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Bulletin of the Seismological Society of America, Vol. 79, No. 1, pp. 51-66, February 1989

DURATION AND DEPTH OF FAULTING OF THE 22 JUNE 1977 TONGA EARTHQUAKE

BY JIAJUN ZHANG AND THORNE LAY

ABSTRACT

The 22 June 1977 (Mw = 8.2) Tonga earthquake has the longest rupture duration ever reported for a normal fault event. The 150-km depth range spanned by aftershocks of the earthquake is also unusually large. There has been substantial controversy over both the depth and duration of faulting for this great event, obscuring its tectonic significance. We study the source process of the Tonga event using long-period Rayleigh waves recorded by the Global Digital Seismo- graph Network (GDSN) and International Deployment of Accelerometers (IDA) networks. For a standard assumption of a Haskell source, a total duration of 84 -+ 4 sec is obtained using a least-squares inversion method. We introduce the use of the spectral amplitude as a weighting factor in measuring the misfit between the data and a given source finiteness model, which reduces the scatter and improves the resolution of source duration determined from data ranging in period from 150 to 300 sec. Using a more realistic shape for the source-time function in the inversion (drawing upon results from body-wave analysis) reveals a much longer (165-sec process time) component of the source process of the Tonga earthquake. The fundamental mode Rayleigh waves do not resolve any horizontal source directivity. However, the centroid depth of the earthquake is well resolved as 96 km with 90 per cent confidence range (93, 104 km). The estimated error in the depth determination due to the uncertainties in the source finiteness and earth models is only a few kilometers. The results indicate that the rupture of the earthquake excited long-period seismic waves at depths somewhat greater than the 70 to 80 km depth range where the primary body- wave radiation occurred, favoring rupture on the steeply dipping plane of the focal mechanism. The fundamental mode Rayleigh waves with periods longer than 150 sec cannot resolve vertical extent of the faulting; however, additional information from body-wave and free oscillation analyses indicates a vertical fault extent of about 50 km with a frequency-dependent variation in seismic radiation with depth.

INTRODUCTION

Determination of the depth and duration of rupture of large normal fault earthquakes has been a challenging task in the study of earthquake source parameters in recent years. Notable examples of large normal fault earthquakes include the 1933 Sanriku, Japan ( M w = 8.4) and 1977 Sumba, Indonesia ( M w = 8.3) earthquakes. Accurate knowledge of the depth extent and source duration of these large normal fault earthquakes can provide key information about the intraplate deformation and characteristics of the strain release process at subduction plate bound- aries. However, there are continuing controversies over both depth extent and source duration for many of the important large normal fault earthquakes.

The Tonga earthquake (22 June 1977, 12:08:33.4UT, 22.88°S, 175.90°W, mb = 6.8, M s = 7.2, M w = 8.2) is one of the largest normal fault earthquakes which have occurred in recent years. Figure I shows that the earthquake occurred in the vicinity of the intersection of the Tonga trench and the Louisville Ridge. The earthquake was followed by aftershocks spanning a depth range of 150 km (Silver and Jordan, 1983), which is unusually large compared with other subduction zone events. This

51

52 JIAJUN ZHANG AND THORNE LAY

20 ° I ~ 9 ~ ~ I

"~4"I0 4 i:,6, ~ ~ =,~ I~

i m ~ - - J 2 --

6/22 / 7 7 ~ ' S ~ y n e - c

o ,~o 1 '~-:., sec ~I

I t ~ ¢ . i@<<,

~w @ ZEALAND

40° ' 180 ° W WO °

FIG. 1. Map of the Tonga-Kermadec region showing the epicenter, focal mechanism (Giardini, 1984), and average source time function obtained by P-wave deconvolution (Christensen and Lay, 1988) of the 22 June 1977 Tonga earthquake. The moment indicated for the time function is the average for the nondiffracted stations used (Christensen and Lay, 1988).

earthquake also has the longest source duration ever reported for a normal fault earthquake with an estimate of 66-sec characteristic time (114-sec duration for a boxcar time function) being determined from free oscillations by Silver and Jordan (1983). Although the Tonga rupture has a seismic moment about a factor of two smaller than the 1977 Sumba earthquake (Given and Kanamori, 1980; Silver and Jordan, 1983; Giardini et al., 1985), the duration of the Tonga event reported by Silver and Jordan (1983) is much longer than that of the Sumba earthquake (43 to 79 sec) determined by various investigators (Furumoto and Nakanishi, 1983; Silver et al., 1986; Zhang and Kanamori, 1988a). We will examine the rupture duration of the Tonga event in detail in this paper.

Determination of the depth extent of the Tonga earthquake is important for our understanding of large-scale intraplate deformation in the southern Tonga subduction zone, where the Louisville Ridge intersects the Tonga trench. A shallow centroid or small depth extent of the earthquake may suggest that the stress state in this region is controlled by subduction of the Louisville Ridge (Christensen and Lay, 1988); whereas a deep centroid and large depth extent may indicate that the rupture of the sinking slab in this region is controlled primarily by gravitational slab pull (Lundgren and Okal, 1988).

Resolving the depth extent of the Tonga earthquake has proved difficult. Lund- gren and Okal (1988) review previous research on the source depth. The hypocentral depth estimated from first arrival times is 65 km by the National Earthquake Information Center (NEIC) and 69 km by the International Seismological Centre

DURATION AND DEPTH OF FAULTING OF THE TONGA EARTHQUAKE 53

(ISC). The centroid depths estimated from long-period waves given by the Harvard centroid moment tensor (CMT) solution (61 km) (Giardini, 1984) and by the total- moment spectra technique of Silver and Jordan (1983) (65 km) are similar to the hypocentral depth of the earthquake. Christensen and Lay (1988) analyzed long- period P-waves recorded by the World Wide Standardized Seismograph Network (WWSSN) and found that the body-wave radiation was concentrated in the 70 to 80 km depth range. However, several other studies of P waves, SH, waves, and excitation of various overtone modes (e.g., radial modes) suggest that the centroid depth of the earthquake is deeper, on the order of 100 km (Masters and Gilbert, 1981; Giardini, 1984; Fukao and Suda, 1987; Lundgren and Okal, 1988), requiring a large vertical rupture extent.

In order to resolve the discrepancies in the depth extent and centroid depth estimates for the Tonga earthquake, further analysis of the seismic waves is necessary. Body waves for the Tonga earthquake are complex and difficult to model (Christensen and Lay, 1988; Lundgren and Okal, 1988). Because most P arrivals were off scale on WWSSN and the Global Digital Seismograph Network (GDSN) stations, only three GDSN stations were available for the body-wave analysis of Lundgren and Okal (1988), and Christensen and Lay (1988) were forced to use horizontal component and diffracted P-wave data. The identification of depth phases in the raw seismograms is difficult because the rupture process is complex, being comprised of two major pulses of moment rate with a combined process time of about 50 sec (Christensen and Lay, 1988), as shown in Figure 1. The moment- rate function simplicity criterion used by Christensen and Lay (1988! to determine the depth of the earthquake is empirical and qualitative, and it is difficult to obtain formal error estimates. Furthermore, for large earthquakes the depth extent determined by waveform analysis of short-period waves may not coincide with that of long-period waves, if the displacement and stress release on a fault plane are not uniform and coherent in space. It thus appears that the use of long-period seismic waves to determine the overall depth of faulting of the Tonga earthquake is necessary.

The accuracy of depth determinations of large earthquakes using long-period surface waves depends on how well the effects of source finiteness, propagation, and excitation of seismic waves are known. Recent progress in analysis of source finiteness (Romanowicz and Monfret, 1986; Zhang and Kanamori, 1988a) and lateral heterogeneity of the earth (Nakanishi and Anderson, 1983, 1984; Woodhouse and Dziewonski, 1984; Tanimoto, 1985, 1986) has made it possible to use surface-wave data to determine accurately the centroid depths of large earthquakes. Spatial and temporal source finiteness can be determined independently from both the earthquake mechanism and earth structure influence on surface-wave excitation using the method of Zhang and Kanamori (1988a) (hereinafter referred to as ZKa). The method used by Zhang and Kanamori (1988b) (hereinafter referred to as ZKb) provides a tool for the direct determination of the depth extent of faulting of large earthquakes.

In this paper, we first study the source duration of the earthquake using a method similar to that of ZKa, and then determine the centroid depth and other source parameters of the earthquake using the method of ZKb.

DATA

The basic data used in this study are phase and amplitude spectra at periods from 150 to 300 see computed from the vertical component fundamental mode Rayleigh


waves. The periods used are 150, 175, 200, 225, 256, 275, and 300 sec. We computed the spectra of 24 Rayleigh wave phases recorded by the GDSN and International Deployment of Accelerometers (IDA) networks. (GDSN (R2 and R3): SNZO, NWAO, MAIO, GUMO, CTAO, ANMO, ZOBO; IDA (R3 and R4): GAR, HAL, NNA, PFO, SUR). The spectral data used here are corrected for propagation phase delay using the laterally heterogeneous earth model M84C obtained by Woodhouse and Dziewonski (1984) and for attenuation using the Q model of Dziewonski and Steim (1982). For each period, observed source spectra are represented by a vector, of which the elements are the real and imaginary parts of the spectrum of every observed Rayleigh-wave phase.

SOURCE DURATION

The temporal finiteness of a seismic source is often represented by two characteristic time constants: the rise time and rupture time. For an earthquake with complex source-time function a convenient measure of the temporal finiteness is the apparent source duration:

1 ~ t~+~ = f ( t ) dt (1) ts /max ~i

where f (t) is the source-time function with height/max and is time-limited within the interval of integration defined by the starting time ti and the process time r, which represents the entire interval of source radiation. The apparent source duration ts is the base of a rectangle with height/max and area equal to the area of the f (t) curve (Papoulis, 1962). If f (t) represents the moment-rate function of the source, the integral in (1) gives the seismic moment Mo: Mo = fmaxts. For a Haskell source, which is a point source characterized by a rise time ~R and rupture time ti, the time function is a trapezoidal function of duration ts = ~R + tr.

We first determine the source duration of the Tonga earthquake using a method similar to ZKa. We invert the spectra of a given period by solving the following system:

B D = V (2)

where D is the solution vector involving products of the moment-tensor elements and corresponding excitation functions, and V the data vector calculated from the observed spectra vector by correcting for the effects of propagation, instrument response, and a given source-finiteness model (see ZKb). The data vector V has 2N elements; N is the number of records obtained from stations with azimuths ¢1, • • ", CN from the source. The source finiteness correction, which can include both temporal and spatial terms, always increases the size of the elements of V, with shorter periods being more strongly affected (ZKb). The column vectors of B (ZKb) span a linear vector space.

The least-squares solution of (2) is the orthogonal projection of the data vector V onto the linear space. The RMS error

e(~) = H (I - B ( B T B ) - I B T ) V I I / 2 ~ / - ~ (3)

measures the size of the projection residual (ZKb), which depends on both the size and direction of the data vector V. The amplification of V due to the correction for


source finiteness tends to increase the RMS error and thus hampers accurate determination of the source finiteness, while the change of the direction of V due to the correction may increase or decrease the error. Therefore, we introduce a weighting factor in measuring the misfit between the data and a given source finiteness model.

The weighted RMS error, which measures the misfit, is given by

a(o~) = e(w)[~ ][~(~)P(~, ¢i),Z/N] 1/2 (4)

where/~ (w) and P (w, ¢ i) represent the effect of the finiteness due to the dislocation time function and due to the rupture propagation, respectively (ZKa). The weighting factor in (4) is proportional to VJ-V-~V, the length of the vector V. For a point source with finite duration the weighted error a(o~) is the same as the RMS error when the observed source spectra are used for V and source finiteness corrections are included in B (Zhang and Lay, manuscript in preparation, 1989). We determined the source finiteness of the Tonga earthquake by minimizing the weighted RMS error a(~) for simplified point source models characterized by temporal finiteness alone.

We first used the simplest source model of ZKa, a Haskell (1964) source, which is represented by a trapezoidal time function of duration t8 = ~R + t~. We calculated the source finiteness effects of the source with a trial source duration t8 assuming rR = ts/lO (Kanamori and Anderson, 1975). We then obtained a data vector V by adjusting the source spectra for the step-time function using the source finiteness corrections. We inverted the data by solving (2) and obtained a RMS error e(~) and a weighted RMS error a(w), both being functions of the assumed source duration. The source duration, t~, which yields a minimum of these functions, is our estimate.

The introduction of the weighting in the determination of source duration reduces the scatter and improves the resolution of the source duration for the data with periods from 150 to 300 sec. Figures 2a and 2b show e(o~) and a(w) curves, respectively, for trial source durations in the range of 0 to 140 sec for the Tonga earthquake. The curves for all periods are of similar shape, with a single minimum. For the shorter periods of 150 and 175 sec the minima are significantly better resolved using the weighted RMS error; whereas for longer periods of 275 and 300 sec the weighted RMS error curves are similar in shape to the normal RMS error curves. This indicates that the improvement of the resolution is significant for short periods and marginal for long periods.

Table 1 lists source duration and 90 per cent confidence intervals obtained using a F test from the RMS error and weighted RMS error for the inversions at various periods. Using the weighted RMS error we obtained estimates of source duration with a large mean value, smaller deviation, a n d smaller range compared with those using the RMS error. The range, which is the difference between the largest and the smallest value in the estimates, is reduced by 35 per cent using the weighted RMS error.

When different earth models are used for the propagation corrections, the mean value of the estimates of the source duration obtained at various periods can change significantly, while the deviation of the estimates remains about the same. The variation of source duration obtained for different Q models is very small. Figure 3a shows the source duration obtained at periods from 150 to 300 sec for two


June 22, 1977 Tonga

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\ \ 4 / / / \ \ _ _ _ \ 2--- ,

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(a) (b)

50 1 O0 50 1 O0 Source Duration (s) Source Duration (s)

FIG. 2. (a) The RMS error and (b) weighted RMS error plotted versus trial source duration for the inversion at the periods of (1) 150, (2) 175, (3) 200, (4) 225, (5), 256, (6) 275, and (7) 300 sec, respectively. The duration corresponding to the minimum error on each curve is taken as the source-process time measured at that period. For the period 256 sec (solid line) the observability of the source finiteness is higher compared with other periods.

TABLE 1

SOURCE DURATION(S) DETERMINED AT VARIOUS PERIODS

Source Period(s) ~ s R

Durations 150 175 200 225 256 275 300

t~ 67 76 78 73 80 84 73 76 5.5 17 (0, 89) (47, 95) (56, 95) (52, 90) (63, 95) (67, 100) (50, 93)

t~' 82 84 83 76 82 86 75 81 4.1 11 (56, 108) (62, 106) (63, 102) (57, 95) (66, 98) (70, 103) (53, 97)

to and t [ are the source duration obtained using the RMS error e and weighted RMS error ~', respectively; R, S, and ~ are the range, 1 S.D., and mean of the estimates of the source duration over the periods used. Numbers in parentheses give the 90% confidence interval obtained using a F test for these estimates.

different phase velocity models, the model M84C and a homogeneous model (HOM) of Gilbert and Dziewonski (1975). Using the weighted RMS error, the first ear th model gives the es t imates with a mean of 81 sec and a S.D. of 4.1 sec, while the la t ter model gives a mean of 71 sec and a S.D. of 5.7 sec.

I f we use the weighted RMS error and the laterally heterogeneous ear th model M84C, the mean value of the es t imates at all periods becomes larger and closer to the corresponding es t imate at 256-sec period compared with those obtained if the R M S error and the model H O M are u s e d (Figure 3a). This is consis tent with the variable observabil i ty of source finiteness effects, 7, calculated for the Tonga ear thquake using the method of ZKa, which has the largest value at 256-sec period over the period range considered indicating this period yields the most reliable results (Figure 3b).

Based on the es t imates of the durat ion a t 256-sec and 275-sec periods, which we considered more reliable t han a t o ther periods, we obtain 84 _ 4 sec for the source durat ion of the Tonga ear thquake. Because the S.D. of the es t imates over all the periods is about 4 sec, we considered this an appropr ia te es t imate of the error bound for the source durat ion of this ear thquake, for an assumpt ion of a t rapezoid m o m e n t


June 22, 1977 Tonga 90 ' I ' I ' |

o, (a) J I ~ 4 t - - _ i i G, M84C / ~ , , /

I . .Q- . 0" . m - - l , ",tl~

"~ 70 , , , , . ~ . G, HOM / / z

m - . , ," |

i 60 g, HOM ~" / t

50

, I ~ I ,

.S

I ~ I 150 200 250 300

Period (s)

FIG. 3. (a) Comparison of the estimated source duration of the Tonga earthquake measured at periods of 150, 175, 200, 225, 256, 275, and 300 sec by the inversion method using the RMS error c and weighted RMS error a for two phase-velocity models; phase velocity is calculated (1) from the average observed normal mode periods compiled by Gilbert and Dziewonski (1975) (HOM) (squares) and (2) from a laterally heterogeneous earth model obtained by Woodhouse and Dziewonski (1984) (M84C) (circles). (b) The observability ~ of source finiteness (Zhang and Kanamori, 1988a) of the Tonga earthquake measured at these periods. A point source with 84-sec duration is used for the source finiteness model.

rate function. If we assume that the moment rate function is a boxcar (TR = 0), we obtain the same estimate of the source duration.

In order for a particular source-finiteness model to be useful for imaging the source process of an earthquake, the model must fit the data as well as or better than any other simple model. For very large earthquakes the seismic moment and the mechanism of the earthquake are usually obtained from long-period surface waves by assuming that the source-time function is a boxcar function (e.g., Kana- mori and Given, 1981; Dziewonski and Woodhouse, 1983). However, body-wave analysis indicates that the source process of many large earthquakes consists of several subevents represented by triangular or trapezoidal time functions. It is desirable to appraise the effect of the assumed moment-rate function on the source- duration estimate for the Tonga event.

The effect of complexity in the shape of the moment-rate function on source- parameter estimates may be tested using the following sine functions:

(4/7r) N F,n=l H0(t)sin[(2n - 1 ) T r t / T ] / ( 2 n - 1) [IN(t) = ~'~.nN_--i 1/(2n -- 1) (5)

where v is process time, and [io(t) = [H(t) - H ( t - T)]/7 is a boxcar function. These functions are expansions of a boxcar function and have similar spectra at low frequencies to the boxcar function. We use these functions as moment-rate functions. Figures 4(a-1), 4(a-2), and 4(a-3) show three examples of simple single- event time functions: the boxcar IIo, its first order expansion (half-sine) II1, and its second order expansion II2, respectively.

Given independent knowledge of the shape of the moment-rate function for a particular earthquake, additional constraints on the long-period source can be


June 22, 1977 Tonga Earthquake

(aq) I I " ~ (b-l) ~ / " ~ V ~ I ~ " e* (c-1) o°~ ( d - l ) . 2 4 1 I (e'l)

o i i

0 100 0 .02 0 .02 200 300 T

(a-2) (b-2) (c-2) (d-2) (e-2)

/2 I • , .2 I 4 o m

0 100 0 .02 0 .02 200 300 T (a-3) (b-3) (c-3) (d-3) (e-3) 4F

1 0 ~ 0 .02 0 .02 ~ ' ° ~ ' E 200 300

'~ (a-4) -- (b-4) ~ (c-4) ~ o .~ (e-4) n" o.. (d-4)

I o , , 0 1 O0 0 .02 0 .02 200 300

(a-5) (b-5) (0-5) (d-5) (e-5)

• e e

I ~ o ' , .2 0 1 O0 0 .02 0 .02 200 300

(a-6) (b-6) (c-6) (d-6) (e-6)

L I o , , I 200 0 .02 0 .02 200 300

Time (s) Frequency (Hz) Period (s)

FIG. 4. (a-l) boxcar function, (a-2) half-sine function, (a-3) second-order expansion of boxcar, (a-4) double half-sine function each with the same process time, (a-5) superposition of a double half-sine and a half-sine functions, (a-6) superposition of a double half-sine function with fixed process time and a half-sine function; (b-l) through (b-6) amplitude spectra for (a-l) through (a-6), respectively. (c-1) through (c-6) phase spectra for (a-l) through (a-6), respectively. (d-l) through (d-6) process times determined with (a-l) through (a-6), respectively. (e-l) through (e-6) weighted errors, p, of the moment tensor inversion for (a-l) through (a-6), respectively.

determined using surface waves. An earthquake with an arbitrary time function can be decomposed into events with these sine functions with various starting time, process time, and seismic moment. For the Tonga earthquake, body-wave studies indicate that the high-frequency source process consists of two major subevents spanning a total process time of 50 sec after the starting time (Christensen and Lay, 1988; Lundgren and Okal, 1988). Figure 1 shows the average source-time function of the Tonga event obtained by Christensen and Lay (1988) from P waves recorded at WWSSN stations. However, because of the limited instrument response at long periods and uncertainties in the Green's functions, later pulses and long- period components of the source-time function of the earthquake cannot be determined unambiguously from body waves. Rather than arbitrarily assigning a total process time and moment for the missing long-period component, as is commonly done, we will objectively determine the long-period characteristics.

We consider several extreme cases for the time history of the long-period source of the Tonga earthquake: three single-event time functions as shown in Figure 4(a-


1), 4(a-2), and 4(a-3), and three multiple-event time functions as shown in Figure 4(a-4), 4(a-5), and 4(a-6). These multiple-events consist of subevents represented by the time function of the half-sine pulses II1. The first multiple-event 4(a-4) consists of two subevents of the same process time r/2 with the second event starting r/2 after the first event. The second multiple-event 4(a-5) consists of three subevents, of which the first has process time r/2; the second starts r/2 after the first event with process time r/2; and the third starts at the beginning of the first event but with process time r. The third multiple-event 4(a-6) also consists of three subevents, of which the first has 25-sec process time; the second starts 25-sec after the first event with process time 25 sec; and the third starts at the beginning of the first event but with process time r. For the multiple-events 4(a-4) and 4(a-5), every subevent has the same moment. For the last multiple-event 4(a-6), the first and second subevents have the same moment, while the third has a 40 per cent of the total moment. The last multiple-event has two pulses resembling the time function of the Tonga earthquake obtained from body waves.

Figures 4(b-1) to 4(b-6) and 4(c-1) to 4(c-6) show the amplitude and phase spectra of the time functions 4(a-1) to 4(a-6), respectively. For the period range 150 to 300 sec events 4(a-1) to 4(a-5) have similar phase spectra, while event 4(a-6) has a much different phase spectrum.

We determined the process time of the long-period component of the source process of the Tonga earthquake using these various source-time functions. We inverted the spectra corrected for the finiteness effects computed for the source time functions of 4(a-1) to 4(a-6) and obtained the best process time for each event by minimizing the error of the inversion of (2). Figures 4(d-1) to 4(d-6) show the process time determined for the time functions of 4(a-1) to 4(a-6) at various periods using the weighted error a. Since a depends only on the phase spectra, we obtained the same estimates of process time (84 sec) for all the events 4(a-1) to 4(a-5). However, for case 4(a-6), which is the only source function which will match the body-wave radiation, we obtained estimates of r with a mean of 140 sec and S.D. of 21 sec at the periods used. The average of estimates obtained at periods of 256 and 275 sec is 165 sec. This is the process time of the long-period component of the time function. The scatter of the estimates obtained at various periods for case 4(a- 6) indicates the uncertainties in resolving a very long-period component of the source process of the event. However, for periods of 256 and 275 sec the minimum weighted error a obtained is about the same for all the time functions considered here, which indicates that these time functions fit the data equally well as far as the phase shift caused by the source finiteness is concerned. The centroid time for case 4(a-6) is 48.3 sec, which is about the same as the 42-sec centroid time for all cases 4(a-1) through 4(a-5). However, the apparent duration for case 4(a-6) is 46.9 sec, which is much shorter than the 165-sec process time, indicating the complexity of the moment-rate function.

The observed source spectra of the Tonga earthquake from long-period Rayleigh waves indicate that the rupture of the earthquake has a component with much longer process time than the 50-sec process time of the two major pulses obtained from body-wave analysis. The long-period component is at least 84-sec long and may be as long as 165 sec. The 84-sec process time is obtained for the simplest source-time functions, but it is known that these models will not match the body- wave radiation. Introduction of source complexities such as those for case 4(a-6), in which the short pulses are dominant, results in much longer estimates for the total process time of the earthquake.


We found that the horizontal source directivity of the Tonga earthquake cannot be determined with confidence, an uncertainty which we attribute to a bilateral rupture over a limited horizontal fault extent. In the following determination of the depth and mechanism of the earthquake, we use the point source model with finite duration.

DEPTH OF FAULTING

Using the solution vector D's obtained by solving (2) at several periods, we determined the moment tensor of the earthquake by solving the following system (ZKb):

r M = A (6)

where r is a matrix of excitation coefficients for a given source depth and earth model, vector M the moment-tensor solution, and vector A is a stack of vector D's obtained by solving (2) using a given source finiteness model. A has 5K elements; K is the number of periods. We solve (6) for the moment tensor by least squares and find the depth that minimizes the weighted RMS error

where n = 5K and

pi = ( A i - ~ ,~ ) / (A~a) 1/2 (i = 1, . . . , n ) (8)

Here Ai and -~i are the observed and predicted components of vector A in (6). In order to compare the minimum error obtained for various source finiteness models, a weighting factor is introduced in (8), and the error p is nondimensional. The rank dispersion test of Siegel and Tukey (1960) can be applied to examine the variability of squared residuals pi 2 (i = 1, • •., n) obtained for various source finiteness models.

To apply the test, the squared residuals for two source finiteness models are combined and indexed as follows: the smallest p2 gets rank 1, the largest two are given ranks 2 an.d 3; then the second and third smallest get 4 and 5, the third and fourth largest are given 6 and 7, and so on. We compare the value of the statistic

= [2IA -- n ( 2 n + 1) + 1] / [n2(2n + 1)/3] 1/2 (9)

with the values of the standard normal variable and obtain the level of significance for differences in variability between the residuals for the source finiteness models. Here I~ is the sum of the indices for one of the models which is greater than n (2n + 1)/2.

Figure 5 shows the p versus depth curves for moment-tensor inversions for two moment-rate functions and two earth models: the average ocean model of Regan and Anderson (1984) (hereinafter referred to as R-A) and PREM (Dziewonski and Anderson, 1981). We consider the model R-A to be more appropriate for the earthquakes in the Tonga region. For the constant moment-rate function (boxcar) (as shown in Figure 4(a-l)) with an 84-sec process time (or duration) the centroid depth is 91 km for R-A and 98 km for PREM, respectively. For the more realistic variable moment-rate function (as shown in Figure 4(a-6)) with an 165-sec process


June 22, 1977 Tonga .5 ~ , I

.4 " - - ' ~ ' ~ ~ " ~ " ~ C-m, R-A -

\ C-m, PREM

P.3 m, PREM I

.2 V-m, R-A ~ _

. 1 I , ~ , ~ I 50 100

Depth (kin)

FIG. 5. The weighted error, p, in the moment-tensor inversion obtained for two source-time functions: (C-m) the constant moment-rate function (Fig. 4(a-l)), boxcar, and (V-m) the variable source-time function (Fig. 4(a-6)). Two earth models are used: (R-A) the average ocean model of Regan and Anderson (1984) and PREM (Dziewonski and Anderson, 1981).

time, the centroid depth is 96 km with a (93, 104) 90 per cent confidence depth range for R-A and 103 km with a (99, 112) 90 per cent confidence depth range for PREM, respectively, found using a Student's t test (ZKb). However, the errors in the inversion using the variable moment-rate function have a lower mean (about 45 per cent smaller) and weaker dispersion with 90 per cent confidence using the Siegel-Tukey (1960) test for both earth models than the errors for the constant moment-rate function. Moreover, the variable moment-rate function fits the body- wave radiation, while the constant moment-rate function does not. The moment- tensor solution obtained by using R-A for a point source at 96-kin depth with the variable moment-rate function is our preferred solution for the Tonga earthquake.

Using the variable moment-rate function and model R-A, we obtained a centroid depth of 96 km with a (93, 104 km) depth range of 90 per cent confidence. This estimate of the centroid depth is somewhat greater than the 65-km depth of the earthquake hypocenter determined from the first P-wave arrivals (NEIC). This indicates that much of the long-period energy of the earthquake is released at depths below the depth of initiation of the earthquake.

The seismic-moment tensor may be decomposed into a major double couple and a minor double couple (Kanamori and Given, 1981) or a best double couple and a linear vector dipole (Dziewonski and Woodhouse, 1983). The best double couple has the same mechanism but a moment smaller than the major double couple. If the minor double couple or linear vector dipole is very small, the moment of the best double couple is about the same as the major double couple.

Our moment-tensor solution has a minor double couple of 5 per cent of the major double couple, which has strike, ¢ = 212 °, dip, 6 = 69 °, and slip, X = 262 ° for the first nodal plane, and ¢ = 53 °, 8 = 23 °, and X = 290 ° for the second nodal plane. This solution is consistent with the results obtained by other investigators (Giardini, 1984; Christensen and Lay, 1988). Figure 6 shows the fit of our model to the data. From the P-wave first motions the steeply deeping nodal plane is well constrained, but the slip angle along the plane may vary from 235 ° to 280 ° (Lundgren and Okal, 1988). Since aftershocks of the earthquake, which span a 150-km depth range,


June 22, 1977 T o n g a

L ' • . . . . I . . . . • ' b . . . . . I '~ ' ' ' l . . . . . L . . . . " 1

2 | T = 2 2 5 s

a ' - 2 •

¶ . . . . . I . . . . . I . . . .

~ 2 0 - •

E • • ~o

~1o E

0 ~ ' ' ' ' . . . . . I . . . . . I , . ° ~ , , , J . . . . . i . . . . 60 120 180 240 3 0 0 3 6 0

A z i m u t h (deg)

FIG. 6. The observed spectra of 225-s Rayleigh waves recorded at GDSN and IDA stations corrected for propagation and instrument response. The predicted spectra of the moment tensor solution obtained in this study are shown as the solid line.

suggest a plane steeply dipping to the west, and the tsunami generation is consistent with vertical movement on the fault, the steeply dipping nodal plane is probably the fault plane. The greater centroid depth of the longer-period waves relative to the body waves also indicates this.

Introduction of complexities in the source-time function has a stronger effect on process time than on the mechanism. The centroid depths obtained using the various source-time functions shown in Figure 4a range from 90 to 96 km. For all the time functions considered here, the best double couples of the moment-tensor solutions have very similar mechanisms.

Because of the long process time of the Tonga earthquake, source spectra for the periods concerned differ considerably from the spectra of a point source with a step- time function. The reliability of the seismic moment obtained depends critically on how accurately we can correct the observed spectra for the source finiteness effects of the Tonga event. The seismic moment of the best double couple obtained for model R-A is 16.6 (in units of 1027 dyne cm) for the constant moment-rate function (boxcar) (Figure 4(a-l)) and ranges from 14.1 to 17.6 for the time functions shown in Figure 4(a-l) through 4(a-5). However, the seismic moment is 21.5 ( M w = 8.2) for the more realistic time function (Figure 4(a-6)).

The scatter of published seismic moments for the Tonga earthquake probably results from the different source finiteness models and centroid depths that are used by various investigators for this event. The seismic moment of the Tonga earthquake obtained from surface waves or free oscillations (in units of 1027 dyne cm) is 14 by Giardini (1984) using a boxcar time function of 50-sec duration for 61- km depth; 23 (total moment) by Silver and Jordan (1983) using a finiteness model equivalent to a boxcar time function of l l4-sec duration for 65-km depth; and 17 by Christensen and Lay (1988) using a boxcar time function of 60-sec duration for 70-km depth. Other estimates of the seismic moment include 17 by Talandier et al.

(1987) from the first passage of Rayleigh waves recorded by a broadband instrument at Papeete, Tahiti 25 ° from the earthquake epicenter and 18-20 by Lundgren and Okal (1988) from the radial modes excited by the earthquake.


The vertical extent of faulting of the earthquake cannot be determined with confidence. Inversion for the vertical rupture extent indicates that the rupture propagates down to 119-km for a uniformly distr ibuted source start ing at 75-km depth, and to 162-km depth for a source start ing from 40-km depth. The fundamental mode Rayleigh waves with periods longer than 150 sec cannot resolve which vertical extent of faulting is more appropriate.

Figure 7(a) shows our preferred model for the moment- ra te function of the Tonga earthquake, which comprises three components with a total process t ime of 165 sec. The 165-sec process t ime of the moment - ra te function est imated here is obtained by assuming tha t the first and second components, start ing at 0 and 25 sec, respectively, have the same process time, 25 sec and the same moment (ml = m2); and the th i rd component , s tart ing at 0 sec, has a moment (m3) 40 per cent of the total moment , ml + rn2 + m3. The assumptions on start ing times, process times, and sizes of the first and second components are based upon the results of the analysis for the body-wave t ime function of the Tonga ear thquake by Christensen and Lay (1988). The size of the moment assigned to the third component , 40 per cent of the total moment , is es t imated by comparing errors, p and a, obtained for various assumed moment ratios.

Figures 7(b) and 7(c) show p, a, process time, and seismic moment obtained for various moment ratios. Here a is obtained at 256-sec period, which we consider more reliable than other periods. For a uni t ratio, which corresponds to a half-sine moment - ra te function shown in Figure 4(a-2), ~, p, and process t ime are about the same as for the boxcar moment - ra te function. For a ratio of about 40 per cent, p is minimized; process t ime is 165 sec; and seismic moment is the maximum, 21.5. For a small ratio (less than 40 per cent) which corresponds to a moment- ra te function with a small th i rd component , the moment - ra te function of any process t ime does not have sufficient phase delay to satisfy the data; and a is larger than tha t for

J u n e 2 2 , 1 9 7 7 T o n g a .5 - ( a )

~ Moment Rate Function

>.,

o

m3 0 ~ ~ ~ , I ~ ~ r ~ ~ ,

0 5o T i m e (s ) 100 150

.4 - - 4 200 -- 25 ' , (b ) ( c ) o a

~o 20-o Eo ~ ml+m2+m3 o

.2 - " , " J (~ ~" 100 " ~ ~ - 15

I I r r ] = I i r 2 , , , , I , , , , 1 0 ~ 0 .5 1.0 .5 1.0

m3/(ml +m2+m3) m3/(ml+m2+m3)

FIG. 7. (a) The moment rate function of the 1977 Tonga earthquake obtained in this study. (b) The errors, p (solid line) and a (256-s period) (dashed line) versus moment ratio curves. (c) the process time, r, (solid line) and total seismic moment, ml + m2 + m3, (dashed line) versus moment ratio curves.


other ratios. The process time and seismic moment shown in Figure 7(c) for a small ratio are for the moment rate which has the maximum phase delay. We found that the moment-rate functions with a small ratio are not acceptable. Our preferred total moment estimate of 21.5 × 1027 dyne cm and the average body-wave moment of 11.9 × 1027 dyne cm in Figure 1 (Christensen and Lay, 1988) indicate that about 45 per cent of the moment is missed by the body waves, which is consistent with our moment-rate parameterization.

Accounting for the realistic shape of the time function of the Tonga earthquake improves the estimates of the seismic moment and source mechanism compared to assumption of a simple boxcar or trapezoid time function. The source finiteness model shown in Figure 7(a) (Figure 4(a-6) as well) gives a better fit to the long- period Rayleigh wave data than any of the other source finiteness models tested. Figures 4(e-1) to 4(e-5) show the errors, p, obtained for the models 4(a-l) to 4(a-5) are about the same. However, model 4(a-6) gives an error about 45 per cent smaller than other models. Although the details of the source finiteness model cannot be determined uniquely, the use of the source-time functions determined from body waves to constrain the source finiteness models improves the moment-tensor solution from long-period surface waves. This observation needs further exploration.

DISCUSSION AND CONCLUSIONS

This analysis of the 22 June 1977 Tonga earthquake has revealed several interesting aspects of the rupture process of this great normal fault event as well as revealing the importance of suitable moment-rate parameterization for source inversion using long-period surface-wave data. Drawing upon results of body-wave deconvolution analysis (Christensen and Lay, 1988), which indicate that the high- frequency radiation occurred in two pulses with a combined process time of 50 sec, we find in this study that the total source process time is as long as 165 sec. The shape of the moment-rate function does, in fact, influence the process-time estimate, and it has a corresponding influence on the moment-tensor inversion. In determin- ing the source duration model it is very useful to weight the error estimate so as to account for amplitude effects of the finiteness.

Although the details of the source process for the Tonga event are not uniquely resolved, we can attempt to provide one possible interpretation which reconciles various seismological observations. The earthquake rupture nucleated near a depth of 65 km with failure of a strong region near the hypocenter. This initial rupture had a time constant of about 12 sec, with a decrease in moment rate prior to failure of a second asperity commencing about 25 sec into the event. The moment release from these two subevents was concentrated in the depth range 70 to 80 km (Christensen and Lay, 1988). A coincident, but longer, process-time component of slip occurred which preferentially radiated only long-period energy. This slip appears to have occurred at greater depth on the steeply dipping fault plane, given the shift to larger centroid depths for long periods. The Rayleigh-wave data (150 to 300 sec period) analyzed in this paper indicate a centroid of 96 kin, while longer-period radial modes indicate a centroid depth of more than 100 km (Lundgren and Okal, 1988). This frequency dependence of centroid depth may correspond to the depth dependence of seismic slip behavior and fault theology. Presumably, depth-dependent variations in failure properties over the fault plane are responsible for variations in the seismic radiation, as suggested for interplate events by Hartzell and Heaton (1988). The existence of the complexity of the source may explain the large tsunami generation for this intermediate-depth event (peak-to-peak tsunami amplitude of 40 cm at Suva and 12 cm at Papeete).


The analysis of fundamental mode Rayleigh waves is improving to the degree that more detailed rupture models can be constrained than previously possible. However, resolution of vertical fault extent is still beyond the capability of such analysis. It appears that combining body-wave moment-rate deconvolution and high-resolution surface-wave analysis is both viable and necessary for investigation of great earthquake rupture process.

ACKNOWLEDGMENTS

We thank L. J. Ruff for many helpful discussions throughout the course of this work and the anonymous reviewer for helpful comments. The IDA data used in this study were made available by courtesy of the IDA project team at the Institute of Geophysics and Planetary Physics, University of California, San Diego. This work was supported by NSF grant EAR-8451715.

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DEPARTMENT OF GEOLOGICAL SCIENCES THE UNIVERSITY OF MICHIGAN ANN ARBOR, MICHIGAN 48109

Manuscript received 5 July 1988

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