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Geophysical Journal InternationalGeophys. J. Int. (2017) 210, 166–183 doi: 10.1093/gji/ggx147Advance Access publication 2017 April 12GJI Seismology

Resolution, uncertainty and data predictability of tomographic Lgattenuation models—application to Southeastern China

Youlin Chen1 and Jiakang Xie2

1Advanced Technology Division, Array Information Technology, Inc., Greenbelt, MD 20770, USA2Air Force Research Laboratory, Albuquerque, NM 87117, USA. E-mail: [email protected]

Accepted 2017 April 27. Received 2017 April 6; in original form 2016 October 26

S U M M A R YWe address two fundamental issues that pertain to Q tomography using high-frequency regionalwaves, particularly the Lg wave. The first issue is that Q tomography uses complex ‘reducedamplitude data’ as input. These data are generated by taking the logarithm of the product of(1) the observed amplitudes and (2) the simplified 1D geometrical spreading correction. Theyare thereby subject to ‘modeling errors’ that are dominated by uncompensated 3D structuraleffects; however, no knowledge of the statistical behaviour of these errors exists to justifythe widely used least-squares methods for solving Q tomography. The second issue is that Qtomography has been solved using various iterative methods such as LSQR (Least-SquaresQR, where QR refers to a QR factorization of a matrix into the product of an orthogonalmatrix Q and an upper triangular matrix R) and SIRT (Simultaneous Iterative ReconstructionTechnique) that do not allow for the quantitative estimation of model resolution and error. Inthis study, we conduct the first rigorous analysis of the statistics of the reduced amplitude dataand find that the data error distribution is predominantly normal, but with long-tailed outliers.This distribution is similar to that of teleseismic traveltime residuals. We develop a screeningprocedure to remove outliers so that data closely follow a normal distribution. Next, we developan efficient tomographic method based on the PROPACK software package to perform singularvalue decomposition on a data kernel matrix, which enables us to solve for the inverse, modelresolution and covariance matrices along with the optimal Q model. These matrices permitfor various quantitative model appraisals, including the evaluation of the formal resolutionand error. Further, they allow formal uncertainty estimates of predicted data (Q) along futurepaths to be made at any specified confidence level. This new capability significantly benefitsthe practical missions of source identification and source size estimation, for which reliableuncertainty estimates are especially important. We apply the new methodologies to data fromsoutheastern China to obtain a 1 Hz Lg Q model, which exhibits patterns consistent with whatis known about the geology and tectonics of the region. We also solve for the site responsemodel.

Key words: Guided waves; Seismic attenuation; Seismic tomography; Site effects; Crustalstructure; Asia.

1 I N T RO D U C T I O N

Lg is typically the most prominent short-period seismic phase ob-served over continental paths at regional and teleseismic distances(Ewing et al. 1957; Nuttli 1973). Arriving with a group velocity ofabout 3.5 km s−1, Lg can be modeled as a sum of higher mode sur-face waves (Knopoff et al. 1973) or multiple supercritical S-wavereflections trapped in the crustal wave guide (Bouchon 1982; Ou& Herrmann 1990). The Lg attenuation coefficient or its inverse,the Lg quality factor (Lg Q), should closely resemble the crustalaverage of shear wave attenuation or Q. Lg Q is known to vary

significantly across major continents, for example, by a factor of10 in Eurasia and a factor of 5 in North America (Xie & Mitchell1990; Phillips et al. 2001; Xie et al. 2006). Tomographic mapping oflateral variations of Lg Q is important for inferring the tectonic evo-lutionary history of the crust, which affects the rheology of crustalmaterials at a microscopic scale and crustal structural variations ata macroscopic scale (Mitchell 1995). In particular, Lg Q values aresensitive to variations in temperature and fluid content in crustalmaterial (Mitchell 1995; Xie et al. 2004; Phillips & Stead 2008;Zhao & Xie 2016). In practice, tomographic Lg Q mapping is alsoimportant for ground motion prediction in engineering seismology

166Published by Oxford University Press on behalf of The Royal Astronomical Society 2017. This work is written by (a) US Governmentemployee(s) and is in the public domain in the US.

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Tomographic Lg attenuation models 167

(Burger et al. 1987; Atkinson & Boore 1995), for inferring sourceproperties such as event type and size (e.g. Nuttli 1973; Phillipset al. 2001; Xie 2002) and for measuring the yield when the sourceis an underground explosion (Nuttli 1986; Ringdal et al. 1992; Zhaoet al. 2012).

It is well known that Q tomography is a difficult problem since ituses seismic amplitude data which not only contains the effects offinite Q, but also the effects of wavefront expansion in 3D velocitystructures (Nolet 1987). The latter effects are typically parametrizedas a ‘Geometrical Spreading Term’ (GST, e.g. Aki & Richards 1980;Mitchell 1995) whose properties are complex and unknown at highfrequencies (>0.1 Hz). Because Q affects amplitudes through anexponential function, the input data for estimation of Q are typi-cally the logarithm of the observed amplitudes after correcting for asimplified 1D GST. We will refer to this input as ‘reduced amplitudedata’ following Menke et al. (2006). Values of Q estimated usingreduced amplitude data are subject to ‘modeling errors’ caused bythe true 3D GST, whose effects remain in the data after the crude1D GST correction is applied. In practice, there are additional un-modeled errors in reduced amplitude data, such as the pre-eventambient noise and the pre-phase noise resulting from the coda ofpreceding seismic phases. We extend the term ‘modeling errors’ toinclude these errors as well. Finally, the effects of ground motionamplitude measurement error are typically ignorable when com-pared to modeling error and thus will not be included here (e.g.Mitchell 1995; Xie 1998; Menke et al. 2006). To date, the statisticalbehaviour of modeling errors has not been systematically studied.Instead, past Q tomography studies have implicitly assumed nor-mally distributed modeling errors, thereby allowing for the use ofleast-squares algorithms to solve for a Q model. These algorithmsinclude widely used linear system solvers such as SIRT (Simul-taneous Iterative Reconstruction Technique), back-projection andLSQR (Least-Squares QR, Pei et al. 2006; Xie et al. 2006; Hearnet al. 2008; Phillips & Stead 2008; Pasyanos et al. 2009; Zhao etal. 2010, 2013; Zhao & Xie 2016), all of which start with an initialQ model and interactively improve the model for better data fit.

The objective of these methods is fast convergence toward afinal Q model, which minimizes the data residual. Nonetheless,the quality of the resulting Q models, including their resolutionsand errors, cannot be formally evaluated because these methodsdo not yield model resolution and covariance matrices. Usually,the resolutions of these models are empirically demonstrated usingthe ‘checkerboard’ test, synthetic ‘point-spreading function’ (Xie &Mitchell 1990; Hearn et al. 2008; Zhao et al. 2013) or approximateelliptical areas that reflect ray sampling density (Zhao & Xie 2016).The errors or uncertainties of the Q models are indirectly measuredby the variance reduction of the fit from the initial to the final model(e.g. Phillips & Stead 2008), ignoring the large and dominant modelerror in the input data.

It is highly desirable to quantify the statistical characteristics ofmodeling errors and how they cause random uncertainties in thetomographic Q model. It is also highly desirable to quantify the res-olution and error of the Q model and examine the inherent trade-offsbetween them (Backus & Gilbert 1970). Only with realistic quan-titative estimates of model resolution and error, we can determinewhich features of the tomographic Q model are truly resolved andtherefore which can be used to infer the causes of lateral variationsin Q (e.g. variations of temperature and fluid/crack contents). Fi-nally, it is only with these quantitative estimates that we can use theQ model for ground motion prediction, source spectral estimation,source type identification and source size estimation with realisticuncertainty.

In this paper we first focus on extracting and analyzing the statis-tics of modeling errors in reduced amplitude data. We then focuson developing a new tomographic methodology to invert and quan-titatively assess Q models. The methodological results are appliedto two-station (TS) based measurements in order to develop a new1 Hz Lg Q tomographic model of southeastern China. Compared tothe ‘coupled Q tomography’ method, in which Q and the source pa-rameters are solved simultaneously, the TS-based methods use dif-ferential amplitudes from pairs of stations aligned with the sources,thus eliminating the source and/or site effects in path Q measure-ments. Although the TS based method uses more restrictive data, itis much simpler than coupled tomography because the latter mustdeal with the non-uniqueness of the Q model caused by the inherenttrade-off between Q and the source parameters (Xie 1998; Menkeet al. 2006).

Our work starts with a quantitative understanding of the statis-tical features of modeling error in the reduced amplitude data. Weshow that after a reasonable screening to exclude large outliers, themodeling error closely follows a normal distribution. On the basisof this conclusion we can solve Q tomography as a least-squaresproblem. We next develop a novel 2D Q tomography method usinga matrix solver approach, in which the PROPACK software package(Larsen 1998) is applied in order to efficiently and accurately de-compose the kernel matrix into singular values and singular vectorswith well-maintained orthogonality. This approach enables us toformally construct model resolution and covariance matrices whensolving for a tomographic Q model. The matrices are used for rig-orous appraisals of the Q model by quantitatively estimating modelresolution and error. As a result, we now use only the reliably recov-ered features of the Q model to infer crustal properties. In addition,when predicting Q values along any new path we are capable offormally estimating the prediction uncertainty. This capability isvery important for practical missions of ground motion prediction,source spectral estimation for event identification and for estimat-ing the source size and yield of an underground explosion. Finally,we extend our singular value decomposition (SVD) methodology toinvert for station site responses. These results can be used in futurecoupled Q tomography studies to constrain site response solutionswhich would otherwise trade-off with Q and source parameters.

2 M E A S U R E M E N T O F Q A N D I T SE R RO R S T RU C T U R E

2.1 Two-station-based Lg Q measurements

The observed Lg spectrum at frequency f can be stochasticallymodeled as:

A ( f ) = S ( f )

4πρβ3G (�) exp

(− π f �

V Q ( f )

)R ( f ), (1)

where � is the epicentral distance, V is the Lg group velocity(∼3.5 km s−1), G(�) is the GST, Q( f ) is the quality factor, andR( f ) is the receiver site response (e.g. Street et al. 1975; Xie 1993;Yang 2002). The term S( f )

4πρβ3 quantifies the excitation of the Lgspectrum, with S( f ) being the source spectrum and ρ and β beingthe crustal average density and shear wave velocity around source,respectively (e.g. Street et al. 1975; Sereno et al. 1988). Ideally, thetrue GST from the 3D structure should be used in eq. (1). However,the true GST will remain unknown and a simplified 1D GST of theform G (�) = G0�

−m is used in reality, with G0 being a constanttypically set at an epicentral distance of �0 = 100 km and m beinga decay parameter set to 0.5 (Street et al. 1975).

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168 Y. Chen and J. Xie

Figure 1. Schematic diagrams illustrating the recording geometries for (a)the two-station (TS), (b) the reversed two-station (RTS) and (c) the reversedtwo-event (RTE) methods. The blue triangles represent seismic stations andthe red stars represent earthquakes. In the data selection there is a smalldifference δθ between the source-to-station azimuths for stations Ri and Rj.

The TS method requires that the Lg spectra of one source berecorded at two stations, i and j, that are approximately aligned withthe source along the same great circle (Fig. 1a). The Q (f) alongthe interstation path can be measured using the ratio of the spectrarecorded at the two stations i and j (e.g. Xie & Mitchell 1990; Xieet al. 2004, 2006; Zor et al. 2007; Gallegos et al. 2014):

Ai ( f ) G(� j )

A j ( f ) G (�i )= Ri ( f )

R j ( f )exp

(π f � j

V Q j ( f )− π f �i

V Qi ( f )

)

≈ exp

(π f �

V Q ( f )

), (2)

where � = (� j − �i ) is the effective distance between the twostations. The main advantage of the TS method is that the sourcespectrum is cancelled, eliminating the trade-off between source andQ parameters. Note that the TS method ignores the non-unity Ri ( f )

R j ( f )

ratios, which will cause errors in the estimated Q( f ). In a practi-cal recording geometry it is difficult to find recording stations thatare exactly aligned with the source (ideal case). A more realisticdata selection criterion is to allow for a small angle δθ between thesource-to-station azimuths as shown in Fig. 1(a). Since Lg propaga-tion in the real 3D crustal structure can deviate from the great-circledirection by as much as 15◦ (Der et al. 1984), Xie et al. (2004) andBao et al. (2011) set this physically based threshold of 15◦ as themaximum allowable δθ in data selection.

The reverse two-station (RTS) method (Chun et al. 1987) is animprovement upon the TS method that introduces an RTS pathgenerated by another source located on the opposite side of the twostations (Fig. 1b). For each source (a and b) we can obtain a spectralratio as defined in eq. (2)

Ai,a ( f ) G(� j,a)

A j,a ( f ) G (�i,a)= Ri ( f )

R j ( f )exp

(π f (� j,a − �i,a)

V Q ( f )

)(3)

A j,b ( f ) G (�i,b)

Ai,b ( f ) G(� j,b

) = R j ( f )

Ri ( f )exp

(π f (�i,b − � j,b)

V Q ( f )

). (4)

The product of the ratios in eqs (3) and (4) gives the followingquantity for estimating Q( f ) between stations i and j:

√(Ai,a ( f ) · A j,b ( f )

A j,a ( f ) · Ai,b ( f )

)(G(� j,a) · G (�i,b)

G (�i,a) · G(� j,b)

)

= exp

(π f

[(� j,a − �i,a) + (�i,b − � j,b)

]/2

V Q ( f )

)

= exp

(π f �

V Q ( f )

), (5)

where � = [(� j,a − �i,a) + (�i,b − � j,b)]/2 is the effective inter-station distance associated with the two sources and two stations.The major advantage of the RTS over the TS method is that the siteresponses of the two stations are cancelled in eq. (5). Additionally,the RTS method can be used to determine the site response ratioRi ( f )R j ( f ) by taking the ratio of eqs (3) and (4)

√(Ai,a ( f ) · Ai,b ( f )

A j,a ( f ) · A j,b ( f )

)(G(� j,a) · G(� j,b)

G (�i,a) · G (�i,b)

)

= Ri ( f )

R j ( f )exp

(π f [(� j,a − �i,a) − (�i,b − � j,b)]/2

V Q ( f )

)

≈ Ri ( f )

R j ( f ), (6)

where the difference of the effective interstation distances,[(� j,a − �i,a) − (�i,b − � j,b)]/2, is small enough to be ignored.While the RTS method improves the precision of the Q( f ) esti-mates, it requires a stricter recording geometry than the TS method,resulting in sparser ray-path coverage.

The reverse two-event (RTE) method (e.g. Bao et al. 2011; Ranas-inghe et al. 2015) is an extension of the RTS method that is appli-cable when the two sources are aligned with the two stations, butare located between the station pair rather than outside of the pair(Fig. 1c). For the RTE method, the expression for estimating Q( f ) isthe same as eq. (5) except � is now the inter-event distance. Unlikethe RTS method, RTE does not yield the measurement of the ratioof station site responses.

The three types of TS-based methods can be expressed via auniform equation that is used to estimate Q( f ):

�

Qe ( f )= V · ln( A)

π f, (7)

where A = Ai ( f )G(� j )

A j ( f )G(�i ) for the TS method and A =√(

Ai,a ( f )·G(� j,a )

A j,a ( f )·G(�i,a ) )(A j,b( f )·G(�i,b )

Ai,b( f )·G(� j,b ) ) for the RTS and RTE methods. In

general, Qe( f ), which represents the estimate of Q( f ), contains anestimation error arising from the replacement of the true 3D GSTby the simplified 1D GST. This estimation error is referred to as‘modeling error’ by Menke et al. (2006). In this paper we extendthe definition of modeling error to include additional errors causedby (1) imperfect data sampling censoring criteria, (2) the unmod-eled part of the station site response, which may be azimuthallydependent and (3) the ambient noise and ‘pre-phase’ noise causedby the coda of the preceding seismic phase. It should be noted thaterrors of this nature also exist in coupled source-Q measurements.

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The statistical properties of these errors will be discussed in thefollowing section.

2.2 Method to study statistical propertiesof Q measurements

Past Q tomography studies have exclusively solved a least-squaresproblem with path-specific Q measurements as input data and thushave implicitly assumed that the errors of these measurements arenormally distributed. This assumption, however, has never beenverified in past studies. Most importantly, the ‘amplitude’ in eq. (7),defined as ln( A), is derived by multiplying a simplified 1D GSTcorrection to the observed amplitude ratio data and then takingthe logarithm. Menke et al. (2006) referred to those manipulatedamplitudes as ‘reduced amplitude data’ and suggested that theirerror is dominated by modeling errors and may not obey a normaldistribution. We are compelled to conduct an overdue study of thestatistics of these reduced amplitudes. Following eqs (A6) and (A7)of Xie et al. (2004), the errors in �

Qe( f ) measured using eq. (7) canbe expressed as

δ

(�

Q ( f )

)= �

Qe ( f )− �

Q ( f )= V

π fδ [ln( A)]

= V

π fln (1 + ε) ≈ V

π fε, (8)

where ε is a random variable that represents the error in ln( A), as-suming the error is small and a first-order approximation is valid.Since Earth’s lithosphere is highly heterogeneous at the frequencyrange of interest (>0.1 Hz), errors in �

Qe( f ) as represented by therandom variable ε should be uncorrelated from measurement tomeasurement, even if they are collected from the same station pairwith slightly varying azimuths (within 15◦) and differing sourcelocations. This lack of correlation arises because the compositionof a local wavefield will change, even when sources are locatedat the same azimuth with varying distances (e.g. Mitchell 1995;Romanowicz 1998; Ji et al. 2005; Gibbons et al. 2010). Our ex-perience has always been that repeated TS Lg Q measurementsrandomly fluctuate from the mean without any pattern, implyinga lack of correlation in ε. This observed random fluctuation isalso consistent with observations from the Middle East (personalcommunication with Eric Sandvol, 2016). Since our observationalexperiences have confirmed that repeated TS Lg Q measurementswill randomly fluctuate in an uncorrelated manner, every measure-ment of �

Qe( f ) provides a sample of the random variable ε. Fora typical data set used in regional Q tomography, there are manyrepeated �

Qe ( f ) measurements between a pair of stations (or events).These repeated measurements provide a subgroup of samples of ε.Assuming the mean of �

Qe( f ) for the subgroup is close to the true

mean �

Q( f ) , we can subtract individual �

Qe( f ) values by the mean to

approximately estimate δ( �

Q( f ) ), yielding the values of individualsamples of ε according to eq. (8). The statistical properties of ε

for the entire dataset, including its distribution and variance Var(ε)given by

Var

(�

Qe ( f )

)= V 2

π 2 f 2Var (ε), (9)

can be calibrated using samples of ε from all repeated paths. InSection 3.2, we make use of the data collected in and around south-eastern China to investigate the statistical behaviours of ε.

3 C O L L E C T I O N A N D S TAT I S T I C A LA NA LY S I S O F DATA F RO MS O U T H E A S T E R N C H I NA

3.1 Data processing and quality control

We used 102 Chinese National Digital Seismic Network (CNDSN)and Global Seismic Network (GSN) stations in and around south-eastern China (Fig. 2) and collected more than 10 000 verticalcomponent seismograms from 109 regional crustal earthquakes(M ≥ 4.5), which occurred between 2009 and 2011. The stations areequipped with various instruments, including the Chinese BBVS-120, BBVS-60, CTS-1 and JCZ-1 serial systems, KS-2000 series(Geotech Instruments, LLC), CMG series (Guralp Systems Lim-ited) and the standard GSN STS-1 and STS-2 systems. In this studywe only analyse Lg spectral amplitude data near 1 Hz, so a narrowbandwidth (0.5–1.5 Hz) with a central frequency of 1.0 Hz is usedto obtain average spectra. The responses of the instruments are flatto input ground velocity in frequency bands that are much widerthan this 1 Hz band. Lg is assumed to form at a minimum epicentraldistance of 200 km (Ou & Herrmann 1990). Lg waveforms are usedas long as they satisfy the signal-to-noise ratio (SNR) thresholdelaborated below, resulting in recording distances as large as a fewthousand kilometres (Xie et al. 2006).

We hand-pick the Lg waveform window, whose typical groupvelocity ranges between 3.6 and 3.0 km s−1 (Fig. 3). We try to avoidthe inclusion of the Sn coda and the beginning of the fundamental-mode Rayleigh wave whenever possible. The Sn coda arrives withina group velocity window of 4.2–3.7 km s−1 and is considered to bethe pre-phase noise of Lg. A fast Fourier transform (FFT) with a10 per cent cosine taper window (Xie 1993) is used to obtain Lgspectra, the spectra of the pre-event noise prior to the first P arrivaland the spectra of the Sn coda. Instrument responses are removedfrom the raw spectra to obtain ground displacement spectra. Forevery set of spectra we take an average in our defined frequencyband to obtain stable 1 Hz spectral estimates. To reduce the effectsof ambient noise, we subtract the Lg power spectra by pre-P powerspectra averaged over ∼10 s pre-P windows (Xie 1993).

We compute a pre-event SNR by dividing the Lg spectral am-plitude by the pre-event amplitude. Similarly, the pre-phase SNRis found by dividing the Lg spectral amplitude by the Sn coda am-plitude. Data screening is done by setting the thresholds of bothSNRs. Choosing these thresholds is complicated because Lg spec-tra in southeastern China can be absent due to Lg blockage causedby the Tibetan Plateau (Ruzaikin et al. 1977). We made a substantialeffort to find appropriate pre-event and pre-phase SNR thresholdsthat can be used to reject blocked Lg. After numerous trials we setthe optimal thresholds of pre-event SNR and pre-phase SNR at 3.0and 1.3, respectively.

Figs 3 and 4 show examples of the data and how they are pro-cessed and screened. Figs 3(b) and (c) show the vertical componentseismograms of an M 5.1 earthquake that occurred in Sichuan,China (Fig. 3a). The waveforms are band-pass filtered between 0.9and 10.0 Hz and sorted according to epicentral distance (we showthe waveforms in a broader band because Lg blockage is inherentlya broad-band phenomenon). Over westerly paths which traverse theTibetan Plateau, the Lg waves are much weaker as a result of Lgblockage. Fig. 4 shows the Lg, pre-event noise and pre-phase noisespectra of two example seismograms recorded by stations QID andDUL for the event shown in Fig. 3. Station QID is located to thenortheast of the event, whereas DUL is located inside the TibetanPlateau (Fig. 3a). Around 1 Hz both spectra pass the pre-event SNR

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Figure 2. A map of the study region showing topography and the distribution of CNDSN and IRIS-GSN stations. The red triangles represent the 102 stationsused in this study. The black solid lines indicate major tectonic blocks and the grey dashed lines indicate faults.

threshold, but the DUL spectrum does not pass the pre-phase SNRthreshold and is therefore screened out. We have found that forthe entire dataset used, the pre-phase threshold is generally moreefficient than the pre-event threshold for screening out blocked Lgspectra.

3.2 Statistics of Q measurement errors

After calculating 1 Hz interstation spectral ratios using the narrow-band averaging described in the prior section, we estimate the re-duced amplitude data (ln( A)) in eq. (7). These reduced amplitudedata are then used to estimate interstation �

Q( f ) , where the frequencyf is set at 1 Hz. We obtain 1431 RTS, 51 RTE and 3297 TS uniquepaths for �

Q( f ) measurements, respectively. Of these paths, thereare 1237 RTS, 50 RTE and 2480 TS paths that are multiply sam-pled, allowing �

Q( f ) to be repeatedly measured. Fig. 5 shows the

distribution of the number of repeated �

Qe ( f ) measurements for in-dividual paths using the RTS method as an example. More than 63per cent of the multiply sampled paths have more than two repeatedmeasurements, with the largest number of repeated measurementsreaching 300.

In Section 2.2 we show that the random errors in the �

Q( f ) mea-surements are fully represented by the random variable ε (eq. 8)and samples of ε can be collected using �

Q( f ) measurements overrepeatedly sampled paths. We denote ε values estimated using theRTS, RTE and TS methods by εRTS, εRTE and εTS, respectively. For

�

Q( f ) values measured using the three methods, the total number of

obtained εRTS, εRTE and εTS are 53 721, 618 and 20 290, respectively(Table 1), and their distributions are shown in Figs 6(a), (c) and (e)while their respective cumulative distribution functions (CDFs) areshown in Figs 6(b), (d) and (f).

The distributions of εRTS, εRTE and εTS are characterized by acentral peak, a feature of the normal distribution. Away from thepeaks, however, the distributions have longer tails in both the pos-itive and negative directions. These distributions are rather sim-ilar to those of the teleseismic P-wave traveltime residuals thatwere discussed by Nolet (1987), who suggested that the distri-bution is actually composed of a dominating normal (Gaussian)distribution function supplemented by a low-amplitude functionthat decays slowly, causing tails. As suggested by Nolet (1987),once an adequate screening is conducted for rejecting large out-liers of the random data, the resultant new data become approxi-mately normally distributed. This screening serves the purpose of re-jecting physically unreasonable traveltime residual measurements,or in our case, physically unreasonable �

Q( f ) measurements (e.g.negative Q).

After attempting several screening trials on our ε samples, webuild a data screening criterion that requires any ε departing fromthe median by more than two standard deviations of ε (i.e., anyε that satisfies |ε − median(ε)| > 2 × std(ε)) to be screened out.Note that we use the deviation of ε from the median (which is knownas the median absolute deviation or MAD), rather than the standarddeviation of ε from the mean to form the screening criteria sincethe former deviation is more robust than the latter in the presenceof large outliers.

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Figure 3. (a) Map showing the location (31.46◦N, 103.96◦E) of an M 5.1 Sichuan earthquake occurring at a GMT time of 2009 June 29, and the resultingray paths. Stations in the northeastern direction and their associated ray paths are annotated in red, and stations in the northwest and west directions and theirassociated ray paths are annotated in blue. (b) Vertical component seismograms recorded at the northeastern stations. The waveforms are band-pass filteredthrough a 0.9–10.0 Hz band and sorted by epicentral distance. The predicted arrival times of the Pn and Lg window are labeled on the seismograms (red lines).(c) The same as (b) but for seismograms recorded at the northwestern and western stations.

For all three ε samples, the number of original samples that arescreened out is less than 15 per cent of the total (Table 1). Thebest-fit normal probability density function (PDF) of the screenedεRTS samples is plotted over the εRTS distribution in Fig. 6(a). Theobserved CDF calculated using the screened εRTS samples is plottedin Fig. 6(b) and compared to the best-fit theoretical normal CDF.The correlation coefficient between the observed and screened CDFand theoretical best-fit normal CDF is 0.9999 (Table 1), which guar-antees the screened εRTS, and therefore the screened RTS δ( �

Q( f ) )measurements, very closely follows a normal distribution function.The same conclusions have also been drawn about the statistical be-

haviour of εRTE and εTS after the same screening criterion is applied(Figs 6c–f and Table 1).

We also apply other commonly used graphical (e.g. normalprobability and quartile–quartile (Q–Q) plots) and numerical (e.g.Kolmogorov–Smirnov (K–S) test, Shapiro–Wilk test, Anderson–Darling test or Lilliefors test) testing methods to the screened ε

samples. All these methods are developed for testing the hypothesisthat random samples are consistent with a normal distribution ata certain confidence level. However, these methods are only validfor εRTE, failing to generate statistically meaningful results for εRTS

and εTS. These failures occur because the tests discussed here are

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Figure 4. Spectra of Lg wave (blue), pre-phase noise (red) and pre-event noise (green) recorded on stations (a) QID and (b) DUL. The instrument responsesof each station are removed from the spectra.

Figure 5. Distribution of the number of repeated RTS measurements overindividual paths.

Table 1. Estimated ε information using the RTS, RTE and TS methods.

RTS RTE TS

Number ofestimates of ε

53 721 618 20 290

Number of ε

within 2σ

87.46 per cent 90.94 per cent 85.87 per cent

Correlationcoefficient ofECDF andNCDF

99.99 per cent 99.95 per cent 99.98 per cent

K-S test – Accept nullhypothesis ofnormality at95 per centconfidence

–

Std(ε) 0.2524 0.2643 0.3779Var(ε) 0.0637 0.0699 0.1428

designed for relatively small (no more than a few thousand) samplesizes. It has been well documented that with very large sample sizes,these tests will suggest non-normality when there is even a trivialdeparture from perfect normality (e.g. Amaratunga & Cabrera 2004,Park 2003). In our case, the εRTS and εRTE samples closely follow a

normal distribution, but fail these tests because of their exceedinglylarge sample sizes (more than 46 000 and 16 000 for the screenedεRTS and εTS, respectively). On the other hand, the K–S test per-formed on 562 randomly selected εRTE subsamples indicates that thenull hypothesis of normality can be accepted at a confidence level of95 per cent (Fig. 6d).

The distributions of εTS, εRTS and εRTE have the same means atzero, but different variances (Table 1). εTS has a much larger vari-ance than εRTS and εRTE, that is, Var(εTS) > Var(εRTS) and Var(εRTE),which results in a wider distribution than εRTS and εRTE. The trivialdifference between Var(εRTS) and Var(εRTE) is very likely causedby the different number of samples for repeated measurements col-lected by each individual method. The major difference between theRTS or RTE methods and the TS method is that the RTS and RTEmethods eliminate a significant (azimuthally independent) portionof the station site responses (eq. 5), while the TS method does notreduce the effects of site response (eq. 2; also see Ford et al. 2008).The estimated εTS contain errors arising from these effects, causingthe variance of εTS to be larger than the variances of εRTS and εRTE.Using eq. (9), the Var(ε) (where ε can be defined as εRTS, εRTE or εTS)is used to calculate Var( �

Qe( f ) ) over all individual paths generatedby the three TS-based methods. This information will be used laterto formulate our Q tomography problem described in the followingsections.

4 M E T H O D O L O G Y O F Q T O M O G R A P H Y

4.1 Estimation of resolution and error of Q modelusing SVD

Lateral variations of Lg wave Q in a study area can be solved using a2D attenuation tomography. The laterally varying Q is parametrizedby dividing the study area into M small cells, where the Q valueis assumed to be constant in each cell (m = 1, 2, 3, . . . , M). Withthe Q measurements over N individual paths collected, we have thefollowing linear equation for each path

�n

Qn ( f )=

M∑m=1

�nm

Qm ( f ), (10)

where n = 1, 2, . . . , N denotes the nth ray path and m = 1, 2, . . . ,M the mth model cell. Qn( f ) is the estimated Q( f ) over the nthpath, which has a length of �n and intersects the mth cell with a

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Figure 6. (a) Distribution of ε estimated from RTS data. The black lines mark the value of |ε − median(ε)| = 2 × std(ε), which we use as criteria for subsequentdata screening. The best-fit theoretical normal PDF (red) of the screened εRTS samples is plotted over εRTS distribution. (b) The CDF of the screened εRTS

samples (blue) and the best-fit theoretical normal CDF (red). The grey lines indicate lower and upper 95 per cent confidence bounds for the CDF. (c) and (d)are the same as (a) and (b), but for RTE data. The maximum distance, D, from the empirical CDF to the theoretical normal CDF is labeled in (d), which is usedin the K-S test. (e) and (f) are the same as (a) and (b), but for TS data.

distance segment �nm . It is noteworthy that eq. (10) deals with 1/Q,rather than Q. This is analogous to traveltime tomography, whereone solves for a model of slowness rather than velocity. Introducingthe model and data vectors

m =[

1

Q1 ( f ),

1

Q2 ( f ), · · · , 1

QM ( f )

]T

(11)

and

d0 =[

�1

Q1 ( f ),

�2

Q2 ( f ), · · · , �N

QN ( f )

]T

, (12)

where superscript T denotes the transpose, we can then writeeq. (10) into a matrix form

A0m = d0, (13)

where A0 is an N by M matrix. Note that if any nth path is repeatedlysampled, the average of the screened measurements is used as �n

Qn ( f )

in eq. (12). The variances of individual �nQn ( f ) have been estimated

using eq. (9) as described in the previous sections. For an nth paththat is repeatedly sampled an L number of times, the averaged �n

Qn ( f )measurement should have a variance that is reduced by L, that is,

Var

(�n

Qn ( f )

)= V 2

π 2 f 2 LVar (ε). (14)

To better reflect the uncertainties quantified by changing vari-ances among �n

Qn ( f ) measurements, we introduce an N by N diagonaldata covariance matrix Cd, whose nth diagonal element is given byVar( �n

Qn ( f ) ). We then normalize eq. (13) by dividing the square-rootof Cd on both sides:

C−1/2d A0m = C−1/2

d d0 (15)

or

Am = d, (16)

where A = C−1/2d A0 and d = C−1/2

d d0. Since the number of cellscrossed by a ray for each n-path is much smaller than M, matrix

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A is sparse. If the data d is subjected to an error that is normallydistributed, one can use a least-squares algorithm to invert for modelm from data d using eq. (16). Since we have shown in the prior sec-tions that the error is indeed normal, the use of a least-squares al-gorithm is justified and we will develop an alternative least-squaresalgorithm making use of SVD in this section.

In past Q tomography studies, the least-squares problem has beensolved without solving the inverse matrix of A, which generallyrequires a large computer memory. Instead, various iterative tech-niques, such as SIRT, backprojection (e.g. Xie & Mitchell 1990;Xie et al. 2004, 2006; Zor et al. 2007) and LSQR (e.g. Phillipset al. 2005; Pei et al. 2006; Hearn et al. 2008; Zhao et al. 2013;Gallegos et al. 2014; Ranasinghe et al. 2015) have been adopted.These techniques focus on iteratively updating a trial solution muntil the predicted data misfit stops improving. However, they donot yield any quantitative estimates of the resolution and error ofthe final model. Rather, they only make use of empirical tests suchas inverting synthetic checkerboards or spike-like Qm( f ) modelswith simulated error (e.g. Humphreys & Clayton 1988, Hearn et al.2008) to find the approximate resolving power and error of themodel. Checkerboard tests are widely used and visual, but they arenot quantitative and have intrinsic limitations (Leveque et al. 1993;Liang et al. 2004; Nolet et al. 1999; Zhang & Thurber 2007). Anad-hoc interpretation of the path coverage density has also beenused to crudely estimate the model resolution (Zhao et al. 2013).

In contrast to these iterative techniques, the SVD technique solvesleast-squares problems in a more straightforward manner, provid-ing formal quantitative estimates of model resolution and error.This technique solves least-squares problems by calculating the in-verse matrix of A, which can be decomposed using the well-knownproperty of

AN×M = UN×M SM×M VTM×M , (17)

where S is a diagonal matrix consisting of singular values, and Uand V are matrices consisting of associated singular vectors withorthogonal properties UT U = I and VT V = V VT = I. Ideally, theinverse matrix of A is A−1 = VM×M S−1

M×M UTM×N . In reality, SM×M

will often contain zero diagonal elements so that S−1M×M cannot be

found. In this situation a pseudo-inverse of SM×M must be used toobtain a pseudo-inverse matrix A† (Section 4.2). The model canthen be estimated using the matrix operation

m = A† d = A†Am = Rmm, (18)

where

Rm = A†A (19)

is the model resolution matrix that quantifies how any element of theestimated model m is connected to the elements of the true model mthrough a weighted averaging (smearing) process (Aki & Richards1980; Menke 2012). Along with the model resolution matrix, wecan also obtain the covariance matrices of the random errors �dand �m, contained in the data vector d and the estimated model m,respectively. According to eq. (18; also see Aki & Richards 1980)the model covariance matrix is given by

Cm = �m · �mT = [A†�d] [A†�d]T

= A† [�d · �dT ] (A†)T = A† Cd(A†)T , (20)

where Cd = �d · �dT is the data covariance matrix constructedusing eq. (14).

An important practical usage of Cm is that it enables a formalestimate of the uncertainties of any predicted Q (or 1/Q) over a new

path (Ballard et al. 2016) to be made using eq. (10), which we canrewrite as

1

Q= 1

�

∑ �l

Ql=

∑ �l

�

1

Ql

=∑

al1

Ql= a · m (l = 1, 2, . . . , M), (21)

where Q is the predicted value, al = �l�

is the weighted segmentthat the new path traverses in the lth cell of the tomographic 1/Qmodel and vector m is also defined in eq. (11). Note that many al arezeros. The predicted 1/Q is a weighted linear summation of normalvariables 1

Qland hence a normal variable itself. Its variance is thus

given by

Var

(1

Q

)= Var [(a · m) (a · m)T ]

= aVar(mmT ) aT = a CmaT , (22)

where Cm is the model covariance matrix given by eq. (20). WhenCm contains off-diagonal elements that are negative, eq. (22) pre-dicts a smaller Var(1/Q) than if Cm is simplified to be diagonal(Ballard et al. 2016). Note that the square-root of Var(1/Q) givesStd(1/Q), the standard deviation of the predicted 1/Q as a normalvariable. Formal uncertainty estimates of predicted 1/Q can be ex-pressed in terms of n × Std( 1

Q ), corresponding to confidence levelsof 68 per cent (n = 1), 95 per cent (n = 2) and 99.7 per cent (n =3) in terms of the ‘68–95–99.7’ rule in statistics. The uncertaintiesof the predicted 1/Q can be translated to the estimated uncertaintyof the predicted log-source spectra at different confidence levels asshown below

δln [Se ( f )] � δSe ( f )

Se ( f )� π f �

V× n × Std

(1

Q ( f )

). (23)

Using the normal probability function, we can replace n by anynon-integer number between 1.0 and 2.0, thereby providing otherapplicable confidence levels. Note that a first-order approximationhas been used to derive eq. (23) since the values of Std( 1

Q( f ) ) aretypically small (less than about 7 per cent in Fig. 11).

Obtaining the quantitative model resolution matrix (eq. 19) aswell as the model covariance matrix (eq. 20), and by extension theformal uncertainty estimates (eqs 22 and 23), represents a majoradvantage of the SVD technique as compared to the iterative tech-niques, which only provide empirical estimates of the resolution anderror. While the SVD technique provides the advantages describedabove, this technique also usually requires much more computermemory than the various iterative methods, prohibiting its use forlarge-scale data and modeling. In order to overcome this issue, weadopt the PROPACK package developed by Larsen (1998) to effi-ciently compute the singular vectors and values of the kernel matrixA, drastically reducing memory requirements. PROPACK is basedon the Lanczos bidiagonalization (LBD) process (Paige & Saunders1982), in which a matrix breaks down into a sequence of Lanczosvectors and a lower bidiagonal matrix through LBD iteration. SinceSVD for a lower bidiagonal matrix is much easier and faster than itis for A, the LBD provides a low-cost method to decompose matrixA; however, the derived Lanczos vectors quickly lose their orthog-onality within a limited number of LBD iterations. In PROPACK,a partial reorthogonalization algorithm is applied to the LBD pro-cess in order to efficiently keep the orthogonality of the Lanczosvectors as the process iterates. The purpose of PROPACK is simplydecomposing matrix A. It chooses a random vector to start the LBDprocess which allows the decomposition process to be completelyindependent of data d. PROPACK can be applied to any matrix,

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Figure 7. Coverage of RTS and RTE paths over the study region. Path colours represent the 1 Hz Q values estimated for the paths.

but is particularly efficient for highly sparse matrices such as thoseencountered in seismic tomography. The PROPACK package hasbeen successfully applied to 3D traveltime tomography to obtain aresolution matrix (Zhang & Thurber 2007).

It is noteworthy that while the LSQR method also starts with theLBD process, it significantly differs from PROPACK. In its sim-ple LBD, the LSQR method does not apply reorthogonalization ofLanczos vectors and uses data vector d as the initial vector of theLBD process. Therefore, the most important difference in practiceis that the LSQR makes use of the conjugate gradient algorithmto iteratively solve for the model without constructing the pseu-doinverse A† (Paige & Saunders 1982) and hence does not providethe model resolution and covariance matrices. This is in contrastwith PROPACK, which is a matrix-based operation that does leadto the generation of these matrices from explicitly constructed A†.We note that in practice, Q models obtained using the LSQR andSVD-PROPACK methods are not expected to differ significantly(Zhang & Thurber 2007). The detailed implementation of the SVDalgorithm on a seismic tomographic problem is elaborated below.

4.2 Practical implementation of the SVD algorithm

A typical seismic tomographic problem in the form of eq. (16) isintractable because it is ill-conditioned and rank deficient, owingto insufficient observations and noise contamination. To tackle thisissue, various regularizations in the form of κLm = 0 are imposed,where L is called a Tikhonov regularization matrix (e.g. Aster et al.2013) and κ is a weighting parameter that controls how stronglythe regularization should be imposed. One regularization is spa-tial smoothing of the model m, which eliminates unphysical abruptjumps between adjacent model cells. For our 2D tomography prob-lems, we use a second-order Tikhonov regularization that convolves

the model m with a 2D Laplacian operator L and is weighted bysmoothing parameter λ, which we will denote as λLm = 0.

Another issue that occurs when m is solved for is that the inverseof the diagonal singular value matrix S may not exist because someelements of S are zero or close to zero (Section 4.1). To obtain anapproximate, pseudo-inverse matrix A†, we need to obtain a pseudo-inverse of S in which division by small, near zero S elements isavoided. We choose to form the pseudo-inverse of S by replacingnear zero elements with elements defined as Si/(S2

i + α2), whereα is a damping parameter, effectively making our SVD techniquea ‘damped least-squares’ method. The value of Si/(S2

i + α2) isvirtually unchanged from 1/Si when Si � α and becomes Si/α

2

when Si � α, thus avoiding division by very small Si values. Notethat this technique is usually designated a damped least-squaresproblem and an alternative way to introduce damping is to formallyadd a regularization of the form αIm = 0, where the Tikhonovregularization matrix L turns into an identity matrix I and α cannow be treated as a weighting parameter (Appendix A).

In general, the damping focuses on (1) seeking for the model withthe minimal complexity as measured by the L-2 norm (Aster et al.2013) and (2) tapering very small singular values. The smoothingaims to discourage spurious features in the resulting model. Theexpressions of the model solution, resolution and covariance ma-trices for the damped and smoothed problems that append eq. (16)are given in Appendix A. The choices of smoothing parameter λ

and damping parameter α profoundly affect the error, roughnessand resolution of the estimated model. Although some basic criteriato objectively determine optimal damping and smoothing param-eters have been suggested in some studies (e.g. Sambridge 1990;Zhang & Thurber 2007; Nolet 2008), the choice of these parame-ters is rather subjective for the specific problem. We will discussthe details of how to choose α and λ for our amplitude tomography

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in Section 5.1 with realistic data. We implement our tomographicmethodology discussed in Section 4 into a new program package.

5 A P P L I C AT I O N : T O M O G R A P H I C 1 H zQ M O D E L F O R S O U T H E A S T E R N C H I NA

5.1 Choice of damping and smoothing parameters

We apply the novel matrix-solver tomographic method developedin Section 4 to southeastern China, where the modeling errors ofthe collected data have been thoroughly analysed in Section 3.2. Toobtain the 1 Hz Lg Q tomographic model, as well as its resolutionand uncertainty, we only make use of the RTS and RTE measure-ments, which are free of both source and site response effects. Fig. 7shows 1482 RTS and RTE unique paths with colours representingtheir 1 Hz Q values. We discretize the Q model by dividing the studyarea into 0.5◦ × 0.5◦ cells based on crude checkerboard tests.

We employ both damping and Laplacian smoothing to solve thematrix problem defined in eq. (16). To select damping parameterα, we adopt the commonly used ‘L-curve’ criterion. The L-curve isconstructed using values of ||Am − d||2 versus ||m||2 with varyingα. The optimal α value is given by the point along the curve closestto the origin of the coordinate axes, balancing the trade-off betweenthe data residual norm and model norm. The L-curves used for α

selection are shown in Fig. 8(a), where each L-curve is constructedwith a specific value of smoothing parameter λ. To select an optimalλ, we add another criterion requiring a minimal occurrence of phys-ically implausible model parameters corresponding to very small1/Q values (Nolet 2008). When the 1/Q values are very small (e.g.1/Q ∼ 1/1000), attenuation in that sub-area is very weak and theestimated values in the model become unstable, potentially leadingto very large or even negative values of Q. We seek for λ valuesthat lead to a minimum number of model cells within which 1/Q isbelow a threshold (set here to ∼1/1000). Fig. 8(b) shows that thepercentage of 1/Q values less than 1/1000 varied with differing λ

and α values. When λ is larger than ∼1000–2000, the percentageof cells with small 1/Q values is below 4 per cent at most α values(Fig. 8b). We therefore set our optimal smoothing parameter λ tobe 2000. We then use the respective L-curve (Fig. 8a) to obtain anoptimal α of 750.

5.2 Lateral variations of 1 Hz Q

Our study area in southeastern China includes the southernmostpart of the Sino-Korean Craton (also known as the North ChinaBlock) to the north, the Songpan-Ganzi and Sanjiang fold belts(tectonic terranes in the Tibetan Plateau) to the west and the entireSouth China Block, consisting of the Yangtze Craton, CathaysiaBlock and South China fold belt, to the south and east. We choosesoutheastern China as an experimental field for our methodologiesbecause this region contains contrasting tectonic structures.

Our tomographic Q model (Fig. 9) contains distinct and interest-ing patterns of lateral Q variations that correlate with these varyingtectonic blocks. Prior studies (e.g. Xu et al. 2007; Yu et al. 2010)have suggested that inside the Cathaysia Block there may be twodistinct terranes separated by a roughly east–west boundary: theWuyishan in the northeast, which consists of Precambrian base-ment, and the Nanling-Yunkai in the southwest, which consistsof Meso-Neoproterozoic crust that has been extensively reworkedat various stages during the Caledonian (450 Ma), Indosinian(240 Ma) and early Yanshanian (160 Ma) thermal events. Lg Qvalues are high (≥800) in the Wuyishan terrane. By contrast, they

Figure 8. (a) ‘L-curve’ showing the trade-off between the residual misfit||Am − d||2 and the model norm ||m||2 with a varying damping parameterα. Each curve is constructed for a different smoothness parameter λ, whosevalues are indicated by different colours and are labeled near the curves.(b) Percentage of model cells with estimated 1/Q < 1/1000, plotted as afunction of smoothness parameter λ. Each curve is constructed for a specificdamping parameter α, whose values are indicated by different colours andlabeled near the curves.

are relatively low (∼350) in the Nanling-Yunkai terrane. Interest-ingly, the high Q in the northeast of Cathaysia coincides with theexposed Precambrian basement while the lower Q values in thesouthwestern part of Cathaysia coincide with the reworked crust inthat area.

The amalgamation of the Yangtze Craton, an area consistingof Precambrian rock overlain by a sedimentary succession, withthe Cathaysia block occurred early on [estimated ages range be-tween ∼1.0–0.85 Ga (Pirajno 2013) and ∼860–800 Ma (Wang et al.2007)], resulting in the Jiangnan orogenic belt, which separates thetwo units. During the later Permian to early Triassic, the northernedge of the Yangtze Craton subducted northward. This subductionled to the collision of the South China Block and the Sino-KoreanCraton, which in turn formed the Qinling-Dabie orogenic belt, amajor late Triassic collision zone consisting of the Qinling foldsystem and the Dabie Mountains. These complex tectonic eventshave resulted in the distinct Q patterns crossing the two collisionzones. There is a high Q zone starting from the west of the Qinlingportion of the Qinling-Dabie belt and stretching southward to theSichuan Basin. The Q along the Dabie Mountains is ∼400, flanked

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Figure 9. Tomographic map of 1 Hz Lg Q obtained by inverting combined RTS and RTE data.

by the low Q (∼150–250) regions of the Weihe Graben (a trans-extensional basin) and the North China Plain to the north, and theHanzhong Plain to the south. The Qingling-Dabie orogenic belthas remained stable since late Jurassic times, whereas the plainsto the north and south have been affected by more recent tectonicactivities. Overall, these complex Q variations are related to thecomplicated and largely unresolved tectonic evolutionary history ofthe collision zones, and to the unsolved mechanism through whicha competent western margin of eastern China strongly resists thewestward pushing of the Tibetan Plateau.

An interesting feature of our model is that Q in the Sichuan Basinexhibits a significant east–west variation, with high Q (∼300–600)in the east and low Q (∼100–150) in the west. The values of high Q inthe eastern basin are similar to the Q values estimated by Zhao et al.(2013) for the entire Sichuan Basin. Such a pattern demonstrates thatour data coverage and new methodology are capable of resolvinga previously unseen east–west variation of Q in the Sichuan Basin(Fig. 10). The high Q values in the eastern part of the Sichuan Basincorrelate well with exposed Precambrian basement (Wei et al. 2015)whereas the thick and younger sediments found in the western partof the basin (e.g. Meng et al. 2005) correspond well with low Lg Q.These results also compare well with seismic velocity tomographystudies conducted using both body and surface waves (e.g. Xieet al. 2013; Jiang et al. 2014). These studies suggest significantcrustal velocity changes from west to east, particularly in the uppercrust. The high Q values observed in the eastern Sichuan Basinextend much further north, forming a high Q (∼600–800) belt thatpasses between the Longmanshan fold belt and the Qinling foldsystem, ultimately terminating at ∼35◦N. This high Q zone roughlycoincides with a north–south tectonic belt under which the Moho isrelatively flat at depths of ∼40–45 km (e.g. Jiang et al. 2014; Chen& Niu 2016). This belt separates the rapidly westward thickeningcrust of the easternmost Tibetan Plateau and the rapidly eastward

thinning crust of eastern China. Under this high Q belt there is apersistent pattern of abnormally high crustal P-wave velocities atall depths (Jiang et al. 2014).

Finally, along with the Sichuan Basin, the Songpan-Ganzi terraneand the Sanjiang fold belt show the lowest Lg Q (∼100–150) valuesin our model, which is in agreement with the results of previousQ measurements for the Tibetan Plateau (Xie et al. 2004, 2006;Phillips et al. 2005; Zhao et al. 2013). Persistent low Lg Q values(<200) in the plateau have been attributed to high temperatureand/or mid-crustal melt bearing materials (e.g. Xie et al. 2004).

5.3 Model resolution, uncertainty and data predictability

We use eqs (19), (A6), (A7) and (A17) to compute the resolu-tion matrix Rm of the model discussed in the prior sections. Aspreviously mentioned, Rm quantifies the relationship between theestimated model and the true tomographic model of 1/Q. As in a typ-ical seismic tomography problem, our Rm is not diagonal, meaningthe estimated model is a smeared version of the true model. Morespecifically, the resolved 1/Q value in any cell is a weighted av-erage of the true value contained within that cell and the valuesof the adjacent cells, causing a limited spatial resolution. It is in-convenient to view the resolution of a tomographic model directlyfrom off-diagonal elements of Rm, so we use the Backus–Gilbertspread function (SF) to measure the smearing distance caused bythe resolution matrix. For the ith model element, the SF is definedas (Menke 2012)

SFi =√√√√ M∑

j=1

d2i j R2

i j

/M∑

j=1

R2i j , (24)

where j denotes cells for which elements of the resolution matrixRm, Ri j , are non-zero. di j is the distance between the centres of

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Figure 10. Spread functions of the tomographic Q model, which give approximate resolutions at the locations of each model cell. The unit of the colour bar isin kilometres.

the cells i and j (in km). The SF is a root-mean-square distancewith weighting controlled by elements of the resolution matrix Rm,resulting in an approximate measure of the finite resolution of theestimated model of 1/Q (Backus & Gilbert 1970). The SF values areplotted in Fig. 10 for all the model cells. The minimum SF valuesapproaches the cell size (∼50 km). The smallest values (highestresolutions) are found in the northwestern Qinghai Province, wherestation coverage is dense (Fig. 2). In most of our study area, the SFvalues are less than 250 km, so the resolution of our 1/Q model isno worse than ∼250 km.

The uncertainty of the 1/Q model is fully quantified by the modelcovariance matrix (eqs 20, A8–A10 and A18–A19). To the first or-der, this uncertainty is equal to the relative uncertainty of Q, δQ/Q,whose standard deviation can be approximately estimated usingthe square root of the diagonal elements of the covariance matrix.Fig. 11 shows the relative uncertainty δQ/Q values (percentage)in units of one standard deviation. Although we do not plot theoff-diagonal elements of the model covariance matrix, we find thatthey contain non-zero and sometimes negative values, similar tothe covariance matrix found by Ballard et al. (2016) for traveltimetomography. Because the non-zero off-diagonal values indicate acorrelation between model parameters, any calculation of the pre-dicted 1/Q variance along a new path must involve the full modelcovariance matrix (eq. 22). Note that from eq. (22), the uncertaintyof 1/Q is a random error quantified by Cm which is linearly mappedfrom the data covariance matrix Cd. Since the data are normallydistributed (Section 3.2), the uncertainty of 1/Q is also a normalrandom variable whose variance can be estimated.

In this case, the approximate model uncertainties at one standarddeviation illustrated by Fig. 11 for our 1 Hz Q model are less thanabout δQ/Q = 7 per cent. In practice, this information can be used

to determine the relative uncertainty of 1 Hz source spectral esti-mates via eq. (23). For example, for an upper limit δQ/Q (withinone standard deviation) of ∼7 per cent and a moderate 1 Hz Q es-timate of 450, the uncertainties of estimated δSe( f )/Se( f ) (eq. 23)would be about 7 per cent at 500 km and 14 per cent at 1000 km at a68 per cent confidence level. The uncertainties double if the confi-dence level is set at 95 per cent.

5.4 Inversion of site response

In the final part of our study, we investigate the site responsesdetermined using the RTS technique. Logarithms of all site responseratios at 1 Hz measured using eq. (6) can be used to set up a linearequation system

ln(Ri ) − ln(R j ) = bk (i, j = 1, . . . , M ; k = 1, . . . , N ),

(25)

where data bk are measured differential site responses between sta-tions i and j. If there are sufficient measurements resulting in ahigh-rank linear system, eq. (25) can be used to solve for individualln(Ri ) values. A caveat is that since all measurements give only dif-ferential estimates (ln(Ri ) − ln(R j )), absolute values of ln(Ri ) arenot resolvable. To overcome this problem, we impose a constraintthat the average of the site responses is unity:∑M

i=1 ln (Ri )

M= 0. (26)

In southeastern China, there are 101 stations (out of the totalof 102 used stations) that can provide site response ratios. Weform a matrix problem of the same form as eq. (16) to solve for

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Figure 11. Relative uncertainty (δQ/Q) at each model cell derived from the diagonal elements of the covariance matrix for 1/Q. The colour bar shows thevalues of δQ/Q in percentages.

ln(Ri ) (i = 1, . . . , M) using the SVD method. The stations are sowell sampled by RTS paths that only one singular value is found tobe close to zero. It means the problem is virtually full rank and weonly need to damp one singular value to solve for the ln(Ri ) terms.The inverted site responses are shown in Fig. 12. In a statisticalanalysis we estimate that more than 97 per cent of the site responsesrange from 0.25 to 3.25. There is a tendency for larger responsevalues (>1.0) to occur inside basins. The logarithms of these siteresponses have zero mean and a standard deviation of 0.36. Therange of site responses in our study region is consistent with thesite responses obtained for other regions in the world (e.g. Drouetet al. 2008; Pasyanos et al. 2009). The site terms do not appearto correlate well with tectonic settings and hence may reflect verylocal structural effects at the surface. Nevertheless, there is a crudetrend for the site responses to be higher than 1.0 within sedimentarybasins, such as the Sichuan Basin, Qaidam Basin and HanzhongPlain, and for the responses to be lower than 1.0 in the south Chinafold belts and the Tibetan Plateau, where Bao et al. (2011) alsofound low responses.

6 D I S C U S S I O N A N D C O N C LU S I O N S

This study focuses on two profound issues of attenuation (Q) to-mography. One issue is that Q tomography does not use observedground motion as input data. Rather, the tomography uses ‘reducedamplitude data’ derived by multiplying the observed amplitude withan oversimplified 1D geometrical spreading correction and then tak-ing the logarithm of the resulting quantity. The error in such data isdominated by ‘modeling errors’ arising from 3D structural effects,pre-event and pre-phase noises and azimuthally varying station site

responses. Previous works on Q tomography have implicitly as-sumed a normal distribution of the error in reduced amplitude data inorder to justify solving the tomographic Q model as a least-squaresproblem. A rigorous justification of the normality assumption hasnever been attempted. Motivated by the lack of statistical analysisof the reduced amplitude error, we conduct the first ever rigorousstatistical analysis of this error. Under a first-order assumption, wedevised a new scaled random variable, denoted by ε, which is lin-early connected to the reduced amplitude error through a factor ofVπ f , but is not explicitly dependent on path length. ε is a singlerandom variable for an entire set of reduced amplitude data acrossall paths involved. Samples of ε can be easily collected and thus wetransform the difficult task of statistically analysing reduced ampli-tude data error into the more tractable task of analysing ε instead.Detailed analysis shows that the distribution of ε is dominated bya peak that mimics a normal distribution supplemented by outliers,which cause weak long-tails to spread outward on either side ofthe peak. Interestingly, this statistical property of the reduced am-plitude data error is very similar to that of unscreened teleseismictraveltime residuals analysed by Nolet (1987). We develop a datascreening criterion in which large outliers, with ε values differingfrom the statistical mean by more than two-standard deviations, areremoved. This screening results in a new data set that closely followsthe normal distribution, thus justifying solving Q tomography as aleast-squares problem. We note that the modeling error addressedin this paper is also present in coupled source spectra-path Q to-mography (Menke et al. 2006), which will be addressed in futurestudies.

The second issue of Q tomography is that it has been solvedusing various iterative methods that are aimed at finding Q mod-els that rapidly improve data fit, but are unable to obtain formal

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Figure 12. Site responses estimated for 101 stations in southeastern China.

quantitative estimates of model resolution and errors. Empiricalsynthetic checkerboard and impulsive model tests can only crudelyestimate the model resolution and error. As a result, the predic-tion of Q values along new paths can be done but without formaluncertainty estimates. To address this issue, we develop a new tomo-graphic method by performing efficient SVD on the kernel matrixusing the PROPACK software package. With this matrix-solver ap-proach, we obtain not only the Q model, but the formal resolutionand covariance matrices associated with the model. The estimatedcovariance matrix in particular allows for the estimation of the for-mal uncertainty of any 1/Q predicted for a new path. This newcapability is especially important for the practical mission of eventidentification and source size estimation.

This new methodology is successfully used to invert for a 1 HzLg Q tomographic model in southeastern China. The results demon-strate the capability of our new methodology in providing (1) quan-titative model appraisals, (2) a solid basis for using only the reliablyrecovered features of the Q model to infer crustal properties and(3) path-specific Q predictions with formal uncertainty estimation.Finally, our methodology can be applied to other high-frequencyregional phases such as Pg, Pn and Sn.

The main results of this study can be summarized as the follow-ing:

1. Statistically, the original distribution of the reduced ampli-tude data is dominated by a peak that is characteristic of a normaldistribution, with the addition of long tails caused by outliers.

2. We developed a data screening criterion by which outliers areremoved, resulting in a new dataset that closely follows a normaldistribution. The outliers constitute 14 per cent of the unscreeneddata. The normalized random variable ε, sampled over repeated

path measurements using each of the RTS, RTE and TS methods,is normally distributed and has a variance of 0.0637, 0.0699 and0.1428, for the RTS, RTE and TS methods, respectively. The fact thatVar(εTS) > Var(εRTS) and Var(εRTE), can be largely attributed to theeffects of uncorrected site response ratios in the TS measurements.

3. We implement a new software package to solve for the tomo-graphic Q model and its inverse, model resolution and covariancematrices by explicitly decomposing the large and sparse kernelmatrix using the PROPACK package. We implement a smooth-ing and damping procedure that not only optimizes the trade-offbetween model smoothness and data fit, but also minimizes thatoccurrence of very large Q (>1000) parameters that are physi-cally non-plausible and numerically unstable. Our matrix-solverapproach enables us to formally construct resolution and covari-ance matrices while solving for a tomographic Q model.

4. The 1 Hz Lg Q tomographic map of southeastern China sug-gests that Lg Q values coincide well with the distinct tectonic blocksin this region, likely reflecting the most recent tectonic history ofthese blocks. Specifically, the Cathaysia Block exhibits a contrast-ing pattern, with high Q (≥800) in the northeast and relatively lowQ (∼350) in the southwest, corresponding to the Wuyishan and theNanling-Yunkai units, respectively. The Q values in areas with thicksedimentary formations tend to be low, particularly in the westernSichuan Basin where Q is as low as ∼180. The Songpan-Ganziterrane and the Sanjiang fold belt exhibit the lowest Lg Q (∼100–150), reflecting the unusually low Q structure and Lg blockage inthe easternmost Tibetan Plateau. The relatively stable Late Juras-sic age Qinling-Dabie orogenic belt collision zone, which separatesthe South China Block and the North China Block (Sino-KoreanCraton), exhibits distinctly higher Q values than either block, wheremore recent tectonic activities have occurred.

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5. The resolution of the Lg Q tomographic model has been es-timated using the SF derived from the full-resolution matrix. Theresolution of the study area ranges between ∼50 and ∼250 km.

6. We invert the site responses of 101 stations in southeasternChina. We find that ∼97 per cent of these responses lie between∼0.25 and 3.25, consistent with previous studies. The sedimentaryregions tend to have site responses higher than 1.0, with rathererratic variations reflecting localized structural effects.

7. The quantitative uncertainty of the tomographic model allowsus to obtain formal error estimates associated with any path-specificQ predictions. These error estimates can be used to establish con-fidence levels for event identification and source size estimationwhen path Q corrections are used.

A C K N OW L E D G E M E N T S

We would like to thank Z. Huang, R. Liu, S. Liang and other staffmembers at the CENC for providing valuable data processing ser-vices and technical support. We would also like to thank AndreaGallegos for her help in structuring and polishing the manuscript,G. Eli Baker and Huining Kang for their suggestions and discus-sions on the statistical testing of the data error distribution, W. ScottPhillips, Xiwei Xu, Guihua Chen, Shunping Pei and Jiankun He fortheir insightful discussions on our Lg Q tomographic model andZhenkang Shen, Chun-Yong Wang and Mingjie Xu for providinginformative geological knowledge of our study region.

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A P P E N D I X : DA M P E D L E A S T - S Q UA R E SP RO B L E M W I T H S M O O T H N E S SC O N S T R A I N T S

Both damping and smoothness constraints are usually required ingeophysical inverse problems to deal with rank-deficient and ill-conditioned linear systems. They make such linear systems solv-able based on SVD. Generally, both damping and smoothing canbe expressed in a matrix form as κLm = 0, where κ is a weightingparameter that controls how strongly the regularization should beimposed. This matrix form is then appended to the original linearsystem. Examples of L include Tikhonov regularization matriceswith different orders (e.g. Aster et al. 2013). The L matrices used assmoothness constraints contain off-diagonal elements and smooththe model parameters with operators such as a 2D Laplacian ora Gaussian. In a damped least-squares problem, L degrades to anidentity matrix. In this appendix we derive expressions of the modelsolution, resolution and covariance matrices when smoothness con-straints and damping constraints are used.

A1 Model solution, resolution and covariance matriceswith smoothness constraints

For a linear system with smoothness constraints, we work on an

augmented kernel matrix G = [ AλL] and solve new equations

Gm = [d0] . The SVD of G is

G =[

AλL

]= P · � · QT , (A1)

where � is the singular value matrix, and P and Q are associatedsingular vectors for G in matrix form. The matrix P can be writtenin a partitioned form

P =[

P1

P2

], (A2)

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where matrix P1 corresponds to the NA data rows associated withA, and P2 corresponds to the NL regularization rows associated withL (Vasco et al. 2003). The partition for P allows us to express theorthogonality of P as

P = PT P = [PT

1 PT2

] ·[

P1

P2

]= PT

1 P1 + PT2 P2 = I (A3)

and rewrite matrix A as

A = P1 · � · QT . (A4)

When the inverse of G exists and can be written as G−1 =Q · �−1 · PT , the SVD solution for the model parameters is

m = Q · �−1 · PT ·[

d0

]= Q · �−1 · PT

1 · d. (A5)

Because the original linear system Am = d still holds, inputtingeq. (A4) into (A5) lets us obtain the model resolution matrix in theform

Rm = Q · [�−1 · PT

1 · P1 · �] · QT (A6)

or

Rm = Q · [I − �−1 · PT

2 · P2 · �] · QT , (A7)

in which the orthogonal relationship shown in eq. (A3) is adopted.The choice of using eq. (A6) or (A7) to compute Rm dependson the relative size of matrices P1 and P2. It is computationallyadvantageous to choose the matrix with a smaller size to computeRm.

According to the definition of the data covariance matrix in eq.(20), where the pseudo-inverse A† is replaced with G−1, we canexpress Cm in terms of the SVD form of G

Cm = [Q · �−1 · PT

1

] · Cd · [P1 · �−1 · QT ]. (A8)

Note that the NL zeros inserted into the data vector donot have errors, so eq. (A8) only involves the original datavariance Cd and the matrix P1 corresponding to NA datarows.

If the linear system is normalized by C−1/2d as shown in eqs (15)

and (16), the variance of the ‘new’ data on the right-hand of eq.(15) reduces to an identity matrix. Eq. (A8) can thus be simplifiedto

Cm = Q · �−1 · [PT

1 P1

] · �−1 · QT (A9)

or

Cm = Q · �−1 · [I − PT

2 P2

] · �−1 · QT

= Q · [�−2 − �−1PT

2 P2�−1

] · QT . (A10)

Similar to the resolution matrix, the orthogonal properties of Pare adopted in (A10) and choosing (A9) or (A10) to compute Cm isdependent on the relative size of matrices P1 and P2.

A2 Model solution, resolution and covariance matriceswith damping

The smoothness constraints cannot guarantee the full rank of G,and thus cannot guarantee the existence of G−1 either. The damp-ing is specifically designed to deal with a rank-deficient problemof a linear system. Damping can be introduced by introducing anapproximation, or pseudo-inverse, of the singular matrix S with

a damping parameter α (Section 4.2). Alternatively, it can be in-troduced formally as a regularization by appending a new set ofequations αIm = 0 to the original equations:

Gm =[

AαI

]m =

[d0

], (A11)

where I is the identity matrix and α is the damping parameter. Aslong as α is nonzero, the last M rows (number of model parameters)in eq. (A11) are linearly independent. Eq. (A11) is thus full rankand can be solved in the sense of least squares by the method ofnormal equations

[ AT αI ] ·[

AαI

]m = [ AT αI ] ·

[d0

], (A12)

(e.g. Aster et al. 2013) which can be simplified to

(AT A + α2I)m = AT d. (A13)

These equations mean that the least-squares damped problemcan be solved by working on the original kernel matrix A directly,instead of on the augmented matrix G. Employing the SVD ofA = USVT , we expand eq. (A13) to(

VST UT USVT + α2I)

m = VST UT d. (A14)

Repeatedly using UT U = I, VT V = VVT = I, ST = S andST S = S ST = S2 on eq. (A14), we have the following deductions(VS2VT + α2I

)m = VST UT d

V(S2 + α2I

)VT m = VST UT d (A15)

(S2 + α2I

)VT m = ST UT d

m = V · S(S2 + α2I

)−1 · UT · d. (A16)

Eq. (A16) defines a pseudo-inverse of A as A† = V ·S(S2 + α2I)−1 · UT by replacing S−1 with S−1(S2 + α2I)−1. Thisresult is why the regularization matrix approach here has the sameeffect as simply finding a pseudo-inverse of S (Section 4.2) with-out specifically introducing regularization. Obviously, the dampingensures the stability of the solution of the linear system by taperingvery small values in S through parameter α.

In terms of the definitions of the model resolution matrix ineq. (19) and the covariance matrix in eq. (20), the Rm and Cm forthe damped problem can be respectively expressed as

Rm = V · (S2 + α2I)−1S2 · VT (A17)

and

Cm = [V · (S2 + α2I)−1

S · UT ] · Cd · [U · S(S2 + α2I)−1 · VT ],

(A18)

respectively. If the data variance degrades to an identity matrix,equation (A18) can be simplified to

Cm = V · S2(S2 + α2I)−2 · VT (A19)

In the damped problem, the singular vectors of the original ma-trix are not changed. Therefore, if damping and smoothing areboth adopted in a tomography problem, the inverse of the sin-gular values �−1 in eqs (A5)–(A10) should all be replaced by�(�2 + α2I)−1.

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