10:45 AM - 12:15 PM
[SSS07-P17] Derivation of adjoint equations for estimating seismic and tsunami wavefields
Keywords:Seismic waves, Tsunami waves, Adjoint equations
The estimation of seismic- and tsunami wavefields as spatially continuous quantities based on tight-lattice observation records has been studied from the viewpoints of both wave propagation problem in heterogeneous media and real-time forecast of earthquake ground motions and tsunamis. In particular, the optimal interpolation method, which is a kind of data assimilation method that predicts a state one step ahead by numerical simulation and corrects and interpolates it with observation records, is being utilized for the real-time problems. Numerous studies have shown that this method is computationally efficient and useful in practical applications. However, this method is essentially an interpolation method from data at the current time, and although it uses predictions from past states to the current time, these predictions are destructively updated by residuals from the observed values at the observation points. Thus, the resulting spatio-temporal wavefield does not always perfectly fit the governing wave equations. Recently, I proposed a method for obtaining initial conditions of the wave field for the linear long wave equation system of tsunami that minimizes the residuals between the numerical model and the observations over a finite time interval, and showed through numerical experiments that the method can be used consistently for the immediate prediction problem. In this study, we generalize the proposed method and present a mathematical framework for deriving a method that can be applied to wave propagation problems and its application to elastic bodies.
In this study, an adjoint equation is derived based on optimal control theory. This method is a generalized variational method in which the objective function is the integral over a finite time interval of the square of the evaluation function between the theoretical prediction and the observation at discrete observation points, and the initial state of the wave field is estimated such that this objective function is minimized. If the evaluation function is the difference between observation and prediction, this is the same problem set-up as the inversion, but this method does not require the observation to be expressed as a convolution of a finite number of wave sources or epicenters with a Green's function. Instead, the adjoint equation of the governing equation is utilized to estimate the differential coefficients of the objective function with respect to the initial state of the wavefield. Once the differential coefficients are obtained, successive updates can be performed in the direction of minimizing the objective function based on the gradient method.
For continuum equations, the adjoint equations often have a shape similar to the original governing equations. For linear long wave tsunamis and linear perfect elastic SH waves, the original equations and the adjoint equations are in perfect agreement. Note that a single force term is added to the volume conservation law in the linear long wave equation and to the equation of motion in the SH wave equation to radiate the residual between the observed and predicted values from the observation point. Therefore, solving the adjoint equations backward in time is closely related to back-propagation calculations based on observation point records.
It should be noted that a variable transformation is required to make the adjoint equation and the governing equation identical. This means that the solution to the adjoint equation obtained will be used to update the initial conditions of the original variables via the inverse of this variable transformation. In the case of the linear long wave equation and SH waves, this relationship corresponds to a mere multiplication of the adjoint variables by a factor, but the value is not a uniform constant and varies from place to place in an inhomogeneous medium. The adjoint equations for a three-dimensional perfect elastic body are also of the same type as the original equations of motion and constitutive equations, but the stress equivalent component of the adjoint equations contributes to the real-space stress via the elastic compliance tensor. The stress equivalent component in the adjoint equation contributes to the stress in real space via the elastic compliance tensor. This implies that when seismic waves are back-propagated to the source region, it is not the stress equivalent value obtained but the strain equivalent value obtained by applying it to the elastic compliant tensor that corresponds to the stress variation at the source.
In this study, an adjoint equation is derived based on optimal control theory. This method is a generalized variational method in which the objective function is the integral over a finite time interval of the square of the evaluation function between the theoretical prediction and the observation at discrete observation points, and the initial state of the wave field is estimated such that this objective function is minimized. If the evaluation function is the difference between observation and prediction, this is the same problem set-up as the inversion, but this method does not require the observation to be expressed as a convolution of a finite number of wave sources or epicenters with a Green's function. Instead, the adjoint equation of the governing equation is utilized to estimate the differential coefficients of the objective function with respect to the initial state of the wavefield. Once the differential coefficients are obtained, successive updates can be performed in the direction of minimizing the objective function based on the gradient method.
For continuum equations, the adjoint equations often have a shape similar to the original governing equations. For linear long wave tsunamis and linear perfect elastic SH waves, the original equations and the adjoint equations are in perfect agreement. Note that a single force term is added to the volume conservation law in the linear long wave equation and to the equation of motion in the SH wave equation to radiate the residual between the observed and predicted values from the observation point. Therefore, solving the adjoint equations backward in time is closely related to back-propagation calculations based on observation point records.
It should be noted that a variable transformation is required to make the adjoint equation and the governing equation identical. This means that the solution to the adjoint equation obtained will be used to update the initial conditions of the original variables via the inverse of this variable transformation. In the case of the linear long wave equation and SH waves, this relationship corresponds to a mere multiplication of the adjoint variables by a factor, but the value is not a uniform constant and varies from place to place in an inhomogeneous medium. The adjoint equations for a three-dimensional perfect elastic body are also of the same type as the original equations of motion and constitutive equations, but the stress equivalent component of the adjoint equations contributes to the real-space stress via the elastic compliance tensor. The stress equivalent component in the adjoint equation contributes to the stress in real space via the elastic compliance tensor. This implies that when seismic waves are back-propagated to the source region, it is not the stress equivalent value obtained but the strain equivalent value obtained by applying it to the elastic compliant tensor that corresponds to the stress variation at the source.