Advances in modelling and inversion of seismic …seismain/pdf/2010_h019z_2009.pdfAdvances in modelling and inversion of seismic wave propagation V. Hermann, N.D. Pham, A. Fichtner,

Advances in modelling and inversion of seismicwave propagation

V. Hermann, N.D. Pham, A. Fichtner, S. Kremers, H.-P. Bunge,H. Igel

Abstract We report on progress in modelling and inversion of seismic waveforms.This involves in particular the simulation of wave propagation through Earth mod-els with complex geometries (i.e., internal interfaces or topography) using numericalsolutions based on tetrahedral meshes. In addition, efficient solvers in 3-D based ona regular-grid spectral element method allow for the simulation of many Earth mod-els and for the inversion (i.e., for the fit) of observed seismograms using adjointtechniques. We present an application of this approach to the Australian continent.Furthermore results are presented on exploiting ideas fromreverse acoustics to esti-mate finite source properties of large earthquakes and to constrain crustal scatteringthrough modeling joint observations of rotational and translational ground motions.

1 Introduction

Computational seismology has become an increasingly important discipline of seis-mology and will become even more relevant as the observed data volumes increase.After several years of code developments in the field of computational wave prop-agation, now the focus is on solving scientific problems withthe tested and bench-marked research codes. Here we report results based on two flavours of numericalmethods: the spectral element method and the discontinuousGalerkin method.

Within the last few decades a number of different numerical methods for mod-elling and inversion of seismic waveforms has been developed, including basic finitedifference schemes ([44, 60]), Fourier pseudospectral methods ([16, 29]), finite el-ement approaches ([12, 47]) and spectral element methods (SEM) ([54, 36]). TheDiscontinuous Galerkin (DG) method ([20]) has been developed within the last few

Verena Hermann, Nguyen Dinh Pham, Andreas Fichtner, Simon Kremers, Hans-Peter Bunge,Heiner Igel, Department of Earth and Environmental Sciences, Ludwig-Maximilians-Universityof Munich, Germany; e-mail: [email protected]

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2 V. Hermann, N.D. Pham, A. Fichtner, S. Kremers, H.-P. Bunge, H. Igel

years and has already become a well-established numerical method. It is suitable forthe simulation of seismic wave propagation in general and able to handle complexgeometries as well as heterogeneous media using an arbitrary high approximationorder in space and time (ADER-DG). Many simulations have been accomplished onthe SGI Altix 4700 machine (HLRB II) of Leibniz-Rechenzentrum. One of the mod-els, e.g., serves as a demonstration for handling quite complex structures. It containsdifferent materials of strongly varying velocities including a salt dome which is ofgreat interest in exploration industry. In this example, the computational domain iscomposed of 3.1 million elements. For a simulated time of 10s, the program wasrunning about 20h on 512 processors. In this study, the ADER-DG method is usedto examine topographic effects on wave amplification. This subject was first intro-duced over 30 years ago and is still an ongoing topic as shown by recent publications(e.g. [43]).

In the field of seismic tomography we are now moving into a new era: while sofar seismic data were reduced to a few travel time observations of some seismicphases (e.g., P and S waves) the aim is now to use the information contained inthe complete waveforms of the observations. In the section on seismic tomographywe show the first application of this concept to tomography ona continental scale.The tomographic inversion of the subsurface structure under Australia demonstratesfor the first time that - given the current computational power - inversion based oncomplete 3-D wave propagation solvers is feasible and leadsto much improvedtomographic models ([24],[25]).

We use a simular concept to investigate whether it is possible to recover earth-quake source characteristics by time reversing seismic wavefields. In theory, timereversing the complete seismograms as source injected at the receiver locations con-stitutes the adjoint field and leads to a first update for the seismic source process([59]). The results shown indicate that the concept works inprinciple but that a veryhigh station density is necessary to quantitatively recover source characteristics.

Finally, the discontinuous Galerkin method is used to directly model an entirelynew type of observation: rotational motions (in seismologywe usually observe threecomponents of translational motions). It turns out that this new observable - in com-bination with collocated translational observations - allows the recovery of informa-tion on the scattering properties of the Earth’s crust ([52]).

All the results presented indicate that the 3-D modelling tools for seismic wavepropagation can now be used to model actual observations. The parallelized algo-rithms based on implementations with Fortran-MPI have beenused on O(100) pro-cessors, and show good scaling behaviour. However, for the next generation hard-ware, more tests and optimization is required to render the algorithms performant onO(100k)-core hardware. We intend to accomplish this through collaborations withinthe MAC (Munich Centre for Advanced Computing) project.

computational seismology 3

2 Topographic effects on seismic waves

Amplification effects on seismic waves due to strong topographic gradients havebeen the subject of thorough analysis for more than 30 years ([50]). A number ofnotable earthquakes have led to repeated and consistent observations of amplifiedground motion on top of rising topographic features such as mountains, ridges andcliffs. According to Boore [13], surface topography can be neglected when seismicwavelengths are much larger than topographic irregularities and the slopes of theirregularities are small. If these conditions do not hold, considerable amplificationcan be observed (e.g. [13, 14, 19]).

At Mount Hochstaufen (see [39]), SE-Germany, we observe topographic ampli-fication up to a factor of 4.4 compared to a station at the mountain base using thespectral ratio method. The obtained amplification increases with rising receiver al-titude.

In order to examine this effect more intensely, we try to model this amplificationpattern. Therewith we are prepared to extend the study theoretically by simulat-ing constructed topographies. For the simulation we use high-performance compu-tational methods: The ADER-DG method offers the possibility of handling verycomplex geometries and achieving arbitrary high orders of accuracy within theframework of numerical wave propagation ([20]). This ability is needed for sucha rugged topography as the Staufen Massif displays. Our simulations were runningas a 128-processor partition on the Intel Itanium2 Montecito Dual Core machine(called HLRB II) of the Leibniz-Rechenzentrum in Garching.

For the model setup we employ a digital elevation model of thearea around theStaufen Massif(Ω = [0;18.5km]× [0;17.5km]× [−6km;1.5km]) kindly providedby the Bavarian Geologic Survey. The three dimensional unstructured tetrahedralmesh contains about 1.6million elements with a spacing of 200m at the surface andgrowing with depth to 350m (see Fig. 1). Throughout the whole domain we keepthe material properties constant(vp = 4.7km

s ). As we want to involve only teleseis-mic events we choose an inflowing plane p-wave inz-direction as a source term.Having finished the simulation we can deconvolve the input from the computedseismograms and obtain the Green’s function of our model at some defined receiverlocation which can be used for ongoing analysis.

Comparing real to synthetic data, the time domain amplification factors arewithin the same range whereas due to the many simplifications, data were not uti-lizable for a profound spectral analysis. We examine the amplification factors asratios of groundmotions at a receiver location in the valleyaround Hochstaufen andone on top of the mountain. Depending on the component (N, E and Z) we obtainvalues of 1.25 (N), 3.81 (E) and 3.28 (Z) for the observed seismograms. Averag-ing over only 7 events the standard deviation of 25% is acceptable. The spectraldistribution of amplification shows a larger difference in the amplification of theE- and Z-components compared to the N-component with risingfrequency. The E-component is amplified most, the Z-component has only slightly smaller values, andthe N-component suffers least amplification, which might berelated to the east-westelongated shape of Mt. Hochstaufen.


Fig. 1 Unstructured tetrahedral mesh for the area around Mt. Hochstaufen. The unit is meter.

For the numerically computed data time domain amplificationshows average valuesof 1.88 (N), 7.94 (E) and 1.21 (Z). The standard deviation on average accounts for22%. The trend in amplification could be partially extractedfrom the simulation, asthe E-component displays the largest amplification for bothdata sets. There appearsrising amplification with altitude and stronger topographic gradients.

Concluding, one can say, that this study was a first step towards a very interestingsubject of topographic effects at Mt. Hochstaufen. Furtherapplications incorporatehigher topographic resolution, different slopes and altitudes. We intend to analyzethe frequency dependence of the amplification factor in detail.

3 Seismic tomography using waveforms

Recent progress in numerical methods and computer science allows us today to sim-ulate the propagation of seismic waves through realistically heterogeneous Earthmodels with unprecedented accuracy (e.g. [22, 37, 20]). These new capabilitiesmust now be used to further our understanding of the Earth’s 3D structure. De-tailed knowledge of subsurface heterogeneities is essential for studies of the Earth’sdynamics and composition, for reliable tsunami warnings, the monitoring of the


Comprehensive Nuclear Test Ban Treaty and the exploration for resources includ-ing water and hydro-carbons.

Full waveform tomography is a tomographic technique that takes advantage ofnumerical solutions of the elastic wave equation (e.g. [38,57, 24, 25]). Numericallycomputed seismograms automatically contain the full seismic wavefield, includingall body and surface wave phases as well as scattered waves generated by lateralvariations of the model Earth properties. The amount of exploitable information isthus significantly larger than in tomographic methods that are based, for example,on measurements of surface wave dispersion or the arrival times of specific seismicphases. The accuracy of the numerical solutions and the exploitation of completewaveform information result in tomographic images that areboth more realistic andbetter resolved [25, 57].

We developed a novel technique for full 3D waveform tomography for radiallyanisotropic Earth structure. This is based on the combination of spectral-elementsimulations of seismic wave propagation and adjoint techniques. The misfit betweenobserved seismograms and spectral-element seismograms isreduced iteratively us-ing a pre-conditioned conjugate-gradient scheme.

The application of our method to the upper mantle in the Australasian regionallows us to explain the details of seismic waveforms with periods between 30sand 200s (Fig. 2), and it provides tomographic images with unprecedented resolu-tion (Fig. 3). In the course of 19 conjugate-gradient iterations the total number ofexploited waveforms increased from 2200 to nearly 3000. Thefinal model,AM-SAN.19, thus explains data that were not initially included in the inversion. Thisis strong evidence for the effectiveness of the inversion scheme and the physicalconsistency of the tomographic model. Our model of the shearwave speed in theupper mantle (Fig. 3) reveals the deep structure of the Australasian region: TheCoral and Tasman Seas, located east of mainland Australia, are characterized by apronounced low-velocity zone, centred around 140km depth. It indicates the pres-ence of a flow channel where temperatures are several hundreddegrees higher thanin the surrounding mantle. A similar low-velocity zone is not present under conti-nental Australia. A low-velocity band extends along the eastern continental margindown to at least 200km depth. This is associated with high surface heat flow, re-cent volcanism and some mild seismic activity. The continental lithospheric mantleis confined to depths above 250km, where the velocity perturbations exceed 5%.A region of high wave speeds under the Arafura and Timor Seas is evidence for anorthward continuation of the North Australian craton.

Currently our research focusses on the application of our method to the Europeanupper mantle and regions where more accurate Earth models are needed for theimproved assessment of seismic risk.


Fig. 2 Exemplary waveform comparisons for a variety of source-receiver geometries. Black solidlines are data, black dashed lines are synthetics for the initial model (a smoothed version of aprevious tomography) and red solid lines are synthetics forthe final model shown in Fig. 1. Thedominant period is 30 s. A time scale is plotted in the upper-right corner. While significant discrep-ancies exist between data and the initial synthetics, the final synthetics accurately explain both thephases and the amplitudes of the observations.

4 Time reversal of seismic waves

Time Reversal is a promising method to determine earthquakesource characteristicswithout any a-priori information (except the earth model and the data). It consistsof injecting flipped-in-time records from seismic stationswithin the model to createan approximate reverse movie of wave propagation from whichthe location of thesource point and other information might be inferred. The backward propagation isperformed numerically using a spectral element code. We investigate the potentialof time reversal to recover finite source parameters, test iton a synthetic data set,the SPICE Tottori benchmark model, and on point source moment tensor sources.Time Reversal has been largely used in acoustics in the last twenty years for locatingsound source and scatterers in laboratory experiments ([26]; [63]; [17]; [27]; [28]);however the basic ideas date back to the early 1960’s ([51]).In its simplest form,Time Reversal is used to reconstruct a sound source. This is accomplished in the


Fig. 3 Horizontal slices through the relative lateral variationsof the shear wave speed at differentdepth in the Australasian region

laboratory by measuring acoustic waves emanating from an acoustic source. Thewaves are reversed-in-time, and re-emitted from the detector locations. Due to reci-procity and invariance of the wave equation to time-reversal, they back-propagatefollowing their forward wave paths back onto the source point. When the backwardstep is numerically performed, the method is often referredas Time Reversal Imag-ing and relies on the accuracy of both the numerical scheme and the velocity modelmimicking the actual media. In the past several years Time Reversal has begun to beapplied to seismological questions first related to seismicmigration ([9], [58], [18])and then to source imaging but limited to simple velocity models ([46], [53]). Inthe last decade, the drastic increase of compute power combined with the develop-ment of new methods (beside the classical finite difference method) such as spectralelement has made the modeling of wave propagation in complexvelocity modelsfeasible. Application to actual seismic recordings was first conducted by [40] forthe 24 December 2004, M9.0 Sumatra–Andaman earthquake. Most recently, sev-eral glacial earthquakes involving sudden motion of ice-mass downhill were locatedwith the method [41].

Time Reversal versus adjoint methods Time Reversal is the first step in theiterative inversion problem of source location if the problem is formulated usingthe adjoint technique (e.g., [59]). Therefore it is important to note that when timereversing seismograms we do not actually image the source but estimate a sourceupdate (it would be the source only in the case of an exactly linear problem). Anexample is shown in [59] for a point source of which the location and source timefunction are known and the moment and mechanism are the unknowns. It is shown


that to calculate the gradients the elements of the strain tensor are used. In thisformulation the adjoint sources are simply the time-reversed data.

Synthetic Tests The concepts for time-reversing seismograms recorded close tolarge seismic sources are first illustrated using syntheticmodelling. The forwardmodelling is carried out using a spectral-element approach(e.g., [23]) in Carte-sian coordinates. To investigate the potential to recover quantitative information onfinite-source properties (e.g., source extent, rupture velocities, location of asperities,etc.) we time-reverse synthetic seismograms calculated within the kinematic sourceinversion blind test organized within the European SPICE project ([45]). Resultscan be seen in Fig. 4. While focusing can be observed in all cases, there is alsoa substantial amount of additional energy propagating in the medium originatingfrom phases generated by the adjoint sources (e.g., surfacewaves, also called ghostwaves in the time reversal community).Synthetic test with a point source mo-ment tensor The initial tests with a point source moment tensor served asa controlwhether the code is setup correctly and the time-reversed input is working. In Fig. 5snapshots of the rotational energy for a time series ending at the expected focusingpoint in time are shown in a plane through the source for the SPICE setup with 33injected time-reversed signals. Injection at the stationscan be observed, leading toa sharp focus at source time.

How to improve the results in terms of focussing and source location estimationis matter of ongoing discussion, but valuable approaches seem to be weighting ofthe adjoint sources and choosing the right field (and component) for visualization.Further studies will deal with the problem of time reversingreal earthquake data.

5 Scattering in the Earth’s crust

It was a surprise to discover considerable rotational energy in a time window con-taining the P coda in the teleseismic seismometer records of[33]. To understandthe original of the observations, we study the generation ofP coda rotations underthe assumption of P-SH scattering in the crust by modeling complete theoreticalseismograms created by a plane P wave of dominant period 1s (the same as the pre-dominant period of the observed P waves), propagating upward from the bottom ofa random medium.

A random medium can be described through a spatial distributionu(x) of materialparametersu. This distribution can be expressed as [35]

u(x) = u0(x)+U(x) (1)

whereu0(x) is the mean value ofu(x) andU(x) is a realization of the random quan-tity. In our study, to simplify the modeling, we perform simulations with randommedia in which the wave velocities are randomly perturbed inspace, but the massdensityρ , the ratio between P and S wave velocitiesVp

Vs, and the mean values of

the velocities are kept constant. The realization of the random velocities is calcu-


Fig. 4 Snapshots of a horizontal plane through the source depth (12.5 km) at source time forvarious properties of the reverse field of the SPICE finite source. Receiver locations are indicatedby ‘+‘, the location of the hypocenter by the red circle. The fault is indicated by the black line.

lated in terms of white noise filtered by a spectral filter (see[35]). Since we justconsider high frequencies (i.e., small heterogeneities),we choose the spectral filtercorresponding to the Zero Von Karman correlation function mentioned in [35]:

f (k) = κ(

a−2 + k2)−d/4, (2)

wherea is the Von Karman correlation length,d is the Euclidean dimension of thespace, andκ is a constant corresponding to the given value of perturbation.

To be as realistic as possible, the used model setup is based on the crust modelof [8] at the Wettzell area. The model is 60800m long, 60800m wide and 40850mdeep. The mass density is takenρ = 2.9g/cm3, and the mean values of P and Swave velocities (respectively) areVp = 6600m/s,Vs = 3700m/s. In order to produce


Fig. 5 Snapshots in source depth (12.5 km) at various times for the sum of squared amplitudes(rotations) of the reverse field. Receiver locations are indicated by ‘+‘, the source location by thered circle.

significant scattering energy, we use correlation lengths between 1000m and 15000m(see [2, 56]). Furthermore, the root mean square perturbation of wave velocities istaken in the range from 0% (homogeneity) up to about 11%.

Seismograms are calculated using the ADER-DG method (the combination of aDiscontinuous Galerkin finite element method and an Arbitrary high-order DERiva-tive time integration approach developed by [20]) that was extended to allow out-putting the three components of rotation rate, in addition to the three components oftranslational velocity. The modeling parameters are detailed in Tab. 1. Fig. 6 illus-trates schematically a three-dimensional Von Karman random medium used in thisstudy with correlation lengtha = 2000m, root-mean square perturbation of 6.51%.In addition, synthetic seismograms obtained at three receivers located at differentsites on the surface of the model are shown. For each set of translational veloci-


Mesh type HexahedralElement edge length 950mTotal number of elements 176128Polynomial order inside elements4Number of processors 64Length of seismograms 60sBoundary conditions Free surface (top), inflow (bottom), periodic (sides)Average time step 8.13008×10−3sRun time per simulation ≈ 2 hours

Table 1 The modeling parameters used in this study.

Fig. 6 Schematic illustration of a 3D Von Karman random medium used in this study (correlationlengtha = 2000m, root-mean square perturbation ofVs is 6.51%) and 6-component seismogramsobtained at three different receivers for a plane P wave propagating upward (in a vertical direction)from the bottom of the model. For each set of translational velocities (Vx,Vy,Vz) or rotation rates(Ωx,Ωy,Ωz) amplitudes are scaled.


ties (Vx,Vy,Vz) or rotation rates(Ωx,Ωy,Ωz) the amplitudes are scaled. The figureshows that, as waves travel through the random medium, both rotational and hor-izontal translation components are generated by scattering. The delayed arrival ofthe vertical rotation compared to the onset of the vertical velocity that we noticedin the observations [52] can also be clearly seen in our synthetics. Moreover, thesimulated seismograms differ at each station because the random field is completelythree-dimensional. This allows us to stack aspects of the data from different re-ceivers rather than multiple simulations.

The effects of the perturbation and correlation length on P coda rotations aresystematically investigated. The partitioning of P and S energy and the stabilizationof the ratio between the two are used to compare the simulatedP coda rotationswith the observed ones (see [52] for details). For each pair of scattering parameters(a velocity perturbation and a correlation length), we obtain a value of the energyratio, averaged through all 24 receivers in circular configuration (Fig. 6). Initially,we take a fixed correlation length of 2000m and calculate the average energy ratiosas a function of velocity perturbation. The results are shown as a black line in Fig. 7.As expected, the energy ratio increases as we increase the perturbation. The curve

0 2 4 6 8 10 1210

−11

10−10

10−9

10−8

10−7

10−6

Vel. Perturbation (%)

E(Ω

Z)/

E(V

Z)

Fig. 7 Comparison of observed and simulated energy ratios. Black curve: the (average) energyratio as a function of velocity perturbation calculated from simulated seismograms (correlationlength fixed at 2000m); Squares: the (average) energy ratio as a function of correlation length(from 1000m up to 15000m) obtained from simulations; Horizontal gray band: range ofthe energyratios obtained from observations.


shows a non-linear behavior, with the slope decreasing as the perturbation increases.Similarly, we can calculate the average energy ratio as a function of correlationlength at some fixed values of velocity perturbation. We cover a range of correlationlengths from 1000m up to 15000m and fixed perturbation values of 4.8%, 6.1% and8%. The results are plotted as squares in Fig. 7 and strongly suggest that for ourmodel geometry the energy ratio is much more sensitive to thevelocity perturbationthan to the correlation length.

The results (Fig. 7) demonstrated that the observed signalscan be explainedwith PSH scattering in the crust with scatterers of roughly 5km correlation length(not well-constrained) and root mean square perturbation amplitude of 5% (well-constrained). This result further illustrates the efficacyof rotation measurements,for example, as a filter for SH-type motion. Similar processing steps will be possi-ble for the horizontal components of rotation and the corresponding components oftranslation. It is conceivable that the combination of these various components mightlead to tight constraints on near-receiver structure, results otherwise only availablefrom array measurements.

6 Conclusions

We report several examples of 3-D wave propagation algorithms to recover eitherthe Earth’s structure or the earthquake source characteristics. 1) Using the capabil-ity of the discontinuous Galerkin method to simulate waves through models withcomplex geometries we attempt to model sensational observations made at the Mt.Hochstaufen showing amplifications due to local topographyby a factor of morethan 10. Or results suggest that we cannot reproduce the amplification indicatingthat either more topographic details or additional internal 3-D strucrure is required.2) We show a first tomographic inversion using complete 3-D modeling. This indi-cates that we enter a new era in seismic tomography in which wecan expect to usemuch more information from the observed seismograms and recover sharper imagesof Earth’s interior. 3) The time reversal study indicates that source information canbe recoverd be re-injecting time-flipped seismograms at thereceiver locations butthe recovery of finite source characteristics is difficult. Finally, 4) we are able tomodel observations of rotational motions using ring laser technology with the 3-Ddiscontinous Galerkin method, which allows putting constraints on crustal scatter-ing properties. The key message is that 3-D wave simulation agorithms are about tobecome standard research tools for seismic data processing. The recent funding ofa large EU project in this field (QUEST, www.quest-itn.org, coordinated by LMUseismology) supports the view that we are now looking into a much wider usage of3-D simulation technology for real data analysis.


7 Acknowledgments

We acknowledge funding from the Bavarian Elite Network THESIS, KONWIHR,and the LRZ for providing the computational resources. We also acknowledgethe European Human Resources Mobility Programme (SPICE project), the DFGEmmy-Noether Programme, and the DAAD (N.D. Pham) and the Vietnamese Gov-ernment. HI and SK also acknowledge support from the Los Alamos National Lab-oratory, New Mexico.

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Advances in modelling and inversion of seismic …seismain/pdf/2010_h019z_2009.pdfAdvances in modelling and inversion of seismic wave propagation V. Hermann, N.D. Pham, A. Fichtner,