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Bulletin of the Seismological Society of America, Vol. 86, No. 2, pp. 436-444, April 1996

Application of Genetic Algorithms to an Inversion of Surface-Wave Dispersion Data by Hiroaki Yamanaka and Hiroshi Ishida

Abstract

A new method for inversion of surface-wave dispersion data is introduced. This method successfully utilizes recently developed genetic algorithms as a global optimization method. Such algorithms usually consist of selection, crossover, and mutation of individuals in a population. To facilitate convergence to an optimal solution, we added elite selection, which ensures that the "best" individual with the smallest misfit value is not excluded from the succeeding generation, and dynamic mutation, which contains a generation-variant mutation probability. Using synthetic and observed earthquake data, we examined the applicability of this genetic surfacewave inversion method in deducing an S-wave profile for sedimentary layers from short- and intermediate-period surface-wave dispersion data. W e demonstrated that the method is robust and can be used to interpret surface-wave dispersion data.

Introduction Surface-wave dispersion analysis is one method used to extract a subsurface structural model from records of earthquakes, explosions, or microseisms. A number of studies have explored regional or global subsurface structures and shallow soil profiles (e.g., Horike, 1985; Cara, 1983; Kafka and Reiter, 1987). In inversions of dispersion curves for observed surface waves, a structural model is determined by fitting the observed dispersion curves with a theoretical curve. A dispersion curve for a surface wave propagating in a horizontally layered structure can be calculated by a matrix method (e.g., Haskell, 1953). Generally, a dispersion curve is a nonlinear function of shear- and/or compressional-wave velocities, densities, and thicknesses for each layer. To invert a dispersion curve into these parameters, usually a linearized approximation is used by neglecting higher-order terms in the Taylor series expansion. Then, an optimal solution is obtained by an iterative perturbation process based on linear inverse theory (e.g., Aki and Richards, 1980). However, these linearized inversions have sometimes numerical difficulties in the presence of noise. Furthermore, a final inverted model determined by a linearized inversion inherently depends on an assumed initial model, because of the existence of locally optimal solutions. In general, perturbations in the characteristics of long-period surface waves and free oscillations of the earth vary slowly (Wiggins, 1972); therefore, the dependency of the final inverted model on the assumed initial model is small. However, characteristics of surface waves at periods less than 10 sec are significantly affected by near-surface low-velocity sedimentary layers, which vary laterally and vertically (e.g., Kinoshita et al., 1992; Yamazaki et al., 1992). For sedimentary layers near the Earth's 436

surface, it is sometimes difficult to set up an initial model that is sufficiently close to the real solution. Therefore, the dependency on an initial model is an important problem when inverting short-period surface-wave data. When an appropriate initial model can be generated using a priori information about a subsurface structure, linearized inversions can find an optimal solution that is the global minimum of a misfit function that is defined as the sum of differences between observed and calculated data. However, if a priori information is either scant or unavailable, the inversion may find a local optimal solution. To reduce these difficulties, we examined a recently developed nonlinear optimization method that uses a genetic algorithm (GA), which has been applied in several other fields, such as engineering design (e.g., Goldberg, 1989). Using an analogy to population genetics (i.e., where the operations are selection, crossover, and mutation), these algorithms can simultaneously search both globally and locally for an optimal solution by using several models. Since what is required in using a GA is only to compute the objective functions that are to be minimized, these algorithms are robust. These GA's have already been used in several seismological applications. Stoffa and Sen (1991) used a GA in an inversion of 1D reflection seismograms. Wilson and Vasudevan (1991) estimated static corrections in reflection data processing by using a GA. Furthermore, Sambridge and Drijkoningen (1992) applied a GA to a waveform inversion of reflection seismograms. In an earthquake study, Sambridge and Gallagher (1993) examined a GA-based inversion method of travel-time data for determining hypocenter lo-

437

Application of Genetic Algorithms to an Inversion of Surface-Wave Dispersion Data

cations. Recently, Kobayashi and Nakanishi (1994) obtained fault-plane solutions of large earthquakes using a GA. In this study, we applied a GA to invert group velocities of surface waves in a period range from 2 to 20 sec to deduce an S-wave velocity profile for sedimentary layers lying over a basement. First, we explain the procedure for applying a GA to inversion of surface-wave dispersion data, specifically group velocity of surface waves. Then, we explain the numerical experiments in which we applied this genetic inversion of surface-wave dispersion to synthetic dispersion data. Finally, we applied this inversion method to observed surface-wave data.

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gc(Zi),

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where N is the number of the observed data and a(Ti) is the standard deviation of the observed group velocity at a period of Ti. With the misfit function, we defined the fitness of an individual by

Procedure for Using a Genetic Algorithm to Invert G r o u p Velocity

A basic explanation of the GA can be found in a textbook by Goldberg (1989). Here, we explain the genetic inversion of surface-wave dispersion data using the group velocity of Love wave. The same procedure can be easily applied to phase velocity inversion and to Rayleigh waves. We considered a Love wave propagating in a horizontally layered half-space. In the inversion, we fixed the density of the layers and chose the shear-wave velocities and thicknesses of the layers as the parameters to be determined, because they significantly affect the dispersion curve. In the GA, these parameters are encoded in gene type that is suitable for the genetic operations. We referred to Stoffa and Sen (1991) for coding the parameters. First, we defined search areas for the S-wave velocity and thickness for each layer. Then, each parameter in the search area was digitized with an n-bit binary string of 2n; namely, parameters at the lower and upper limits became (00 . . . 0) and (11 . . . 1), respectively, and any parameter to be examined could then be transferred in a binary string between the two bit strings. Using the genetic analogy, the binary strings for all parameters in a subsurface structural model were concatenated to generate a "chromosome" to be used in the genetic operations. An optimal solution was searched for using such a chromosome that corresponds to a structural model. Using random numbers, we generated an initial population that consisted of Q individuals, i.e., Q different models. Then, the three genetic operations were applied to the initial population to produce a new population with the same population size. In the selection, a new population was reproduced based on a fitness function for each individual. A fitness function is generally defined so as to indicate how a model can explain observed data. When the individual has larger fitness, it has higher probability of being reproduced and passed down into the next generation. Forjth individual, we defined a misfit function, ~bj,that was the average of rootmeans-squares of the differences between the observed group velocity, Uo(Ti), and the calculated group velocity,

~(Zi;

k=l Using this reproduction probability, Q models for the next generation were selected from the current generation by the roulette rule (Goldberg, 1989). When a model has a higher reproduction probability, the number of the model increases in the next generation. These reproduced models were modulated by the crossover and mutation. In the crossover, all surviving individuals were randomly paired. At a given crossover probability, Pc, the pair exchanged a part of each bit string corresponding to each model parameter at a random locus in the string. In the calculation, we generated a random number ranging from 0.0 to 1.0. Then, if the random number was smaller than Pc, the crossover took place, and two new individuals were created and replaced the original individuals. If the random number was greater than Pc, the two individuals in the original pair remained into the next generation. In the mutation, genes in each chromosome were randomly changed from 0 to 1 or vice versa at a mutation probability of P,n. Similar to the crossover, we determined the occurrence of the mutation by generating random numbers. By iteration of these three genetic operations, the initial population approached a global optimal solution. These calculations were iterated until the number of renewed generations reached a predetermined number, which was 100 in our case. Modified GA To make the inversion converge faster on the global optimal solution, we used two additional genetic operations, namely, elite selection and dynamic mutation. In the selection explained above, it is possible that an individual with the lowest misfit value will not be selected for the next generation. Furthermore, the chromosome of the best individual might also be destroyed by the crossover. To assure that the best model appeared in the next generation, we used elite selection in which the worst model in the current generation was replaced by the best individual in the previous genera-

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H. Yamanaka and H. Ishida

tion (Goldberg, 1989). There is some discussion that the elite selection might have a premature convergence problem (Goldberg, 1989). The premature convergence probably occurs, because all individuals in the population concentrate near a model that has a local minimum of the misfit. Thus, the possibility of generating a model near the global optimal solution by the crossover becomes very low. On the other hand, it is possible that mutation can help the inversion avoid the premature convergence. Sen and Stoffa (1992) used a generation-variant mutation probability that is similar to a cooling process in simulated annealing algorithms. Similarly, we introduced a generation-variant mutation probability that was dynamically varied during the inversion. In our dynamic mutation, when most of the individuals had similar chromosome patterns, we setpm higher than that defined initially, and after the variety of chromosomes in the population increased, we set Pm back to its initial value. To determine the level of the variety in the population, we calculated the average coefficient of variation, % for all parameters in all individuals in each generation as

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~ > 0.1 0.02 < 7 < 0.1 , 7 < 0.02

where Pinit is the initial mutation probability. Figure 1 shows a flow diagram of this modified GA.

Figure 1. Flowdiagram of the modified GA-based inversion method.

Table 1 Structural Model Used in the Test and Search Space Assuming a Four-Layer Model

Inversion of Synthetic Data We tested the modified genetic inversion method explained above using synthetic group velocities of a Love wave propagating in a layered half-space. The structural model we used is shown in Table 1. This is a four-layer model that was obtained for the Kanto Plain, Japan, from a dispersion analysis of the Love wave in earthquake records (Kinoshita et al., 1992). The top three layers are Tertiary sediments over the basement that has an S-wave velocity of 2.6 km/sec. Group velocities for the fundamental Love wave were calculated in a period range from 3 to 20 sec. We added noise that had an amplitude of 10% of the group velocity at each period. In the inversion, we searched for an optimal combination of an S-wave velocity and a thickness for each layer. The total number of unknown parameters were four velocities and three thicknesses. They were digitized as 8bit binary strings. The upper and lower bounds in the search spaces of the parameters are also listed in Table 1. We set

True Model

Search Space

Vs (km/sec)

H (kin)

p (g/cm3)

Vs (kin/see)

H (kro)

0.5 0.8 1.2 2.6

0.5 0.6 1.0 ~

1.7 1.9 2.2 2.5

0.4~.8 0.~1.2 0.8-1.8 2.~3.0

0.34.7 0.4-0.8 0.5-1.5 --

the population size Q at 20, pc at 0.7, and Pinit at 0.02. We terminated the iterations at the 100th generation. Since this inversion method is a kind of probabilistic approach using random numbers and finds models near a global minimum solution, we did 20 inversions, where each inversion had different initial random numbers. A final optimal model was determined by averaging the resulting parameters from these 20 inversions. The minimum misfit value obtained at each generation is shown in Figure 2, where each misfit value is the average

Application of Genetic Algorithms to an Inversion of Surface-Wave Dispersion Data

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of the 20 inversions. The misfit value decreased rapidly in the first 20 generations and then gradually in the next 80 generations. This gradual decrease suggests convergence near the minimum solution. For comparison, Figure 2 also shows the misfit values obtained using the simple GA, which did not have elite selection and dynamic mutation. The convergence of the misfit values for the two algorithms was almost the same up to the 20th generation. After that, however, compared with the simple GA, the modified GA method found a better solution as indicated by the lower misfit value at the end of the calculation. This verifies the effectiveness of the two additional operations in the inversion. The structural model obtained from the inversion by the modified GA method is shown in Figure 3. The true S-wave profile was reconstructed well by the inversion. The comparison between the synthetic and inverted group velocities can be seen in Figure 4, showing that the synthetic data were perfectly reproduced by the inversion. It is well known that GA's cannot find an exact optimal solution, but rather a model near the optimal solution, because such a solution has a misfit that is similar to that for the optimal solution. Therefore, the profiles obtained from the 20 inversions were near the optimal solution. In the inversions, the calculation was terminated when the number of renewed generations reached a given number. If we had continued a calculation until a solution was sufficiently close to the optimal solution, then a single run of the program could have found the global optimal solution. However, we did not know in advance how many generations were required to find such a solution. Therefore, it was more effective and economical to try several inversions with different initial random numbers. Inversion o f Observed Data We applied the modified GA inversion method to the surface wave observed during an earthquake near the Izu Peninsula, Japan, on 20 February 1990. Since the depth of

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Figure 3. Subsurfacestructures for the Kanto Basin (Kinoshita et al., 1992). The thick dashed line represents the true model used for testing the inversion method. The thin line indicates the structural model obtained from the modified genetic inversion method for the synthetic dispersion data for a fundamental Love wave. The dotted lines indicate the standard deviation of 20 models obtained from the inversions. The dashed-and-dotted lines show the lower and upper bounds of the search area.

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