Wavefield decomposition using ML-probabilities in modelling single

Geophysical Journal (1988) 93, 197-213

Wavefield decomposition using ML-probabilities in modelling single-site 3-component records

A. Christoffersson," E. S. Husebye'f and S. F. Ingatel' * Department of Statistics and Department of Geophysics, University of Uppsala, P. 0. Box 513, S-75120 Uppsala, Sweden. .t NTNFINORSAR, Kjeller, and Department of Geology, University of Oslo, Norway. $ MIT, ERL, Cambridge, Massachusetts, USA.

Accepted 1987 September 15. Received 1987 August 24; in original form 1986 March 11.

SUMMARY This paper presents a new approach to the analysis of three-component digital seismograms. Earlier approaches used techniques such as Principal Components to estimate particle- motion using models of P and S waves. In this paper the Maximum-Likelihood (ML) estimator is preferred because this allows the use of X2-probabilities to test whether energy of a specific wave type (P, S, Love or Rayleigh) is present. In addition, this analysis allows the joint estimation of azimuth of approach and in cases of P- and SV-waves also apparent angle of incidence (and, hence, information on apparent velocity). For a single three-component seismogram, the covariance matrix provides only six independent observations, thus restricting analysis to rather simple wave models. The technique works satisfactorily for P-waves whereas shear and surface-wave models sometimes prove cumbersome to handle due to correlation between radial and transverse components reflecting complex propagation characteristics in inhomogeneous media. This technique has been tested on synthetic data and in such cases works perfectly for all wave types.

An important aspect of this work has been the visual display of the probability and velocity information as functions of time and azimuth. Displaying the data in this form provides information on the ray path in a manner similar to analysis performed by seismic arrays.

Practical examples on a variety of siesmic data are given to illustrate the viability of the technique: the data cover a broad spectrum of frequencies and applications from broad-band teleseismic data (1 Hz), regional seismic data (10-40 Hz), seismic profiling data (125 Hz) and VSP (500 Hz) recordings.

Key words: Wavefield decomposition, 3-component records, seismic phase identification, signal parameterization, polarized waves

1 INTRODUCTION

Polarization properties of scalar and vector fields have been topics of extensive research in areas ranging from electromagnetics (optics and radio transmission) to physical oceanography and seismology. Although the 3-component technique given in this paper can be generalized to any n-component recording system of polarized phenomenon, the emphasis in this paper is seismological.

Many seismic-data acquisition systems include 3- component instrumentation for the very simple reason that the seismic wavefield comprises both vertical and horizontal ground motions. Seismologists have for many years attempted to exploit the information potential of three- component records for modelling wave-propagation, velocity, structural and source parameters. These efforts have generally been successful when applied to the low frequency

'Now at: Nuclear Monitoring Centre, Bureau of Mineral Resources, Canberra, Australia.

part of the wavefield (less than 0.2Hz). Much work has been invested in extracting similar information in the high frequency band, say of 1-100 Hz, but these efforts have been relatively less successful (c.f. Shimshoni & Smith 1964; Flinn 1965; Montalbetti & Kanasewich 1970; Husebye, Christoffersson & Frasier 1975; Esmersoy, Cormier & Toksoz 1985; Gal'perin 1983). Two reasons for this difficulty may be cited:

(1) High-frequency records are complex due to phase shifts, mode conversions, multipathing and scattering, thus contaminating seismic wavefield polarization properties (e.g. see Booth & Crampin 1986).

(2) Analysis results are often difficult to interpret and fail to produce wavefield parameters such as wave type, slowness, arrival time and amplitude. For example, particle motion plots are inconvenient and difficult to quantify, while polarization filter operators are often non-linear and do little to simplify the original recordings.

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198 A . Christoffersson, E. S . Husebye and S. F. Zngate

In other words, 3-component seismogram analysis techniques have not yet become a viable alternative to those often used for arrays of vertical-component siesmometers such as frequency-wavenumber (f - k) processing (e.g. see Lacoss, Kelly & Toksoz 1969; Capon 1973). Arrays do have disadvantages, namely that they utilize only amplitude and timing information, and that assumptions implied in array beamforming techniques (e.g. that wavefronts are planar and coherent across the array) do not always hold.

The topic of this paper is a new technique for extracting wavefield parameters using a single 3-component seismometer. Practical examples are given to demonstrate that this technique represents a valuable tool for different kinds of seismogram analysis.

2 POLARIZATION CHARACTERISTICS OF SEISMIC WAVES

In a laterally homogeneous medium, the seismic wavefield can be decomposed into three components of particle motion, namely, P , SV and SH. In the horizontal plane, SH is the particle-motion component of the shear wave and can be derived from the scalar wave-equation. P is the compressional wave with the particle motion along the direction of progagation. SV is the other component of the shear wave and represents particle motion orthogonal to P and SH but not necessarily uncorrelated with SH. P and SV interact at the surface so that the apparent angle of incidence differs slightly from the true angle of incidence (see Aki & Richards 1980, p. 190). For SV incidence, however, the SV-P interaction may be complex due to excitation of shear-coupled waves and SP diffracted waves.

In such cases the angle of incidence for SV is not well defined, particle motion is roughly elliptical, and this naturally also applies to the corresponding apparent velocity parameter (Booth & Crampin 1985; Langston & Baag 1985). The three wave types are 1-D so that by numerically rotating the digitized seismometer output (or even physically rotating the seismometer), particle motion will lie along a single axis.

In the case of horizontally layered media, surface waves exhibit two modes of propagation; dispersive Rayleigh (R) waves result from P and SV wave interaction at the surface and exhibit elliptical polarization, i.e. particle motion on the vertical and radial axes (e.g. see Boore & Toksoz 1969). Dispersive Love (L) waves are similar to SH and thus exhibit only transverse particle motion. A scheme for representing the particle-motion displacements in the frequency domain for the wave-types mentioned above are given in Table 1. These models may seem overly simplified, but, on the other hand, more complex models cannot easily be accommodated in single-site three-component analysis because the number of independent observational parameters available are six for power and six for phase angles. For example shear-wave propagation in inhomogeneous and anisotropic media is difficult to model synthetically due to W I S H wave splitting, the mentioned non-linear SV/ P conversions and so forth (e.g. see Booth & Crampin 1985; Langston & Baag 1985). At this stage of development, the simple particle-motion models in Table 1 do not accommodate such wave-propagation phenomena, but this

Table 1. Idealized particle motion properties summarized for principal types of seismic waves at a given site and a given frequency. For body waves and Love waves angles w and cp represent the direction of propagation from the vertical (angle of incidence) and x-axis (azimuth), respectively. For Rayleigh waves =tan-’ e, where e is the ellipicity. Notice, the angle of incidence for body waves as observed at the surface differ from the ‘true’ angle of incidence due to interference between upgoing and downgoing (reflected) waves (e.g. see Aki & Richards 1981, p. 180) which in turn depends on crustal velocities. This effect does not impair the particle motion rectilinearity and thus the validity of our model. The obvious reason for not attempting to estimate ‘true’ angle of incidence directly is that in any case corrections must be introduced to account for local velocity anomalies and the wavelength dependent sampling of crustal velocity. This is most conveniently done by using master events for calibrating average velocity as demonstrated in the result section.

P waves

r(w) = { -sin sin cp] s ( w ) = hp * s(w)

SH and Love waves

-sin I# cos cp

cos * -sin cp

s ( o ) - ~ ~ ~ . s ( o )

SV waves

s (w)=hsv . s (w)

Rayleigh waves

-i sin 1/, cos cp

is not considered to be a severe drawback as simplistic assumptions are implicit in other forms of wavefield analysis such as array processing and other schemes for three- component processing (e.g. see Ingate, Husebye & Christoffersson 1985; Esmersoy et al. 1985). It is conceivable that by using the frequency-domain models in Table 1 at discrete values of frequency unstable results may be produced. In this paper, the three-component technique operates in the time domain with only six independent observational parameters. Regarding the robustness of these particle motion models, practical tests have been performed on synthetic seismograms which will be examined in a later section.

3 THEORY

In this section the particle motion models necessary for processing 3-component seismograms are developed.

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WaveEeld decomposition using ML-probabilities 199

so that y , is the radial, yz the transverse and y3 the (original) vertical component. For P, S, L and R waves, A and @ become:

P: A p = { Z } @ = ($11) sgn # s@

3.1 Modelling the observations

The technique for analysing 3-component data is based on the following model;

y ( t ) = 4 t ) + 4 t ) ,

Y( t ) = [Yl(t), Y2(t), Y 3 ( W

(1) where

is a vector containing the observed 3-component recordings at time t , and * denotes vector transpose.

z ( t ) = [z l ( t ) , . . . 9 Zk( t ) ]*

is a k-dimensional representation of the signal at time t. For the case of a P-wave k = 1, while for an S-wave with SV and SH particle motion, k = 2. Define

as a matrix of unknown constants relating the signal to the observed data, and

E ( t ) = [El(t), E 2 ( f ) 9

is the residual (noise) vector. Four basic assumptions are made:

(1) z ( t ) and ~ ( t ) are uncorrelated. (2) ~ ( t ) has expectation zero, i.e. E { E ( t ) } = 0 (3) The components of z ( t ) are linearly independent, i.e.

(4) All moments up to at least second-order exist.

These basic assumptions imply that the zero-lag second-order moments of y ( t ) can be written as

the signal is k-dimensional.

X ( t ) = A@(t)A* + Y(t),

X ( t ) = E { y ( t ) y ( t ) * ) , Y(t) = E { E ( t ) E ( t ) * } .

(2)

where

@(t) = E { z ( t ) z ( t ) * ) ,

Averaging equation (2) over a time window T l < t < T, yields the following second-order structure

X = A@A* + Y,

where (3)

1 f i 1 % 1 % X = - X ( t ) dt, @ = Jfi @ ( t ) dt, Y = T Y(t) dt.

T I ,

X is the expected 3 x 3 symmetric covariance matrix of recordings, containing six different elements, @ is the signal covariance matrix and Y is the noise covariance matrix and

As an aside, note that the form of equation (3) is similar to many of those arising from modelling experiments in the social sciences: solutions are sought using ordinary factor-analysis techniques, which are described in detail in most text books on multivariate statistics.

The structure of the A and @ matrices depend on the particle motion of the incoming wave. For simplicity, let NS, EW and Z seismometer components be ordered and rotated

T = T2- Tl.

The diagonal elements of the @ matrices are powers in the corresponding dimensions spanned by the signal. For the S-wave model, the first column of A corresponds to the SH-component and the second to the SV-component. The particle motions are orthogonal and are here modelled as being uncorrelated. Rayleigh waves on the other hand are 1-D but with a 90" phase shift between the radial and the vertical components. The phase shift entails that the Rayleigh signal will be 2-D with the radial and vertical components orthogonal. One might also consider combina- tions of these models in order to simulate interfering phases, etc. However, as will be shown below, given single-site three-component observations, this may lead to problems that are partly insoluble. For the body waves P and SV, the angle of incidence can be computed directly from the radial and vertical amplitudes, e.g. A1 and A3 or A12 and A32. This can then be converted to apparent velocity using standard formulae for given crustal velocities (Bullen 1963; Aki & Richards 1980).

3.2 Identification

Before discussing how to estimate the unknown parameters in each of the models, the problem of structure identification should be adressed. In other words, are the unknown model parameters per se identified properly, or can different sets of parameters give rise to the same &matrix? In order to identify the structure there can in general be at most six unknown parameters in A, 0 and Y. It follows that six unknown parameters and only six degrees of freedom, in general, implies perfect fit to data and no means to obtain a test for the validity of the assumed model structure. One possible way to overcome this might be to use an array of 3-component stations to increase the number of degrees of freedom. However, in such cases additional problems are likely to arise, namely those associated with poor phase and amplitude correlation between stations (cf. Ingate et al. 1985), and a requirement for extensive calculations in slowness space to optimize the stacking delays.


200 A. Chrzktoffersson, E. S . Husebye and S. F. Zngate

3.3 Estimation

One way to obtain an estimator for the unknown parameters in equation (3) is to base it on the observed second-order zero-lag moments within a short time-window

(4)

One might also consider a simultaneous estimation of the unknown signal and model parameters. As this gives rise to problems concerning existence and uniqueness (the number of unknowns increases with the number of observations) it seems best to base the estimation on the observed moments S as given in equation (4).

There are several possible estimators that could be used. Most are based on some fitting-function, i.e. minimizing some function of the difference between observed S and the theoretical Z. Some commonly used estimators are:

(1) Maximum likelihood (ML) The ML estimator is based on Gaussian assumptions of noise and minimizes

F = log IZI - tr (SZ-’) -log IS1 - q, ( 5 ) where q is the dimension of the observed data ( q = 3 for three-component data) and tr = trace of the matrix.

This estimator minimizes (2 ) Unrestricted Least Squares (ULS)

F = tr (S - Z)z (6) i.e. the sum of squares of all elements in S - Z.

The GLS-estimator minimizes (3) Generalized Least Squares (GLS)

F = tr (I - S-’Z)’ (7)

(4) Principal Components (PC) The PC-estimator can be obtained by maximizing

F = tr (A*SA) (8) subject to normalizing conditions on A.

3.4 Properties of the estimators

The PC-estimator was suggested by Flinn (1965) and corresponds to an eigenvalue decomposition of the S-matrix. It differs from the others in that it focuses heavily on the diagonal elements of the S-matrix, i.e. powers, whereas other estimators put weight on the off-diagonal elements, i.e. on the structure of the Z-matrix. ML and GLS are asymptotically equivalent and under Gaussian assumptions of white noise it is possible to obtain tests for model fit to observed data. They do require that the observed matrix is positive definite whereas ULS works even with non-Grammain S. ULS may thus be favoured because P-waves with good signal-to-noise rations (SNR) exhibit almost singular S-matrices. However, the problem with almost singular S-matrices may be overcome by attenuating the transverse component, i.e. by adding white noise to the diagonal elements of the S-matrix. In this study the ML-estimator is used mainly because it is possible to test

the validity of the models . These tests are of course strictly valid only under the Gaussian assumptions of white noise. There is an additional problem: The ML-estimator is derived under the assumption of stochastic signal, which is not necessarily valid in seismology. If the signal is stochastic, the distribution of S is the central Wishart which leads to the above ML-estimator. On the other hand, a non-stochastic signal obeys a non-central Wishart distribution. For the latter case, the ML-estimator involves Bessel functions of the eigenvalues of @. Furthermore, the equations defining the estimator are in general extremely difficult to solve. However, Anderson & Rubin (195.6) have shown that for the factor analysis model, i.e. equation (3), the ML-estimators based on central and non-central Wishart-distributions are asymptotically equivalent, and thus the ML-estimator based on equation (5) can be justified.

3.5 Model test

Under the assumption of Gaussian white noise, a model test based on the ML-estimator can be obtained. Let F denote the minimum in equation (5). From the general properties of ML-estimators it follows that ( N - l)F is asymptotically distributed as x’ with degrees of freedom equal to six minus the unknown parameters and N = (Tz - TI + 1). To illustrate how the test is carried out and how a probability measure for different wave types can be constructed, consider a P-wave model where the noise is independent and having the same power on all three components. The following structure is obtained for the P-wave:

Zp = ApOPA,* + yI (9)

Z, = yI. (10)

whereas for noise the structure is

I is the identity matrix and y is a positive constant. Let Fp and F, denote the minimum F for the two models.

It then follows that (N - 1)F, is x’ distributed with five degrees of freedom provided the structure in equation (10) is valid, i.e. the recorded data is just noise. (N - l)Fp on the other hand is X2-distributed with three degrees of freedom if either of the structures in equations (9) or (10) is valid. Furthermore, the difference (N - 1) . (FN - Fp) is xz with two degrees of freedom if the structure in equation (10) is valid. For data generated by a P-wave, the noise model will have poor fit to data and tend to give large X2-values. The drop in x2, (N-l).(F,-Fp) will also be large, whereas the P-model will tend to have small Xz-values. Defining

P ( P ) = Probability of ~ ’ ( 3 ) > (N - l)Fp P ( D ) = Probability of ~ ’ ( 2 ) > (N - l)(FN - F p )

yields a measure of the ‘probability’ that the recorded data are generated by a P-wave by using

P ( P ) * P(PE > 0) = P ( P ) * (1 - P ( D ) ) (11) which can be phrased as ‘the probability of a P-wave multiplied with the probability of the P-wave energy being larger than zero’. The probabilities may be computed as a function of time and azimuth, and plotted as shown in Fig.

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Wavefield decomposition using ML-probabilities 201

period, and overlap by a factor between 33% and 50% as it is moved along the recorded traces. Increments in azimuth are usually in the range of 1" to 5" and reflect the resolution available in the observations. The probabilities are mostly contoured at levels between 0.5 and 0.9. Values below 0.5 are considered insignificant, i.e. reasonable polarized wavelets usually give probabilities well above 0.5. For P and SV the apparent velocity is plotted as a function of time and azimuth subject to the condition that the corresponding probability is larger than a specified lower limit, usually around 0.5. A major problem in analysing seismograms is that of recognizing phases as such and then to correctly associate the phase with a specific path, e.g. P, PP, PcP, etc. In this 3-component analysis scheme, phase association is a relatively simple matter, as illustrated in Figs 1-6. As a first step towards phase association, the particle motion model used explicitly identifies the type of phase arrivals in the seismogram. An exception here is the ambiguity of 180" in apparent azimuth of P and SV waves, that is a P-model analysis of SV would give P-motion azimuth shifted by 180". In addition, the probability contours may be replaced by appropripte velocity estimates. Thirdly, the maximum probability estimated at each time-step can, provide it is above a pre-defined level, be used as a simple weight function for the observed data. In this way we obtain a set of 'probability filtered' traces where the arriving main phases are enhanced. Another simple weighting scheme is to assign the weight 1 to time windows where the apparent velocity is within a specified range and zero otherwise. The latter is used in the figures displaying velocity contours. A further refinement is to combine the probability and velocity estimates so that only waves of a specific type and velocity range are passed within a specified time window. In this way, an effective form of velocity filtering is achieved.

Occasionally, problems in the form of 'false alarms' arise. These false alarms may be seen prior to the main signal arrival. It is important to be aware of this problem, as the 3-component technique may provide interpretable information even for SNRs as low as one, and the possibilities of having weak seismically-generated P-waves in the noise cannot be excluded (the so-called whispering- mantle effect, c.f. Ingate et al. 1985). For the case of NORESS data these were initially considered to be noise detections and thus reflect a certain instability in the method. However, more extended analysis has demonstrated that these triggers to a large extent stem from cultural-noise sources like hydroelectric power plants, saw mills, nearby cities, etc. A general feature of these pre- signal detections is their crustal propagation velocities. A convenient way to eliminate them is to run an STA/LTA detector using a very low threshold value, say 1.5 combined with the above mentioned velocity filter which is also demonstrated in the result section below. For comparison, detector threshold values in array-operations are usually around 4.0 for automated event detection.

1. The probability measures for the other wave-type models can be derived in a similar manner.

For P- and S-waves the assumption of equal noise power on the different components may be relaxed. Apart from making the approach more general, this also simplifies the computations. With equal noise power, minimizing the function F has to be done iteratively, while under the relaxed conditions, the minimum of F and the corresponding probability can be obtained directly. However, in order to estimate velocity, additional restrictions must be imposed; one reasonable set is to assume that the noise levels are the same on the radial and vertical components. The probability measures derived here are rather robust regarding the assumption of Gaussian noise and that of zero autocorrelation. In seismic applications the Gaussian assumption is often reasonable while the autocorrelation may be substantial. Perhaps the best way to cope with the high autocorrelations is to avoid narrow bandpass filters which has a parallel in f - k analysis of array data.

4 PRACTICAL CONSIDERATION A N D APPLICATIONS

In conventional 3-component seismogram interpretation, the analyst is often faced with displays of frames of particle motion trajectories on EW-NS and/or Z-RAD- plane projections or even stereographic projections, (e.g. see Buchbinder 1985; Gal'perin 1983). In other cases where PC analysis techniques are used, a rectilinearity measure is defined on the basis of axis ratios, and this in turn can be used for non-linear filtering operations: For example, if P-wave motion is to be enhanced, the observed traces are multiplied with (1 - AJA,); where A, and A3 are the major and minor axes (eigenvalues) of the particle motion ellipsoid computed over a sliding time-window.

As mentioned earlier apparent velocity may be calculated from angle of incidence via standard formulas for given known crustal velocities at the site. The estimates of apparent velocity depend rather critically on the assumed crustal velocities and on effects of anomalies near to the reciever. Because the angle of incidence depends both on PIS interactions at the surface and on wavelength and hence frequency, crustal P-velocities are assumed to range from 5 km s-' for 3-8 Hz to 7.5 km sK1 for frequencies lower than 0.04 Hz in order to ensure approximately correct velocity estimates. Note that velocity estimates are restricted to body waves, in practice only P-waves. The reason is, as mentioned, that SV particle motion often is complex (non-linear) and the corresponding wave propagation effects are not incorporated in the models under consideration. Hence, if such non-linearities are substantial, the approach may not be able to identify the SV-phase properly and naturally the corresponding velocity estimate will not be correct. Anyhow, velocity biases due to structural anomalies near to the receiver can be compensated for in a way similar to that used for arrays, i.e. by calibrating the station using a set of master events.

For the examples given in this study, the results are given in terms of probabilities of a specific wave occurrence, ranging from zero to unity. The probabilities are calculated for narrow windows of time and azimuth. Typically, the window length (T' - TI + 1) is 1-2 cycles of dominant signal

5 RESULTS

The 3-component technique has been extensively tested on many types of seismological data, ranging from the IRIS prototype broad-band station recordings at Harvard (HRV),


202

intermediate band, short and ultra-short period recordings from NORESS (Norway), Uppsala short-period and broad-band records, Swedish (EUGENO) seismic profiling records and VSP recordings from a MIT/CCG borehole in Michigan. For these data, the sampling rates range from 1 Hz to 500 Hz. In all cases the technique appears to work satisfactorily for P-waves of reasonable SNR. Also, the most stable results are obtained for analyses at relatively low frequencies; Ruud et at. (1988) have shown that estimates of azimuth are accurate to within 2" using 20 s period P-waves. Shear and surface waves have sometimes proved difficult to interpret, due to their interaction with earth heterogeneities and/or aniostropy. For cases with no or very little SH motion in the records, the technique works well. Such situations can be generated artificially by zeroing the transverse component which essentially amounts to making the data 2-D; however, the penalty is loss of resolution in azimuth. More complex particle motion models for S and Rayleigh waves are now being contemplated.

Since this paper focuses upon the methodological aspect of three-component analysis a few practical examples are presented to demonstrate the general applicability of the technique. The first example is tied to analysis of synthetic seismograms for an upper crust dip-slip dislocation source recorded at a distance of 300km. The synthetics were generated using the discrete wave-number approach (Bouchon & Aki 1977; Bouchon 1981) and for this rather idealized case our 3-component analysing technique works perfectly for all types of wave motion. Naturally, this implies that the particle motion models used, which, besides, are similar to those used by others like Esmersoy et aI. (1985), are tenable (as will be discussed below). This and other examples comprise various types of seismic recordings, namely;

(1) Synthetic records: Vertical dip-slip source, distance

(2) NORESS broad-band records of the large Mexico

(3) NORESS short-period records of a Hindu Kush

(4) Uppsala broad-band records of a presumed under-

( 5 ) EUGENO-S wide-angle refraction profiling records

(6) Michigan borehole recording of vibrator, i.e. VSP-

A. Chr&toffersson, E. S . Husebye and S. F. Ingate

300 km (Fig. 1)

earthquake, 1985 Sep 25 (Fig. 2)

earthquake, 1985 Sep 4 (Fig. 3)

ground nuclear explosions in E. Kazakh (Fig. 4)

(Fig. 5 )

seismogram (Fig. 6)

Details of these event recordingsldata are given in Table 2. The figures show original traces together with the probability filtered seismograms and the corresponding Xz-probabilities for the presence of a specific wave-type as a function of time and azimuth. For some of the P-waves the corresponding apparent velocities and velocity-filtered traces are also plotted. Selected phases are marked with crosses and the associated azimuth, time, probability and apparent velocities are given in some of the figures. The seismograms and probabilities/velocities are plotted against the same time-scale so that identification of the phases is made easier. The vertical component is indicated by Z, the East-West by E and the North-South by N. When the horizontal components are rotated, the rotation angle (ROT) is

Table 2. Hypocentre and signal processing parameters for events analysed in this study. Earthquake and underground explosion parameters from USGS PDE.

(1) DT300D: Synthetic seismogram based on discrete wavenumber approach. Vertical dip-slip source. A = 300 km, h = 15 km. Two layered crust with each of the layers 20 km. Corresponding velocities are 6.15 and 6.90 km s-l. Velocity below model is 8.15 km s-' Signal processing parameters: half window length (WLS) = 0.40 s; updating interval (DS) = 0.20 s; smoothing parameter (AV) = 0.10; azimuth interval (AL) = 1.0" sampling rate (SRATE) = 10 Hz

(2) NLO: NORESS broadband recording of the Mexican earthquake 1985 Sep 25 origin time 01 h 37m 13.8s GMT, 17.28"N, 101.67"W, h = 33 km, mb = 6.3, A = 85.2", 0 = 298.6'. Signal processing parameters: WLS = 0.25 m; DS = 0.10 m; AV = 0.10; AL = 0.5"; SRATE = 60 samples/min = 1 Hz NORESS short-period (site C7) recording of a Hindu Kush event 1985 Sep 4, origin time 08h 32m 25.8s GMT, 36.23"N, 71.02"E, h=66km, mb=4.9, A = 44.3", 8 = 95.6'. Signal processing parameters: WLS = 1.0 s; DS = 0.5 S; AV = 0.10; AL = 2.0" SRATE=4OHZ

(4) UA: Uppsala broadland recording of two presumed nuclear explosions in E. Kazakh, USSR, 1985 Jun 15, origin time 00 h 57 m 00.7 s GMT, 49.89"N, 78.89 E, h=0 , mb=6.0, A=35.2", 0=78.4", and 30 June 1985, origin time 02 h 39m 02.7s GMT, 49.96"N, 78.70°E, h = 0, mb = 6.0, A = 35.2", 8 = 78.4". Signal processing parameters (same for both events): WLS = 0.30 s: DS = 0.15; AV = 0.10; AL = 1.0 deg;

(3) NSO:

SRATE = 95.54 Hz I

(5'1 EUG: Refraction orofile 4. shot 7. EUGENO-S. 1984 Jun \ I ~

26, E-W piofile across S. Sweden, A = 205 km (Huh et al. 1986). Signal processing parameters: WLS = 0.10 s; DS = 0.05 s; AV = 0.20; AL = 2.0"; SRATE = 125 Hz MIT/CCG Group Shoot VSP in Michigan, Autumn 1983. Vibrator offset was 600111, tool depth in borehole was 1440 m. The tool is level with and offset by approximately l00m from a Silurian reef. Signal processing parameters: WLS = 0.05 s; DS = 0.02 s; AV = 0.20; AL = 5.0"; SRATE = 500 Hz

(6) B159:

displayed and the radial component is denoted by R, and the transverse by T. The following comments apply to the results of the analysis;

(1) Synthetic seismogram analysis (Fig. 1) The synthetics for an epicentre distance of 300 km are rather complex despite the use of a simple two-layer crustal model (details in Fig. 1 caption). With the dip-slip source located within the upper crust at 15 km depth, a number of pP-type phases are generated but the sampling rate is sufficient to permit proper phase identification through estimated apparent velocities which are within f O . l km s-l of the true ones. Now, in the synthetics the weak P,-phase arrives firstly on correct azimuth and is followed by P* (or Pb) and Pg as evident from the derived velocities in Fig. l(b). However, the first 25s of the wave train is dominated by wavelets with lower crustal velocities like pP*, sP*, PmP and similar reverberations in the source area. Using ray tracing an attempt was made to identify specific wavelets but this was not successful due to time overlap of phases of the above kind. S-wave motions are relatively weak (Fig. lc) although Sn could be identified. The SmS phase on basis of


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Figure 1. DT300D: 3-component analysis of SYNTHETIC seismogram. (a) P-wave probabilities. Original (solid) and filtered (dotted) records; Lower probability (SYNTZ) for filtering/weighting = 0.50; lower (CUTL) and upper (CUTU) velocity passband of 6.0 km s-' and 25 km s-'. (b) P-wave velocities. Original (solid) and filtered (dotted) records; Lower probability (PROBLIM) for computing contours = 0.50; lower (CUTL) and upper (CUTU) velocities for filtering/weighting 6.0 km s-' and 10.0 km s-'. (c) SV-wave probabilities. Original (solid) and filtered (dotted) records; Lower probability (SYNTZ) for filtering/weighting = 0.50; lower (CUTL) and upper (CUTU) velocity passband of 3.5 km s-' and 15.0 km s-'. (d) R-wave probabilities. Original (solid) and filtered (dotted) records; Lower probability (SYNTZ) for filtering/weighting = 0.50; lower (CUTL) and upper (CUTU) velocity passband of 0.0 km s-' and 0.0 km s-'. (Velocity is not defined for the R-model). For further details on event, processing and extracted signal parameters, see Table 2 and result section.

203 Downloaded from https://academic.oup.com/gji/article-abstract/93/2/197/617478by gueston 12 April 2018

204 A. Christofersson, E. S. Husebye and S. F. Ingate

SYNTZ- 0 . 5 0

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conversion is weak. A puzzling feature in the P-phase analysis is that the apparent velocity varies from 22 to 26 km s-l, equivalent to 1" change in angle of incidence, for the first few cycles, in contrast to about 10" in azimuth. Physically, this may be explained in terms of lateral velocity variations, say a curved lithosphere/asthenosphere transition zone. The same phenomena are commonly observed for short-period P-waves and other types of body-wave phases. For long periods, azimuth estimates are often within f 2 " and this combined with distance estimates derived from differential travel-times like dT(P - P P ) permits event locations within 1-2" as demonstrated by Ruud et al. (1988). Assuming a crustal velocity of 7.5 km sK1 and using only information from the P-phase, the error in azimuth is 2.6" while the error in distance is just 1.1"

(3) Short-period teleseismic P-waves (Fig. 3) Short-period P-waves at teleseismic distances are 'coloured' by small-scale source and receiver heterogeneities with the effect that both azimuth and velocity estimates are less precise than those obtainable from analysis of broad-band records. For this event there are two peaks in the first part of the P wave (Fig. 3a) separated by 0.5 s. The earlier peak is 4" off true azimuth and about 1.5" off in distance while the later (having larger probability value) is on the correct azimuth but 4" off in distance.

traveltime calculations is clear but its particle motion is elliptic and thus typical for Rayleigh waves which, in addition, dominate the L, wave train (Fig. Id). As is well known, SV-particle motion may be linear or elliptic depending on the angle of incidence at the reflector. Notice that there is no triggering overlap in time between P, SV, and Rayleigh-type wave motions. In short, even for simple crustal models the synthetic seismograms are complicated due to interfering phases of various kinds and this testifies towards the complexity of conventional seismogram analysis for regional events. The significance of synthetic seismogram analysis (just one example shown here) is that the P, S, R and L particle motion models introduced in sections 2 & 3 work satisfactorily for conventional (homogeneous) Earth structure models. Even slightly inhomogeneous models give P-wave motions which are predominantly linear as demonstrated by Massinon & Mechler (1986).

(2) Broad-band earthquake recorah (Fig. 2 ) At low P-wave frequencies (f -0.04 Hz) Earth structures appear to be rather homogeneous as demonstrated in this example (Fig. 2a, b) where surface multiple reflections of P-waves are clearly observable. Since the P signal-to-noise ratio is high, 'triggering' starts as soon as the sliding time-window moves into the signal. There is no shear motion prior to the S arrival (Fig. 2c) implying that P-to-S

T I M E 1 . 6 2 k Z I l l 301.15

F R O B 1.00 NJELU F 2 . 4 4

R O T 301.15

1 .00 T I l l E ( M ) 19.00

Figure 2. NLO: 3-component analysis of NORESS broadband recording of the Mexican earthquake, 1985 Sep 21. (a) P - W m probabilities. Original (solid) and filtered (dotted) records; Lower probability (SYNTZ) for filtering/weighting = 0.50; lower ( C U n ) and upper (CUTU) velocity passband of 7.5 km s-' and 40.0 km s-'. (b) P-wave velocities. Original (solid) and filtered (dotted) records; Lower probability (PROBLIM) for computing contours = 0.50; lower (CUTL) and upper (CUTU) velocities for filtering/weighting 7.5 km s - ~ and 40.0 km s-'. (c) SV-wave probabilities. Original (solid) and filtered (dotted) records; Lower probability (SYNTZ) for filtering/weighting = 0.50; lower (CUTL) and upper (CUTU) velocity passband of 4.4 km s-' and 25.0 km s-'. For further details on event, processing and extracted signal parameters, see Table 2 and result section.


206 A. Christofersson, E. S . Husebye and S. F. Zngate

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Figure 3. NSO: 3-component analysis of NORESS recording of Hindu Kush event. (a) P-wave probabilities. Original (solid) and filtered (dotted) records; Lower probability (SYNTZ) for filtering/weighting = 0.50; lower (CUTL) and upper (CUTU) velocity passband of 7.8 km s-' and 25.0 km s-'. (b) P-wave velocities. Original (solid) and filtered (dotted) records; Lower probability (PROBLIM) for computing contours = 0.50; lower (CUTL) and upper (CUTU) velocities for filtering/weighting 7.8 km s-l and 25.0 km s-'. For further details on event, processing and extracted signal parameters, see Table 2 and result section.


208 A. Christoffersson, E. S . Husebye and S. F. Ingate

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Figure 4. UA: 3-component analysis of Uppsala broadband recording of two presumed nuclear explosions in E. Kazakh, USSR, on 1985 June 15 and 1985 June 30. (a) P-wave probabilities for the June 15 event. Original (solid) and filtered (dotted) records; Lower probability (SYNTZ) for filtering/weighting = 0.50; lower (CUTL) and upper (CUTU) velocity passband of 6.0 km sC1 and 25.0 km s-'. (b) P-wave probabilities for the June 30 event. Original (solid) and filtered (dotted) records; Lower probability (SYNTZ) for filtering/weighting = 0.50; lower (CUTL) and upper (CUTU) velocity passband of 6.0 km s-' and 25.0 km s-'. For further details on event, processing and extracted signal parameters, see Table 2 and result section.


Wavefield decomposition wing ML-probabilities 209

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210 A. Christoffersson, E. S . Husebye and S. F. Zngate

Scattering contributions, mainly P-to-P conversions, are also noticeable as coda amvals often are triggered out of the azimuth plane or not triggering at all. In general, we find that the P-coda is weak in shear-wave type motions although some scientists claim that P-to-Lg conversions contribute significantly here (A. Dainty, private communication).

(4) SeBmic records from nuclear explosions (Fig. 4 ) P-wave recordings from presumed E. Kazakh underground nuclear explosions are relatively simple, and besides may be recognized as such on the basis of a certain stationarity in the derived probability patterns for nearby events, compare Fig. 4(a) and (b). With this kind of application in mind, we may consider the three-component analysis technique as a variant of the so-called attribute processing (instantaneous phase and amplitude, etc.) now becoming popular in seismic prospecting (e.g. see Vidale 1986). For these two events, presumably located in the vicinity of each other, the difference in estimated azimuth is 1.7" while the difference in apparent velocity of 0.3 km s-l leads to a difference of 2.7" in distance.

(5) Analysis of refraction profiling records (Fig. 5) A common, practical problem in profiling is the deployment of many, independently operated instruments to ensure

adequate spatial sampling when the number of shot points is limited. The analysis results displayed here imply that much signal information can be extracted from single site records (Fig. 2a, b). The Pn-phase is clear while its coda is characterized by very high velocities and a wide azimuth range, features which are not uncommon for interfering phases having different azimuths. Also P* and Pg interfere with each other but are separable using this technique. The late arrival at 8.70 s (V = 6.0 km s-l) may be indicative of an upper crust-reflector.

(6) VSP-profiling records (Fig. 6) A common problem here is to distinguishhdentify the up- and down-going parts of the wavefield. Even at extreme high-frequencies the analysis technique appears to work satisfactorily as demonstrated here. Also, notice that by introducing velocity filtering as discussed above, artificial noise preceding P-wave onsets, can be removed efficiently (compare Fig. 6a and c). The two major portions of linearized particle motions at 0.44s and 0.54s with an azimuth separation of c. 180" may reflect up- and downgoing P-waves or alternatively the second phase may be of the SV-type. (There is an inherent phase shift of 180" between P and SV motions.)

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Figure 6. B159: 3-component analysis of MIT/COG VSP record. (a) P-wave probabilities. Original (solid) and filtered (dotted) records; Lower probability (SYNTZ) for filtering/weighting = 0.60; lower (CUTL) and upper (CUTU) velocity passband of 3.0 km s-l and 100.0 km s-'. (b) P-wave velocities. Original (solid) and filtered (dotted) records; Lower probability (PROBLIM) for computing contours = 0.50; lower (CUTL) and upper (CUTU) velocities for filtering/weighting 3.0 km sC1 and 100.0 km s-'. (c) P-wave probabilities. Original (solid) and filtered (dotted) records; Lower probability (SYNTZ) for filtering/weighting = 0.60; lower (CUTL) and upper (CUTU) velocity passband of 7.0 km s-' and 50.0 km s-'. For further details on events, processing and extracted signal parameters, see Table 2 and result section.


Wavefield decomposition using ML-probabilities 21 1

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212 A . Christoffersson, E. S . Husebye and S. F. Zngate

Result summary

For synthetic seismograms the 3-component analysis technique works very well for all wave types. In case of real records, however, the technique works well for P- and SV-waves but less satisfactory for SH-, R- and L-waves. Not surprisingly, similar problems are encountered in array processing of similar data. For this reason our study has concentrated on the analysis of P-waves and indeed, P-waves are most often used in most types of seismic research. The difficulties may be attributed to wave propagation complexities in an inhomogeneous Earth including scattering and crustal reverberation effects. For example, even shear-motions have been observed in P-waves records from nuclear explosions at teleseismic distances. In order to overcome problems of this kind, more complex multiple-signal models may be introduced in principle as discussed above. Alternatively, simple-signal models may be used by ignoring transverse wave-motions although trade-off with azimuth resolution exists. Still, 2-D particle motion modelling has proved beneficial in analysis of complex event records at regional distance in the sense that the recorded wavefield can be separated into P- and S-type phase arrivals. This kind of information is important for event locations as epicentral distance estimates may be obtained from differential travel-time of dominant P- and S-phases (e.g. see Ruud et al. 1988).

Other applications of the technique lie in scattering or coda wave studies where major problems arise from conversion mechanisms (P-to-P waves or P-to-Lg waves) and also whether source or receiver structural complexities dominate.

6 DISCUSSION A N D CONCLUSION

A long-standing problem in seismology is that of properly decomposing wavefield records into signals (wavelets), then classifying signals according to wave type and finally estimating the associated slowness vector, thereby providing path information. These problems have become acute in recent years with significant advances in synthetic seismogram analysis on the basis of both 3-D ray tracing and full-wave equation modelling. The standard approach to decomposing the wavefield is via deployments of 1-D and 2-D arrays of vertical sensors, which works satisfactorily in many cases. However, for complex wavefields, array analysis often becomes problematic due to the spatial lag between sensors requiring relatively long time-windows combined with decreasing signal correlation. In contrast, single-site 3-component recording does not involve spatial lagging, thus permitting the use of relatively short time-windows, and allowing the exploitation of the phase information in the signals. This combination demonstrably provides a valuable tool for decomposing 3-component wavefield records. Naturally, an ideal case is to use arrays of three-component instrumentation, which in principle would permit more complex wavefield modelling than dealt with here.

The very large number of deployed 3-component instruments testifies towards the importance of these data in geophysics, yet on the other hand, such data are at best only modestly used in seismic research work except at low

frequencies. Earlier attempts at analysing 3-component data appear to have been not entirely successful, and this is attributed in part to difficulties in transforming the original records into easily handled and interpretable domains. This aspect was given due consideration from the very outset of the work in this presentation. An illustrative example is that the azimuth-time display of probability and velocity estimates (Figs 1-6) exhibits many features similar to displays of array data analysis, like sonograms and f - k plots. Also, for other trivial seismogram analysis work, the filtered records are far easier to read than the original ones.

In conclusion, the approach to 3-component seismogram analysis presented works satisfactorily for P and SV-waves. The more complex shear waves have proved difficult to analyse by this technique. An exception here is the case of 2-D composite Sg-Lg wave trains (e.g. see Kennett 1984). Indeed, this is not surprising. The problem with S-waves arises from the use of simplistic particle motion models and/or correlation between SV and SH components. So far no extensive analysis has been carried out on Love and Rayleigh waves. What has been demonstrated is that by exploiting the phase information inherent in three-component records in an objective way, particularly the P-wave part, it is possible to decompose the seismic wavefield into well-defined wavelets, with associated estimates of their slowness vector. The technique is restricted to single-station analysis or a simple linear stack of nearby station traces prior to analysis, and thus somewhat simplified signal modelling ensues. Further research will prompt progress in developing spatial three-component models and estimators with the final goal of integrated analysis of three-component array recordings.

Acknowledgement

We thank Carol Blackway of ERL/MIT for the VSP data, and Won Young Kim, Department of Seismology, University of Uppsala, for the synthetic data.

This research was supported by the Advanced Research Projects Agency of the Department of Defense and was monitored by AFTAC, Patrick Air Force Base, FL 32925, under Contract FO8606-84-C-0002-P00002, the Swedish Natural Science Research Council under Contract G- Gu2728-117, and by the Office of Naval Research, Arctic Programs Office, through Contract N00014-77-C-0266.

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Wavefield decomposition using ML-probabilities in modelling single