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Moment
Tensor
& Source Depth Inversion
Program

__Description of technique__

Instant point source can be described by the moment tensor - a symmetric 3x3 matrix . Seismic moment is defined by equation , where is transposed moment tensor , and . Moment tensor of any event can be presented in the form , where matrix is normalised by condition .

We are considering a double couple instant point source (a pure tangential dislocation) at a depth h. Such a source can be given by 5 parameters: double couple depth, its focal mechanism which is characterising by three angles: strike, dip and slip or by two unit vectors (direction of principal tension T and direction of principal compression P) and seismic moment . Four of these parameters we determine by a systematic exploration of the four dimensional parametric space, and the 5-th parameter - solving the problem of minimisation of the misfit between observed and calculated surface wave amplitude spectra for every current combination of all other parameters.

Under assumptions mentioned above the relation between the spectrum of the displacements in any surface wave and the total moment tensor can be expressed by following formula

(1)

i,j = 1,2,3 and the summation convention for repeated subscripts is
used.
in equation (1) is the spectrum of Green function for the chosen model
of medium and wave type (see Levshin, 1985; Bukchin, 1990), -
source location. We assume that the paths from the earthquake source to
seismic stations are relatively simple and are well approximated by
weak
laterally inhomogeneous model (Woodhouse, 1974; Babich et al., 1976).
The
surface wave Green function in this approximation is determined by the
near source and near receiver velocity structure, by the mean phase
velocity
of wave, and by geometrical spreading. The amplitude spectrum ||
defined by formula (1) does not depend on the average phase velocity of
the wave. In such a model the errors in source location do not affect
the
amplitude spectrum (Bukchin, 1990). The average phase velocities of
surface
waves are usually not well known. For this reason as a rule we use only
amplitude spectra of surface waves for determining source parameters
under
consideration. We use observed surface wave phase spectra only for very
long periods.

__Surface wave amplitude spectra inversion__

If all characteristics of the medium are known the representation (1) gives us a system of equations for parameters defined above. Let us consider now a grid in the space of these 4 parameters. Let the models of the media be given. Using formula (1) we can calculate the amplitude spectra of surface waves at the points of observation for every possible combination of values of the varying parameters. Comparison of calculated and observed amplitude spectra give us a residual for every point of observation, every wave and every frequency . Let be any observed value of the spectrum, i = 1,?,N; - corresponding residual of ||. We define the normalised amplitude residual by formula

_{.
}(2)

The optimal values of the parameters that minimiz*e *eamp
we consider as estimates of these parameters. We search them by a
systematic
exploration of the five dimensional parameter space. To characterize
the
degree of resolution of every of these source characteristics we
calculate
partial residual functions. Fixing the value of one of varying
parameters
we put in correspondence to it a minimal value of the residual eamp on
the set of all possible values of the other parameters. In this way we
define one residual function on scalar argument and two residual
functions
on vector argument corresponding to the scalar and two vector varying
parameters: , and .
The value of the parameter for which the corresponding function of the
residual attains its minimum we define as estimate of this parameter.
At
the same time these functions characterize the degree of resolution of
the corresponding parameters. From geometrical point of view these
functions
describe the lower boundaries of projections of the 4-D surface of
functional on
the coordinate planes.

It is well known that the focal mechanism can not be uniquely determined from surface wave amplitude spectra. There are four different focal mechanisms which will radiate this same surface wave amplitude spectra. These four equivalent solutions represent two pairs of mechanisms symmetric with respect to the vertical axis, and within the pair differ from each other by the opposite direction of slip.

To get a unique solution for the focal mechanism we have to use in
the
inversion additional observations. For these purpose we use very long
period
phase spectra of surface waves or polarities of P wave first arrivals.

__Joint inversion of surface wave amplitude
and
phase spectra__

Using formula (1) we can calculate for chosen frequency range the phase spectra of surface waves at the points of observation for every possible combination of values of the varying parameters. Comparison of calculated and observed phase spectra give us a residual for every point of observation, every wave and every frequency . We define the normalised amplitude residual by formula

. (3)

We determine the joint residual by formula

. (4)

Joint inversion of surface wave amplitude spectra and P wave polarities

Calculating radiation pattern of P waves for every current combination of parameters we compare it with observed polarities. The misfit obtained from this comparison we use to calculate a joint residual of surface wave amplitude spectra and polarities of P wave first arrivals. Let be the residual of surface wave amplitude spectra, - the residual of P wave first arrival polarities (the number of wrong polarities divided by the full number of observed polarities), then we determine the joint residual by formula

_{.
(5)}

Before inversion we apply to observed polarities a smoothing
procedure
which we will describe here briefly.

Let us consider a group of observed polarities (+1 for compression
and -1 for dilatation) radiated in directions deviating from any medium
one by a small angle. This group is presented in the inversion
procedure
by one polarity prescribing to this medium direction. If the number of
one of two types of polarities from this group is significantly larger
then the number of opposite polarities, then we prescribe this polarity
to this medium direction. If no one of two polarity types can be
considered
as preferable, then all these polarities will not be used in the
inversion.
To make a decision for any group of n observed polarities we calculate
the sum ,
where n+ is the number of compressions and
is the number of dilatations. We consider one of polarity types as
preferable
if |m| is larger then its standard deviation in the case when +1 and -1
appear randomly with this same probability 0.5. In this case n+ is a
random
value distributed following the binomial low. For its average we
have ,
and for dispersion .
Random value m is a linear function of n+ such that .
So following equations are valid for the average, for the dispersion,
and
for the standard deviation s of value m

, , and.

As a result, if the inequality is valid then we prescribe +1 to the medium direction if , and -1 if .

__References__

V.M. Babich, B.A. Chikachev and T.B. Yanovskaya, 1976. Surface waves
in a vertically inhomogeneous elastic half-space with weak horizontal
inhomogeneity,
*Izv.
Akad. Nauk SSSR, Fizika Zemli, *4, 24-31.

B.G. Bukchin, 1990. Determination of source parameters from surface
waves recordings allowing for uncertainties in the properties of the
medium,
*Izv.
Akad. Nauk SSSR, Fizika Zemli, *25, 723-728.

A.V. Lander, 1989. Frequency-time analysis. In: V.I. Keilis-Borok (Editor), Seismic surface waves in a laterally inhomogeneous earth. Kluwer Academic Publishers Dordrecht, 153-163.

A.L. Levshin, 1985. Effects of lateral inhomogeneity on surface wave
amplitude measurements, *Annles Geophysicae*, 3, 4, 511-518.

J.H. Woodhouse, 1974. Surface waves in the laterally varying
structure.
*Geophys.
J. R. astr. Soc.*, 90, 12, 713-728.