The mixed discrete‐continuous inverse problem: Application to the simultaneous determination of earthquake hypocenters and velocity structure

Abstract

Some inverse problems are characterized by a model consisting of a piecewise continuous function and a set of discrete parameters. For linear problems of this general type, which we call mixed, we show that when the number of data d is greater than the number of parameters p, it is always possible to construct a set of a least d ‐ p equations that are independent of the values of the discrete part of the model. These equations, which we call the annulled data set, can be used to estimate the continuous part of the model. The discrete part of the model can be estimated from a second set of p equations that relate the discrete and continuous parts of the model. The linearization of the nonlinear travel time functional that enter in the hypocenter location problem leads to a mixed inverse problem. The splitting procedure is natural to this problem if the hypocenters are estimated initially by conventional nonlinear least squares by using travel times calculated from some initial estimate of the velocity model. The annulled data are a set of linear combinations of the residuals that are unbiased by that initial location, and as a result, they can be used directly to estimate a perturbation to the velocity model by a Backus‐Gilbert procedure. This makes an iterative algorithm possible that consists of a conventional hypocenter location followed by estimating a perturbation of the velocity model from the annulled data set. The uniqueness of the final velocity model is assessed via the linear resolution analysis of Backus and Gilbert (1968, 1970). We also construct a set of Frechet derivatives that relate perturbations of each hypocenter component to perturbations of the velocity model. These kernels are used to assess the possible error of the hypocenters due to inadequate knowledge of the velocity structure by an application of the generalized prediction approach of Backus (1970a). Good results are obtained when the procedure is applied to a simple synthetic data set.

Three‐dimensional velocity structure of the Kilauea Caldera, Hawaii

“…Examples of the application of the technique can be found in the analysis made by Benz et al, (1996) In order to make the inversion tractable, slowness and hypocenter perturbations are separated (Pavlis and Booker, 1980) and the resulting system of equations is solved using a least square algorithm (Paige and Saunders, 1982). This approach avoids a full matrix inversion, but does not allow a direct assessment of the resolution of the system of equations.…”

Three‐dimensional seismic velocity structure of Mauna Loa and Kilauea volcanoes in Hawaii from local seismic tomography

“…The direct determination of the V p ∕V s structure using S−P times is more robust than simply taking the ratio of the 3-D V p and V s models for body wave tomography [Eberhart-Phillips, 1990]. The simul2000 code uses a combination of parameter separation [Pavlis and Booker, 1980;Spencer and Gubbins, 1980] and damped least squares inversion to solve for model perturbations. The appropriate damping parameters are usually selected by using a data misfit versus model variance trade-off analysis.…”

Earthquake Swarm Recorded by an Ocean Bottom Seismic Array in Southwest Offshore of Taiwan in October, 2005

“…However, this iterative inversion approach does not guarantee convergence. Here we adopt the method of Pavlis and Booker (1980) which is able to simultaneously determine the velocity structure and earthquake hypocenters. We set the inversion of 1-D velocity model as a linear problem consisting of a continuous function and a set of discrete parameters.…”