We present a new formulation of the inverse problem of determining the temporal and spatial power moments of the seismic moment rate density distribution, in which its positivity is enforced through a set of linear conditions. To test and demonstrate the method, we apply it to artificial data for the great 1994 deep Bolivian earthquake. We use two different kinds of faulting models to generate the artificial data. One is the Haskell-type of faulting model. The other consists of a collection of a few isolated points releasing moment on a fault, as was proposed in recent studies of this earthquake. The positions of 13 teleseismic stations for which P- and SH-wave data are actually available for this earthquake are used. The numerical experiments illustrate the importance of the positivity constraints without which incorrect solutions are obtained. We also show that the Green functions associated with the problem must be approximated with a low approximation error to obtain reliable solutions. This is achieved by using a more uniform approximation than Taylor's series. We also find that it is necessary to use relatively long-period data first to obtain the low- (0th and 1st) degree moments. Using the insight obtained into the size and duration of the process from the first-degree moments, we can decrease the integration region, substitute these low-degree moments into the problem and use higher-frequency data to find the higher-power moments, so as to obtain more reliable estimates of the spatial and temporal source dimensions. At the higher frequencies, it is necessary to divide the region in which we approximate the Green functions into small pieces and approximate the Green functions separately in each piece to achieve a low approximation error. A derivation showing that the mixed spatio-temporal moments of second degree represent the average speeds of the centroids in the corresponding direction is given.
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