Analysis of the trade-off between hypocentral depth and source time function

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Abstract
Focal depth determination for shallow (≤60 km) intermediate-size (Ms ≈ 6.5 to 7.5) earthquakes continues to be problematic. For these events, the depth phases (i.e., pP and sP) are not distinctly separated in time, and thus synthetic modeling of wave shapes becomes necessary. Unfortunately, the uniqueness of the solution is not easily determined because of the trade-off between various focal parameters. We have investigated the trade-off between depth and source function both theoretically and experimentally. The changes in the Green's function caused by an incorrect assumed depth can almost always be compensated by changes in the source function, producing the same “observed” seismogram. Thus, there is considerable trade-off between depth and the source function. Perturbing the depth from the “true” depth leads to a more “complicated” source function. The “simplest” source function then pinpoints the true depth. We have used two different measures of simplicity, the varimax norm (Vmax) and the half-absolute moment time (T12⁠); both measures successfully characterize the complications that appear in the source functions for incorrect assumed depths. The true depth of an event can be interpreted from the resultant Vmax versus depth and T12 versus depth curves. Thus, we have a technique that determines the depth range of faulting in a systematic quantitative fashion. Our observations from the synthetic study have been applied to an outer-rise event off the coast of central Chile (16 October 1981; Ms = 7.2), to an event in southern Mexico (24 October 1980; Ms = 7.0), and to the Borrego Mountain event in southern California (9 April 1968; Ms = 6.4) resulting in faulting depth ranges of 0 to 20 ± 5 km, 60 to 70 ± 5 km, and 10 to 14 ± 5 km, respectively.