Apparatus and method for calculating efficient 3D traveltime by using coarsegrid mesh for shallow depth source

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First Claim
1. An apparatus for calculating efficient 3D traveltime by using coarsegrid mesh for a shallow depth source, the apparatus comprising:
 a Green'"'"'s function calculation unit configured to calculate a Green'"'"'s function for a homogeneous halfspace medium;
an equivalent source vector calculation unit configured to calculate an equivalent source vector equivalent to an original point source vector by using a wavefield vector sampled at coarsegrid points calculated by the Green'"'"'s function calculation unit;
a wavefield vector calculation unit configured to calculate the wavefield vector by using the equivalent source vector calculated by the equivalent source vector calculation unit;
a wavefield vector'"'"'s partial derivative generation unit configured to generate a partial derivative of the wavefield vector calculated by the wavefield vector calculation unit; and
a firstarrival traveltime calculation unit configured to calculate a firstarrival traveltime by a suppressed wave equation estimation of traveltime (SWEET) algorithm with both the wavefield vector calculated by the wavefield vector calculation unit and the partial derivative of the wavefield vector generated by the wavefield vector'"'"'s partial derivative generation unit,wherein the Green'"'"'s function calculation unit calculates the Green'"'"'s function by the following equation 1;
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Abstract
The present invention relates generally to an apparatus and method for calculating efficient 3dimensional (3D) traveltime by using coarsegrid mesh for a shallow depth source. More particularly, the present invention relates to an efficient 3D traveltime calculation method for a shallow depth source by combining a suppressed wave equation estimation of traveltime (SWEET) algorithm and an equivalent source distribution (ESD) algorithm, wherein the SWEET algorithm is a traveltime calculation algorithm using an damped wave equation and the ESD algorithm is for equivalently distributed sources; and to an apparatus and method for calculating efficient 3D traveltime by using coarsegrid mesh for a shallow depth source which may need less calculation time compared with that of a conventional SWEET algorithm.
1 Citation
No References
METHOD AND APPARATUS FOR TIMEDOMAIN REVERSETIME MIGRATION WITH SOURCE ESTIMATION  
Patent #
US 20120051179A1
Filed 06/20/2011

Current Assignee
Seoul National University RDB Foundation

Sponsoring Entity
Seoul National University RDB Foundation

10 Claims
 1. An apparatus for calculating efficient 3D traveltime by using coarsegrid mesh for a shallow depth source, the apparatus comprising:
a Green'"'"'s function calculation unit configured to calculate a Green'"'"'s function for a homogeneous halfspace medium; an equivalent source vector calculation unit configured to calculate an equivalent source vector equivalent to an original point source vector by using a wavefield vector sampled at coarsegrid points calculated by the Green'"'"'s function calculation unit; a wavefield vector calculation unit configured to calculate the wavefield vector by using the equivalent source vector calculated by the equivalent source vector calculation unit; a wavefield vector'"'"'s partial derivative generation unit configured to generate a partial derivative of the wavefield vector calculated by the wavefield vector calculation unit; and a firstarrival traveltime calculation unit configured to calculate a firstarrival traveltime by a suppressed wave equation estimation of traveltime (SWEET) algorithm with both the wavefield vector calculated by the wavefield vector calculation unit and the partial derivative of the wavefield vector generated by the wavefield vector'"'"'s partial derivative generation unit, wherein the Green'"'"'s function calculation unit calculates the Green'"'"'s function by the following equation 1;  View Dependent Claims (2, 3, 4, 5)
 6. A method of calculating efficient 3D traveltime by using coarsegrid mesh for a shallow depth source, the method comprising:
calculating, by a Green'"'"'s function calculation unit, a Green'"'"'s function for a homogeneous halfspace medium; calculating, by an equivalent source vector calculation unit, an equivalent source vector equivalent to an original point source vector by using a wavefield vector sampled at coarsegrid points calculated at the calculating the Green'"'"'s function; calculating, by a wavefield vector calculation unit, the wavefield vector by using the equivalent source vector calculated at the calculating the equivalent source vector; generating, by the wavefield vector'"'"'s partial derivative generation unit, a partial derivative of the wavefield vector calculated at the calculating the wavefield vector; and calculating, by a firstarrival traveltime calculation unit, a firstarrival traveltime by using a SWEET algorithm with both the wavefield vector calculated at the calculating the wavefield vector and the partial derivative of the wavefield vector generated at the generating the partial derivative of the wavefield vector, wherein at the calculating the Green'"'"'s function, the Green'"'"'s function is calculated by the following equation 6;  View Dependent Claims (7, 8, 9, 10)
1 Specification
The present application claims priority to Korean Patent Application No. 1020170026726, filed Feb. 28, 2017, the entire contents of which is incorporated herein for all purposes by this reference.
The present invention relates generally to an apparatus and method for calculating efficient 3dimensional (3D) traveltime by using coarsegrid mesh for a shallow depth source. More particularly, the present invention relates to an efficient 3D traveltime calculation method for the shallow depth source by combining a suppressed wave equation estimation of traveltime (SWEET) algorithm and an equivalent source distribution (ESD) algorithm, wherein the SWEET algorithm is a traveltime calculation algorithm using a damped wave equation and the ESD algorithm is for equivalently distributed sources; and an apparatus and method for calculating efficient 3D traveltime by using coarsegrid mesh for a shallow depth source which may need less calculation time compared with that of a conventional SWEET algorithm.
Recent advances of 3D seismic survey led to the development of the data processing technique for 3D seismic data. In particular, the use of 3D reversetime migration to study subsurface structures has attracted much attention (Abdelkhalek et al., 2009; ArayaPolo et al., 2009; Kim et al., 2011; Yoon et al., 2003). As the reversetime migration method became more popular, a more accurate velocity model was needed. As a result, recent research in data processing areas has focused on accurate velocity model building, and full waveform inversion is one of the available techniques. Recently, research on 3D full waveform inversion is carried out by many geophysicists (BenHadjAli et al., 2009; Plessix, 2009; Pyun et al., 2011b; Son et al., 2014). Under these circumstances, a practical problem has arisen in that the reversetime migration requires enormous computational costs to verify the usefulness of inversion results. Therefore, a costeffective migration technique is needed to verify the waveform inversion results. Although Kirchhoff migration is not as accurate as reversetime migration, it is efficient enough to verify 3D inversion results.
To carry out Kirchhoff migration, underground traveltime information is necessary, wherein the traveltime can be calculated by many algorithms such as various ray tracing methods (Coultrip, 1993), Eikonal solvers (Vidale, 1988; Vidale, 1990) and waveequationbased algorithms (Shin et al., 2002; Shin et al., 2003; Qin et al., 2005), etc. Although ray tracing methods or Eikonal solvers are more efficient than waveequationbased methods, waveequationbased methods can properly handle caustics and other problems related to ray theory. In addition, waveequationbased methods can compute amplitudes simultaneously, thus a waveequationbased algorithm called the SWEET (Yang et al., 2003) is used, wherein the SWEET solves the wave equation in the Laplace domain. In this algorithm, a seismic trace is considered as a series of weighted spikes. By solving the wave equation in the Laplace domain, all the spikes except the firstarrival event are attenuated and become negligible. Thus, a seismic trace originating from a series of weighted spikes can be approximated by a single spike.
As a result, the firstarrival traveltime can be extracted from the solution of wave equation in the Laplace domain. To solve the Laplace domain wave equation, a standard finiteelement method is used, wherein the method exploits the combination of consistent and lumped mass matrices. The resultant matrix equation is solved by preconditioned conjugate gradient method. A numerical modeling by the Laplace domain wave equation shows less numerical dispersion error than timedomain or frequencydomain modeling (Shin et al., 2002; Shin and Cha, 2008). This characteristic enables the numerical modeling to adopt large grid spacing. However, this choice hinders exact simulation of a shallow depth source for the modeling with a coarsegrid in acoustic media. Cha and Shin (2010) used adaptive meshes to solve this problem, but implementation thereof is very complicated and the algorithm does not converge well when an iterative solver is used.
A traveltime calculation method using a conventional SWEET algorithm will be described below. The SWEET algorithm, which was suggested by Shin et al. (2002), calculates the firstarrival travel time by solving the Laplace domain wave equation. Yang et al. (2003) applied the SWEET algorithm to a 3D problem using a direct large sparse matrix solver. To calculate a firstarrival traveltime, the SWEET algorithm makes use of an assumption that a seismic signal is equal to a series of weighted spikes (see
where u is a Laplace domain wavefield, S is a Laplace domain variable, and S_{opt }is an optimal Laplace damping constant.
The optimal Laplace damping constant S_{opt }can be determined by the following equation 2 (Shin et al., 2002):
where, ν_{ave }is an average velocity of a given model, Δ is a grid spacing, and G is a number of grid points per pseudowavelength.
The Laplace domain wavefield is obtained by solving the 3D acoustic wave equation in the Laplace domain by the following equation 3:
where, v is a propagation velocity in the medium, and f is a source function in the Laplace domain.
Using the finiteelement method, equation 3 can be expressed as the linear algebraic system (Marfurt, 1984) by the following equations 4 and 5:
Su=f (Equation 4)
with
S=K+s^{2}M, (Equation 5)
where, S is the impedance matrix, u is the wavefield vector in the Laplace domain, f is the source vector in the Laplace domain, K is the stiffness matrix, and M is the mass matrix.
In the mean time, perfectly matched layer (PML) boundary condition is applied to eliminate unwanted edge reflections (Cohen, 2002).
To efficiently solve the equation 4, the preconditioned conjugate gradient method (Pyun et al., 2011b) is used. The partial derivative of wavefield in equation 1 is calculated by a backpropagation algorithm (Shin et al., 2002) by the following equation 6:
The traveltime calculation method configured as above using the conventional SWEET algorithm can accurately calculate a wavefield when a source is located at a grid point with an assumption that grid spacing is small enough to avoid numerical dispersion. In particular, the Laplace domain wave equation allows accurate modeling for relatively large grid spacing when the source is located at a grid point. However, when a real source is located at shallow depth close to free surface, a problem arises that the wavefield cannot be accurately calculated by using coarsegrid spacing.
The foregoing is intended merely to aid in the understanding of the background of the present invention, and is not intended to mean that the present invention falls within the purview of the related art that is already known to those skilled in the art.
 (Patent Document 1) Korean patent No. 1172506
 (Patent Document 2) Korean patent No. 1182838
 (NonPatent Document 1) Abdelkhalek, R., Calandra, H., Coulaud, O., Roman, J., Latu, G., 2009. Fast seismic modeling and reverse time migration on a GPU cluster. High Performance Computing and Simulation, Leipzig, Germany, 3644.
 (NonPatent Document 2) ArayaPolo, M., Rubio, F., Cruz, R., Hanzich, M., Cela, J. M., Scarpazza, D. P., 2009. 3D seismic imaging through reversetime migration on homogeneous and heterogeneous multicore processors. Sci. Program. 17, 185198.
 (NonPatent Document 3) BenHadjAli, H., Operto, S., Virieux, J., 2009. Velocity model building by 3D frequencydomain, fullwaveform inversion of wideaperture seismic data. Geophysics 73, VE101VE117.
 (NonPatent Document 4) Cha, Y. H., Shin, C., 2010. Twodimensional Laplace domain waveform inversion using adaptive meshes: and experience of the 2004 BP velocityanalysis benchmark data set. Geophys. J. Int. 182, 865879.
 (NonPatent Document 5) Cohen, G. C., 2002. Higherorder numerical methods for transient wave equations. Springer, Berlin.
 (NonPatent Document 6) Coultrip, R. L., 1993. Highaccuracy wavefront tracing traveltime calculation. Geophysics 58, 284292.
 (NonPatent Document 7) Kim, Y., Min, D., Shin, C., 2011. Frequencydomain reversetime migration with source estimation. Geophysics 76, S41S49.
 (NonPatent Document 8) Marfurt, K. J., 1984. Accuracy of finitedifference and finiteelement modeling of the scalar and elastic wave equations. Geophysics 49, 533549.
 (NonPatent Document 9) Plessix, R. E., 2009. Threedimensional frequencydomain fullwaveform inversion with an iterative solver. Geophysics 74, WCC149WCC157.
 (NonPatent Document 10) Pyun, S., Shin, C., Chung, W., 2011a. Equivalent source distribution for efficient 3D acoustic wave equation modeling in the Laplace domain. Geophys. J. Int. 186, 740750.
 (NonPatent Document 11) Pyun, S., Son, W., Shin, C., 2011b. 3D acoustic waveform inversion in the Laplace domain using an iterative solver. Geophys. Prospect. 59, 386399.
 (NonPatent Document 12) Qin, Y., Zhang, Z., Shin, C., 2005. A robust and accurate traveltime calculation using a frequencydomain twoway waveequation modeling algorithm. J. Seism. Explor. 13, 227245.
 (NonPatent Document 13) Shin, C., Cha, Y. H., 2008. Waveform inversion in the Laplace domain. Geophys. J. Int. 173, 922931.
 (NonPatent Document 14) Shin, C., Ko, S., Kim, W., Min, D., Yang, D., Marfurt, K. J., Shin, S., Yoon, K., Yoon, C., 2003. Traveltime calculations from frequency domain downward continuation algorithms. Geophysics 68, 16481655.
 (NonPatent Document 15) Shin, C., Min, D., Marfurt, K. J., Lim, H. Y., Yang, D., Cha, Y., Ko, S., Yoon, K., Ha, T., Hong, S., 2002. Traveltime and amplitude calculations using the damped wave solution. Geophysics 67, 16371647.
 (NonPatent Document 16) Son, W., Pyun, S., Shin, C., Kim, H.J., 2014. Laplacedomain waveequation modeling and full waveform inversion in 3D isotropic elastic media. J. Appl. Geophys. 105, 120132.
 (NonPatent Document 17) Vidale, J., 1988. Finitedifference calculation of traveltimes. Bull. Seismol. Soc. Am. 78, 20622076.
 (NonPatent Document 18) Vidale, J., 1990. Finitedifference calculation of traveltimes in three dimensions. Geophysics. 55, 521526.
 (NonPatent Document 19) Yang, D., Shin, C., Marfurt, K. J., Kim, J., Ko, S., 2003. Threedimensional traveltime and amplitude computation using the suppressed wave equation estimation of traveltime (SWEET) algorithm. J. Seism. Explor. 12, 75101.
 (NonPatent Document 20) Yoon, K., Shin, C., Su, S., Lines, L. R., Hong, S., 2003. 3D reversetime migration using the acoustic wave equation: An experience with the SEG/EAGE data set. Lead. Edge 22, 3841.
Accordingly, the present invention has been made keeping in mind to resolve the above problems occurring in the related art, and the present disclosure proposes: an efficient 3D traveltime calculation method for a shallow depth source, and an apparatus and method for calculating efficient 3D traveltime by using coarsegrid mesh for a shallow depth source which may need less calculation time compared with that of a conventional SWEET algorithm.
In order to achieve the above object, according to one embodiment of the present invention, an apparatus for calculating efficient 3D traveltime by using coarsegrid mesh for shallow depth source, the apparatus includes a Green'"'"'s function calculation unit configured to calculate Green'"'"'s function for a homogeneous halfspace medium; an equivalent source vector calculation unit configured to calculate an equivalent source vector equivalent to an original point source vector by using a wavefield vector sampled at coarsegrid points calculated by the Green'"'"'s function calculation unit; a wavefield vector calculation unit configured to calculate a wavefield vector by using the equivalent source vector calculated by the equivalent source vector calculation unit; a wavefield vector'"'"'s partial derivative generation unit configured to generate a partial derivative of the wavefield vector calculated by the wavefield vector calculation unit; and a firstarrival traveltime calculation unit configured to calculate a firstarrival traveltime by the SWEET algorithm by using the wavefield vector calculated by the wavefield vector calculation unit and the partial derivative of the wavefield vector generated by the wavefield vector'"'"'s partial derivative generation unit.
According to one embodiment of the present invention, an apparatus for calculating efficient 3D traveltime by using coarsegrid mesh for shallow depth source, the Green'"'"'s function calculation unit can calculate the Green'"'"'s function by the following equation 7:
where G(s,ν_{0},r_{g},r_{s},r′_{s}) is a Green'"'"'s function, S is a Laplace domain variable, ν_{0 }is a propagation velocity for the homogeneous halfspace medium, r_{g }is a position vector of the source, and r′_{s }is a position vector of an imaginary source.
According to one embodiment of the present invention, an apparatus for calculating efficient 3D traveltime by using coarsegrid mesh for shallow depth source, the equivalent source vector calculation unit can calculate the equivalent source vector by the following equation 8:
f^{equi}=Sũ, (Equation 8)
where f^{equi }is a new equivalent source vector for the homogeneous halfspace, S is an impedance matrix, and ũ is the wavefield vector sampled at coarsegrid points from the analytical solution of equation 7.
According to one embodiment of the present invention, an apparatus for calculating efficient 3D traveltime by using coarsegrid mesh for shallow depth source, the wavefield vector calculation unit can calculate the wavefield vector by the following equation 9:
u^{equi}=S^{−1}f^{equi}, (Equation 9)
where u^{equi }is the wavefield vector generated from the equivalent source vector.
According to one embodiment of the present invention, an apparatus for calculating efficient 3D traveltime by using coarsegrid mesh for shallow depth source, the wavefield vector'"'"'s partial derivative generation unit can generate a partial derivative of the wavefield vector by the following equation 10:
According to one embodiment of the present invention, an apparatus for calculating efficient 3D traveltime by using coarsegrid mesh for shallow depth source, the firstarrival traveltime calculation unit, calculate the firstarrival traveltime by the following equation 11:
where t^{equi }is a firstarrival travel time, and s_{opt }is optimal Laplace decay coefficient.
In order to achieve the above object, according to another embodiment of the present invention, a method for calculating efficient 3D traveltime by using coarsegrid mesh for shallow depth source, the method includes the steps of: (a) calculating, by the Green'"'"'s function calculation unit, Green'"'"'s function for a homogeneous halfspace medium; (b) calculating, by the equivalent source vector calculation unit, an equivalent source vector equivalent to an original point source vector by using a wavefield vector sampled at coarsegrid points calculated by the (a); (c) calculating, by the wavefield vector calculation unit, a wavefield vector by using the equivalent source vector calculated by the (b); (d) generating, by the wavefield vector'"'"'s partial derivative generation unit, a partial derivative of the wavefield vector calculated by the (c); and (e) calculating, by the firstarrival traveltime calculation unit, a firstarrival traveltime by the SWEET algorithm by the wavefield vector calculated by the (c) and the partial derivative of the wavefield vector generated by the (d).
According to another embodiment of the present invention, a method for calculating efficient 3D traveltime by using coarsegrid mesh for shallow depth source, the Green'"'"'s function can be calculated in (a) by the following equation 7:
where G(s,ν_{0},r_{g},r_{s},r′_{s}) is a Green'"'"'s function, S is a Laplace domain variable, ν_{0 }is a propagation velocity for the homogeneous halfspace medium, r_{g }is a position vector of the source and r′_{s }is a position vector of an imaginary source.
According to another embodiment of the present invention, a method for calculating efficient 3D traveltime by using coarsegrid mesh for shallow depth source, the equivalent source vector can be calculated in (b) by the following equation 8:
f^{equi}=Sũ, (Equation 8)
where f^{equi }is a new equivalent source vector for the homogeneous halfspace, S is an impedance matrix, and ũ is the wavefield vector sampled at coarsegrid points from the analytical solution of the equation 7.
According to another embodiment of the present invention, a method for calculating efficient 3D traveltime by using coarsegrid mesh for shallow depth source, the wavefield vector can be calculated in (c) by the following equation 9:
u^{equi}=^{−1}f^{equi}, (Equation 9)
where u^{equi }is the wavefield vector generated from the equivalent source vector.
According to another embodiment of the present invention, a method for calculating efficient 3D traveltime by using coarsegrid mesh for shallow depth source, the partial derivative of the wavefield vector can be generated in (d) by the following equation 10:
According to another embodiment of the present invention, a method for calculating efficient 3D traveltime by using coarsegrid mesh for shallow depth source, the firstarrival traveltime can be calculated in (e) by following equation 11:
where t^{equi }is a firstarrival travel time, and s_{opt }is optimal Laplace decay coefficient.
According to the embodiment of the present invention, an apparatus and method for calculating efficient 3D traveltime by using coarsegrid mesh for a shallow depth source has an excellent effect of providing an efficient calculation method for the shallow depth source, and requiring less calculation time compared to a conventional SWEET algorithm, wherein the apparatus and method are configured to: calculate, by the Green'"'"'s function calculation unit, Green'"'"'s function for a homogeneous halfspace medium; calculate, by the equivalent source vector calculation unit, an equivalent source vector equivalent to an original point source vector by using a wavefield vector sampled at coarsegrid points calculated by the Green'"'"'s function calculation unit; calculate, by the wavefield vector calculation unit, a wavefield vector by using the equivalent source vector calculated by the equivalent source vector calculation unit; generate, by the wavefield vector'"'"'s partial derivative generation unit, a partial derivative of the wavefield vector calculated by the wavefield vector calculation unit; and calculate, by the firstarrival traveltime calculation unit, a firstarrival traveltime by the SWEET by using the wavefield vector calculated by the wavefield vector calculation unit and the partial derivative of the wavefield vector generated by the wavefield vector'"'"'s partial derivative generation unit.
That is, according to the embodiment of the present invention, an apparatus and method for calculating efficient 3D traveltime by using coarsegrid mesh for shallow depth source is configured to: combine the SWEET and ESD algorithms, whereby the combination of SWEET and ESD algorithms can be successfully used for the traveltime calculation under the condition of a shallow depth source; and use coarsegrid mesh, wherein the algorithm using coarsegrid mesh has an excellent effect in that less computational time is required than the conventional SWEET algorithm using relatively finegrid mesh, and wherein the SWEET algorithm is a traveltime calculation algorithm using a damped wave equation, and the ESD algorithm is a method to define a set of distributed nodal sources that approximate a point source at the internodal location in a velocity model with large grid spacing.
The patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawing(s) will be provided by the Office upon request and payment of the necessary fee.
The above and other objects, features and other advantages of the present invention will be more clearly understood from the following detailed description when taken in conjunction with the accompanying drawings, in which:

 100: Green'"'"'s function calculation unit
 200: Equivalent source vector calculation unit
 300: Wavefield vector calculation unit
 400: Wavefield vector'"'"'s partial derivative generation unit
 500: Firstarrival traveltime calculation unit
Hereinbelow, exemplary embodiments of the present invention will be described in detail with reference to the accompanying drawings. Throughout the drawings, the same reference numerals will refer to the same or like parts.
First, the ESD algorithm applied in the present invention will be described. The ESD algorithm is a method to define a set of distributed nodal sources that approximate a point source at the internodal location in a velocity model with large grid spacing and an algorithm to perform more efficient and accurate modeling of the 3D Laplace domain wave equation for a coarsegrid velocity model.
According to an embodiment of the present invention, an apparatus for calculating efficient 3D traveltime by using coarsegrid mesh for a shallow depth source, as illustrated in
The Green'"'"'s function calculation unit 100 calculates Green'"'"'s function for a homogeneous halfspace medium. The Green'"'"'s function calculation unit 100 calculates the Green'"'"'s function by the following equation 7:
where G(s,ν_{0},r_{g},r_{s},r′_{s}) is a Green'"'"'s function, S is a Laplace domain variable, ν_{0 }is a propagation velocity for the homogeneous halfspace medium, r_{g }is a position vector of the source and r′_{s }is a position vector of an imaginary source.
The equivalent source vector calculation unit 200 plays calculates the equivalent source vector equivalent to an original point source vector by using the wavefield vector sampled at coarsegrid points of the Green'"'"'s function calculated by the Green'"'"'s function calculation unit 100 and the impedance matrix for the coarsegrid mesh. The equivalent source vector calculation unit 200 calculates the equivalent source vector by the following equation 8:
f^{equi}=Sũ, (Equation 8)
where f^{equi }is a new equivalent source vector for the homogeneous halfspace, S is an impedance matrix, and ũ is the wavefield vector sampled at coarsegrid points from the analytical solution of equation 7.
The wavefield vector calculation unit 300 calculates, as shown in the equations 4 and 6, the wavefield vector by using the equivalent source vector calculated by the equivalent source vector calculation unit 200. The wavefield vector calculation unit 300 calculates the wavefield vector by the following equation 9:
u^{equi}=S^{−1}f^{equi}, (Equation 9)
where u^{equi }is the wavefield vector generated from the equivalent source vector.
The wavefield vector'"'"'s partial derivative generation unit 400 generates a partial derivative of the wavefield vector calculated by the wavefield vector calculation unit 300. The wavefield vector'"'"'s partial derivative generation unit 400 generates a partial derivative of the wavefield vector by the following equation 10:
The firstarrival traveltime calculation unit 500 calculates the firstarrival traveltime by the SWEET algorithm by using the wavefield vector calculated by the wavefield vector calculation unit 300 and the partial derivative of the wavefield vector generated by the wavefield vector'"'"'s partial derivative generation unit 400. The firstarrival traveltime calculation unit 500 calculates the firstarrival traveltime by the following equation 11:
where t^{equi }is a firstarrival travel time, and S_{opt }is optimal Laplace decay coefficient.
Meanwhile, the Green'"'"'s function calculation unit 100, the equivalent source vector calculation unit 200, the wavefield vector calculation unit 300, the wavefield vector'"'"'s partial derivative generation unit 400, and the firstarrival traveltime calculation unit 500 described above can be configured into one terminal device (e.g., a notebook, a personal computer, a PMP, etc.).
Hereafter, according to the embodiment, configured as above, of the present invention, a method for calculating efficient 3D traveltime by using coarsegrid mesh for a shallow depth source will be described.
First, the Green'"'"'s function calculation unit 100 calculates the Green'"'"'s function (S10), by equation 7.
Then, the equivalent source vector calculation unit 200 calculates the equivalent source vector equivalent to an original point source vector by using the wavefield vector sampled at coarsegrid points of the Green'"'"'s function calculated by step S10 and the impedance matrix for the coarsegrid mesh (S20), wherein the equivalent source vector is calculated by equation 8.
In step S30, the wavefield vector calculation unit 300 calculates the wavefield vector by using the equivalent source vector calculated by step S20, wherein the wavefield vector is calculated by equation 9.
In step S40, the wavefield vector'"'"'s partial derivative generation unit 400 generates a partial derivative of the wavefield vector calculated by step S30, wherein the wavefield vector'"'"'s partial derivative generation is performed by equation 10.
In step S50, the firstarrival traveltime calculation unit 500 calculates the firstarrival traveltime by the SWEET algorithm by using the wavefield vector calculated by step S30 and the partial derivative of the wavefield vector generated by step S40, wherein the firstarrival traveltime calculation is performed by equation 11.
In the meantime, according to the embodiment of the present invention, described above, the method for calculating 3D traveltime focuses on the shallow depth sources located close to free surface. However, when sources are located at a deep area, the wavefield (or traveltime) can be simulated by using coarsegrid mesh without the ESD algorithm. In addition, by employing an interpolation algorithm (e.g. trilinear interpolation), the source can just be distributed to the neighboring grid points. Meanwhile, the ESD algorithm can be straightforwardly applied to deep sources as well.
For the verification of the method for calculating 3D traveltime according to the embodiment of the present invention, numerically calculated traveltimes and analytically calculated traveltimes for a homogeneous halfspace medium will be compared hereafter.
The constant velocity of the homogeneous model is 2000 m/s, and the model size is 10 km×10 km×10 km with a grid spacing of 50 m. The source point is located at the center of XY plane and at 10 m deep in Zdirection from the free surface. The optimal Laplace damping constant was calculated by using equation 2. In equation 2, 25 of G, which is the number of grid points per pseudowavelength, was used, and numerical dispersion errors were confirmed as being less than 0.4% with G=25 from the dispersion analysis.
In
One is the SWEET algorithm with an original point source and the other is the SWEET algorithm with equivalently distributed point sources.
When the equivalently distributed point sources are not used, a point source which is mislocated at a depth of 50 m should be used, which leads to incorrectly calculated firstarrival traveltimes (
However, the numerical traveltimes obtained by the SWEET and ESD algorithms agree with the analytical traveltimes due to the appropriate representation of the point source at 10 m depth (
For the verification of the method for calculating 3D traveltime of the present invention for a complicated heterogeneous medium, the firstarrival traveltime for the SEG/EAGE 3D salt model (
Unlike the homogeneous model examples, the coarsegrid wavefield vector should be sampled from the exact finegrid solution. However, the traveltime calculation for the heterogeneous medium requires more computational time to obtain the finegrid solution. To overcome this difficulty for a heterogeneous medium, the wavefields for a heterogeneous medium were simulated by using the equivalent source for a homogeneous halfspace. This demonstrated that the equivalent source for a homogeneous velocity model can be applied to a heterogeneous velocity model without losing accuracy.
Therefore, the heterogeneous medium (i.e. SEG/EAGE 3D salt model) was simulated by using the equivalent source for a homogeneous halfspace. The model size was 13.6 km×13.6 km×4.2 km and the source point was located at the center of XY plane and at 20 m deep in Zdirection from the free surface.
Dashed lines illustrate the firstarrival traveltimes calculated by using the SWEET algorithm and the equivalently distributed point sources with 40 m grid spacing.
Solid lines illustrate the firstarrival traveltime calculated by using the SWEET algorithm and one point source with 20 m grid spacing.
The traveltimes using 20 m grid spacing (as a reference) were used to compare with the results of the algorithm of the present invention using 40 m grid spacing.
From
The computational time of the SWEET algorithm is improved by about 15 times when doubled grid spacing (40 m instead of 20 m) with the ESD algorithm is used. Although the waveequationbased traveltime calculation is not as efficient as certain traditional traveltime algorithms, the calculation can preserve the benefit of waveequationbased algorithm (e.g. there is no shadow zone) while simultaneously enhancing the efficiency of traveltime calculation.
According to the embodiments of the present invention, an apparatus and method for calculating efficient 3D traveltime by using coarsegrid mesh for a shallow depth source may provide an efficient calculation method for the shallow depth source, and reduce calculation time needed compared with that of a conventional SWEET algorithm, wherein the apparatus and method are configured to: calculate, by the Green'"'"'s function calculation unit, Green'"'"'s function for a homogeneous halfspace medium; calculate, by the equivalent source vector calculation unit, an equivalent source vector equivalent to an original point source vector by using a wavefield vector sampled at coarsegrid points calculated by the Green'"'"'s function calculation unit; calculate, by the wavefield vector calculation unit, a wavefield vector by using the equivalent source vector calculated by the equivalent source vector calculation unit; generate, by the wavefield vector'"'"'s partial derivative generation unit, a partial derivative of the wavefield vector calculated by the wavefield vector calculation unit; and calculate, by the firstarrival traveltime calculation unit, a firstarrival traveltime by the SWEET by using the wavefield vector calculated by the wavefield vector calculation unit and the partial derivative of the wavefield vector generated by the wavefield vector'"'"'s partial derivative generation unit.
That is, according to the embodiment of the present invention, an apparatus and method for calculating efficient 3D traveltime by using coarsegrid mesh for shallow depth source is configured to: combine the SWEET and ESD algorithms, whereby the combination of the SWEET and ESD algorithms can be successfully used for the traveltime calculation under the condition of a shallow depth source, and use a coarsegrid mesh, wherein the algorithm using a coarsegrid mesh may require less computational time than the conventional SWEET algorithm using a relatively finegrid mesh, and wherein the SWEET algorithm is a traveltime calculation algorithm using a damped wave equation, and an ESD algorithm is a method to define a point source at the internodal location in a velocity model with large grid spacing as a set of distributed nodal sources.
Although optimal embodiments are disclosed and specific terminologies are used in the drawings and the specification, these are used for illustrative purposes only but not to limit the meaning of them or the scope of the invention described in claims. Therefore, those skilled in the art will appreciate that various substitutions, changes, and modifications are possible without departing from the scope and spirit of the present invention. Accordingly, the real technical protection scope of the present invention should be defined by the technical concept of the attached claims.