=Paper=
{{Paper
|id=Vol-2267/302-306-paper-57
|storemode=property
|title=Different approaches for elastic imaging using multiprocessor computing systems
|pdfUrl=https://ceur-ws.org/Vol-2267/302-306-paper-57.pdf
|volume=Vol-2267
|authors=Vasiliy. I. Golubev,Alena V. Favorskaya
}}
==Different approaches for elastic imaging using multiprocessor computing systems==
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Proceedings of the VIII International Conference "Distributed Computing and Grid-technologies in Science and
             Education" (GRID 2018), Dubna, Moscow region, Russia, September 10 - 14, 2018




DIFFERENT APPROACHES FOR ELASTIC IMAGING USING
      MULTIPROCESSOR COMPUTING SYSTEMS*
                                Golubev V.I. 1,2, a, Favorskaya A.V. 1,2 b
      1
          Moscow Institute of Physics and Technology, 9 Institytsky Pereylok st., Dolgoprudny, Moscow
                                      Region, 141700 Russian Federation
          2
              Scientific Research Institute for System Studies of the Russian Academy of Sciences, 36(1)
                               Nahimovskij av., Moscow, 117218 Russian Federation

                                E-mail: a w.golubev@mail.ru, b aleanera@yandex.ru


One of the important problems of the seismic survey process is the seismic migration. The result
provides enough information for the delineation of oil and gas deposits. In this article two independent
approaches for solving the migration problem in elastic media are discussed. Main formulas for the
Born approximation are provided and advantages of this algorithm are represented. Limitations of the
used background model were overcome with the full-wave elastic migration approach. It uses the grid-
characteristic method on structured meshes of high-orders. The possibility of the single geological
crack identification was tested. A wide range of crack plane orientations was covered.

Keywords: seismic imaging, numerical simulation, Born approximation, grid-characteristic method

                                                                © 2018 Vasiliy. I. Golubev, Alena V. Favorskaya




*
    The research was supported by the grant of the President of the Russian Federation No. MK-1831.2017.9.

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             Education" (GRID 2018), Dubna, Moscow region, Russia, September 10 - 14, 2018




1. Introduction
         One of the first paper about the seismic migration was published in 1954 [1]. Later on, Dr.
Claerbout postulated the finite-difference algorithm for solving the migration problem as a scalar wave
equation [2]. Numerous different algorithms were constructed by now: Kirchhoff integral method [3],
the frequency wavenumber migration algorithm [4], the Born approximation [5], the Reverse Time
Migration [6]. All of these methods were suffered from the limitations imposed on the background
model. A ray-Born method was proposed in [7] to take into account a non-uniform background
medium. In most cases authors used the simple acoustic approximation to describe the behavior of the
geological medium.
         In the paper [8, 9] the Born method and the corresponding migration algorithms were
extended to the elastic wavefields as well. Further, the adjoint operator approach [10] and the grid-
characteristic method [11] was successfully applied to the elastic migration problem [12].
         In this work we initially presented the main formulas of the elastic Born migration algorithm
and carried out numerical experiments to highlight advantages of it. Further, we applied the full-wave
grid-characteristic migration algorithm to estimate the possibility of the single geological crack
identification on the migration image.


2. Elastic migration with Born approximation
         The Lame equation for the elastic medium can be written in the operator form as
                                             𝜕2𝐮        1
                                      𝚲𝐮 −          = − 𝐟 e, 𝚲 = 𝑐𝑝2 ∇∇ ⋅ −𝑐𝑠2 ∇ × ∇ ×,                               (1)
                                             𝜕𝑡 2       𝜌
         where 𝑐𝑝 , 𝑐𝑠 are the pressure and shear wave velocities; 𝜌 is a mass density of the medium; 𝐟 e
is a strength of the external force per unit volume applied to the elastic body; 𝐮 is a displacement field.
         For the case of the constant background model the integral relationship between the field in
the volume and displacements on the day surface is written as
                                                 +∞
                          𝐮𝛼𝑠 (𝐫 ′ , 𝑡 ′ ) = ∫𝑉 ∫−∞ {Δ𝚲(𝐫)[𝐮𝑖 (𝐫, 𝑡) + 𝐮 𝑠 (𝐫, 𝑡)]} ⋅ 𝐆𝐿𝛼 (𝐫 ′ , 𝑡′|𝐫, 𝑡)𝑑𝑡 𝑑𝑉,        (2)
                  𝐿
         where 𝐆𝛼 is the Green’s tensor for the homogeneous space. In the Born approximation we
neglect the scattered field with respect to the background field in the volume 𝑉 and transform the
equation as
                                                      +∞
                          𝒖𝛼𝑠,𝐵 (𝒓′ , 𝑡 ′ ) = ∑𝛽 ∫𝑉 ∫−∞ 𝛥𝑐𝛽2 (𝒓)𝛻 2 𝒖𝛽𝑖 (𝒓, 𝑡) ⋅ 𝑮𝐿𝛼 (𝒓′ , 𝑡′|𝒓, 𝑡)𝑑𝑡 𝑑𝑉.              (3)
         According to the adjoint operator approach for the inverse problem solution we obtain
                              2                       +∞
                          𝛥𝑐𝛼,𝑚𝑖𝑔𝑟       (𝒓) = ∫𝑆 ∫𝑇 ∫−∞ {𝛻 2 𝒖𝑖𝛼 (𝒓, 𝑡)} ⋅ 𝑮𝐿 (𝒓′ , 𝑡 ′ |𝒓, 𝑡) ⋅ 𝒅(𝒓′ , 𝑡′)𝑑𝑡 𝑑𝑡′ 𝑑𝑆. (4)
         To demonstrate the advantage of the elastic approach over the acoustic approach we used the
simple 2D one layered model with the homogeneous spherical inclusion. Parameters of the
background model were set as 𝐶𝑃 = 2500 𝑚/𝑠, 𝐶𝑆 = 1250 𝑚/𝑠, 𝜌 = 2500 𝑘𝑔/𝑚3 in the rectangle
10 km x 2.5 km. The contrast of the spherical inclusion was only 1 to stay in the Born
approximation. Zero-offset seismograms with the 25 Hz peak frequency were used. For comparison
we limited the measurements with only the vertical component of displacements for all cases. We
parallelized the computational algorithm with the OpenMP standard and ran all calculations on the 12-
30 cores system with the shared memory.




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             Education" (GRID 2018), Dubna, Moscow region, Russia, September 10 - 14, 2018




               a




               b




               c
Figure 1. Migration images for the homogeneous spherical inclusion. The acoustic image (a) and elastic images
                                  for pressure waves (b) and shear waves (c)
        The analysis of these three images shows us that with the acoustic approach we obtain less
intensive artefacts on the P-wave image and also reproduce steeper geological boundaries on the S-
wave image.


3. Full-wave elastic migration
        In general, the dynamic behavior of the geological medium can be precisely described with the
elastic medium approach. It consists of the movement equation and the rheological relationship
between the stress tensor and the strain tensor. They form the hyperbolic system of equations in partial
derivatives:
                                     𝜕𝑣𝑖       𝜕𝜎𝑖𝑥       𝜕𝜎𝑖𝑦       𝜕𝜎𝑖𝑧
                                 𝜌         =          +          +          , 𝑖 = 𝑥, 𝑦, 𝑧                 (5)
                                   𝜕𝑡          𝜕𝑥          𝜕𝑦        𝜕𝑧
                                 𝜕𝜎𝑖𝑗                     𝜕𝜀𝑘𝑙                              𝜕𝜀
                                      = λδ𝑖𝑗 δ𝑘𝑙       + μ(δ𝑖𝑘 δ𝑗𝑙 + δ𝑖𝑙 δ𝑗𝑘 ) 𝑘𝑙 ,                    (6)
                                  𝜕𝑡               𝜕𝑡                         𝜕𝑡
        here 𝜎 – the stress tensor, 𝜀 – the strain tensor, 𝑣𝑖 – the component of the velocity vector.
        The solution of this system can be found numerically with different methods: the partial
derivatives approximation [13], the continuous or discontinuous Galerkin method [14], the grid-
characteristic method [11]. In this paper we used the last one which was successfully applied
previously in many direct seismic problems [15-17].
        The migration (inverse) problem may be written as a problem of the residual functional
minimization [10]. It was formulated mathematically as
                                           1
                                 𝜒(𝒎) = ∑𝑟 ∫‖𝒔(𝒙𝑟 , 𝑡; 𝒎) − 𝒅(𝒙𝑟 , 𝑡)‖2 𝑑𝑡,                            (7)
                                           2
        here 𝒙𝑟 – receiver positions, 𝒔 – synthetic data, 𝒅 – field data.
        With this approach the sequence algorithm for solving the general migration problem in an
arbitrary elastic medium can be proposed as
                                 𝐾𝑍 = 𝐾𝜌 + 𝐾𝜅 + 𝐾𝜇 ,                                                   (8)
                                                      †
                                 𝐾𝜌(𝒙) = 𝜌(𝒙) ∫ 𝒗 (𝒙, −𝑡)𝒗(𝒙, 𝑡) 𝑑𝑡,                                   (9)
                                                            †
                                 𝐾𝜅 (𝒙) = −𝜅(𝒙) ∫[𝛻 ⋅ 𝒔 (𝒙, −𝑡)][𝛻 ⋅ 𝒔(𝒙, 𝑡)] 𝑑𝑡,                     (10)
                                 𝐾𝜇 (𝒙) = −2𝜇(𝒙) ∫ 𝐃† (𝒙, −𝑡): 𝐃(𝒙, 𝑡) 𝑑𝑡,                            (11)
                                           ∇𝒔+(∇𝒔) 𝑇       ∇⋅𝒔
                                 𝐃=           − 𝐈, 𝐃† : 𝐃 = ∑𝑖,𝑗 𝐷𝑖𝑗† 𝐷𝑖𝑗 ,                              (12)
                                           2     3
                                 𝒇† (𝒙, 𝑡) = ∑𝑟[𝒔(𝒙𝑟 , −𝑡) − 𝒅(𝒙, −𝑡)]𝛿(𝒙 − 𝒙𝑟 ),                        (13)

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Proceedings of the VIII International Conference "Distributed Computing and Grid-technologies in Science and
             Education" (GRID 2018), Dubna, Moscow region, Russia, September 10 - 14, 2018



         here 𝒔 and 𝒗 - displacement and velocity vectors and the sign † means the complex
conjugation operation.
         This algorithm was used in this work to estimate the possibility of the single geological crack
identification on the elastic migration image. The homogeneous medium 8 km x 3 km with 𝐶𝑃 =
4000 𝑚/𝑠, 𝐶𝑆 = 2300 𝑚/𝑠, 𝜌 = 2500 𝑘𝑔/𝑚3 was used. The fluid-filled crack with the 200 m length
was submerged on the 2000 m depth. The set of numerical experiments was carried out with the
different orientation of the crack. The angle from the horizontal line was varied in wide range from
10 to 80. On the Figure 2 two migration images are presented.




                a




                b
        Figure 2. Migration images for fractured geological medium. Orientations: 10 (a) and 80 (b)
        The analysis of the results of numerical simulations indicates that the visibility degree of the
fracture plane is higher for sub-horizontal cracks and is lower for sub-vertical cracks. Also, all images
have spherical artefacts and their amplitudes decrease with the increase of the angle of the crack.

4. Conclusion
        In this work two different approaches for creating seismic images were considered. As the
base of them the system of the linear elasticity was used. For constant background model the formulas
for Born approximation were written. Advantages of the described algorithm over the acoustic
approximation were shown. To overcome restrictions imposed on the background model the full-wave
approach may be applied. We estimated the possibility of the identification of single geological crack
inside the homogeneous space. The visibility degree of the fracture plane is higher for sub-horizontal
cracks and is lower for sub-vertical cracks. The further direction of the research may be the attempt to
remove obtained artefacts from migration images.


References
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[2] Claerbout J.F. Toward a unified theory of reflector mapping // Geophysics. 1971. V. 36, I. 3. P.
467–481
[3] French W.S. Computer migration of oblique seismic reflection profiles // Geophysics. 1975. V. 40,
I. 6. P. 961–980

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             Education" (GRID 2018), Dubna, Moscow region, Russia, September 10 - 14, 2018



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1206-1215




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