Geophys. J . Int. (1993) 115, 991-1011

Inference of core-mantle boundary topography from ISC PcP and PKP traveltimes

Arthur Rodgers and John Wahr Department of Physics and Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, CO 80309-0390, USA

Accepted 1993 May 2. Received 1993 May 2; in original form 1993 February 20

SUMMARY We invert ISC PcP and PKP absolute and differential traveltimes in an attempt to infer the long-wavelength topography of the core-mantle boundary (CMB). The data selection and processing methods are described and evaluated. These travel- time data are very noisy and the geographic distribution of the data is highly non-uniform, inhibiting reliable inference of CMB topography. Spatial averaging enhances the coherent component of the residual variance (related to heteroge- neity), however, the random component of the variance is much larger than the coherent component. We show that for PcP data the coherent signal due to mantle heterogeneity overshadows that arising from the CMB, and that the effects of mantle heterogeneity are mapped into our inferred CMB solutions. The PcP data are not correlated across the spatial averaging bins and seem to have a strong bias due to small-scale structure and/or noise. The non-uniform geographic sampling of the data plays a role in the mapping of mantle heterogeneity onto the CMB. Spatial patterns of CMB models inferred from different phases do not agree. Amplitudes of seismically inferred CMB undulations vary greatly. The sensitivity of inferred CMB models to the processing, spatial averaging procedure, and inversion techniques are investigated. Topographic amplitudes increase strongly with increasing input re- sidual variance. The power spectrum of inferred topography indicates that there are unmodelled heterogeneities that must be described with spherical harmonics of degree 6 and higher. Based on this work, we conclude that reliable inference of long-wavelength CMB topography is not likely with the current ISC data set or with a spherical harmonic expansion truncated to degree and order 6.

Key words: core-mantle boundary, seismic tomography.

1 INTRODUCTION

In recent years the determination of the Earth’s large-scale aspherical structure has been a major focus of research in global seismology. Seismic traveltime inversion (also referred to as seismic tomography) has inferred a great deal of lateral heterogeneity within the Earth (earlier works by Dziewonski, Hager & O’Connell 1977; Clayton & Comer 1983; Dziewonski 1984, have inferred lower mantle heterogeneity). A number of researchers have attempted to determine the long-wavelength topography of the core- mantle boundary (CMB) using body wave arrival times reported to the International Seismological Centre (ISC) (e.g. Creager & Jordan 1986; Gudmundsson, Clayton & Anderson 1986; Morelli & Dziewonski 1987a; Wahr, Rodgers & Billington 1987; Doornbos & Hilton 1989).

Topography of the core-mantle boundary (CMB) is defined as the deviation of that boundary from the hydrostatic ellipsoidal shape caused by Earth rotation. CMB topog- raphy probably results primarily from dynamical processes within the mantle (e.g. mantle convection). The seismic velocities change abruptly across the CMB. Short-period seismic body waves are reflected and refracted by the CMB and the presence of topographic relief perturbs the traveltimes of those waves that interact with the CMB.

CMB topographic models produced by investigators using different subsets of the ISC data and different inversion schemes are not in good agreement. Also the amplitudes of the seismically inferred topographies are much larger (as much as 10 times) than those inferred from other geophysical fields such as geoid modelling. (See, for example, Hager et al. 1985, which infers CMB topographic

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undulations to be less or equal to 1.5 km. However, the amplitudes of CMB topography inferred from geoid modelling are strongly dependent on the viscosity of the lowermost mantle and the density jump across the CMB.) Global traveltime data are noisy and sample the Earth highly non-uniformly. The variance of traveltime residuals is much greater than would be expected from plausible CMB topographic amplitudes (say, consistent with geoid modell- ing). It is known that seismic traveltime data contain errors due to: inaccurate reading of arrival times, misidentification of phases, poor location of events, and station biases (Grand 1989). Also, there is signal-generated noise which inhibits isolation of the CMB signal such as: delays due to unmodelled structure above the CMB, aliasing by small-scale structure, and complications of arrivals due to diffraction and interference not modelled in ray theory (Wielandt 1987; Neuberg & Wahr 1991). Gudmundsson, Davies & Clayton (1990) found measures of random noise and mantle heterogeneity in the ISC P residuals. They found that significant heterogeneity exists in the crust, upper mantle and D" region (the lowermost 200-350km of the mantle). We have found evidence in the ISC PcP data that is consistent with their findings. This and previous studies of CMB topography have not attempted to remove the effects of upper mantle heterogeneity. Failure to account for this heterogeneity probably obscures images of deep Earth structure.

Further inhibiting CMB inference is the fact that global traveltime data sample the CMB highly non-uniformly due to the clustering of seismic events near plate boundaries and the absence of seismic stations on ocean floors. Pulliam & Stark (1993) and Stark & Hentgartner (1993) show that gaps in the geographic coverage of the data and noise lead to large errors in estimates of CMB topography when a low-order spherical harmonic expansion is used to parameterize the structure. These problems are distinct in that even if the noise level is lowered, poor geographic coverage could prevent reliable inference of CMB topography, and, conversely, if the coverage is improved, noise could hamper the effort. The combination of these two factors make the problem even more severe. For example, random noise would tend to average out in heavily sampled areas. Thus, inferred CMB topography could well exhibit correlations with the sampling pattern: small topography in well-sampled areas, and large topography in regions of poor sampling.

These facts motivate a thorough evaluation of the ISC global seismic traveltime data and the methods used to infer lateral heterogeneity within the Earth. In this paper we present such an evaluation using the ISC PcP and PKP traveltimes, emphasizing CMB topography. The next section describes the preliminary processing of the traveltime data and discusses selection criteria. A statistical analysis and discussion of the processed data is presented in Sections 3 and 4. The inversion approach is described in Section 5. In Section 6 we present the results of inversions of different phases of the ISC data. Experiments were performed on PcP and PKPbc data sets in order to evaluate how inference of CMB topography is impacted by data selection, processing procedures and inversion techniques. Results of these experiments are presented in Sections 7 and 8.

A . Rodgers and J . Wahr

2 PRELIMINARY D A T A PROCESSING

We extracted the traveltimes of P, p P , PcP and PKP phases from the ISC catalogue for the years 1964 to 1989. ISC event locations, origin times and phase identifications were determined using the Jeffreys-Bullen tables (Jeffreys & Bullen 1940; Adams, Hughes & McGregor 1982; ISC 1987). We used the imp92 spherical Earth model (Kennett & Engdahl 1991) to account for the spherically symmetric Earth structure and phase arrival times. Phases were re-identified in an iterative procedure developed and run by Bob Engdahl (described in van der Hilst & Engdahl 1993). In his algorithm each event is relocated using P and p P arrivals and then the phase arrivals are checked to see if their arrival times lie within a prescribed window of the theoretical arrival times. This time window is successively reduced at each iteration and phase arrivals are re-identified along the way. The theoretical traveltimes are computed as a sum of an imp91 traveltime, a hydrostatic ellipticity correction and a station correction (Toy 1990). The ellipticity correction we used is consistent with the iasp91 earth model and is based on the method of Dziewonski & Gilbert (1976). The depth phase, p P , is used to better constrain the event depths (see, for example, van der Hilst & Engdahl 1993). If the theoretical traveltimes of two arrivals overlap (e.g. crossing phases), the data are discarded. This procedure reidentified many arrivals. To ensure that the events were well located, we required at least 10 teleseismic observations per event, and that the open azimuth at the event be less than 180". In the end, we used PcP and/or PKP arrival data from 48,280 events reported from 2,575 stations. The geographic distribution of the events and stations is shown in Figs l(a) and (b), respectively.

In the inversion of traveltime data it is necessary to remove the effects of presumed known structure in order to isolate the signal due to unknown structure of interest. Traveltime residuals of the core phases were computed using the earthquake relocations described above and by accounting for spherical imp92 structure, hydrostatic ellipticity, station corrections and mantle heterogeneities. We have investigated traveltime corrections derived from two aspherical mantle models: L02.56 of Dziewonski (1984); and isc5-11 of Pulliam, Vasco & Johnson (1992). Note that L02.56 is a model of aspherical velocity perturbations to the PREM model (Dziewonski & Anderson 1981) spanning the lower mantle from a depth of 670 km to the CMB (depth of 2891 km) and that isc5-11 is a model of aspherical velocity perturbations to the imp92 model (Kennett & Engdahl 1991) spanning the crust, upper and lower mantle from the surface (0 depth) to the CMB (depth of 2889 km). We found no compelling evidence that either model greatly removes the effects of heterogeneity using the statistical analysis described below. However, in the hope of removing possible biases in the data, we removed the effects of L02.56 on each residual for the CMB inversions.

Selecting the data to be used for an inversion requires some analysis. One must balance the advantages of including as many rays as possible (to improve the geographic coverage and averaging statistics) and the disadvantages of introducing data which are possibly misidentified or are more likely to be contaminated by the

CMB topography from ISC traveltimes 993

Figure 2. Ray paths of PcP, PKPab, PKPbc and PKPdf used in this study.

effects of unmodelled structure and errors. There are several criteria that must be considered when selecting the data for such an inversion. Two such criteria are the epicentral distance range and the cut-off time. The distance range is chosen in order to eliminate possible phase misidentifica- tions, as is discussed below. The cut-off time is a window around the mean residual within which data are kept for inversion. We discuss the cut-off, and its implications for inversions, in the following sections.

The phases used for our CMB topography inversions were: PcP, PKPab, PKPbc and PKPdf. The ray paths of these phases are shown in Fig. 2. There are three phases which pass through the outer core because the outer core P-wave velocity is lower than that of the lowermost mantle (i.e. a triplication in the traveltime curve). We also analysed and inverted PKPab-PKPdf and PKPbc-PKPdf differen- tial traveltimes, which are less sensitive to errors in event location and station biases.

PcP data were initially included in the epicentral distance range 25" to 85", but we later restricted this range. Fig. 3(a) shows over 81,000 PcP residuals plotted as a function of the observed epicentral distance, A. The plot shows con- siderable scatter, particularly at distances greater than about 70". Direct P, p P and PcP arrivals become poorly separated at these distances, and possibly lead to misidentifications. The observed scatter may be due to the fact that rays at such distances (A > 70") sample more of the D" region, which is known to be more heterogeneous than the overlaying mantle (Dziewonski 1984; Lay 1989; Gudmundsson et a f . 1990). Also, at distances greater than 70" precursors of PcP have been observed which are consistent with a reflection above the CMB, (Davis & Weber 1990; Weber & Korning 1990; Neuberg & Wahr 1991). Thus we discarded rays that were observed beyond 70". In addition, there appears to be a band of scatter in the distance range of about 40" to 45". In this distance range the PcP arrival is nearly coincident with

994 A . Rodgers and J . Wahr

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Figure 3. PcP residuals after application of iusp91, ellipticity, station and L02.56 aspherical mantle corrections. (a) (top) residual time, 6t, versus epicentral distance, A. The data have been binned into 0.1 s by 1.0 degree bins, and are plotted at different gray shades corresponding to the percentage of the entire data set each bin comprises. (b) (Middle) residual time mean and rms as a function of epicentral distance. Data have been binned in 1.0 degree bins. (c) (Bottom) histogram of residual data binned in 0.1 second bins. Also plotted are Gaussian and Lorentzian distributions of equal mean and rms.

the PP and PPP arrivals. One can also see this as a kink in the running mean versus A plot in Fig. 3(b). To avoid possible misidentifications of arrivals and to reduce contamination of the desired CMB signal by lowermost mantle structure, we included for our inversions only data in the distance ranges: 30" to 40" and 45" to 70".

The traveltimes of several thousand PKP rays, corrected to surface focus, are shown in Fig. 4. Fig. 4(a) shows the distance range 140" to 180". Fig. 4(b) shows the detail of the triplication in the distance range 144" to 156". Also shown in

Fig. 4 are the theoretical traveltimes given by iasp91 and the caustics of the traveltime curves. PKP phases must be selected carefully as the arrivals of different branches can be nearly coincident. It is important to not misidentify the PKP phases as the ray paths of the various phases are quite different. From Fig. 4(b), it is clear that at distances in the range 144" to 153" it is difficult to separate PKPab, PKPbc and PKPdf. We used PKPbc observations in the distance range 146" to 155". This is similar to what was done by Morelli & Dziewonski (1987a and b). They found evidence

C M B topography from ISC traveltimes 995

around 40°, has a generally negative slope in the mean versus distance. The PKPdf has a positive offset and PKPab has a smaller negative offset. In our CMB inversions we solved for baseline time offsets and perturbations in the depth of the CMB, which can be related to trends in the running mean versus distance. An evaluation of the iasp91 earth model, the traveltimes of core phases and global lateral heterogeneity is currently underway (Rodgers et al. 1992).

Histograms of the number of individual residuals for all phases, binned in 0.1 s intervals, are shown in Figs 3(c), 5(c), 6(c) and 7(c). The traveltime residuals of the identified phases vary up to 10.0 s. One (1.0) second of traveltime can correspond to a great deal of CMB topography: 5.6 km for PcP, and 9.6 km for PKP. Thus the variance of the residuals is much greater than traveltime signal consistent with other geophysical inference of CMB topography (e.g. geoid modelling). That such a large traveltime variance is observed indicates that the CMB signal will be difficult to isolate and that the noise level present in the data is significant.

These histograms can be useful for determining the cut-off time. The cut-off is a window around the mean residual within which data are kept for inversion and outside of which data are discarded. Most of the residuals lie fairly close to the mean. Obviously, the cut-off affects the size of the variance of the residuals. We performed several inversions of PcP data with different cut-offs to investigate the effect on the inferred CMB topography. For reference, Gaussian and Lorentzian distributions of equal mean and standard deviation (and thus equal area) are plotted in Fig. 3(c). The distributions of each data set matches a Lorentzian distribution much more closely than it does a Gaussian distribution. To account for the non-Gaussian distribution of the PcP residuals, we have performed maximum likelihood inversions in addition to solution by generalized inverse techniques. Results are discussed in Section 7.

Data from events as deep as 670 km were included in the inversions. Traveltime data from shallow events are more likely to be contaminated by depth phases ( p P , sP, pPcP, pPkP, etc.). This study and others (Gudmundssun et a f . 1990) found that residuals from shallow events had a larger variance than residuals from deep events. This is consistent with depth phase contamination as well as the observation that larger amplitude heterogeneity is concentrated in the upper mantle and crust.

We now consider the spatial distribution of the data. A map of where the individual PcP, PKPab, PKPbc and PKPdf data were reflected or refracted by the CMB is shown in Figs 8(a)-(d), respectively. These will be referred to as 'hit maps'. Note that the patterns of hits are very non-uniform, with small regions of very dense coverage as well as large regions lacking any data, particularly in the Southern Hemisphere and in the Pacific, Indian and Atlantic Oceans. Clearly, if one attempts to seek a global solution from such a non-uniformly distributed data set, it is necessary to determine the possible biases which may arise from the geographic distribution of data.

To avoid over-weighting the data in well-covered areas, we constructed composite residuals similar to Spakman (1988) and Hager & Clayton (1990). Note that composite rays are different from summary rays (Dziewonski 1984) in

1100 140 145 150 155 160 165 170 175 180

Epicentral distance A (degrees)

1220 , , , I , , , , , , , I I , I

(b)

P 144 146 148 150 152 154 156

1160 ua-LL Epicentral distance A (degrees)

Figure 4. Surface focus corrected traveltimes of P K P . The three branches of the P K P triplication (ab, bc and df) are apparent. The caustics (A, B, C, D and F) are marked where applicable. (a) For the distance range 140" to 180". (b) For the distance range 144" to 156".

to support the argument of Anderssen & Cleary (1980) that 85 per cent of arrivals in this distance range are PKPbc phases. They argue that this is due to the low amplitude of PKPab and PKPdf phase arrivals relative to PKPbc. However, it is still likely that the PKPbc identifications are severely contaminated by PKPdf arrivals. The phase identification algorithm of Engdahl exercises care while attempting to identify PKPbc. It uses a time window centred on the theoretical arrival time that is narrow at smaller distances ( f l .O s at 146") and grows with distance (to f5.0 s at 155"). We have used PKPbc data identified by Engdahl in this manner and our CMB modelling results are different from those we have obtained following Morelli & Dziewonski's identification scheme. We note that, by using Engdahl's scheme for identifying PKPbc , we may somehow be systematically discarding PKPbc data, however, those phases that we do identify as PKPbc are likely to be PKPbc indeed. For PKPab and PKPdf phases, we used data in the distance range 155" to 180". These phase arrivals are quite well separated in time over this distance range, and it is very unlikely that these phases are misidentified.

Figures 5(a), 6(a) and 7(a) show the branches of PKP plotted as a function of epicentral distance, A. The running mean and rms as a function of A are plotted in Figs 5(b), 6(b) and 7(b) for PKPab, PKPbc and PKPdf, respectively. These statistics are fairly constant over the range of distances plotted, indicating that the iasp91 earth model produces the traveltimes of these phases rather well, with the exception of PcP which, disregarding the PP cross-over

996 A . Rodgers and J . Wahr

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Figure 5. Same as Fig. 3 , but for PKPub data. We only plotted the distance 155" to 170" because there are very few PKPub observations beyond 170".

that composite rays retain the sensitivity of each ray to the sought-after structure. (A summary ray is a single ray which represents a number of individual rays from a given source-to-station region. See Spakman 1988, for a discussion of these spatial averaging techniques.) The Earth's surface was partitioned into 448 roughly equal-area source and receiver regions, 10" by lo" at the equator. These regions are approximately 1100 km square at the Earth's surface and 600 km square when projected onto the CMB. Residuals of rays that originated from the same source region and were observed in the same receiver region were averaged to form a composite residual. Rays from the same source and

receiver regions constitute a bundle of rays that sample roughly the same long-wavelength heterogeneity. By doing this procedure one hopes to enhance the signal common to rays along that bundle and average out the effects of small-scale heterogeneity and random errors. The Fresnel zone for 1 s period P waves is typically 400 km square. This region is smaller than the composite ray regions projected onto the CMB (approximately 600 km square), as well as the smallest wavelength of CMB undulations sought in our inversions (approximately 3600 km). Thus, the residuals averaged in this manner will contain a common long- wavelength signal, which we hope is enhanced by the

C M B topography from ISC traveltimes 997

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averaging procedure. We will discuss our efforts to procedures. For N residuals, {a t i } where i = 1, 2 , 3, . . . , N, understand this and another possible spatial averaging we compute the mean, at: scheme in later sections. I N

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3 DATA ANALYSIS A N D MODELLING THE COMPOSITE RESIDUAL VARIANCE and variance, a:,:

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We now present our analysis of the corrections applied to

corrections) and the traveltime residuals before inversion for CMB structure. Various statistics of the residuals and the The rms (standard deviation), a,,, is the square root of the corrections applied to the residuals were found at each stage variance. These statistics are computed for individual and in the processing in order to evaluate the processing composite residuals. The objective at this stage will be to

998 A . Rodgers and J . Wahr

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note whether there is a significant change in the mean or rms after a correction is applied.

To characterize the spatially averaged data we computed the composite residual variance of Morelli & Dziewonski (1987a), o',. For composite residuals computed by averaging N individual residuals, { G t ~ " } where i = 1, 2, 3, . . . , M,, this is defined as:

(3)

where st(,) is the mean of composite residuals with N individual residuals. Thus, u', is the estimated variance of

all the composite residuals which are composed of N individual residuals.

The value of u', depends on N. If the residuals of the individual rays were purely random and uncorrelated, then .", would take on the form: a; = &IN, where u', is the variance of the individual residuals. If, on the other hand, there were no noise or errors and the variance was due only to lateral heterogeneity of spatial scales greater than the composite ray grid size, then u', would be a constant: .', = a i , independent of the averaging procedure. That is, each individual ray in a composite ray would be affected the same amount by the heterogeneity, and so the variance of a composite residual would not decrease with an increase in

CMB hit points for PcP

CMB hit points for PKPab

CMB hit points for PKPbc

CMB hit points for PKPdf

Figure 8. Geographic distribution of the sampling of the CMB (hit maps): (a) (top) PcP, (b) PKPub, (c) PKPbc, (d) (bottom) PKPdf. All data were selected with a 4.0s cut-off and correspond to the data sets described in Tables 1 and 2. The maps are Mollweide projections.

C M B topography f rom ISC traveltimes 999

7 m T - PcP composite residuals. 4 0 s rut-off

Obefore corrections (lasp91) f f R = 177 s uw = 103 s

uR = 169 s r8 = 052 s

nbefore inversion (lasp9I. hyd, stal & mantle) :I.-- * / 0 ' 0 I 0 5 10 15 20

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1 5 2 2 5 uH, coherent component ( s )

Figure 9. (a) (Top) composite ray variance, uk, for PcP, 4.0s cut-off data. There are two cases: one in which only the imp91 correction has been applied, and the second case in which all corrections (iusp91, ellipticity, station and mantle heterogeneity) have been applied. Also shown are the fits to the data, following

= u ~ N + u:,. The fit parameters are given in the figure. (b) (Bottom) the 95 per cent confidence intervals for the estimation of the fit parameters shown above. Also shown is the 95 per cent confidence interval for the fit parameters after solving for and removing a CMB topographic model. We discuss this result in Section 6.

the number of individual rays per composite ray. In that case u', would be a statistical measure of the heterogeneity. Fig. 9(a) shows the composite residual variance for PcP residuals with a 4.0 s cut-off. There are two cases shown in the figure, one is the PcP imp91 residuals, the other is the PcP residuals before inversion, i.e. after hydrostatic ellipticity, station and aspherical mantle corrections have been applied. Fig. 9(a) indicates that u', is made up of contributions due to both random as well as deterministic processes. Following Morelli & Dziewonski (1987a), we fit the composite residual variance to:

The fit parameters (uR and (I") are given in the figure. The parameters are fit by means of a maximum likelihood technique where we explicitly compute the probability density function on a dense grid of points in uR-uH space and search for the (global) maximum. We found this technique to be superior to simple least-squares fitting because it is robust (i.e. more appropriate for non-Gaussian distributed data) and it allows us to determine confidence

1000 A . Rodgersand J . Wahr

Table 1. Processing statistics for the residuals and the corrections performed on PcP, PKPab, PKPbc and PKPdf traveltimes, all with a 4.0s cut-off. The rms, u,,,, and the coherent component of the composite residual variance, u,,, are defined in the text. We include the per cent change in the composite residual rms, Au,,,, and the coherent component of the composite residual variance, Au,,. A negative value of A u indicates a reduction. The superscripts ind. and comp. correspond to the rms for individual and composite residuals, respectively. The superscript corr. corresponds to the rms for the corrections themselves.

PCP Correction

imp9 1 ellipticity station mantle (L02.56)

PKPab Correction

iasp91 ellipticity station mantle (L02.56)

PKPbc Correction

imp91 ellipticity station mantle (L02.56)

PKPdf Correction

imp91 ellipticity station mantle (M2.56)

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AUH

_ _ -50.0% -27.1% 10.2%

A'SH

-_ -34.0% -8.7% -1.6%

intervals and trade-offs between the fit parameters. We describe this and other techniques for modelling residual data in detail in Rodgers & Wahr (1993b). In Fig. 9(b), we show the 95 per cent confidence intervals on the estimates of the fit parameters for the cases shown in Fig. 9(a). Typical

uncertainties for these parameters, for all phases, (at the 95 per cent level) are f 0 . 2 0 ~ for uR and f 0 . 1 0 ~ for uH. Clearly the coherent component of the variance is reduced by the application of the hydrostatic ellipticity, station and aspherical mantle corrections. Also shown in Fig. 9(b) are the fit parameters for the PcP residuals after fitting for and removing a CMB solution, which we will discuss in Section 6.

The statistics for the corrections to the absolute traveltime data sets, with a 4.0s cut-off, are summarized in Table 1. The row entries mean the following: ellipticity are the resulting statistics after the imp91 residual was corrected for ellipticity; station are the resulting statistics after the imp91 residual was corrected for ellipticity and station corrections; mantle (L02.56) are the resulting statistics after the imp91 residual was corrected for ellipticity, station and aspherical mantle corrections computed from L02.56 (Dziewonski 1984). The best measure of the success or failure of a given correction is probably the change in the composite residual rms, AU::"'~. The change in the coherent component of the composite residual variance, A o H , represents how much of the non-random component of the composite residual variance was successfully modelled. This statistic is, generally, consistent with the rms reductions, i.e. when the composite residual rms decreased so did the coherent component of the composite residual variance. (Negative signs indicate that the rms or coherent component of the composite residual variance was reduced.) One can see, in Figs 9(a) and (b), that the coherent component, uH, of a: is reduced following application of the ellipticity, station and aspherical mantle corrections. The ellipticity and station corrections are clearly significant, reducing both u:ymP and uH. The removal of mantle heterogeneity does not greatly affect the rms of the data. In fact in some cases, the coherent component of the composite residual variance increases (e.g. the mantle correction to the PKPub and PKPbc data), indicating that a coherent signal is being added to the residuals rather than removed. However, these increases are probably not statistically significant as will be discussed below along with the error estimates.

The statistics for all data, after all corrections and a 4.0s cut-off have been applied, are given in Table 2. We note that, generally, we use much more data than previous studies. This is because we used the entire ISC catalogue between 1964-1989, as well as an improved algorithm for selecting the data. Also, the residual data rms of each phase tends to be less than those reported in other studies. This is

Table 2. Statistics for data after all corrections have been performed and a 4.0s cut-off has been applied, but before CMB inversions: the number of individual rays used, mean and rms; the number of composite rays used, mean and rms; and the fit to the composite residual variance. The superscripts ind. and comp. correspond to the statistics for individual and composite residuals, respectively. The parameters used to model the composite residual variance, uR and a,,, are defined in the text. The errors for these parameters are ~k0.20 s for oR and 10.10 s for u,,.

- Phase ind. rays tind. okd comp. rays 8tcomp O g m P O R OH

(sec) (sec) (sec) (sec) (sec) (sec)

23,616 0.52 1.54 2,560 0.52 1.38 1.69 0.52 PCP PKPab 31,752 -0.30 1.44 929 0.17 1.13 1.30 0.83

178,159 -0.09 1.13 2,027 -0.04 0.87 1.19 0.47 PKPbc 55,216 0.73 1.62 1,489 0.66 1.32 1.71 0.80 PKPdf

PKPab-PKPdf 10,883 -0.28 1.45 714 -0.15 1.17 1.44 0.61

C M B topography from ISC traveltimes 1001

un and uH, for PcP residuals before and after inversion, using both conventional composite and patch rays, and for both 8.0s and 4.0s cut-offs. It is evident that the coherent part of the variance, uH, is smaller in the case of patch rays than for the case of composite rays, for a given cut-off. However, the error estimates for these parameters are 0.10 s for uH and 0.20s for uR. So, it is not clear if the difference in uH for composite and patch rays is significant. (We will show below that there is another way to parameterize the patch residual variance which allows easier interpretation of the results.) Sometimes the azimuthal distribution of the individual rays in a patch ray is poor, i.e. all the rays come from the same source region and are observed in the same station region. In these cases, the patch residuals are nearly the same as the composite residuals, and the goal of trying to eliminate the long-wavelength mantle structure signal by averaging at the patch, is not achieved. Also note that the coherent part of the variance, uH, is smaller for the 4.0s cut-off case than in the 8.0s cut-off case for both spatial averaging methods. This indicates that the further outlaying residuals ( \& I> 4.0 s) contribute to the coherent component of each composite residual. This suggests that our estimates of uH are biases upward due to noise in the data, since it is unlikely that CMB topography could cause residuals with 16tl> 4.0 s. Note that by comparison, the random component of the variance increases much more strongly as the cut-off is increased.

The values of uH for the patch residuals could still be affected by large-scale mantle heterogeneity. For example, suppose that only a few composite rays hit any given patch, and that these composite rays travel along neighbouring paths in the mantle (i.e. the azimuthal sampling for the patch is poor). Then, long-wavelength mantle structure in the vicinity of these composite rays would tend to affect all the individual rays the same amount, and this would cause an increase in uH for the patch.

We have developed two statistical techniques for separating the variance due to CMB topography from the variance due to mantle structure. For the first technique, suppose a patch ray is composed of N individual rays organized into M composite rays. Let u’, be the variance of the patch ray. If the individual data were entirely random noise with variance, o$, then we would expect a;= &IN. If the data were entirely CMB signal with wavelength much greater than the patch dimension, then u’, would be independent of N and M: u: = ugMB where u’,,, is the variance of the individual ray residuals caused by CMB topography. As a third option, suppose the residuals were due to structure in the mantle at wavelengths long compared with the composite ray bundle, and, suppose the mantle signals for each of the M composite rays were uncorrelated. Then, we would expect: u’, = d,/M.

Thus, the PcP patch residual variance arises from three contributions: random processes, mantle heterogeneity and CMB topography. We represent the variance of any patch residual as:

probably because the event relocation/phase re- identification algorithm of Engdahl is able to relocate the events with much greater accuracy as well as reducing phase misidentifications. The composite residual variance fit parameters, and uH, indicate that the random component of the variance exceeds the coherent component by a factor of two to three. The coherent signal as measured by uH is between 0.39 and 0.89s. This is certainly large enough to be related to lateral heterogeneity. However, as we will show, we cannot distinguish between the effects of mantle heterogeneity and CMB structure with this parameterization of the composite residual variance.

4 MODELLING THE PATCH RESIDUAL VARIANCE

The goal of constructing composite rays is to enhance whatever long-wavelength coherent signal each individual residual possesses and to reduce the random and small-scale structure contributions by averaging. One problem that composite rays do not solve in the present application is how to distinguish between mantle heterogeneity and CMB structure. Composite residuals are an average of all residuals from a bundle of rays that sample long-wavelength mantle heterogeneity as well as the CMB. The coherent component of the composite residual variance could arise from mantle structure as well as from CMB topography.

For PcP phases it is possible to use a different spatial averaging scheme in which all residuals from rays that hit the same region (patch) on the CMB are averaged together. We will refer to the components of this spatial averaging scheme as ‘patch’ rays and ‘patch’ residuals. In this case, it is possible that residuals from rays that sample different long-wavelength heterogeneity in the mantle are averaged together. Thus, the effects of long-wavelength mantle heterogeneities on the residuals should be averaged out when the number of rays contributing to a patch residual is sufficiently large and the azimuthal and epicentral distance coverage is uniform. An analysis similar to that done above with the conventional composite rays can be done here, where we solve for the patch residual variance, a;, and use a maximum likelihood technique to fit for the parameters:

and uH, as in eq. (4). Table 3 shows the fit parameters,

Table 3. Composite residual fit parameters, uR and uI1 (defined in eq. 4) for PcP 4.0 and 8.0s cut-off data. There are two cases shown: data before inversion and data after inversion and removal of the inferred solution. The errors for these parameters in both cases are k 0 . 2 0 ~ for uR and f 0 . 1 0 ~ for u,,.

PcP oR & oH cut-off averaging OR OH (sec) scheme (sec) (sec)

Before Inversion

After Inversion 1 , 0

1, *I

I, 0 1

4.0 4.0 8.0 8.0

4.0 4.0 8.0 8 .O

composite patch

composite patch

composite patch

composite patch

1.69 0.52 1.72 0.49 3.42 0.62 3.63 0.49

1.69 0.39 1.84 0.24 3.43 0.47 3.59 0.30

We used a maximum likelihood technique to solve for the variances a:, u’, and u2CMB, (Rodgers & Wahr 1993b)

1002 A. Rodgers and J . Wahr

- x

Table 4. Patch residual-fit parameters, uR, u, and acMB (defined in eq. 5 ) for fcf 4.0 and 8.0 s cut-off data. Shown are the fits to the data before inversion and after inversion and removal of the inferred solution. The errors for these parameters in both cases can be inferred from the 95 per cent confidence interval plots (Fig. 10). They are: f0.30 s for 0,; f0 .20 s for a,; and f0.20 s for ucMB.

Pcp uR. uM & urnB cut-off avera@g UR DM ocm (sec) scheme (sec) (sec) (sec)

Before Inversion 4.0 patch 1.79 0.73 0.15 I , IS 8 .O patch 3.56 0.96 0.10

After Inversion 4.0 patch 1.72 0.48 0.00 I, I, 8.0 patch 3.43 0.64 0.00

a before inversiom * after inversion -

1 - I

explicitly computing the probability density on a dense grid in the (3-D) parameter space. Table 4 shows the results for these fits with the PcP 4.0 and 8.0 s cut-off data, before and after inversion. The signal due to mantle heterogeneity overshadows that due to the CMB by nearly five times in the 4.0 s cut-off case, and by nearly 10 times in the 8.0 s cut-off case. The uncertainties were estimated similarly to what we did for the conventional composite residual variance. In Fig. 10 we show the 95 per cent confidence intervals for the three parameters, for the PcP data with a 4.0 s. cut-off. They are

95Z confidence int.rrviils

I - - -4.. d3 15 2 5

a' 0 1 1 ' ' ' ' ' ' ' ' 1 ' ' ' ' 6 , ( r x sigma random (s)

1 5 , , , , , , , , , , , , , , , , , / , , , ,

0 2 4 6 8 ow, sigma mantle (s)

i: 12 1 4

1 1 5 2 25 3 o., sigma random (s)

Figure 10. 95 per cent confidence intervals for the estimates of the fit parameters to the patch residual variance. They are each plotted for a pair of parameters while the third is held at its most likely value. There are two cases shown, one is the fit to the fcf patch residual data before inversion (the triangle surrounded by the dotted contour line) and the other is the fit to the fcf patch residual data after inversion and removal of the inferred solution (the asterisk surrounded by the dashed contour line).

each plotted for a pair of parameters while the third is held at its mostly likely value. There are two cases shown, one is the PcP patch residual data before inversion (the triangle surrounded by the dotted contour line) and the other is the PcP patch residual data after inversion and removal of the inferred solution (the asterisk surrounded by the dashed contour line). We discuss the results of the second case in Section 6. Note from the shape of the confidence interval that there appears to be a trade-off between the two parameters uM, and ucMB.

A weakness in this approach is the assumption that the effects of mantle structure on different composite rays are uncorrelated. To avoid this assumption, we have used a second technique, where the patch residual variance is represented in terms of a correlation length, and a random and a coherent variance (A, a: and a',, respectively). We do not distinguish directly the effects of CMB topography and mantle structure. Instead, we focus on the correlation length, A. That parameter is intended to represent the characteristic scale length of heterogeneity sampled by the PcP rays. It is defined so that if two rays are separated by an angular distance greater than A, the residuals are effectively uncorrelated. That is, we assume that the linear correlation between the ith and jth residuals is given by:

pi, = exp [ -+] where A, is the angular distance between points on the ith and jth rays. We introduce two ways to measure Aij below. The random variance represents the incoherent (noise) component of each residual and is measured in seconds. The coherent variance models the amount of CMB topography, mantle heterogeneity, or any other long-wavelength signal that could arise from the non-random component of each residual and is measured in kilometres of equivalent CMB topography. The patch residual variance, a:, is then represented in terms of the parameters A, uR and uc, in an expression of the form:

(7)

where N is the number of individual rays hitting a patch. We used a multivariate maximum likelihood technique where we compute the probability density function on a dense grid in the 3-D parameter space and search for the global maximum. The details of this technique and eq. (7) are described in Rodgers & Wahr (1993b).

We use two different definitions for the distance, A,, between rays i and j . In the first, (case A), Ai, is the angular distance between the two reflection points on the CMB. In the second (case B), Ai, is the angular distance between the points where each ray intersects the Earth's outer surface. In this case we use either the event-event distance or the event-station distance, whichever is smaller.

The interpretation of I is different in these two cases. For case A, the underlying assumption is that the coherent variance is due to CMB topography, and is not affected by mantle structure. In this case, A represents the correlation length of CMB topography, and u', describes the variance of that topography. If A is small then the CMB is likely to be dominated by short-wavelength topography.

CMB topography from ISC traveltimes 1003

For case B, the value of A provides insight into whether the residuals are dominated by CMB topography or by mantle structure. Suppose, for example, that long- wavelength CMB topography provides the dominant signal. Then any two individual rays in a patch ray will be perfectly correlated, regardless of where the two rays intersect the Earth's outer surface. In this case, the correlation length will approach infinite, and .'c represents the variance of the CMB topography. On the other hand, suppose the signal is dominated by unmodelled structure in the crust and upper mantle. Then, different individual rays will be affected differently by that structure, and A will represent the wavelength of the heterogeneity and u: its variance. For structure deeper in the mantle, d will be larger than the wavelength of the structure, and will approach infinity as the inhomogeneities approach the CMB.

The parameters were fit for several data sets, measuring Aij at the CMB (case A) and at the Earth's surface (case B), in order to test the method and assess the relationship between the PcP residuals and CMB topography. The data sets (and a brief descriptive title) are: (1) PcP residuals using a 4.0s cut-off (PcP) ; (2) random numbers, of mean, rms and distribution equal to that of the PcP residuals (Random); ( 3 ) synthetic residuals derived from a smooth, long-wavelength (maximum spherical harmonic degree L = 5) CMB topography of moderate amplitude (Smooth); and (4) synthetic residuals derived from a smooth, long-wavelength (L = 5) CMB topography of moderate amplitude added to random numbers resulting in a data set distributed similarly to the PcP residuals (Synthetic). Recall that the inversions seek long-wavelength CMB topography, so that results of the Synthetic case provide an example of what the real data should look like if they are related only to long-wavelength CMB topography and contain a noise component.

The results of the fits are presented in Table 5. We discuss the uncertainties in the estimates of these parameters below. First consider case A, where Aij was measured at the CMB. In the Smooth case the data fit a correlation length of 15.6" and a very small random component. This tells us that if CMB topography could be completely described with spherical harmonics up to degree L = 5 , and there were no

Table 5. The results of fits for 1, uR and uc. The cases and uncertainties are explained in Section 4 of the text.

data set case h OC OR (degrees) (km) (sec)

Smooth A 15.6' 4.6 0.4 Synthetic A 10.5' 3.0 1.7 PCP A 2.4' 8.0 1.8 Random A 0.0' 0.5 1.4

Smooth B > 360.0' 4.3 0.5 Synthetic B 30.0' 3.0 1.7 PCP B 6.1' 7.7 1.8 Random B 3.1' 0.6 1.5

noise in the data, then the fit parameters would take on these values. This value is consistent with what we have estimated analytically (Rodgers & Wahr 1993b). If we allow noise to be added to the Smooth data set we then have the Synthetic data set. These data preferred a smaller correlation length and a larger random variance. This data set is a more realistic approximation to the real PcP data as there is both a coherent and a random component to each residual, however, the coherent component is composed of only long-wavelength CMB structure. In this case though, we find that d is reduced from the value inferred with the Smooth data set. This indicates that we cannot completely distinguish the parameters and is related to the uncertainties in the estimates discussed below. The PcP data are not correlated across the spatial averaging patches (which are 5" square). This is mostly likely because small-scale structure and noise dominate the long-wavelength CMB signal. The Random case shows that completely uncorrelated residual data fit A = 0" and aKoh,.

The results of the fits for case B, where Ajj was measured at the Earth's surface, show the same trend as case A. For the Smooth data set we obtain the expected result that A approaches infinite (i.e. the coherent signal arises deep in the Earth). When noise is added (Synthetic data set) 1 is reduced, but still larger than that obtained in case A. The

I'cP case B

5 2

c

l , , , , l ~ , , , l , , , , i , , , , l , , , , l ~ ~ ~ l l

3 , , , , , / 1 , , / 1 , , , 1 , , , 1 , , , 1 , / , / , , , ( , , , ~ , , 1

0 100 150 200 250 300 350 50

c A correlation length (degrees)

Q i

c 2 4 6 8 10 12 14 16 18 20

oC sigma correlated (km)

15

P 1 4 1 ! l l ! r # 8 1 1 t # t I f l 8 t l L 5 I C l 8 1

0 100 150 200 250 300 350 A correlation length (degrees)

50

Figure 11. Confidence intervals for the estimates of 1, uc and uR for the PcP data set, case B (where A,, is measured at the Earth's surface). The best fitting values are indicated with an asterisk. They are each plotted for a pair of parameters while the third is held at its most likely value. The inner contour is the 63 per cent confidence interval and the outer contour is the 95 per cent confidence interval.

1004 A . Rodgers and .I. Wuhr

PcP data set prefers a small correlation length of 6.1". This is only slightly larger than that obtained in case A for PcP. However, we must consider the uncertainties.

The uncertainties of these estimates can be inferred from confidence interval plots, similar to that done for the patch residual variance fits. In Fig. 11 we show the 63 per cent and 95 per cent confidence intervals for the estimates of A, uc and uR for the PcP data set case B. The error in uR is f0.5 s. The error in uc is f2 .0 kilometres. The error in A is quite large for both confidence levels. Note that A = m is possible at the 95 per cent confidence interval. The large errors for A arise because we are trying in model the effects of heterogeneity at all spatial scales with a single parameter, A. However, the probability density is fairly strongly peaked at the maximum value, so the PcP data seem to prefer a small correlation length.

5 INVERSION APPROACH

After the pre-processing described above, the traveltime residuals were inverted for CMB topography. Only the longest wavelength topography was sought in the inversion. For a detailed description of the principles and assumptions of seismic traveltime inversion, see Nolet (1987).

CMB topography, 6r(6, $), was parameterized with a spherical harmonic expansion, truncated at degree and order 1 = L:

where 8 is the colatitude, + is the east longitude, and the functions p;l(cos 6) are normalized Legendre functions, defined as:

(Morelli & Dziewonski 1987b). The P;"(cos 6 ) are Associated Legendre polynomials (Abramowitz & Stegun 1964). The maximum harmonic degree, L, we considered was 6, which corresponds to a horizontal wavelength of about 3600 km, on the CMB. The a t term corresponds to a uniform radial perturbation of the CMB. In all inversions we simultaneously solved for a time constant, to, in addition to CMB parameters a;l and b y (including a:;).

Ray paths were computed using the spherical earth model iusp91. Kernels relating the effect of a given component of topography, {a;", b y } , on the traveltime of given ray were computed. Linearized relations for the effect of small boundary topography on body wave traveltimes have been found by Dziewonski & Gilbert (1976) and Rodgers & Wahr (1993a). The traveltime of a PcP phase, which is reflected from the mantle side of the CMB, depends on the topography, 6r(B, $J), in the following way:

For refracted PKP rays the dependence is:

In (10) and (11) to is a time offset, vtl- is the P-wave velocity of the medium directly above/below the refraction point (taken from iasp91), i+ / - is the incident angle at the refraction point, p is the ray parameter, and ro is the unperturbed spherically averaged CMB radius (taken from iasp91 to be 3482 km). Since PKP is refracted by the CMB upon entry and exit from the core, the sensitivity of PKP traveltime to CMB topography must be considered at both points. The method for correcting the traveltimes for aspherical structure parameterized by a spherical harmonic expansion was taken from Backus (1964) and Dziewonski (1984).

Several inversions of the different data sets were performed, including several inversions of PcP to understand how the data selection, processing and inversion techniques impact the CMB solutions. The amplitudes and spatial characteristics of different models were compared by finding the rms (standard deviation) of the inferred topographic amplitudes (ahr) and degree correlations (r,) between models, respectively.

A model is determined by its spherical harmonic parameters: {a?, b y } The degree variance of a given map (related to the power in a given degree) is defined by:

m =O

The variance of the topography is defined to be:

The rms of the topography, a,,, is the square root of the topographic variance. To compare the spatial patterns of two models, denoted with superscripts (1) and (2), we computed the degree correlation, r,, defined as:

I

The 1 = 0 terms cannot be included in this analysis. Confidence intervals were found using the Student t test. This method for comparing spherical harmonic sets is taken from Eckhardt (1984).

6 CMB INVERSION RESULTS

Maps of the CMB topographies inferred from the four absolute traveltime data sets (PcP, PKPab, PKPbc and PKPdf ) are shown in Fig. 12. Table 6 shows various statistics of the data sets and inferred solutions: the per cent rms reduction after removing the inferred solution; AuZmp; the per cent reduction in uH (computed using composite rays); and the rms of inferred topography, ubr. It can be seen that the inferred CMB solutions modelled only a small amount of the total composite residual rms. Reductions in the coherent component of the composite residual variance were significant. However, as was mentioned above, there is no way of distinguishing between the effects of long- wavelength mantle heterogeneity from that of long- wavelength CMB topography with composite rays. We do not know from the reduction of uH whether we have

C M B topography f rom ISC traveltimes 1005

CMB topography from PcP, 4.0 s, k 5 , SVD

CMB topography from PKPab. 4.0 s, k 5 . SVD

CMB topography from PKPbc, 4.0 s, k 5 SVD

10 8 6 4 2

-2 0 ;

-4 -6 -8

- 10

16 12

8 4

k O m

-4

-8 -12 -16

12 9 6 3 O k

m -3 -6

-13 -9

32 24 16 8

0 : -8

-16 -24 -32

CMB topography from PKPdf, 4.0 s, L=5 SVD

Figure 12. CMB topographies inferred from (a) (top) PcP, (b) fKPub, (c) P K f b c and (d) (bottom) PKPdf. The processing, selection criterion and inversion technique are identical for each data set, (see text for details). The 1 = 0 terms are not plotted. Note that the scales are different for each plot.

isolated the signal due to CMB topography, or inadvertently absorbed mantle heterogeneity into our CMB solution. Below we discuss results which suggest the latter.

This study infers large topographic amplitudes, as much as 30.0 km (absolute amplitude), though these amplitudes depend on the cut-off used, as discussed below. We did not correct PKPdf data for inner core anisotropy, modelled by PKPdf traveltimes (Morelli, Dziewonski & Woodhouse 1986; Shearer, Toy & Orcutt 1988; Creager 1991), which may be the cause of the large coherent variance and subsequent large-amplitude topography inferred by PKPdf.

There are few independent constraints on the size of CMB topography. If geoid anomalies arise solely from CMB topography, long-wavelength CMB topography must be less than 3.0 km (Bowin 1986). Studies which allow mantle

Table 6. The results of CMB inversion and removal of the inferred solutions: the per cent reduction of uh,; the per cent reduction in uF,; and the rms of inferred topography, uh,. The processing and inversion procedures are the same for each phase and are described in Section 6.

A 0 H 0 8 r

(sec) (km) Phase cu t-off AD$””’.

PCP 4.0 -3.5% -24.7% 4.14 PKPab 4.0 -6.0% -28.8% 5.33 PKPbc 4.0 -2.8% -21.7% 3.86 PKPdf 4.0 -6.3% -22.3% 11.62 PKPab-PKPdf 4.0 -5.6% -13.4% 3.68

density anomalies to drive flow and deflect the CMB suggest the topography might be less than 1.5km peak to peak (Hager et al. 1985; though see also Forte & Peltier 1989), but that conclusion depends critically on assumptions about the D“ viscosity and about the density jump directly across the CMB. Observed decade-scale variations in the Earth’s rotation rate have been used in the past as an indication that CMB topography should be no more than a few kilometres at most. But, recent work by Jault & Le MouGl (1990) argues that those conclusions are probably not justified.

One reliable, independent constraint is provided by Earth tide and nutation observations. Those observations con- strain the a: component of CMB topography to be between about 0.5 km and 1.0 km, and probably closer to 0.5 km (Gwinn, Herring & Shapiro 1986; Neuberg, Hinderer & Zurn 1987; Wahr & de Vries 1989). (In the parameteriza- tion we have adopted this corresponds to the a: = 1.77 km.) We have complied the inferred a: coefficients and error estimates (Aai;) for the four absolute traveltime data sets in Table 7. The results show that the formal error estimates are probably not reliable: the scatter of the inferred ax coefficients is much larger than the error estimates. The error estimates indicate that the PcP data are best able to

Table 7. Results for the a: components of CMB topography and errors in the estimates, Au:, inferred from the different data sets. FCN is the constraint from free core nutation modelling taken from Wahr & de Vries (1989). MDZ is the result inferred by Morelli & Dziew- onski (1987a) in a combined inversion of ISC Pcf and f K P b c residual data. The coefficients are normalized to eqs (8) and (9) in the text.

Constraint a: Aa: (km)

PCP 2.59 0.11 PKPab -0.80 0.37 PKPbc -0.83 0.33 PKPdf -6.95 2.06 MDZ * 0.14 0.24 FCN ** 1.77 ?

1006 A . Rodgers and J. Wahr

resolve this component of the topography, however, the value exceeds 0.5 km. The PKPab and PKPbc data prefer the a; component to be nearly equal to the constraint set by tidal and nutation observations. The PKPdf inference greatly exceeds the constraint, however, as mentioned above, these data are probably contaminated by inner core anisotropy. In fact our map of CMB topography inferred from PKPdf residuals is consistent with the pattern of residuals found by Morelli et al. (1986), Shearer et al. (1988), and Creager (1991), i.e. paths along the Earth's rotation axis are fast and those along the equator are slow.

Models of CMB topography inferred from different phases of the ISC data are found to be in poor agreement: just a glance at Fig. 12 shows few common features in the maps. Degree correlations between various CMB models are shown in Figs 13(a) and (b). The data processing, selection criteria, and inversion algorithm were identical for each map: iasp91, hydrostatic ellipticity, station, mantle heterogeneity corrections, and a 4.0 s cut-off were applied; the individual residual mean was removed; the maximum harmonic degree of topography sought was 5; and a singular value decomposition inversion algorithm was used with eigenvalues less than one-twentieth of the largest eigenvalue discarded. The degree corelations are near zero or negative for nearly all maps at all degrees, with a few exceptions. The fact that the models are so poorly correlated indicates that unequivocal inference of the CMB, with the ISC data set and a truncated spherical harmonic expansion, it not possible.

-5k / 1 harmonic degree. 1

- - PCP PKPbC

- - PCP PKPdf c 5 1

T O M /\' (b) - 1 1 1 1 5 I 1 I ( ' I 1 ' " I I I I I 1 1 1 1 1 1 I I 1 1 ' 1 I I

0 1 5 harmonic degree. 1

Figure 13. Degree correlations, r,, between CMB topographies inferred from PKP, (a) (top) and PcP, (b) (bottom) phases of the ISC data. The processing, selection criterion and inversion technique are identical for each data set, (see text for details). For evaluation the 90 per cent confidence interval, rw,%, is also plotted.

L

P P

M

1 2 3 4 5 6 harmonic degree. I

Figure 14. Degree correlations, r,, between CMB topographies from, (a) PcP and (b) P K P phases of the ISC data, inferred by this study and Morelli & Dziewonski (1987a). In these cases the maximum harmonic degree is 4. For evaluation the 90 per cent confidence interval, rW%> is also plotted.

Degree correlations for L = 4 models of this study and that of Morelli & Dziewonski (1987a, 1987b), are shown in Figs 14(a) and (b). Morelli & Dziewsonki obtained moderately good agreement between their PcP and PKPbc maps, especially for degrees 2 and 3, as can be seen in Fig. 14(b). We obtained poor correlation between our PcP and PKPbc maps, for both the L = 4 and L = 5 cases. Our PKPbc model is poorly correlated with the PKPbc model of Morelli & Dziewonski. However, we selected the PKPbc data in a much different manner (see Section 2) and this probably has led to the differences in the maps. Our PcP map is moderately well correlated with that of Morelli & Dziewonski. In general, we obtained poor correlations between all these maps at nearly all degrees.

We inverted PKPab-PKPdf and PKPbc-PKPdf differential traveltimes for CMB topography. The PKPbc- PKPdf data inferred extremely large topographic amplit- udes. These data are probably very poor because the two arrivals are nearly coincident and are extremely difficult to pick accurately. We decided that they are not worth analysing. Results of these inversions are also summarized in Table 6. Degree correlations between these phases and their absolute traveltime counterparts are shown in Fig. 15. The PKPab-PKPdf and PKPab maps are fairly well correlated.

To evaluate the solution obtained with the PcP residuals, we fit the composite and patch residual variance, after removal of the inferred solution, as described in Section 4.

CMB topography from ISC traveltimes 1007

traveltime effects of these parameters were removed from the data, the trend in the mean versus distance (apparent in Fig. 3b) was diminished. Perhaps there is an error in the imp91 and we are modelling it with these constant terms. However, we cannot infer exactly where the error arises. (An evaluation of spherical earth models is undertaken in Rodgers et af . 1992.)

I.

1 , , , I , , I , / , , I I , , , I , , , , , , , , , / , , , , , , , - PKPab-PKPdr PXPab-

- - PKPab-PKPdl PKPdf - 5 - -

i - 5 [ ,ryT,d,L

- 1 0 1 2 3 4 5

harmonic degree, 1

3 ,,,l,,,,l,,,,i

6 7 8

Figure 15. Degree correlations, r,, between CMB topographies from differential traveltimes of PKPab-PKPdf and the correspond- ing absolute traveltime data sets. In these cases the maximum harmonic degree is 5. For evaluation the 90 per cent confidence interval, is also plotted.

The results of the two parameter fits to the variance (aR and a") are given in Table 3 and the corresponding confidence intervals in Fig. 9. The random component changes, but only by a few per cent. Note, however, that the coherent component decreases significantly, by more then 20 per cent. The confidence intervals for the two cases (before and after inversion) overlap, but only slightly, see Fig. 9. Clearly the CMB solution models a significant part of the coherent variance, however, with these statistics, we have no way of knowing if we are modelling the effects of CMB topography. The three parameter fits to the patch residual variance (uR,aM and qMB) are given in Table 4 and the corresponding confidence intervals for the 4.0 s cut-off case in Fig. 10. In this case we should be able to separate the effects of mantle heterogeneity from those of the CMB. Here the results are compelling. We are able to model the entire coherent component due to CMB topography: acMB is reduced to zero in both the 4.0 and 8.0s cut-off cases. However, the coherent component due to mantle heterogeneity is also reduced by about 33 per cent for both cases. In absolute terms the reduction in uM is 1.5 to 2 times the reduction we obtained in a,,,. As seen in Fig. 10, the mantle heterogeneity component is significantly reduced after the inferred solution is removed. This indicates that the effects of mantle heterogeneity are not fully removed and are being mapped onto the CMB by our inversion. In fact, we are probably modelling more mantle heterogeneity than CMB topography as AuM is larger than AacMB.

As mentioned above, we solved for perturbations in the spherically averaged CMB radius and a time constant [the a: and t,, terms in (11) and (12)]. We solved simultaneously for these parameters and the aspherical terms. When solving for these parameters we followed the same processing and inversion procedures as outlined for the models shown in Fig. 12. When we did not remove the mean from the residuals before inversion, it was essentially absorbed into the to term; when we did remove the mean, t,, was essentially zero. The residuals generally preferred a slight, but not statistically significant, perturbation in the CMB radius. Unfortunately the uncertainties in the estimates of the spherically averaged CMB radius were quite large. The PcP data preferred a:: = 0.52 km and to = -0.42 s . When the

7 INVERSION EXPERIMENTS

Clearly the fact that the CMB hit points are non-uniformly distributed around the globe affects inversion results. When simple least-squares fit (LSF) inversions were performed, we obtained anomalously large features in regions with sparse coverage. This is because the solution is not constrained in regions that are sampled by no data. Also, the fact that large amplitude topography is inferred where the hit coverage is sparse can be understood by noting that the variance a:, for composite residuals composed of few individual residuals is much larger than that for composite residuals composed of very many individual residuals. This result is demonstrated in Fig. 9(a). Thus composite residuals are large in regions where the coverage is sparse and the least-squares inversion infers large topography. Further, where the coverage is dense composite residuals tend to be smaller and smaller amplitude topography is inferred.

To compensate for the uneven distribution of the data, one can use a singular value decomposition (SVD) inversion method. SVD eliminates the poorly determined degrees of freedom in the parameter space, which correspond to the most poorly resolvable features of the model. This method provides a smoother model than does LSF, as the amplitudes of features are similar in regions of good and poor geographic coverage. For the maximum harmonic degree of L = 5, the system of equations had 36 degrees of freedom. Typically, we discarded eigenvalues that were less than one-twenteth of the largest eigenvalue. This usually lead to discarding about 5 degrees of freedom.

The PcP residuals are not Gaussian distributed (see Fig. 3c), as they should be for generalized inversion techniques. We performed a maximum likelihood (ML) inversion on the PcP data set. This method accounts for the non-Gaussian distribution of residuals. Results of maximum likelihood inversions were similar to those obtained by the LSF method.

Figure 16 shows four maps of CMB topography derived from Pep, 4.0 s cut-off data: 16(a) is a map of the raw data, 16(b) is the LSF solution, 16(c) is the SVD solution and 16(d) is the ML solution. The first map (Fig. 16a) is simply the residuals multiplied by the appropriate CMB topography kernel, binned in 5" by 5" patches on the CMB, and averaged together. In this case we have discarded patches which were hit by fewer than five rays. Fig. 16(a) is a representation of the raw residual data, and not an inversion. The inferred patch elevations vary a great deal, as much as 20 km. There is some coherence in the raw data, notably in the Western Pacific, Andean South American, Siberia, north-eastern Africa and the North Atlantic. Notice how these regions are resolved on the SVD map better than on the LSF map. The amplitude of the LSF solution is larger than that of the SVD solution. Also, the largest anomalies of the LSF map are in regions where the data coverage is

1008 A . Rodgers and J . Wuhr

CMB topography from PcP. 4 0 s. k 5 , ML

Figure 16. CMB topography inferred from PcP, 4.0s cut-off: (a) (top) residual data are converted t o CMB topographic elevation by multiplying by the appropriate kerncl then averaged together in 5" bins on the CMB; (b) results of LSF spherical harmonic fit; (c) results of SVD spherical harmonic fit, we have discarded 5 degrees of freedom; and (d) results of ML inversion. The maps are Mollweide projections.

poor e.g. the large negative anomaly in the East Central Pacific and the positive anomaly in the south-east Pacific. The ML solution infers large amplitude anomalies in regions of poor coverage similar to the LSF solution. The SVD solution evidently yields the smoothest solution.

To test the sensitivity of CMB models on the cut-off time. we inverted PcP data sets with 2 .0s , 4 .0s , 6 . 0 s and 8.0s cut-offs. We found the spatial characteristics of each of the inferred models to be similar. This can be seen in Fig. 17(a), where we have plotted degree correlations between the various inferred topographies. Excellent correlation exists for degrees 2 and 4. We believe that degrees 1 and 3 are poorly determined because of the non-uniform sampling.

1

Figure 17. (a) (Top) degree correlations, r,, between CMB maps inferred from PcP data using 2.0, 4.0, 6.0 and 8.0s cut-offs. For evaluation the 90 per cent confidence interval, rwBw, is also plotted. (b) (Bottom) Rms of inferred topography, a,,, plotted as a function of the rms of residuals, a,,, for maps inferred from PcP data with 2.0, 4.0, 6.0 and 8.0s cut-offs. This is plotted for both individual and composite residuals.

That the spatial characteristics of these maps are so similar indicates that there is a systematic signal in the traveltimes, regardless of cut-off. The rms of the residuals, a,,, varies for the data sets of different cut-offs, as does the rms amplitude of the inferred topography, abr. In Fig. 17(b) we have plotted a,, versus a,, for both composite and individual residuals. From the fairly linear relationship between these quantities we see that the variance of the inferred topography scaled as the variance of the residuals of the set used. This indicates that the more outlying residuals contribute to increasing the amplitude, and to a lesser extent altering the spatial characteristics of the map.

To investigate the power spectrum of the inferred topography, we inverted the 4.0s PcP data for various maximum harmonic degrees, L. The results are summarized in Fig. 18, in which we plot the normalized power as a function of the harmonic degree. It is clear that the degree two power is large for all cases. Numerous seismological inferences of lateral heterogeneity within the Earth have realized large degree two components (see, for example, Su & Dziewonski 1991; however, also see Lay 1991). For the L = 6 case, the power appears constant (non-decreasing) for degrees 4, 5 and 6. This indicates that truncation of the spherical harmonic expansion is not justified by the data. It is very likely that smaller scale structure is not being modelled by the parameterization we have chosen.

CMB topography from ISC traveltimes 1009

_ _ 6 0 k mpul model -

: o b

L

b

d -.5

- 1

Max harmonic degree

0 L = 3

0 L = 4 . L = 5 - L = 6

>--=- -- - - _ .-_

; , , , , , , , , , , , / ( , , , , (4 1 -

mox

0 1 2 3 4 5 6 7 8

4

i 4

0 1 2 4 5 6 7 Harmonic Degree

Figure 18. The normalized power as a function of harmonic degree for PcP inversions with maximum harmonic degree, L, equal to 3, 4, 5 and 6.

8 SYNTHETIC EXPERIMENTS

We performed several synthetic experiments in which we inverted artificial data sets composed of coherent signals with random noise of varying levels. We have shown in Sections 3 and 4 that the variance of ISC traveltime residual data has a large random component. It is the goal of these experiments to investigate how noise impacts the inferred models of CMB topography.

In these experiments we used the geographic distribution of the ISC P c P 4 . 0 ~ cut-off data shown in Fig. 8(a). We then compute for each ray the traveltime residual that would arise due to an artificial (input) CMB topography model of moderate amplitude. The input model is not correlated with the models inferred by the PcP data and the power is equally distributed among all the harmonic degrees. We have used an input topographic model with a,,=2.2 km, and absolute amplitudes of f6 .0 km. The traveltime residuals for this model had an rms of 0.11s, which is consistent with what we found in Section 4 (see Table 4). To these CMB residuals we add Lorentzian distributed noise of 3.8s rms and zero mean. Thus, each synthetic residual is composed of two parts: a coherent CMB signal and a random (noise) signal. Composite residuals are constructed using 1.0 s , 2.0 s, 4.0 S , 6.0 s and 8.0 s cut-offs. The synthetic data are then inverted for CMB topography and the resulting models are compared to the input model. We then examine how the spatial patterns and amplitudes of the models inferred from synthetic data are affected.

The results of these experiments are summarized in Fig. 19. Fig. 19(a) shows the degree correlations between the input model (coherent signal only) and the models inferred from the synthetic data (coherent plus noise signal). The spatial pattern of the input model is rapidly deteriorated as noise is added. Fig. 19(b) shows the inferred topographic rms for the synthetic data sets. As the composite residual rms increases, the amount of noise relative to the coherent signal increases, and the inferred topographic rms increases. The results of these experiments demonstrate that the recovery of the input amplitude and spatial pattern is strongly inhibited by the addition of noise to the coherent signal. Recall that the random component of the composite residual variance for the PcP 4.0s cut-off data was about

1 8 0 & lnput model

harmonic degree. 1

5--v

summary residual rms (src)

Figure 19. Synthetic experiments. (a) (Top) degree correlations between the input model and the models inferred from the synthetic data as a function of composite residual rms. (b) (Bottom) rms of topography inferred from synthetic data sets as a function of composite residual rms.

1.7s. In the synthetic experiments, noise of this strength greatly affects the inferred amplitudes and spatial patterns.

9 CONCLUSION

In this paper we have presented results of our analysis of ISC PcP and P K P traveltimes and inversions for long-wavelength CMB topography. The events were relocated and phases were identified with greatly improved quality by Bob Engdahl. The residuals were computed by removing the effects of spherical imp91 structure, hydrostatic ellipticity, station corrections and mantle heterogeneities from the observed traveltimes. The ellip- ticity and station corrections modelled a significant portion of the residual variance. We found that neither of the aspherical mantle models we evaluated (L02.56 or isc5-11) is able to reduce a significant part of the coherent composite residual variance. After these corrections were applied there remained a fairly strong coherent signal in the residual data. We found that the coherent signal probably is due more to mantle heterogeneity rather than to CMB topography. Also, we found that the data are probably affected by heterogeneity of all spatial scales, including scales smaller than the shortest wavelength of CMB topography we sought on our inversions (3600km) and the patches we used to spatially average the data (660 km).

The inversions revealed contradictory results for CMB topographic models. The spatial patterns and amplitudes of

1010 A . Rodgers and J . Wahr

models inferred from different phases (PcP and three branches of P K P ) did not agree. Furthermore, the amplitudes of the models were not consistent with amplitudes derived from other geophysical observables. Our models are poorly correlated with those inferred by Morelli & Dziewonski (1987a and 1987b). We found that singular value decomposition inversion algorithms result in the smoother models of CMB topography. Least-squares fit and maximum likelihood algorithms lead to anomalously large amplitude topography in regions of poor or no sampling. Truncation of the spherical harmonic series expansion of CMB topography at degree 6 is probably not justified, as there are signals that arise from smaller spatial wavelengths. The geographic distribution of the data, however, is probably too poor to retrieve reliably such higher order structure. The amplitudes of CMB topography are sensitive to the residual variance through the cut-off time (the window around the mean residual within which data are kept for inversion). The amplitudes of inferred CMB models strongly increase as the cut-off is increased. Experiments in which we inverted synthetic data sets revealed that noise, on the scale that exists in the ISC residual data, can severely obstruct reliable inference of both the spatial pattern and the amplitudes of long-wavelength CMB topography of moderate amplitude.

We find no evidence to believe that CMB structure can be reliably inferred from the current ISC data set with a spherical harmonic parameterization. For CMB topography to be reliably inferred, future inversions must:

reduce the noise in the data; improve the geographic coverage of the data to be able to

resolve higher order structure; remove the effects of currently unmodelled mantle and

crustal heterogeneity from traveltime data sets in order to isolate the signal due to CMB structure;

reduce the effects of unmodelled small-scale structure.

Perhaps using long-period waveform data will improve the noise and small-scale structure problems. However, the coverage will not improve. The coverage may be improved by installing seismic stations in the Southern Hemisphere and on the ocean bottoms (Purdy & Dziewonski 1989).

ACKNOWLEDGMENTS

We are indebted to Bob Engdahl for running his event relocation and phase re-identification program to provide us with the ISC data. The results of his work greatly improved the quality and quantity of traveltime data. This work has benefited from discussions with Selena Billington and Michael Ritzwoller. We thank David Taylor of CSS for the use of his ellipticity correction code. We thank Jay Pulliam for making his models of aspherical mantle structure available to us prior to publication. We thank Andrea Morelli for providing us with his CMB topography models. Many plots were made using the publically available GMT system for which we thank Paul Wessel and Walter Smith. This work was supported in part by NSF grant EAR-8804791,

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