Here is a new 9-point scheme for finite difference solution of acoustic waves in frequency domain. The algorithm honors both accuracy and computational efficiency.

1. 2D Frequency-Domain Finite Difference Solution of the Scalar
Wave Equation through Plane Wave Solution Interpolation

2. Abstract No. Th_R06_15
Saeed Izadian , Heriot –Watt University, si63@hw.ac.uk
Kamal Aghazade, University of Tehran, Aghazade.kamal@ut.ac.ir
Navid Amini, University of Tehran, navidamini@ut.ac.ir

3. INTRODUCTION
Frequency Domain Finite Difference
Proposed solutions:
(i) Employing high-order finite-difference schemes or high-points stencil
 Computationally expansive
(ii) Optimization strategy
 Not straightforward
(iii) Optimized high-order finite-difference schemes (i & ii)
 Combined challenges of (i) and (ii)
(iv) Exact finite difference (EFD) approaches by utilizing the exact analytic solutions to build FD schemes
 Not straightforward in high dimensions (2D & 3D)
 Suffers form numerical dispersion error due to the discretization.
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 One of the popular approaches for solution of seismic wave equations
 Supports considering frequency-dependent attenuation mechanisms and multi-source
modeling
 Suitable for multi-scale strategy
 Desirable for RTM and FWI problems.

4. The Goal and Idea behind This study
GOAL: Developing an efficient Finite-Difference Scheme for 2D
Frequcy Domain Acoustic Wave Equation which honors:
 Accuracy
 Computational efficiency
 Straightforward procedure
IDEA:
i. Exploiting the advantages of the EFD method in solving the 2D Helmholtz equation
(Accuracy aspect)
ii. Linear combination of plane waves in pre-defined directions of 9-point scheme
(Computationally efficient)
iii. Avoiding optimization challenges by direct calculation of the FD coefficient (Straightforward)
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𝜕2
𝑝(𝑥, 𝑧)
𝜕𝑥2
+
𝜕2
𝑝(𝑥, 𝑧)
𝜕𝑧2
+ 𝑘2𝑝 𝑥, 𝑧 = 0

5. Theory
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Step1: Starting from the 2D Helmholtz equation
Step2: Employing 9-point scheme
𝑝𝑚,𝑛: 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑤𝑎𝑣𝑒𝑓𝑖𝑒𝑙𝑑 𝑎𝑡 𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛 𝑥𝑚 = 𝑚ℎ, 𝑧𝑛 = 𝑛ℎ
k: wave number
∆𝑥 = ∆𝑧 = ℎ
𝑐𝑖: 𝐹𝐷 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑠
𝜕2
𝑝(𝑥, 𝑧)
𝜕𝑥2
+
𝜕2
𝑝(𝑥, 𝑧)
𝜕𝑧2
+ 𝑘2𝑝 𝑥, 𝑧 = 0
𝑐1 𝑝𝑚,𝑛+1 + 𝑝𝑚,𝑛−1 + 𝑝𝑚+1,𝑛 + 𝑝𝑚−1,𝑛
+𝑐2 𝑝𝑚+1,𝑛+1 + 𝑝𝑚+1,𝑛−1 + 𝑝𝑚−1,𝑛+1 + 𝑝𝑚−1,𝑛−1
+𝑐3𝑝𝑚,𝑛 = 0

6. Theory
Step 3: Linear combination of plane-waves solutions along eight
directions of the 9-point scheme
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𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 2𝑐1 1 + cos 𝑘ℎ + 4𝑐2 cos 𝑘ℎ + 𝑐3 +
𝑎5 + 𝑎6 + 𝑎7 + 𝑎8 4𝑐1 cos
2
2
𝑘ℎ + 2𝑐2 1 + cos 2𝑘ℎ + 𝑐3 = 0
𝑎i: 𝐶𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛
Step 4: Calculating FD coefficients (𝑐𝑖)
2𝑐1(1 + cos 𝑘ℎ + 4𝑐2 cos 𝑘ℎ + 𝑐3 = 0
4𝑐1 cos
2
2
𝑘ℎ + 2c2 1 + cos 2𝑘ℎ + 𝑐3 = 0
𝑐3 = 2 1 + cos 𝑘ℎ + 4cos(𝑘ℎ)
1 + cos 2𝑘ℎ − 2cos(𝑘ℎ)
1 + cos 𝑘ℎ − 2 cos
2
2 𝑘ℎ
subtracting 𝑐1 =
1 + cos 2𝑘ℎ − 2cos(𝑘ℎ)
1 + cos 𝑘ℎ − 2 cos
2
2
𝑘ℎ
𝑐2 =
1
ℎ2

7. PWI Scheme (Main Characteristic)
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In PWI scheme, plane waves satisfy the wave equation in pre-defined
directions, and thus, the dispersion error will be zero in eight pre-defined
directions for 9-point scheme.

8. Dispersion Analysis
Dispersion curves for diffèrent propagation angles
(𝜃 = 0°, 15°, 30°, 45°)
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(a) Optimized 9-point (Jo et al. 1996) scheme
(b) Optimized 25-point (Shin and Sohn, 1998) scheme
(c) Proposed PWI scheme

9. Dispersion Analysis (Main Results)
proposed method outperforms than even 25-point scheme in terms of dispersion error.
 The effect of the interpolation procedure has constrained the dispersion error to be less
than % 0.4 in the worst case scenario (i.e. 2.5 grid per minimum wavelength).
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10. Numerical Examples
Case 1: 2D homogenous model
 2D homogenous model with P-wave velocity of 625 m/s
 Model size: 200 × 200 grids
 Grid interval with ℎ = 10 𝑚 (𝐺 = 2.5)
 Source wavelet: Ricker wavelet with dominant frequency of 10 Hz
and Maximum frequency of 25 Hz
 Source location: Center of the model
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11. Numerical Examples
2D homogenous model (Simulation)
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12. Numerical Examples
2D homogenous model (Comparison with analytical solution)
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comparing numerical solutions in different
angles (with respect to z-axis and 400 m
away from the source) with analytical
solution.

13. Numerical Examples
2D homogenous model (Main Results)
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 The 9-point scheme suffers from grid dispersion error more
than the others.
 In comparison with the proposed PWI scheme, the 25-point
scheme still suffers from dispersion errors in four main
Cartesian coordinate directions.
 Comparison with the analytical solution and NME plot
indicate higher accuracy aspect of the proposed PWI
scheme.

14. Numerical Examples
Case 2: Marmousi model
 P-wave velocity range: 1.56 𝑡𝑜 4.7 𝑘𝑚/𝑠
 Model size: 400 × 800 grids with ℎ = 10 𝑚 (𝐺 = 2.6)
 Source wavelet: Ricker wavelet with the dominant frequency of
20 𝐻𝑧 and maximum frequency of 60 𝐻𝑧
 Source location: top center of model at the surface

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15. Numerical Examples
Marmousi Model (Simulation)
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(a) P-wave velocity model
(b) Snapshot of propagated wavefield

16. Conclusions
 In order to calculate FD coefficients, we proposed interpolation of the plane wave
solutions in eight pre-defined directions of the 9-point stencil in a straightforward
manner.
 Numerical dispersion will be zero in those directions and the proposed plane wave
interpolation (PWI) approach suppresses numerical errors in other directions as
well.
 According to the dispersion error analysis and simulation, the proposed scheme is
more accurate than 9-point and 25-point schemes.
 The proposed approach employs 9-point stencil, which honors computational
efficiency.
 This approach is completely applicable to highly heterogeneous and complicated
media.
In this study, a new compact 9-point finite-difference scheme was proposed for
the numerical solution of the 2D Helmholtz equation in the frequency-domain.
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17. Thank You