A technique is presented which determines bounds on velocity structure from travel-time residuals. The problem is first linearized and then discretized using a series expansion method. The solutions derived are extremal in that they minimize or maximize a property of the model. For a given data set, the solutions are unique to the linearized problem. Examples are presented in which bounds are placed on the total velocity anomaly and on the extent of the anomalous body. The method also provides a mapping from bounds on the data to bounds on the velocity distribution. Furthermore, the technique is applicable to one-, two-, or three-dimensional velocity distributions.
As an example, the technique is applied to a set of teleseismic relative residuals recorded at Long Valley caldera, California. Two methods are proposed to deal with negative travel-time residuals. Upper bounds and lower bounds are placed on velocity perturbations at depth. The upper bounds indicate that teleseismic residuals cannot adequately constrain the velocity structure in Long Valley caldera. Lower bounds indicate where velocity perturbations must occur. The presence of a region of low velocity in the northwest portion of the Long Valley caldera, at a depth of 15 to 20 km, is supported. Also a minimum width of 18 km is derived for any anomalous body which could give rise to the observations.