard deviation (SD) of the error ratio, Er, as defined in Eq. (4-12). It should be noted that the total number of estimated values for mean and standard deviation (SD) calculations is n × N_sim. It is clear that the true values of the model parameters are known. However, those of the settlement have to be estimated. Eq. (4-14) is used for

calculating both true values and estimated values of the settlement at any time tk, using the true values and estimated values of the model parameters, respectively.

Figure 4-10 illustrates the plots of the mean and SD of Er for the model parameter estimation and settlement prediction at the last observation time step (the day 1000^th) against observation time, until which the observation data are used in estimation. It is assumed that ratios of auto-correlation distance to spacing, s/η = 0.25. Clearly, SD of Er for the cases of considering the spatial correlation structure are lower than those of ignoring spatial correlation structure, regardless of the observation time. This confirms that the estimation can be improved by taking into account the spatial correlation structure. The fact that the difference is larger at the earlier stage of observation emphasizes the advantage of using the proposed method for the estimation at an early time. This trend is the same for both model parameters and the settlement estimation.

To investigate the sensitivity of this improvement to the changes of spatial correlation structure, the same calculations at different values of s/η ratio are performed. Only 11 time steps of the observations from the day 10^th to 100^th are selected to use in the calculations. The mean and SD of Er for the model parameter estimation and the settlement prediction at the day 1000^th are determined and illustrated in Figure 4-11.

It can be seen from Figure 4-11 that, when the spatial correlation is considered, the SD of Er

for the model parameter and settlement estimation reduces with the decrease of s/η ratio. This leads us to conclude that, by the proposed method, a stronger spatial correlation gives a better estimation.

Clearly, this improvement becomes significant when observation spacing is shorter than half of the auto-correlation distance, i.e. s/η ≤ 0.5. Note that, in both Figure 4-10 and 4-11, the means of Er are close to zero at any cases. This implies that the bias of these estimations is negligible.

4.2.2.3 Estimation of settlement at an arbitrary location

This section illustrates the ability of the proposed method for settlement estimations at any arbitrary locations. As shown in section 3.2.1 that the component of the unknown parameters at any unobserved points, i.e. xn+1, xn+2, ... , xm, are included in the formulation and the estimates of the parameters at these points will be calculated, at the same time with those at the observation points, by the optimization process based on Bayesian approach. Then, the future settlement at these unobserved points can be predicted using Eq. (4-14).

To investigate the level of error for this type of estimation, the similar calculations with what have been done in the previous section are performed, but, for each calculation, one of the observation points is removed from consideration. Then, the model parameters and the settlement at this removed observation point will be estimated using only the simulated data of the remaining observation points. Due to the fact that the true values of model parameters at each removed observation point are unknown in this case, the estimation errors are determined by comparing the estimated values with the parameter values which are generated from the corresponding random sampling with the other observation points, the data of which is used for the estimation. Figure 4-12 and 13 show the plots of these estimation errors against observation time and s/η ratio, respectively.

For comparison purpose, the calculations presented in Figure 4-12 and 4-13 are analogous with those shown in Figure 4-10 and 4-11 in that s/η = 0.25 is assumed in Figure 4-12 and the data from the day 10^th to 100^th is used in Figure 4-13. The number of simulations for each trial is also 50 (N_sim = 50). The means and SD of the error ratio, E_r, are calculated based on Eq. (4-12), taking into account the estimations at all observation points for all simulations. The values of other parameters are also the same as those assigned in Section 4.2.2.2.

It can be observed from Figure 4-12 that the estimation error, which directly relates to SD of Er, reduces with the observation time in the case with consideration of spatial correlation. In other words, the more observation data we have, the better estimation we obtain. Figure 4-13 clearly shows that the error is significantly higher if the weaker spatial correlation structure is assumed, especially for the settlement estimation. This emphasizes the advantage of the proposed approach in case the strong spatial correlation structure of the parameters is found. Clearly, the case with considering spatial correlation structure provides the more accurate estimation than the case without considering it, in that it gives lower SD of Er especially for the settlement estimation. The bias of estimation, which directly relates to mean of E_r, is found in the calculations without considering spatial correlation, but this also becomes negligible when spatial correlation is considered.