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The Optimization

Recall that in this case I chose to design the survey as a collection of three orthogonal surveys, the definition of each requiring only 5 parameters: source and receiver interval, source and receiver line interval and number of receiver lines per recording patch. Other considerations such as number of shots per salvo, number of stations to roll-along in the inline direction and number of receiver lines to roll-along in the x-line direction are also design parameters but I fixed them to be the number of shots between two adjacent receiver lines, the number of receivers between two adjacent source lines and one, respectively. The upshot is that the model space has only 5 parameters for each zone and an exhaustive search can be employed among all candidate geometries. In more irregular geometries, we may wish to invert for the parameters of each salvo and a micro-genetic algorithm Alvarez (2002a) will be a better choice.

The fitness function has two components: one to minimize the objectives and one to guarantee that the constraints are honored.  
f_i=(1-\lambda)\sum_{j=1}^m \delta_jo_{ij}+\lambda\sum_{j=1}^{n}\epsilon_{j}c_{ij}\end{displaymath} (1)
where i is the index that represents every trial geometry, $\lambda$ is the factor balancing the two contributions to the fitness function, m is the number of objectives, oij is the figure of merit of the jth objective for ith geometry, $\delta_j$ is the relative weight of the jth objective, n is the number of constraints, $\epsilon_{j}$ is the relative weight of the jth constraint and cij is the figure of merit of the jth constraint for the ith geometry. In this case I chose $\lambda=0.5$ which means that I am giving equal weight to the minimization of the objectives and to the satisfaction of the constraints.

The main objective, as mentioned before, is uniformity of target illumination, which requires minimization of the total distance that the emergence ray positions had to be moved to conform with each geometry. Also, since this is a land survey, the main factor in the cost of the survey is the number of shots. Therefore, I used the minimization of the number of shots as the second objective of the optimization. Finally, I used the total receiver- and source-line cut as additional, though less important, objective.

Notice that the constraints are not linear and that they may be partially fulfilled with partial penalties applied. The figures of merit assigned to the objectives and the constraints are normalized between 0 and 1, except when a constraint is completely violated, for example if the required number of channels is larger than the maximum number of available channels, as mentioned before. I made no attempt to differentiate cost between the available recording equipments although in practice this is likely to be an important issue.

The relative weights on each objective and on each constraint for the three zones are summarized on Table 4.

Table 4: Weights for the objectives and constraints applied in each zone: $\delta_1$ is for illumination, $\delta_2$ is for the number of shots, $\delta_3$ is for receiver- and source-line cut, $\epsilon_1$ is for maximum-minimum offset, $\epsilon_2$ is for number of available channels, $\epsilon_3$ is for aspect ratio and $\epsilon_4$ is for fold of coverage.
Zone $\delta_1$ $\delta_2$ $\delta_3$ $\epsilon_1$ $\epsilon_2$ $\epsilon_3$ $\epsilon_4$
1 0.7 0.25 0.05 0.4 0.2 0.3 0.1
2 0.6 0.3 0.1 0.3 0.2 0.3 0.2
3 0.6 0.3 0.1 0.1 0.4 0.2 0.3

next up previous print clean
Next: Geometry with the Proposed Up: The Proposed Approach Previous: Logistic Constraints
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