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

Mechanical properties of the ground are crucial information for safe construction. In particular, seismic analysis is an important stage of structural analysis in areas with high earthquake risk. This includes estimation of a structure response to seismic motion. Specific topics here include performance-based seismic design [

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

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

As mentioned by Boore [

The first-arrival front can be computed by numerically solving the eikonal equation on a regular grid using finite-difference method [

Similar approach to addressing first arrivals is the so-called SWEET algorithm for computing first-arrival traveltimes and amplitudes using the damped wave solution in the Laplace domain as suggested in [

Efficient method of computing the first-arrival waveforms can be beneficial in the reverse-time migration for constructing seismic images [

In this paper we first describe our approach for computing first-arrival waveforms in a narrow computational band following the moving front of the first arrivals. Then we briefly recall the method of the wave-equation tomography and the reverse-time migration which can take advantage of our modeling approach. Finally we present examples showing the speedup that we get in seismic imaging and velocity model update. In this paper we consider only 2D isotropic models to illustrate the concept.

We further develop this idea of computing waveforms in a running window and apply it in tomographic velocity model building which can be done in different ways. The most straightforward approach is ray tomography which requires only traveltime picks of the first arrivals.

The key step is regrading the computational arrays in the time increasing order that naturally comes out from the fast marching method [

Let us list main steps of our hybrid modeling approach. For simplicity we consider the 2D wave equation (but it is straightforward to generalize it to the 3D case and elastic wave equation).

The windowed speedup technique proposed by Vidale [

Solving the eikonal equation to get traveltime at each grid node.

Calculating the solution of the wave equation in the shifted window around the first-arrival front that is defined using the traveltime field from the previous step.

The mathematical background of the windowed calculation technique is obvious: the

During the second wave-equation-solution step at each time level all grid points can be updated or held unchanged. The FD calculations take place only for updated points that form the shifted zone around the first-arrival front. The solution in unchanged points is taken from the previous time level. The conventional way (Kvasnička and Zahradnřandék [

The update condition (

As usual, the numerical solvers provide the

In principle any eikonal FD solver or other computational techniques can be used to obtain traveltime field

Here we give a very brief description of original

Note that if the

Let us consider the one-to-one mapping:

We use mapping (

The proposed time increasing reordering is natural for the windowed speedup calculations. Each time window is set up by its beginning number

In principle any time-domain wave-equation solver can be modified for windowed calculations, based on the introduced time increasing reordering. The space FD stencil for the node

Note that one needs to store some additional indexation-mapping arrays for the window and stencils neighborhoods. As an alternative, the first-arrival wavefront propagation should be done simultaneously with wave-equation solution just ahead of the computational window. The FM approach provides the possibility of such “on the fly” traveltime computations. However one should keep mapping (

The proposed windowed techniques, based on the time increasing array reordering, provide the opportunity of storing the “history” of the wavefield. At every time level

Consider the eikonal equation in isotropic medium:

“Upwind” approximation.

The set of all grid points at each step of FM is divided into three sets:

In the present paper 2D acoustic wave equation is considered:

We use a staggered grid [

The considered FD scheme uses the staggered grid, so that

Staggered grid FD stencil.

We show the benefits of the proposed hybrid calculation-and-storing windowed technique for speeding up the reverse-time migration [

In the reverse-time migration for each shot gather (corresponding to

In the wave-equation tomography for each shot gather (corresponding to

Here the adjoint wavefield

While summing up the results for all shot gathers (for different

Similarly, we sum up all shot gather sensitivity kernels (for different

Iterative methods can be used to invert for the velocity model

The key step of constructing migrated image (

Thus there is a need to store the entire history of forward source field

Let us present an example of windowed wavefield computations. The smooth Marmousi velocity model is shown in Figure

Velocity model (smooth Marmousi) and acquisition geometry.

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

For this example our hybrid computational approach is 20 times faster than the standard approach (full wavefield computation). Also it takes 17 times less memory to store the source wavefield within narrowband following first arrival (compared to storing full wavefield at each time step). These speedup and memory economy will increase with increasing velocity model size (in terms of dominant wavelength).

Let us consider synthetic example motivated by near-surface applications. For this example we use gradient model with high velocity square anomaly as shown in Figure

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

Next example is motivated by applications in seismology. We consider true gradient velocity model with two anomalies as shown in Figure

Seismologic example: true velocity model (a) and model after tomographic inversion (b).

Here we consider the reverse-time migration example. Synthetic data was computed for Marmousi model shown in Figure

Smooth migration model (a) and true velocity model (b).

We use our “windowed” hybrid approach to modeling the source field

Reverse-time migrated images computing source field by standard full wavefield (a) and our “windowed” approach (b).

Examples show that our “windowed” approach allows storing “windowed” source wavefield in memory for any 2D model while fitting “full” source wavefield in memory is not feasible in most cases. Overall speedup of using our “windowed” approach to wave-equation tomography and reverse-time migration is close to 3. For standard approach the source field does not fit memory and thus one requires three standard “full” wavefield computations [

We want to mention potential problems in applying our “windowed” modeling approach for contrast models [

We have presented a new hybrid kinematic-dynamic finite-difference (FD) method for computing and storing the first-arrival seismic waveforms. In this “windowed” computational approach at every time step we perform FD computations only in a narrow region following the first-arrival wavefront. This wavefront is precomputed using fast marching eikonal solver. We have proposed a new effective strategy for computations and wavefield storing which is based on time increasing reordering.

We showed several examples of successful use of our “windowed” modeling approach in wave-equation tomography and migration. Proposed approach showed 20–30 times speedup in computing and 17–20 times less memory for storing the source wavefield when compared to standard “full” wavefield computing and storing. It allows overall 3 times speedup for implementing wave-equation tomography or reverse-time migration.

In this paper we consider only 2D isotropic models for illustrating the concept. Extension to 3D models is straightforward. Generalization of the approach to anisotropic models requires efficient anisotropic eikonal solver algorithm availability. It is a topic of our future research.

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

Research was partly supported by the Russian Foundation for Basic Research (Grant 14-05-00862). Authors are grateful to the unknown reviewer’s comments which have helped to improve the paper.