JingBo Chen^{*}  
Key Laboratory of Petroleum Resources Research, Institute of Geology and Geophysics, Chinese Academy of Sciences, China  
Corresponding Author :  Chen JB Key Laboratory of Petroleum Resources Research Institute of Geology and Geophysics Chinese Academy of Sciences, China Tel: +8601082998156 Email: [email protected] 
Received October 30, 2014; Accepted December 20, 2014; Published December 27, 2014  
Citation: Chen JB (2015) A 19Point Averagederivative Optimal Scheme for 3D Frequencydomain Scalar Wave Equation. J Geophys Remote Sensing 4:140.doi:10.4172/21690049.1000140  
Copyright: ©2015 Chen JB. This is an openaccess article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. 
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A 27point optimal scheme for 3D frequencydomain scalar wave equation was recently developed. This scheme includes 9 optimization coefficients, and is relatively complicated. To simplify this scheme, a 19point averagederivative optimal scheme for 3D frequencydomain scalar wave equation is constructed in this paper. This new 19point scheme includes only 5 optimization coefficients, but maintains similar advantages of the 27point optimal scheme. Compared to the classical 7point scheme, the number of grid points per shortest wavelength is reduced from approximately 13 to approximately 4 by this 19point optimal scheme for equal directional sampling intervals and unequal directional sampling intervals as well. Two numerical examples are presented to demonstrate the theoretical analysis.
Keywords 
Seismic modeling; Frequency domain; Three dimension; Average derivative; Optimization 
Introduction 
Full waveform inversion (FWI) is a fullwavefieldmodelingbased datafitting process to extract structural information of subsurface from seismograms [1]. FWI can be implemented either in timedomain [2 4] or in frequencydomain [57]. 
Forward modeling is an important part of FWI. In line with FWI, forward modeling can be divided into two categories: time domain modeling and frequencydomain modeling. Compared to timedomain modeling [8], frequencydomain modeling has its advantages: convenient manipulations of a single frequency, multishot computation based on a direct solver, and easy implementation of attenuation [9]. In addition, no wavefieldstorage issue occurs when constructing the gradient of FWI for frequencydomain modeling. This is not the case when constructing the gradient of FWI for timedomain modeling. This is because the forward source wavefield and backward receiver wavefield are computed in opposite time direction [10,11]. 
The main disadvantage of frequencydomain modeling is that it can only be performed in an implicit way by solving a set of linear equations. In comparison with the timedomain modeling, this disadvantage is particularly obvious when it comes to 3D computation. Therefore, reducing the number of grid points per shortest wavelength is in great demand in particular when direct solution techniques are employed. Using a rotated coordinate system, Jo et al. [9] developed a 9point optimal scheme for 2D scalar wave equation. This 9point scheme reduces the number of grid points per wavelength to approximately 4, and leads to significant reductions of computer memory and CPU time. Hustedt et al. [12] and Operto et al. [13] generalized the rotatedcoordinate method to variable density case and 3D case, respectively. 
To overcome the disadvantage of the rotated optimal 9point scheme, Chen [14] developed a new 9point finitedifference scheme for 2D scalar wave equation based on an averagederivative approach [15,16]. This new scheme imposes no restriction of equal directional sampling intervals, and reduces the number of grid points per shortest wavelength to approximately 4 for both equal and unequal directional sampling intervals. Chen [17] further generalized the averagederivative method and developed a 27point optimal scheme for 3D scalar wave equation. Compared to the classical 7point scheme, the number of grid points per shortest wavelength is reduced from approximately 13 to approximately 4 by this 27point optimal scheme for both equal and unequal directional sampling intervals within the relative phase error of one percent. 
The 27point scheme Chen [17] includes 9 optimization coefficients, and has a relatively high complexity. By reducing the number of optimization coefficients from 9 to 5, a simplified 19point average derivative optimal scheme for 3D scalar wave equation can be achieved. In the next section, I will present this new 19point scheme. This is followed by the optimization of coefficients and a numerical dispersion analysis. Numerical examples are then presented to demonstrate the theoretical analysis. 
A 19Point Scheme for 3D Wave Equation 
The 3D frequencydomain scalar wave equation can be written as 
(1) 
where P is the pressure wavefield, ω is the angular frequency, and v(x, y, z) is the velocity. 
An averagederivative optimal 19point scheme for equation (1) can be obtained as follows (Figure 1): 
(2) 
where 
where α, β, γ, c, and d are coefficients which are to be optimized, and 
Here, and Δx , Δy , and Δz are directional sampling intervals in the xdirection, ydirection and zdirection, respectively 
In equation (2), the approximation of the derivative in one direction involves an average of wavefield values from remaining two directions. In this way, a family of approximations to the derivative is obtained, which depends on a free parameter (α, β, or γ). The free parameters c and d from the average of the mass acceleration term play the same role. Therefore, based on optimization techniques, the optimization approximation can be chosen to reduce dispersion errors. The averagederivative optimal 19point scheme (2) also includes the classical 7point scheme as a special case, because when α=0, β=0, γ=0, c=1, and d=0, the scheme (2) reduces to the classical 7point scheme: 
(3) 
Optimization and Dispersion Analysis 
Substituting into equation (2) and assuming a constant v, one obtains the discrete dispersion relation 
where 
(5) 
From equation (4), one can obtain the normalized phase velocity 
(6) 
where V_{ph} k_{x}= k sinθ cosφ, k_{y}=k sin θ sin Ï k_{z}=k cosθ ,,θ is the propagation angle, Ï is the azimuth angle, and 
The coefficients α, β, γ, c, and d are obtained by minimizing the phase error: 
where 
The optimization coefficients for different and when Δz = max{Δx,Δy,Δz} are listed in Table 1. The optimization coefficients for the cases where , and are listed in Tables 2 and 3, respectively. 
Now I perform numerical dispersion analysis. First, I consider the case where r1=1 and r2=1, which corresponds to the equal directional intervals Δx=Δy=Δz. Figure 2 shows normalized phase velocity curves of the classical 7point scheme (3) and the averagederivative optimal 19point scheme (2) for fixed azimuth angle Ï and different propagation angles θ. Figure 3 shows normalized phase velocity curves of the classical 7point scheme (3) and the averagederivative optimal 19point scheme (2) for fixed propagation angle θ and different azimuth angles Ï. From these figures, one can conclude that within the phase velocity error of 1%, the classical 7point scheme (3) requires approximately 13 grid points per shortest wavelength, while the averagederivative optimal 19point scheme (2) requires approximately 4 points. 
In order to obtain an overall estimation of the phase velocity errors varying with Ï and θ, the following relative phase velocity error is considered: 
(9) 
Figure 4 shows the relative phase velocity errors for the classical 7point scheme and the averagederivative optimal 19point scheme for different 1. With increasing 1, the relative phase velocity errors for the classical 7point scheme increase. On the other hand, the relative phase velocity errors for the classical averagederivative optimal 19point scheme are within 1% as increases. 
Figures 57 show the results for the case where r1=1 and r2=2. In this case, one can draw the same conclusion: within the phase velocity error of 1% and for equal and unequal directional sampling intervals, the classical 7point scheme (3) requires approximately 13 grid points per shortest wavelength, while the averagederivative optimal 19point scheme (2) requires approximately 4 grid points. For other cases on r1 and r2, similar analysis can be made. 
Numerical Examples 
In this section, two numerical examples are presented to verify the theoretical analysis on the classical 7point scheme (3) and the averagederivative optimal 19point scheme (2). First, I consider a homogeneous velocity model with a velocity of 4000 m/s (Figure 8a). Horizontal and vertical distances are x=2 km, y=2 km, and z=1 km, respectively. A Ricker wavelet is placed at the center of the model as a source. The receivers are placed at a depth of 250 m. PML absorbing boundary conditions are applied along the edges of the model. In this numerical example, a monochromatic wavefield of 20 Hz is computed. According to the criterion of 4 grid points per smallest wavelength, horizontal sampling interval is determined by dx=4000/20/4 m=50 m. Set dy=dx and dz=dx/2. Accordingly, horizontal and vertical samplings are nx=ny=nz=41. In this case, the optimization coefficients for the averagederivative optimal 19point scheme (2) are α=0.098871, β=0.098869, γ=0.095057, c=0.458533, and d=0.090244. In this example, analytical solution is available to make comparisons with numerical solutions. For the analytical solution, the following formula is used 
(10) 
where ï is the Fourier transformation of the Ricker wavelet, and 
where is the source position. 
Figure 8b shows the real part of the 20 Hz monochromatic wavefield computed with the analytical formula (10), the classical 7point scheme (3) and the averagederivative optimal 19point scheme (2). The simulation result with the averagederivative optimal 19point scheme (2) is in good agreement with the analytical result while the result with the classical 7point scheme (3) exhibits large errors due to numerical dispersion. 
Second, I consider a heterogeneous velocity model. Figure 9a shows a salt dome velocity model. The velocity of the salt dome is 5000 m/s, and the velocity of the overburden is 4000 m/s. The distances, sampling numbers, the Ricker wavelet, and the frequency used in this example are the same as those used in the homogeneous velocity model. For the top boundary, a freesurface condition is used, and for other edges of the model, PML absorbing boundary conditions are applied. The source is located at a depth of 200 m, and the receivers are placed immediately under the free surface. In this example, analytical solution is not available. Therefore, I consider the 27point optimal scheme in Chen [17] for a comparison. Figure 9b shows the real part of the 20 Hz monochromatic wavefield computed with the 27point optimal scheme, the classical 7point scheme (3) and the averagederivative optimal 19point scheme (2). The simulation result with the averagederivative optimal 19point scheme (2) is in good agreement with the result of the 27point scheme. On the other hand, the result with the classical 7point scheme (3) exhibits large discrepancies with the result of the 27point scheme due to numerical dispersion. 
Conclusion 
To simplify the 27point optimal scheme, a 19point averagederivative optimal scheme for 3D frequencydomain scalar wave equation is developed. The optimization coefficients are obtained by minimizing the phase velocity errors, and they vary with the ratios of directional sampling intervals. Compared to the classical 7point scheme, within the phase velocity error of 1%, this new 19point optimal scheme reduces the number of per shortest wavelength from 13 to 4 for both equal and unequal directional sampling intervals. Two numerical examples confirm the theoretical analysis. 
Acknowledgements 
This work is supported by National Natural Science Foundation of China under grant Nos. 41474104 and 41274139. 
References 

Table 1  Table 2  Table 3 
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