research article hybrid kinematic-dynamic approach to seismic wave-equation...

Research ArticleHybrid Kinematic-Dynamic Approach to SeismicWave-Equation Modeling, Imaging, and Tomography

Alexandr S. Serdyukov1 and Anton A. Duchkov1,2

1 Institute of Petroleum Geology and Geophysics SB RAS, Pr. Ac. Koptyuga 3, Novosibirsk 630090, Russia2Novosibirsk State University, Street Pirogova 2, Novosibirsk 630090, Russia

Correspondence should be addressed to Alexandr S. Serdyukov; [email protected]

Received 15 April 2015; Revised 1 July 2015; Accepted 6 July 2015

Academic Editor: Yuming Qin

Copyright © 2015 A. S. Serdyukov and A. A. Duchkov. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

Estimation of the structure response to seismicmotion is an important part of structural analysis related tomitigation of seismic riskcaused by earthquakes. Many methods of computing structure response require knowledge of mechanical properties of the groundwhich could be derived from near-surface seismic studies. In this paper we address computationally efficient implementation ofthe wave-equation tomography. This method allows inverting first-arrival seismic waveforms for updating seismic velocity modelwhich can be further used for estimating mechanical properties. We present computationally efficient hybrid kinematic-dynamicmethod for finite-difference (FD) modeling of the first-arrival seismic waveforms. At every time step the FD computations areperformed only in a moving narrowband following the first-arrival wavefront. In terms of computations we get two advantagesfrom this approach: computation speedup and memory savings when storing computed first-arrival waveforms (it is not necessarytomake calculations or store the complete numerical grid). Proposed approach appears to be specifically useful for constructing theso-called sensitivity kernels widely used for tomographic velocity update from seismic data. We then apply the proposed approachfor efficient implementation of the wave-equation tomography of the first-arrival seismic waveforms.

1. Introduction

Mechanical properties of the ground are crucial informationfor safe construction. In particular, seismic analysis is animportant stage of structural analysis in areas with highearthquake risk. This includes estimation of a structureresponse to seismic motion. Specific topics here includeperformance-based seismic design [1], dynamic stress con-centration in tunnels and underground structures duringearthquakes, slope stability analysis, and design of powertransmission tower-lines.Manymethods of computing struc-ture response require knowledge of mechanical properties ofthe groundwhich could be derived fromnear-surface seismicstudies. It is known that subsurface geologic irregularitiesmay have serious effect on actual site response to groundmotion which can be modeled for complicated subsurfacestructures [2]. Thus it is important to construct a detailedseismic model from near-surface seismic studies.

In this paper we address computationally efficient imple-mentation of the wave-equation tomography for construct-ing seismic velocity models. This method requires multiplenumerical computation of seismic wavefield propagation(specifically first-arrival waveforms) which is a computation-ally challenging problem, especially in 3D.

Numerical methods of seismic wavefield modeling(finite-difference (FD), finite-element, etc.) are widely usedin seismic data imaging and inversion. They can be usedfor computing wavefield propagation in complex subsurfacestructures but usually are computationally expensive. For thisreason a lot of work is done for speeding up these methods.

As mentioned by Boore [3] there is no need to computethe wavefield in some region of the computational grid beforethe first-arrival front passes this region (it is uniformly zero inthis case).This approach known as expanding computationaldomain was reintroduced by Karrenbach [4]. Furthermore,if only first-arrival waveforms are of interest, then numerical

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 543540, 8 pageshttp://dx.doi.org/10.1155/2015/543540

2 Mathematical Problems in Engineering

computations are required only for the grid points in a narrowregion following the first-arrival wavefront. Mentioned ideawas introduced by [5] and further developed by Kvasnickaand Zahradnrandek [6] and Hansen and Jacobsen [7].

The first-arrival front can be computed by numericallysolving the eikonal equation on a regular grid using finite-difference method [8, 9] or the fast marching (FM) method[10]. Note that the cost of computing first-arrival traveltimesis negligible compared to numerically solving the waveequation.

Similar approach to addressing first arrivals is the so-called SWEET algorithm for computing first-arrival travel-times and amplitudes using the damped wave solution inthe Laplace domain as suggested in [11]. This algorithm canbe implemented for large velocity models by reducing thenumber of grid points per wavelength with an out-of-coremultifrontal algorithm [12]. The advantage of this algorithmis that it computes most energetic wave arrivals. However, itis different from the previously mentioned methods becausethe Laplace damping is distorting the first-arrival waveforms.Thus it provides stable estimate of finite-frequency travel-times and amplitudes to be used in Kirchhoff type imagingrather than waveforms which can be used in wave-equationmigration/inversion type algorithms.

Efficient method of computing the first-arrival wave-forms can be beneficial in the reverse-time migration forconstructing seismic images [13, 14] and the wave-equationtomography (WT) for velocity model building of Luo andSchuster [15] (finite-frequency generalization of the con-ventional ray-based traveltime tomography). Both methodsconsider each shot gather independently and imply computa-tion of two wavefields: forward computation of the so-calledsource field and adjoint (time-reversed) computation of theso-called receiver field. Construction of seismic images orvelocity update kernels requires time integration of a productof these twowavefields.Thus there is a need to store the entirehistory of forward field in the computer memory which isusually unfeasible. Alternatively, one can store the historyof the forward field at the boundaries and then recalculateit backward in time simultaneously with the adjoint com-putation [16]. This can reduce memory requirements forstoringwavefields but require extra computations to bemade.Computation of the source wavefield only in a narrowbandaround first arrivals seems to be optimal for mentionedapplications as it requires considerably less memory.

In this paperwefirst describe our approach for computingfirst-arrival waveforms in a narrow computational bandfollowing themoving front of the first arrivals.Thenwebrieflyrecall the method of the wave-equation tomography andthe reverse-time migration which can take advantage of ourmodeling approach. Finally we present examples showing thespeedup that we get in seismic imaging and velocity modelupdate. In this paper we consider only 2D isotropic modelsto illustrate the concept.

2. Hybrid Forward Modeling Technique

We further develop this idea of computing waveforms in arunning window and apply it in tomographic velocity model

building which can be done in different ways. The moststraightforward approach is ray tomography which requiresonly traveltime picks of the first arrivals.

The key step is regrading the computational arrays inthe time increasing order that naturally comes out from thefast marching method [10]. Such strategy allows setting allcomputational windows before starting FD calculations.Thuswe save computer time needed for the window identificationand memory pointer shifts outside the window at each FDtime step. Also the resulting windowed wavefield can bestored in RAM at every time step using the proposed strategy.

2.1. Hybrid Approach to Forward Modeling. Let us list mainsteps of our hybrid modeling approach. For simplicity weconsider the 2D wave equation (but it is straightforward togeneralize it to the 3D case and elastic wave equation).

Thewindowed speedup technique proposed by Vidale [5]and illustrated by Kvasnicka and Zahradnrandek [6] includestwo steps:

(i) Solving the eikonal equation to get traveltime at eachgrid node.

(ii) Calculating the solution of the wave equation in theshifted window around the first-arrival front that isdefined using the traveltime field from the previousstep.

The mathematical background of the windowed calcula-tion technique is obvious: the region of influence of the wave-equation solution is determined by the characteristics.

During the second wave-equation-solution step at eachtime level all grid points can be updated or held unchanged.The FD calculations take place only for updated points thatform the shifted zone around the first-arrival front. Thesolution in unchanged points is taken from the previous timelevel. The conventional way (Kvasnicka and Zahradnrandek[6], Vidale [5]) to determine the updated zone is to check thecondition:

𝑡𝑓 − 𝑡left ≤ 𝑛Δ𝑡 ≤ 𝑡𝑓 + 𝑡right, (1)

where 𝑛 is the number of the time levels, Δ𝑡 is the FDtime spacing, 𝑡𝑓 is the first-arrival traveltime, and 𝑡left, 𝑡rightare the constants. These constants can be defined, using thedominant wavelength.

The update condition (1) should be checked for everygrid point at every time level during the wave-equation FDsolution that may cost significant computer time. We wouldlike to show how this checking can be avoided and proposemore efficient new strategy for the windowed calculationsthat is based on traveltime increasing array reordering.

2.2. Numerical Grid Upwind Ordering (from Traveltime Solu-tions). As usual, the numerical solvers provide the viscositysolution of eikonal equation, which corresponds to the first-arrival traveltimes. At every node (𝑖, 𝑗) one has first-arrivaltraveltime 𝑇𝑖,𝑗. The resulting array can be sorted in theincreasing order, so the result would be a vector (timeincreasing list):

𝑇 (𝑘) , 𝑇 (𝑘 + 1) ≥ 𝑇 (𝑘) . (2)

Mathematical Problems in Engineering 3

In principle any eikonal FD solver or other computa-tional techniques can be used to obtain traveltime field 𝑇𝑖,𝑗

for the windowed calculations. We are going to illustratethe approach using the multistencils fast marching method,introduced by [17], which is the more recent and robustmodification of classic FM.

Here we give a very brief description of original fastmarching method. The set of all grid points at each step of FMis divided into three sets: accepted points, narrowband, andfar points. The main feature of FM algorithm is the specialorder of traveltimes computation in the grid nodes similarto Dijkstra’s algorithm. If point (𝑖, 𝑗) is accepted, then thetraveltime value in it𝑇𝑖,𝑗 becomes frozen and does not changeany more. Every point that is not accepted but has an acceptedpoint in its neighborhood should belong to the narrowband;the traveltimes in this set are recomputed. All other pointsare 𝑓𝑎𝑟. At every iteration the point from narrowband withthe minimum value of traveltime becomes accepted. For theefficiency narrowband is stored in the heap structure.

Note that if the fastmarching solver is used, the rank of thegiven grid point in the time increasing list (2) is the numberof acceptances coming out from the FM.

Let us consider the one-to-one mapping:

(𝑖, 𝑗) ←→ 𝑘, (3)

where (𝑖, 𝑗) is the pair of Cartesian indexes and 𝑘 is the rankof the 𝑇𝑖,𝑗 in the traveltime increasing list (2); 𝑖 = 1, . . . , 𝑁𝑧;𝑗 = 1, . . . , 𝑁𝑥; 𝑘 = 1, . . . , 𝑁𝑥 × 𝑁𝑧.

We use mapping (3) to reorder all the arrays, used forthe wave-equation FDmodeling. For example, let us considervelocity model matrix 𝑉(𝑖, 𝑗); it can be stored in the form ofa vector𝑉(𝑘). Note that mapping (3) should be remembered,as soon as there is a need for getting back to the Cartesiangrid. For instance, one can use two integer vectors: 𝐼(𝑘) = 𝑖,𝐽(𝑘) = 𝑗. Actually almost the same mapping (3) happenswithin the fast marching method for the narrowband, storedin the time-sorted heap structure.

2.3. Computing and Storing Widowed Wavefield. The pro-posed time increasing reordering is natural for the windowedspeedup calculations. Each time window is set up by itsbeginning number 𝑘begin and the end number 𝑘end that aredefined using list (2).Thus thewave-equation FD calculationsat every time level 𝑛 are done in the cycle from 𝑘 = 𝑘begin(𝑛)to 𝑘 = 𝑘end(𝑛) and there is no need for checking the “update”condition (1).

In principle any time-domain wave-equation solver canbe modified for windowed calculations, based on the intro-duced time increasing reordering. The space FD stencil forthe node (𝑖, 𝑗) includes someneighborhood points, at least (𝑖±1, 𝑗 ± 1) (if one is using the second-order central differences).As soon as all the arrays are reordered, one should implementmapping (3) several times to get the ranks 𝑘 for all the stencilnodes. The more efficient strategy is to introduce additionalarrays for the ranks of neighborhood points. For instance, therank of the “left” node (𝑖−1, 𝑗) for the node (𝑖, 𝑗)with the rank𝑘 can be remembered in the array 𝑘left(𝑘).

Note that one needs to store some additional indexation-mapping arrays for the window and stencils neighborhoods.As an alternative, the first-arrival wavefront propagationshould be done simultaneously with wave-equation solutionjust ahead of the computational window. The FM approachprovides the possibility of such “on the fly” traveltimecomputations.However one should keepmapping (3) (at leastbackward).

The proposed windowed techniques, based on the timeincreasing array reordering, provide the opportunity of stor-ing the “history” of the wavefield. At every time level 𝑛 =

1, . . . , 𝑁 the wavefield can be recorded just one by one atthe current time level 𝑛 from 𝑘 = 𝑘begin(𝑛) to 𝑘 = 𝑘end(𝑛)in the one large array, containing the entire “history” of thewindowed wavefield. After the computations, one can getthe access to the wavefield history by reading the wavefieldin time level cycle, using the stored widowed limiting ranks𝑘begin(𝑛) and 𝑘end(𝑛) and using mapping (3) for getting tothe Cartesian grid. Note that one can produce the wavefieldreading in a reversed time.

2.4. Fast Marching Method. Consider the eikonal equation inisotropic medium:

(∇𝜏)2=

1

𝑐2(x)

, (4)

where 𝜏(x) is the traveltime in the point x = (𝑥, 𝑧) and 𝑐(x) isthe seismic wave velocity.Wewill give here a brief descriptionof original fastmarchingmethod in the two- dimensional case.Let us consider the rectangular Cartesian grid: 𝑥𝑖 = 𝑥0 + 𝑖ℎ,𝑧𝑗 = 𝑧0 + ℎ ∗ 𝑗, 𝑐𝑖𝑗 = 𝑐(𝑥𝑖, 𝑧𝑗), and 𝜏𝑖𝑗 = 𝜏(𝑥𝑖, 𝑧𝑗). Theoriginal FMmethod [10] is based on the first-order “upwind”FD approximation of eikonal equation (4):

max (max (𝐷−𝑥𝑖𝑗 𝜏, 0) , −min (𝐷+𝑥𝑖𝑗 𝜏, 0))2

+max (max (𝐷−𝑧𝑖𝑗 𝜏, 0) , −min (𝐷+𝑧𝑖𝑗 𝜏, 0))2

= 𝑐−2𝑖𝑗 ,

(5)

where

𝐷±𝑥𝑖𝑗 𝜏 = ±

𝜏𝑖±1,𝑗 − 𝜏𝑖𝑗

ℎ,

𝐷±𝑥𝑖𝑗 𝜏 = ±

𝜏𝑖,𝑗±1 − 𝜏𝑖𝑗

ℎ.

(6)

The essence of “upwind” approximation (5) is shown inFigure 1.The grid nodes for FD stencil are choosing principlethat is based on traveltime increasing “causality”; that is, thedirection of “wind” is the direction of wavefront expansion.

The set of all grid points at each step of FM is dividedinto three sets: accepted points, narrowband, and far points.The main feature of FM algorithm is the special order oftraveltimes computation in the grid nodes similar toDijkstra’salgorithm. If point (𝑖, 𝑗) is accepted, then the traveltime valuein it 𝑇𝑖,𝑗 becomes frozen and does not change any more.Every point that is not accepted but has an accepted point


Direction ofwave prop.

Front

i, j i, j

Figure 1: “Upwind” approximation.

in its neighborhood should belong to the narrowband; thetraveltimes in this set are recomputed. All other points arefar. At every iteration the point from narrowband with theminimum value of traveltime becomes accepted. For theefficiency narrowband is stored in the heap structure.

2.5. The Virieux Staggered Grid FD Scheme. In the presentpaper 2D acoustic wave equation is considered:

1

𝑐2

𝜕2𝑝

𝜕𝑡2− 𝜌∇ ⋅ (

1

𝜌∇𝑝) = 0, (7)

where𝑝(x) is pressure, 𝜌(x) is the density, and 𝑐(x) is the wavespeed.

We use a staggered grid [18] to solve the wave equation(7). In order to implement this scheme the acoustic equationshould be rewritten as two first-order equations:

𝜕𝑝

𝜕𝑡= 𝑐2𝜌(

𝜕V𝑥𝜕𝑥

+𝜕V𝑧𝜕𝑧

) ,

𝜕V𝑥𝜕𝑡

=1

𝜌

𝜕𝑝

𝜕𝑥,

𝜕V𝑧𝜕𝑡

=1

𝜌

𝜕𝑝

𝜕𝑧,

(8)

where (V𝑥, V𝑧) is the particle velocity vector.The considered FD schemeuses the staggered grid, so that

V𝑥 is attached to the points (𝑖, 𝑗) (𝑥𝑖 = 𝑥0 + 𝑖ℎ, 𝑧𝑗 = 𝑧0 + 𝑗ℎ)

of a Cartesian grid, V𝑧 is attached to the “half-integer” nodes(𝑖 + 1/2, 𝑗 + 1/2), and pressure 𝑝 is attached to the points (𝑖 +1/2, 𝑗). Also the velocity and pressure are attached to differenttime layers. The staggered grid FD stencil for acoustic waveequation is shown in Figure 2. The FD equations are

(V𝑥)𝑛+1

𝑖,𝑗 − (V𝑥)𝑛

𝑖,𝑗

Δ𝑡=

𝑝𝑛+1/2𝑖+1/2,𝑗

− 𝑝𝑛+1/2𝑖−1/2,𝑗

𝜌𝑖,𝑗ℎ,

(V𝑧)𝑛+1

𝑖+1/2,𝑗+1/2 − (V𝑧)𝑛

𝑖+1/2,𝑗+1/2

Δ𝑡=

𝑝𝑛+1/2𝑖+1/2,𝑗+1

− 𝑝𝑛+1/2𝑖+1/2,𝑗

𝜌𝑖+1/2,𝑗+1/2ℎ,

𝑝𝑛+1/2𝑖+1/2,𝑗

− 𝑝𝑛−1/2𝑖+1/2,𝑗

Δ𝑡= (𝑐2𝜌)𝑖+1/2,𝑗

(

(V𝑥)𝑛

𝑖+1,𝑗 − (V𝑥)𝑛

𝑖,𝑗

ℎ

+

(V𝑧)𝑛

𝑖+1/2,𝑗+1/2 − (V𝑧)𝑛

𝑖+1/2,𝑗−1/2

ℎ) .

(9)

i, j

i, j

n + 1

n + 1/2 n + 1/2

i − 1/2, j i + 1/2, j

n

nn

n

i + 1/2, j + 1/2

i + 1, j

i + 1/2, j − 1/2

z

t

x

�x�zp

Figure 2: Staggered grid FD stencil.

3. Application to Wave-Equation Imagingand Tomography

We show the benefits of the proposed hybrid calculation-and-storingwindowed technique for speeding up the reverse-timemigration [13, 14] and the wave-equation tomography [15].For both methods we consider shot gathers independentlyand for each of them we compute two wavefields: the so-called source field 𝑝𝑠(x, 𝑡) is a result of forward computationand the so-called adjoint field𝑝𝑟(x, 𝑡) is a result of the reverse-time computation; that is, time-reversed data is plugged in assource signature in the receiver positions. Interaction of thesetwo wavefields is producing seismic image or the velocitymodel update.

In the reverse-time migration for each shot gather (cor-responding to x𝑠) we get an image as a time integration of aproduct of these two wavefields:

𝑔 (x) = ∫𝑝𝑠 (x, 𝑡) 𝑝𝑟 (x, 𝑡) d𝑡. (10)

In the wave-equation tomography for each shot gather(corresponding to x𝑠) we also construct sensitivity kernels asan interaction of two wavefields:

𝛾 (x) = 1

𝑐3(x)

∫ 𝑝𝑠 (x, 𝑡) 𝑝𝑟 (x, 𝑡) d𝑡. (11)

Here the adjoint wavefield 𝑝𝑟(x, 𝑡) is not just time-reversed data but time-reversed pseudo-residual [15]:

𝛿𝜏 (x𝑟, 𝑡) = −2

𝐸 (x𝑟)𝑝obs (x𝑟, 𝑡 + 𝛿𝜏) Δ𝜏 (x𝑟) , (12)

where 𝑝obs(x𝑟, 𝑡) are observed data at receivers x𝑟,

𝐸 (x𝑟) = ∫𝑝obs (x𝑟, 𝑡 + Δ𝜏) 𝑝𝑠 (x𝑟, 𝑡) , (13)


and Δ𝜏 are traveltime misfits estimated by cross-correlatingobserved data 𝑝obs(x𝑟, 𝑡) and synthetic data 𝑝𝑠(x𝑟, 𝑡) directlyfrom waveforms (maximum of the cross-correlation):

Δ𝜏 (x𝑟) = argmin𝜏

∫𝑝obs (x𝑟, 𝑡 + 𝜏) 𝑝𝑠 (x𝑟, 𝑡) d𝑡. (14)

While summing up the results for all shot gathers (fordifferent x𝑠), we get final migrated image:

𝑔 (x) = ∑

𝑠

𝑔 (x; x𝑠) . (15)

Similarly, we sumup all shot gather sensitivity kernels (fordifferent x𝑠) in order to get the velocity model update:

𝛾 (x) = ∑

𝑠

𝛾 (x; x𝑠) . (16)

Iterative methods can be used to invert for the velocitymodel 𝑐(x) in several steps:

𝑐𝑘+1 (x) = 𝑐𝑘 (x) + 𝛼𝑘𝛾𝑘 (x) , (17)

where 𝛾(x)𝑘 is the sensitivity kernel at 𝑘th iteration and 𝛼𝑘 isupdating step length.

The key step of constructingmigrated image (10) or veloc-ity model update (11) is to compute the source field 𝑝𝑠(x, 𝑡)and store it for further cross-correlation with the adjoint field𝑝𝑟(x, 𝑡). For example, precomputed 𝑝𝑠(x, 𝑡) is required forestimating the pseudo-residual used for initiating the adjointfield (cf. (12)).

Thus there is a need to store the entire history of forwardsource field 𝑝𝑠(x, 𝑡) in the computer memory which is usuallyunfeasible. Alternatively, one can store the history of theforward field at the boundaries and then recalculate it back-ward in time simultaneously with the adjoint computation.This can reduce memory requirements for storing wavefieldsbut require one extra waveform computation to be made.Computation of the source wavefield only in a narrowbandaround the first arrivals seems to be optimal for mentionedapplications as it requires considerably less memory. Furtherwe plan to apply the proposed approach only to the sourcefield𝑝𝑠(x, 𝑡)while computing the adjoint receiver field𝑝𝑟(x, 𝑡)in a standard mode (note that it does not need to be stored inmemory).

4. Examples

4.1. Forward Modeling. Let us present an example of win-dowed wavefield computations. The smooth Marmousivelocity model is shown in Figure 3. Receivers are shownby white triangles and source locations are shown by stars.Ricker wavelet with 20Hz central frequency was used as asource function. Computed pressure data for source 1 areshown in Figure 4: “full” wavefield computed by standardFD scheme in panel (a) and “windowed” computation by theproposed hybrid kinematic-dynamic method in panel (b).Note that the first-arrival waveforms look the same. One cansee differences for late arrivals where refracted waves canbe seen on “full” data. For source 2 computed “full” and

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3500

3000

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Source 1

Source 2

Receivers

x (m)

z(m

)

Figure 3: Velocity model (smooth Marmousi) and acquisitiongeometry.

“windowed” data look pretty much the same because thereare no refracted waves in this case (for this reason we do notshow corresponding gathers).

For this example our hybrid computational approach is20 times faster than the standard approach (full wavefieldcomputation). Also it takes 17 times less memory to store thesource wavefield within narrowband following first arrival(compared to storing full wavefield at each time step). Thesespeedup and memory economy will increase with increasingvelocity model size (in terms of dominant wavelength).

4.2. Wave-Equation Tomography. Let us consider syntheticexample motivated by near-surface applications. For thisexample we use gradient model with high velocity squareanomaly as shown in Figure 5(a). For this true velocity wecomputed synthetic data using 60Hz source wavelet (sourcesand receivers are placed every 5m at the top of the model).For these data we performed velocity model update usingwave-equation tomography (initial model is a model withlinear gradient). Source wavefield was computed using ourhybrid “windowed” approach.The result of velocity inversion(16) is shown in Figure 5(b). One can see that the velocityanomaly is recovered.

Next example is motivated by applications in seismol-ogy. We consider true gradient velocity model with twoanomalies as shown in Figure 6(a). Source positions areshown by black stars and receivers were located regularly atthe surface. Synthetic data were computed for 10Hz Rickerwavelet. We performed velocity model update using wave-equation tomography (initial model is a model with lineargradient). Source wavefield was computed using our hybrid“windowed” approach. The result of the velocity inversion isshown in Figure 6(b). One can see that the velocity anomaliescan be recovered.

4.3. Reverse-Time Migration. Here we consider the reverse-time migration example. Synthetic data was computed forMarmousi model shown in Figure 7(b); 50 shot gathers for30Hz Ricker sources were placed every 60m. Migrationvelocity model is shown in Figure 7(a).

We use our “windowed” hybrid approach to modelingthe source field 𝑝𝑠(x, 𝑡); “windowed” modeling is 33 timesfaster than “full” waveform modeling (1.5 and 50 seconds


0.8 1.0 1.2 1.4 1.6 1.8 2.0t (s)

(a)

0.8 1.0 1.2 1.4 1.6 1.8 2.0t (s)

(b)

Figure 4: Synthetic data for source 1 computed by standard FD scheme (a) and “windowed” approach (b).

100 200 300 400 500 600 700 800 900 1000

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z(m

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(a)

100 200 300 400 500 600 700 800 900 1000

50100150200

270028002900300031003200

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z(m

)

(b)

Figure 5: Near-surface example: true velocity model (a) and modelafter tomographic inversion (b).

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2800300032003400360038004000

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Figure 6: Seismologic example: true velocity model (a) and modelafter tomographic inversion (b).

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)

x (m)

(a)

200400600800

1000

40003500300025002000

z(m

)

0 2500200015001000500 3000x (m)

(b)

Figure 7: Smooth migration model (a) and true velocity model (b).

correspondingly on Intel Core i7-4770 3.4GHz CPU). Italso needs 20 times less memory to store the first-arrivalwavefield, that is, 0.8 Gb versus 17Gb. Reverse-timemigratedimages for 50 shot gathers (15) are shown in Figure 8: using“full” waveform computation (a) and using our “windowed”computation (b). Both migration results look identical. Notethat the receiver wavefield 𝑝𝑟(x, 𝑡) was computed in a “full”waveform mode in both cases.

5. Discussion

Examples show that our “windowed” approach allows storing“windowed” source wavefield in memory for any 2D modelwhile fitting “full” source wavefield in memory is not feasiblein most cases. Overall speedup of using our “windowed”approach to wave-equation tomography and reverse-time


200400600800

10000 500 1000 1500 2000 2500 3000

z(m

)

x (m)

(a)

0 500 1000 1500 2000 2500 3000

200400600800

1000

z(m

)

x (m)

(b)

Figure 8: Reverse-time migrated images computing source field bystandard full wavefield (a) and our “windowed” approach (b).

migration is close to 3. For standard approach the sourcefield does not fit memory and thus one requires threestandard “full” wavefield computations [16] for implementingtomography update or migration. Our “windowed” approachdramatically reduces time required for the source field com-putation and it can be stored inmemory for further use.Thuscomputational cost of our “windowed” approach is close toone “full” wavefield computation of the receiver field.

We want to mention potential problems in applying our“windowed” modeling approach for contrast models [6]. Theeikonal solver provides first arrivals which may correspondto weak head waves while energetic waves useful for imagingmay arrive later. Computational window following the first-arrival front may miss these waves. However, the class ofvelocity models in which our approach works is large enoughto make it useful for practical applications.

6. Conclusions

We have presented a new hybrid kinematic-dynamic finite-difference (FD) method for computing and storing the first-arrival seismicwaveforms. In this “windowed” computationalapproach at every time step we perform FD computationsonly in a narrow region following the first-arrival wave-front. This wavefront is precomputed using fast marchingeikonal solver. We have proposed a new effective strategy forcomputations and wavefield storing which is based on timeincreasing reordering.

We showed several examples of successful use of our“windowed” modeling approach in wave-equation tomogra-phy and migration. Proposed approach showed 20–30 timesspeedup in computing and 17–20 times less memory forstoring the source wavefield when compared to standard“full” wavefield computing and storing. It allows overall

3 times speedup for implementing wave-equation tomogra-phy or reverse-time migration.

In this paper we consider only 2D isotropic models forillustrating the concept. Extension to 3D models is straight-forward. Generalization of the approach to anisotropic mod-els requires efficient anisotropic eikonal solver algorithmavailability. It is a topic of our future research.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

Research was partly supported by the Russian Foundationfor Basic Research (Grant 14-05-00862). Authors are gratefulto the unknown reviewer’s comments which have helped toimprove the paper.

References

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[2] D. Komatitsch and J.-P. Vilotte, “The spectral element method:an efficient tool to simulate the seismic response of 2D and3D geological structures,” Bulletin of the Seismological Societyof America, vol. 88, no. 2, pp. 368–392, 1998.

[3] D. Boore, “Finite-difference methods for seismic wave propa-gation in heterogeneous materials,” Methods in ComputationalPhysics, vol. 2, pp. 1–36, 1972.

[4] M. Karrenbach, “Combining eikonal solutions with full wave-form modeling,” in Proceedings of the SEG Annual Meeting,2000.

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[6] M. Kvasnicka and J. Zahradnrandek, “A combined kinematic-dynamic method for fast computations of the first-arrivingwaveforms,” 1996, http://sw3d.mff.cuni.cz/papers.bin/r4mk1.pdf.

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[9] P. Podvin and I. Lecomte, “Finite difference computation oftraveltimes in very contrasted velocity models: a massivelyparallel approach and its associated tools,” Geophysical JournalInternational, vol. 105, no. 1, pp. 271–284, 1991.

[10] J. Sethian, Level Set Methods and Fast Marching Methods: Evolv-ing Interfaces in Computational Geometry, Fluid Mechanics,Computer Vision, and Materials Science, Cambridge UniversityPress, 1999.

[11] C. Shin, D.-J. Min, K. J. Marfurt et al., “Traveltime and ampli-tude calculations using the damped wave solution,” Geophysics,vol. 67, no. 5, pp. 1637–1647, 2002.


[12] D. Yang, C. Shin, K. J. Marfurt, J. Kim, and S. Ko, “Three-dimensional traveltime and amplitude computation using thesupressed wave equation estimation of traveltime (sweet) algo-rithm,” Journal of Seismic Exploration, vol. 12, pp. 75–101, 2003.

[13] E. Baysal, D. D. Kosloff, and J. W. C. Sherwood, “Reverse timemigration,” Geophysics, vol. 48, no. 11, pp. 1514–1524, 1983.

[14] G. A. McMechan, “Migration by extrapolation of time-dependent boundary values,” Geophysical Prospecting, vol. 31,no. 3, pp. 413–420, 1983.

[15] Y. Luo andG. T. Schuster, “Wave-equation traveltime inversion,”Geophysics, vol. 56, no. 5, pp. 645–653, 1991.

[16] O. Gauthier, J. Virieux, and A. Tarantola, “Two-dimensionalnonlinear inversion of seismic waveforms: numerical results,”Geophysics, vol. 51, no. 7, pp. 1387–1403, 1986.

[17] M. S. Hassouna and A. A. Farag, “Multistencils fast marchingmethods: a highly accurate solution to the Eikonal equation onCartesian domains,” IEEE Transactions on Pattern Analysis andMachine Intelligence, vol. 29, no. 9, pp. 1563–1574, 2007.

[18] J. Virieux, “P-SV wave propagation in heterogeneous media:velocity-stress finite-difference method,”Geophysics, vol. 51, no.4, pp. 889–901, 1986.

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