The convergence to the minimum of misfit value, using the Full-SHM method was achieved, see Figures 4.15 and 4.16. We used the same 1024 models which previously were used to derive the misfit regression equation, as the initial models (ni) required for using NA. During the search for the minimum, 2880 models were evaluated including 96 (ns) models at each subsequent iteration (for 30 iterations, and with nr = 48). Again the same 1024 models were utilized, but this time to initialise the parallel search. When the parallel version of the neighbourhood algorithm was applied to search the parameter sub-volumes a similar result to the Full-SHM approach was achieved but using a much smaller number of model evaluations. This time we also used ns = 96 and nr = 48 (to be comparable with the Full-SHM method) but the number of required iterations was just 2. Then only 192 models were called for convergence compared to 2880 models used in the Full-SHM (see Figure 4.15a and b for the two misfit scenarios).
After dividing the parameter space we at most needed to deal with a 2D problem when searching parameters in parallel. Accordingly, the tuning parameters for the neighborhood algorithm can be set for such situation. Another run of the Parallel-SHM was carried out with ns = 8 and nr = 4 for 2 iterations. The same result was obtained but this time with much smaller number of model evaluations of 16 models. The convergence of this case is illustrated in Figure 4.15a.
Using the Serial-SHM, the parameters are searched in order of decreasing importance as determined from the coefficients and their effects on the correlation of the regression
equation predictions against the true misfits. The 2D sub-volume containing barriers ‘e’
and ‘g’ was searched first, and then the other eight 1D volumes were searched in turn.
The NA tuning parameters were set for ni = 32, ns = 16 and nr = 8 for 2D sub-volume, and ni = 16, ns = 8, and nr = 4 for 1D sub-volumes, and the process was carried out with 10 and 5 iterations for the 2D and 1D sub-volumes, respectively. Convergence was achieved in 552 models which is one sixth of the number of models required for the SHM approach. In Figure 4.16 the serial convergence is shown together with Full-SHM convergence results. Nevertheless, this result was not as attractive as the Parallel-SHM approaches which was accomplished with a 70% speedup of the convergence rate (if we count over initial models). In the Serial-SHM method when the nieghbourhood algorithm was applied to sub-volumes sequentially, each time the parameter was searched, there was a need to initialise with the appropriate sampling consistent with the dimension of the sub-volumes. This was not required for the Parallel-SHM method which starts searching with the same initial models used to obtain the misfit regression model.
In Figure 4.17 convergence of the 10 parameters towards the best solution using the Parallel-SHM approach is compared with that of the Full-SHM. The Parallel-SHM resulted in a much faster convergence. In Figure 4.18 the evolution of models in convergence of the Serial-SHM method are shown but only for each serial application of neighbourhood algorithm (i.e. where that parameter was changed).
Figure 4.17a shows that for the Full-SHM method to converge we required 2000 models at least (more than 20 search iterations using NA) to obtain the convergence of transmissibility multipliers of barriers ‘f’ and ‘b’ in the ‘seismic only misfit’ scenario, and for the barriers ‘a’, ‘d’, ‘g’ and ‘j’ in the ‘seismic plus well misfit’, see Figure 4.17a. The model parameter evolution showed a potential bi-modal solution for barrier
‘b’ until convergence to solution is reached, i.e. where the minimum misfit does not improve any more in the ‘seismic only misfit’ scenario (Figure 4.17a). Further, the barriers ‘d’ and ‘j’ in the ‘seismic only misfit’ scenario, and the barriers ‘c’, ‘f’ and ‘h’
in the ‘seismic plus well misfit’ scenario were very much non-convergent using Full-the SHM method. However, using Parallel-SHM and Serial-SHM there was apparent convergence after the first iteration of each search routine (Figure 4.17 and 4.18).
a) ‘total 4D seismic misfit only’ scenario
b) ‘seismic plus injector misfit’ scenario
Figure 4.15: Comparison of misfit convergence of the Parallel-SHM approach to the Full–
SHM: a) ‘total 4D seismic misfit only’ scenario, for two cases of using 'ns = 96, nr = 48' and 'ns = 8, nr = 4', and b) ‘seismic plus injector misfit’ scenario. The solid line indicates the base case simulation misfit. The dark blue symbols represent the evolution of the models during converge to minimum misfit using the Full-SHM method, and pink symbols show the models for Parallel-SHM method (plus the initial models- also used in Full-SHM). Green symbols (marked in red circle) show Parallel-SHM result for the case with 'ns = 8, nr = 4'.
Misfit Convergence
565 615
0 2000 4000
Model
Misfit X1000
Full-SHM
Parallel-SHM (ns=96, nr=48)
Parallel-SHM (ns=8, nr=4) base model
Misfit Convergence
700 900
0 2000 4000
Model
Misfit X1000
Full-SHM Parallel-SHM base model
initial models (ni)
initial models (ni)
a) ‘total 4D seismic misfit only’ scenario
b) ‘seismic plus injector misfit’ scenario
Figure 4.16: Convergence to the solution by applying serial ‘one sub-volume at a time’ or Serial-SHM for: a) the ‘total 4D seismic misfit only’ scenario, and b) the ‘seismic plus injector misfit’ scenario. The solid line indicates the base case simulation model misfit. Each
565 615
0 2000 4000
Misfit x1000
Model
Misfit Convergence
Full-SHM exg
i d a f j b
h c final base model
700 900
0 2000 4000
Misfit x1000
Model
Misfit Convergence Full-SHM
cxi b e d g j a f h final base model
a) ‘total 4D seismic misfit only’ scenario
b) ‘seismic plus injector misfit’ scenario
Figure 4.17: Convergence of the parameters by Parallel-SHM (pink symbols) and Full-SHM (dark blue symbols) for: a) the ‘total 4D seismic only misfit’, and b) the ‘seismic plus injector misfits’ scenarios. The x-axis is the model index and y-axis is log10 of the modifier applied to barrier transmissibility.
a) ‘total 4D seismic only misfit’ scenario
b) ‘seismic plus injector misfit’ scenario
Figure 4.18: Convergence of the parameters when the Serial-SHM is applied to the two scenarios of: a) the ‘total seismic misfit only’, and b) the ‘seismic plus well misfits’. We only show the models as the parameters are modified. The x-axis is the model index and y-axis is log10 of the modifier applied to barrier transmissibility.
barrier a
Chapter 5: Divide and Conquering combined with Experimental Design