In tsunami waveform inversion using the conventional Green's function technique, an optimal solution is sometimes difficult to obtain because of various factors. This study proposes a new method to both optimize the determination of the unknown parameters and introduce a global optimization method for tsunami waveform inversion. We utilize a genetic algorithm that further enhanced by a pattern search method to find an optimal distribution of unit source locations prior to the inversion. Unlike the conventional method that characterized by equidistant unit sources, our method generates a random spatial distribution of unit sources inside the inverse region. This leads to a better approximation of the initial profile of a tsunami. The method has been tested using an artificial tsunami source with real bathymetry data. Comparison results demonstrate that the proposed method has considerably outperformed the conventional one in terms of model accuracy.

Direct observation of sea surface deformation after the occurrence of an earthquake is still difficult to obtain; therefore, its estimation is often performed by consideration of relevant seismic information or the hydrodynamic response of the sea determined from recorded tsunami waveforms. Determination of sea surface deformation generated by earthquakes is crucial to the success of tsunami modeling. One of the most frequently used methods for determining sea surface deformation is to presume it from a fault model (Mansinha and Smylie, 1971; Okada, 1985). A more realistic approach was proposed by Satake (1987) who analyzed recorded waveforms to infer earthquake source parameters or particularly, coseismic slip, using the Green's function technique. Even though the fault model is still required, the division of a fault into smaller sub-faults allows the slip to be estimated in a heterogeneous manner, which leads to better approximation of sea surface deformation. A simpler method was actually introduced earlier by Aida (1972) for which no prior assumption of a fault model was needed. This study is in line with that of Aida because we are more interested in estimating sea surface deformation than a slip on the fault plane. The motivation behind this is that tsunami excitation can sometimes occur as a result of various factors that are independent of the associated seismic characteristics (Geist, 2002).

More recently, several studies using tsunami waveform inversion to estimate sea surface deformation without fault model assumptions have been widely developed. The basic premise is to replace the fault model by an auxiliary basis function on unit sources, which is equivalent to the sub-fault approach by Satake (1987). For instances, Baba et al. (2005) used a simplified fault model by disregarding actual earthquake parameters to produce the initial profile on each unit source, whereas Satake et al. (2005) proposed a more direct approximation using a pyramidal shape with a flat top. Other studies by Liu and Wang (2008) and Saito et al. (2010) demonstrated attempts to use Gaussian function, whereas Wu and Ho (2011) adopted a top-hat small unit source to represent the initial profile. The same approach was proposed by Tsushima et al. (2011) and Yasuda and Mase (2013) for the more practical purpose of a tsunami early warning system. However, this inversion method for tsunami waveforms possesses a limitation, in that the inverse matrix does not always exist because of the non-uniqueness of the solution. In addition to the large number of unknown parameters, which might produce many local optima on the misfit function measure, the search towards optimality is confined by the uniform distance of unit sources used in the regular Green's function.

Tsunami waveform inversion sometimes falls into an ill-posed problem, in which small errors in the observed waveforms are exceptionally amplified in the solution. Therefore, both the uniqueness and the stability of solutions are sometimes difficult to attain without appropriate treatments. The most frequently employed techniques to maintain a stable solution is to use a smoothing constraint (Gusman et al., 2010, 2013; Saito et al., 2010). Other than that, Koike et al. (2003) suggested reducing the unknown parameters using the wavelet base to guarantee the uniqueness of the solution. However, they found later that the selection of the wavelet base was not straightforward. Another effort to overcome the issue was discussed by Voronina (2011). The study promoted a method to control numerical stability for the ill-posed problem in tsunami waveform inversion by means of singular value decomposition and r-solutions techniques. In this paper, we proposed a new approach to tackle the same problem by determining the optimal position or spatial distribution of unit sources located around the tsunami source or epicenter. A genetic algorithm (GA) as a global optimization method, combined with a pattern search (PS) method, is employed to search the mentioned positions prior to the inversion. As the selected positions are probably located in between the initial unit sources, interpolations are performed during the optimization. Therefore, the Green's function evolves dynamically at each generation of the GA and PS iteration.

Generally, the characteristics of tsunami propagation in deep water are
linear. According to Satake (1987), even in shallow coastal areas, the first
leading waves recorded at coastal tide gauges are still well simulated by
the linear long wave model. Therefore, a typical linear non-dispersive
shallow water equation is used in the forward modeling to compute time
histories of sea surface elevation at the specified observation points:

The Green's function

The ultimate purpose of a global optimization method is to find the extreme value of a given non-convex function in a certain feasible region. Following the growth of computer science, new types of optimization based on natural processes and artificial intelligence have been developed extensively and used by scientific and engineering communities. The reason for this is that the new optimization methods possess the interesting feature of being able to avoid local optimum solutions, which is something classical methods fail to do.

The use of global optimization methods in tsunami waveform inversion is not new; relevant discussions can be found in Piatanesi and Lorito (2007) and Romano et al. (2010). They used a simulated annealing technique to solve the inverse problems. Here, we proposed a different algorithm based on a hybrid optimization of GA and PS. The hybrid technique is preferred because global optimization methods, such as GA, are capable of exploring broader search space, but not as good in fine tuning the approximation of the expected solution. Therefore, PS is employed as a local search algorithm to locate other nearby solutions that could possibly be better than the result of the previous search by GA (Payne and Eppstein, 2005; Costa et al., 2010).

The hybrid algorithm proposed in this study works by simply treating the output of the GA optimization result as the initial condition for the PS optimization. The technique is proven effective even though more fitness function evaluation is required; hence, it costs extra computational efforts. However, parallelization of either GA or PS can be easily implemented to expedite the computing time and gain substantial performance enhancement.

The formulation of an optimization problem can be expressed as follows:

In this study, we develop the proposed algorithm with two different design
parameters. The first is simply to search the water elevation of each unit
source without including a search of the optimum locations, thus similar to
that of the ordinary least squares method. By rearranging the Eq. (

Example of a unique identification of the forward model computational grid used for the second design parameters.

GA is an optimization method that searches for an optimal value of a complex function by adopting the process of natural evolution (Goldberg, 1989). It can be categorized as a type of stochastic optimization method and as a part of artificial intelligence. In GA, the model parameters or decision variables in the optimization are first transformed into a chromosome-like data structure that later evolves to form a better individual. The most common representation of design parameters in GA is their encoding into a binary string. There are three basic genetic operators in GA: selection, crossover, and mutation.

As described in Wetter and Wright (2003), in GA, finite lower and upper
bounds of

Study area and bathymetry profile. Red dots indicate unit sources located throughout the inverse region. Green triangles with numbers are artificial observation stations.

Similar to GA, PS is an optimization method that does not require the gradient (derivative-free) of the problem to be optimized. The method was first introduced by Hooke and Jeeves (1961). Later, Torczon (1997) conducted studies to prove the convergence of PS using the theory of positive bases. The algorithm of PS used in this study is similar to that of Wetter and Write (2003), while the design parameters are identical to the GA optimization.

For the same optimization problem as formulated in Eqs. (

The overall procedures of PS optimization can be elaborated as follows:

We have conducted numerical experiments using an artificial tsunami source
propagated on an actual bathymetry profile. An area extending from
140–145

Gaussian basis function.

Waveform interpolation. Left-hand figure shows selected location of a unit source indicated by a black square. Blue dots represent the four nearest unit sources used in the interpolation. Right-hand figures are comparisons of waveforms between numerical model (solid black line) and interpolation (dashed red line) at the artificial gauges.

The selection of a cost function or misfit function is essential because it
directs the fate of the optimization towards the optimal solution. Here, we
use a combination of root mean square error (RMSE) and Pearson correlation
coefficient (

Flowchart of model development procedure.

The fitness evaluation is subject to noise from various factors that might
lead the optimization towards unexpected solutions. The easiest technique to
overcome such problems is by means of explicit averaging over a number of
samples to smooth the cost function (Jin and Branke, 2005). As the best
solution or closest fit is indicated by RMSE

Sea surface deformation. Gray dots indicate the centroid of unit
sources.

A Gaussian shape with 1 m amplitude is used as the basis function for each
unit source (Liu and Wang, 2008). Providing

Summary of statistical evaluations based of the proposed cost function.

For the first design parameters, the optimization is performed merely to search the water elevation or initial amplitudes of each unit source. For this case, the Green's function is constructed based on the initial 28 unit sources, separated by a uniform distance of 60 km, which is identical to that used in the least squares inversion. Hereafter, the first model will be termed the Genetic Algorithm Pattern Search for uniform source distribution (GAPSu). The purpose of this model is simply to compare the performance of the proposed global optimization method with the traditional least squares method in the same model design and environment.

The second design parameters aim to find the optimal locations of unit
sources that, at the initial state, are distributed randomly around the
tsunami source. The second model will be termed the Genetic Algorithm
Pattern Search for random source distribution (GAPSr). In the GAPSr model,
the amplitudes are computed using the least squares inversion; therefore,
the second model is actually a combination of a deterministic and stochastic
optimization. However, the selected source locations might not lay precisely
in the initial 28 unit sources and thus, interpolation is required to
produce the sea surface fluctuations at the observation stations.
Consequently, in the GAPSr model, the Green's function evolves during the
optimization. Nearest neighbor-weighted interpolation is performed for
estimating the phases and amplitudes of the waveforms originating from the
selected locations based on the four nearest unit sources (Mulia and Asano, 2014). An example of the
interpolation results, complete with statistical evaluations in terms of
RMSE and

An artificial tsunami source is used to test our method (Fig. 6a). Instead
of using a simpler profile produced by the Okada's solution, we generate a
more complex shape from a superposition of 10 unit sources with random
amplitudes and positions located inside the inverse region. We purposely
limit the number of unit source to avoid generating shorter wavelengths than
the prescribed long wave assumption. For the regular Green's function (first
design parameter), using only 10 unit sources is insufficient to reconstruct
the target profile. Consequently, the number of unit source should be
increased. The decision of using 28 unit sources in both design parameters
is for the purpose of model performance comparisons that will be further
discussed in the Results and discussion section. The approximation of this
initial profile is performed using unconstrained, traditional least squares
inversion, GAPSu, and GAPSr. However, GAPSr is the most important part in
this study; therefore, we focus our discussion on the GAPSr model. The
development of the GAPSr model depicted in the flowchart (Fig. 5) can be
summarized as follows:

Construct the initial Green's function.

Initialize the GAPSr model by randomly distributing the unit source locations. The search of the optimal location is bounded by the area of the inverse region.

Perform interpolation and update the Green's function.

Evaluate the fitness by performing the least squares inversion.

After reaching the stopping criteria, the forward numerical modeling is run again for each of the optimized unit sources to avoid errors generated from the interpolation result. Subsequently, the inversion is performed for the final time.

Overall, all models can produce a relatively good estimation of the targeted sea surface deformation. This is likely because they are applied to an ideal case with artificial conditions and settings, except for bathymetry. For instance, the target source was generated from the same Gaussian shape as that used to construct the Green's function. Therefore, the task is more straightforward as fewer complexities are encountered. In the real case, however, alternative distributions to represent the initial water height should be considered because the tsunami source does not always follow the Gaussian distribution. This may have a profound effect on the result of the first design parameter, but less significant for the second design parameter as a superposition of unit sources with random locations allows to approximate any surface profile regardless of their shape. Nevertheless, in this study, the use of the ideal case has allowed us to assess the advantage of the proposed method in a more detailed manner.

The GAPSu model yields a slightly better fit of waveforms compared with the least squares method (Table 1). This means that the global optimization method locates a better minimum value in the cost function, which is situated beyond the reach of the least squares method. However, the slight refinement by the GAPSu model over the least squares method makes it difficult to gauge the benefits of employing the method. This should not be a surprise because the model is applied to determine the coefficients in a linear system, which is relatively easy to solve using a conventional method. Moreover, the waveforms used to invert the initial sea surface deformation are generated from an artificial tsunami source instead of real measurements. Therefore, the linearity is well conserved and thus, the use of more advanced methods becomes redundant and unnecessary. In the study by Piatanesi and Lorito (2007), a global optimization method was successfully promoted for the case of tsunami waveform inversion. This was because the optimization method was applied to a nonlinear inverse problem of actual measurement data. Accordingly, the appraisal of such a method can be clearly defined.

Comparison of waveforms at gauges. Gray bar above the time axis indicates the time range for the inversion.

Spurious uplifts and subsidence of the water surface profile are generated in both the least squares and GAPSu model results (see Fig. 6b and c). One may argue that the specified spatial resolution of the unit sources is too coarse to represent the complete form of the target source. A denser distribution of unit sources should improve the results; however, it might also introduce other problems. A large number of model parameters (unknown parameters), which are proportional to the degrees of freedom in the optimization, are liable to cause the solution to become easily entrapped in a local optimum. Without a smoothing constraint, the result of the tsunami waveform inversion might be bumpy and non-physical, especially for cases with high spatial resolution (Wu and Ho, 2011). In other studies on tsunami waveform inversion by Baba et al. (2005) and Wu and Ho (2011), an equality constraint was imposed to maintain the smoothness of the inverted parameters to satisfy the long wave assumption, while by Saito et al. (2010) and Gusman et al. (2010) the constraint was used to obtain stable solutions. However, such constraint might restrict the exploration throughout the feasible search space and render the discovery of an optimum solution more difficult. Another plausible explanation for the unsatisfactory results of the least squares and GAPSu model is simply that the equidistant unit sources in the regular Green's function confines the search for optimality. This can be proven by the result of the GAPSr model, for which the same number of model parameter (28 unit sources) with random locations yields a much better estimation.

Scatter plots of each inversion method with respect to the waveforms of the target source at all gauges.

The different design parameters in the GAPSr model have considerably
improved the inversion accuracy. For instance, at Gauge 1, where the best
fit of the waveform is attained, the measurement of accuracy as RMSE is
0.0260, 0.0256, and 0.0094 m, and as

The search for the optimal locations of the unit sources allows the least squares method to find the unique and optimal solution. Such an approach is difficult to achieve deterministically using conventional gradient methods, because there is the possibility that the constructed design parameters in the GAPSr model produce a discontinuous or non-differentiable error surface owing to the random characteristics exhibited by the artificial tsunami source (target source). The same characteristic is very likely to occur in nature. A high degree of uncertainty has been observed in tsunami sources leading to significant variations in the nearshore tsunami amplitude (Geist, 2005). Therefore, the use of the model in real case applications is encouraged to reveal the underlying dynamics in tsunami generation. However, as with typical stochastic methods, the proposed model cannot ensure a constant optimization response time. The solution and convergence are strongly dependent on the random initial state.

Estimations of tsunami sea surface deformation using a global optimization method with a stochastic nature have been conducted. Our numerical experiments using the GAPSu model revealed that the use of such methods for a linear system with standard design parameters, as in ordinary tsunami waveform inversions, is redundant and promotes trivial improvements. In contrast, the different design parameters in our proposed method (GAPSr), which was applied to determine the optimum location of the unit sources prior to the inversion, demonstrated considerable improvements in the accuracy. The random location of unit sources permitted the inversion to produce a more precise approximation of the initial sea surface deformation without violating the general assumption of long wave theory.

The involvement of stochastic processes in the optimization increased the ability to reveal uncertainties in the tsunami source, which are difficult to discern using deterministic approaches. However, as the signature of typical stochastic optimizations, the optimization response time is erratic because of the strong dependency on the initial state. Using a current standard desktop computer, the required computing time varied from 5 to 10 min. Thus, a more sophisticated computer would be needed to ensure the effectiveness of the method when applied in a real-time application. Nevertheless, the results have demonstrated the efficacy of the method for post-event studies of tsunamis, because it can provide better estimations of the coseismic sea surface deformation compared with traditional tsunami waveform inversion methods.Edited by: I. Didenkulova Reviewed by: A. Gusman and two anonymous referees