The discrete general linear inverse problem reduces to a set of m equations in n unknowns. There is generally no unique solution, but we can find k linear combinations of parameters for which restraints are determined. The parameter combinations are given by the eigenvectors of the coefficient matrix. The number k is determined by the ratio of the standard deviations of the observations to the allowable standard deviations 2. Particular solution. We seek a particular set of parameters P• + AP'• that will satisfy the observations within their variance.3. Resolving power. By using the reparameterization determined in par• 1 we wish to find the resolution of the observations in parameter space. Since we cannot determine the corrections for individual parameters uniquely, our approach to studying resolution is to try to find the smallest groups of parameters for which the average value can be determined. Iniormation distribution. Generally, each particular observation doesno• contribute independent information about the model. We shall determine the distribution of information among the observations. Such knowledge can then be used as a constrain• on data acquisition and smoothing operations. THEORETICAL DISCUSSION In the introduction, I indicated how the general inverse problem can be reduced to the solution of a set of linear simultaneous equations. I shall begin the discussion of the properties of the solution by introducing a matrix notation' Ap • : ß ß ß ß ß ß •b,C t = ---var {AP', } var {Ap'} = vat { AP', } ß ß v•r { AP', } _vat (3) ]c,/oP, ocdoP, ... oc/oP. ß OC'•/OP• ' LOC•/OP• ... Notice that each row of A' corresponds to one particular observation and that each column corresponds to a particular parameter. 254 RALPH A. WIGGINS Example: To help fix ideas, I shall illustrate •he s•eps for constructing the solution to a general inverse problem by considering the specific problem of determining the shear-wave velocity structure of the earth from Rayleigh wave phasevelocity observations. There are two principal types of surface waves generated by earthquakes and explosions: Love waves and Rayleigh waves. The depth of penetration of such waves depends on their period; longer-period waves penetrate deeper than shorter-period waves. In a radially heterogeneous medium, both types of waves are dispersed. The amount of dispersion for Rayleigh waves depends on the shear-wave velocity/•(z), the compressional-wave velocity a(z), and the density p(z) over the range of depths z to which the wave penetrates. The dispersion for Love waves is controlled by only/•(z) and p(z). This example assumes that a(z) and p(z) are known and that we are seeking to determine/•(z). When surface waves travel around the earth several times, various frequencies interfere constructively and destructively to produce discrete modes of free oscillation. Oscillations corresponding to Love-wave interference are called toroidal; oscillations corresponding to Rayleigh-wave interference are called spheroidal. (See Bullen [1965., chapters 5 and 8] or Ga...

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We consider the deconvolution of a suite of teleseismic recordings of the same event in order to separate source and transmission path phenomena. The assumption of source uniformity may restrict the range of azimuths and distances of the seismograms included in the suite. The source shape is estimated by separately averaging the log amplitude spectra and the phase spectra of the recordings. This method of source estimation uses the redundant source information contained in secondary arrivals. The necessary condition for this estimator to resolve the source wavelet is that the travel times of the various secondary arrivals be evenly distributed with respect to the initial arrivals. The subsequent deconvolution of the seismograms is carried out by spectral division with two modifications. The first is the introduction of a minimum allowable source spectral amplitude termed the waterlevel. This parameter constrains the gain of the deconvolution filter in regions where the seismogram has little or no information, and also trades-off arrival time resolution with arrival amplitude resolution. The second modification, designed to increase the time domain resolution of the deconvolution, is the extension of the frequency range of the transmission path impulse response spectrum beyond the optimal passband (the passband of the seismograms). The justification for the extension lies in the fact that the impulse response of the transmission path is itself a series of impulses which means its spectrum is not band-limited. Thus, the impulse response is best represented by a continuous spectrum rather than one which is set to zero outside the optimal passband. This continuity is achieved by a recursive application of a unit-step prediction operator determined by Burg's maximum entropy algorithm. The envelopes of the deconvolution are used to detect the presence of phase shifted arrivals. IntroductionThis paper examines the problem of the deconvolution of source functions from teleseismic recordings. The first step in any deconvolution process is the estimation of the source time function. The next section of this paper deals with this problem.A standard technique in exploration seismology is to decompose the sowce from the seismogram autocorrelation function. This method assumes that the source is minimum phase and that the impulse sequence behaves like white noise (Robinson 1967). Since neither of these assumptions appeared to be generally valid for teleseismic recordings, the technique was not used. Source estimation by homomorphic transformation has also been suggested (Ulrych 1971), and this method is discussed in some detail. However, the low quefrency assumption of this method was found to not be generally valid, and there is also a problem of phase instabilities with the homomorphic transform itself.Both of the above methods attempt to estimate the source from a single seismogram. We decided that on the basis of a single recording of an event it would be difficult, if not impossible, to devise a general method of estimati...

Body‐wave observations from nuclear events at the Nevada Test Site and several earthquakes near the western edge of the United States have been used to construct a model of the upper mantle along profiles extending toward the Great Lakes. The Cagniard‐de Hoop technique for computing synthetic seismograms for laterally homogeneous earth models was used to fit both the amplitudes and travel times of the observations. The model obtained exhibits a prominent low‐velocity zone, sharp discontinuities of velocity at about 430 and 660 km, and a discontinuity in the slope of the velocity at about 550 km.

Seismometer operation for 21 days at Tranquillity Base revealed, among strong signals produced by the Apollo 11 lunar module descent stage, a small proportion of probable natural seismic signals. The latter are long-duration, emergent oscillations which lack the discrete phases and coherence of earthquake signals. From similarity with the impact signal of the Apollo 12 ascent stage, they are thought to be produced by meteoroid impacts or shallow moonquakes. This signal character may imply transmission with high Q and intense wave scattering, conditions which are mutually exclusive on earth. Natural background noise is very much smaller than on earth, and lunar tectonism may be very low.