10:45 AM - 12:15 PM
[SSS07-P18] Seismic wavefield estimation based on the adjoint equations towards early forecast: Numerical experiments of the 2D SH wave case
Keywords:Early forecast, Adjoint equations, Data assimilation
In this study, we estimate the initial state of the seismic wavefield when it is radiated from the source without any information of the source data, based on adjoint method, one of the data assimilation methods, for future application to the early forecast of seismic ground motion. The adjoint method sequentially estimates the initial state of the wavefield so that the residuals between the theoretical predictions and the observation within a certain time range are minimized. This method does not require the Green's function calculated previously as in the traditional inversion. In addition, compared to the optimal interpolation method, one of the data assimilation methods used to estimate the current state of the wavefield, the adjoint method requires the wavefield to satisfy the wave equation within the time range.
The wavefield estimation based on the adjoint method consists of several steps. First, we forecast the wavefield from an initial condition, which is already estimated in previous steps or assumed to be zero at the very first step. Then, the squared sum of the residual between predicted and observed particle velocity traces at each recording station over the entire time period are estimated. This residual is defined as the objective function. In the adjoint method, we solve an optimization problem to minimize this objective function. Next, the adjoint equation is solved in a time-reversal manner from the zero initial condition. In the case of SH waves, the adjoint equation is shown to be equivalent to the wave equation with body force term at stations with the forecasted-observed residuals. The solution of the adjoint equations corresponds to the partial derivative of the objective function with respect to the particle velocity and stress tensor components at the first time of the time period, i.e. initial condition. The initial state of the particle velocity and stress thus can be adjusted and estimated successively by the iteration of forecasting the waveform at the stations and backpropagating the residuals from there.
Based on this theory, we performed numerical experiments with the finite-difference method. In the experiment, we used the two-dimensional inhomogeneous velocity structural model with spatial dimensions 400 km in horizontal and 100 km in vertical directions, respectively. A basin-like low-velocity region was added to the 1D depth-dependent velocity structure model. A point strike-slip source was placed at depth of 30 km. Stations were placed at every 20 km on the ground surface. The SH wavefield was calculated by means of the staggered-grid finite-difference method with the precisions of fourth-order in space and second-order in time. Grid sizes were 0.2 km in space and 0.025 s in time, respectively.
After a large number of iterations, initial conditions were estimated to be the observed record almost perfectly, and the concentration of the wavefield almost agreed with the assumed epicenter location. However, the number of iterations for highly accurate estimation was very large, about 1000 or more in the case of this experiment, which suggests that the direct application to the problem of real-time forecast is difficult. We further made a number of trials by adjusting various parameters used in the numerical simulations. As a result, we could drastically reduce the number of iterations, and successfully reduced it to about 1/7. However, the iterations ecame unstable and the results sometimes diverged after the convergence. Since the boundary between divergence and convergence is characterized by the weight parameter, which defines the amount of modification of the wave field in the gradient method, the selection of the optimal value is open to future studies.
The wavefield estimation based on the adjoint method consists of several steps. First, we forecast the wavefield from an initial condition, which is already estimated in previous steps or assumed to be zero at the very first step. Then, the squared sum of the residual between predicted and observed particle velocity traces at each recording station over the entire time period are estimated. This residual is defined as the objective function. In the adjoint method, we solve an optimization problem to minimize this objective function. Next, the adjoint equation is solved in a time-reversal manner from the zero initial condition. In the case of SH waves, the adjoint equation is shown to be equivalent to the wave equation with body force term at stations with the forecasted-observed residuals. The solution of the adjoint equations corresponds to the partial derivative of the objective function with respect to the particle velocity and stress tensor components at the first time of the time period, i.e. initial condition. The initial state of the particle velocity and stress thus can be adjusted and estimated successively by the iteration of forecasting the waveform at the stations and backpropagating the residuals from there.
Based on this theory, we performed numerical experiments with the finite-difference method. In the experiment, we used the two-dimensional inhomogeneous velocity structural model with spatial dimensions 400 km in horizontal and 100 km in vertical directions, respectively. A basin-like low-velocity region was added to the 1D depth-dependent velocity structure model. A point strike-slip source was placed at depth of 30 km. Stations were placed at every 20 km on the ground surface. The SH wavefield was calculated by means of the staggered-grid finite-difference method with the precisions of fourth-order in space and second-order in time. Grid sizes were 0.2 km in space and 0.025 s in time, respectively.
After a large number of iterations, initial conditions were estimated to be the observed record almost perfectly, and the concentration of the wavefield almost agreed with the assumed epicenter location. However, the number of iterations for highly accurate estimation was very large, about 1000 or more in the case of this experiment, which suggests that the direct application to the problem of real-time forecast is difficult. We further made a number of trials by adjusting various parameters used in the numerical simulations. As a result, we could drastically reduce the number of iterations, and successfully reduced it to about 1/7. However, the iterations ecame unstable and the results sometimes diverged after the convergence. Since the boundary between divergence and convergence is characterized by the weight parameter, which defines the amount of modification of the wave field in the gradient method, the selection of the optimal value is open to future studies.