Frequency-domain finite-difference amplitude-preserving migration

Abstract

S U M M A R YA migration algorithm based on the least-squares formulation will find the correct reflector amplitudes if proper migration weights are applied. The migration weights can be viewed as a pre-conditioner for a gradient-based optimization problem. The pre-conditioner should approximate the pseudo-inverse of the Hessian of the least-squares functional. Usually, an infinite receiver coverage is assumed to derive this approximation, but this may lead to poor amplitude estimates for deep reflectors.To avoid the assumption of infinite coverage, new amplitude-preserving migration weights are proposed based on a Born approximation of the Hessian. The expressions are tested in the context of frequency-domain finite-difference two-way migration and show improved amplitudes for the deeper reflectors.Migration algorithms produce the depth locations and relative amplitude behaviour of reflectors in the Earth from measured seismic data. The classic imaging principle (Claerbout 1971; Esmersoy 1986) for shot-based migration states that reflectors are located where the forwardpropagated wavefield from the source correlates with the backward-propagated wavefield of the receiver data. To find a suitable approximation for the reflector amplitudes, the correlation is divided by the square of the incident field. This approach leads, among others, to the classic reversed-time migration algorithm (Baysal et al. 1983).In the 1980s it was observed that the imaging principle can be obtained by formulating migration as an inverse problem based on a least-squares functional (Lailly 1983; Tarantola 1984;Beylkin 1985). This opened up research on true-amplitude or amplitude-preserving migration. Beylkin (1985) derived an amplitude-preserving migration algorithm by viewing migration as an inverse Radon transform under the assumption of infinite and continuous receiver coverage using high-frequency asymptotes. Lailly (1983) and Tarantola (1984) started with a least-squares error functional that measures the difference between observed and synthetic data. They found that the gradient of this functional with respect to the underlying model corresponds to a migration image. This approach has the advantage that it can be used independently of the numerical scheme that models the synthetic data and independently of the acquisition geometry. In this setting, amplitude-preserving migration is obtained by scaling the gradient by the pseudo-inverse of the Hessian of the least-squares functional. The least-squares approach differs from the imaging principle approach or Beylkin's approach because it can be based on the full data set, resulting in migration weights that are different for multishot multi-offset data. Also, irregular acquisition geometries can be taken into account.The main difficulty with the least-squares approach is the size of the Hessian, the matrix of the second derivatives of the error functional with respect to the model parameters, which is too large to be directly used in practical applications because an approximation of...

Coupling Modeling and Migration for Seismic Imaging

“…These results are consistent with those obtained by Ren et al (2011). From these illustrations, it appears that a good approximation of the Hessian, as for example proposed by Plessix & Mulder (2004), should take into account …”

Coupling Modeling and Migration for Seismic Imaging

“…These results are consistent with those obtained by Ren et al (2011). From these illustrations, it appears that a good approximation of the Hessian, as for example proposed by Plessix & Mulder (2004), should take into account …”

Frequency domain wave equation based angular Hessian for amplitude correction

“…From the compensation result shown in Figure 2, we can see the subsalt areas have higher resolution and the faults in the subsalt area are better imaged. This result can be compared to the previous works (Ren et al, 2009a, Plessix andMulder 2004).…”

An efficient illumination compensation method for reverse time migration

“…Thus, the whole illumination compensation operator (Equation 13) can be efficiently implemented during wavefield extrapolation. It is interesting to observe that our proposed illumination compensation operator shares similar forms with the classical Type 3 migration weights as proposed by Plessix and Mulder (2004). The small differences are the specific inverse trigonometric functions for receiver side's operators and an additional cos 𝛽 s term for source side's operators.…”

Coupling Modeling and Migration for Seismic Imaging

“…These results are consistent with those obtained by Ren et al (2011). From these illustrations, it appears that a good approximation of the Hessian, as for example proposed by Plessix & Mulder (2004), should take into account …”

Frequency domain wave equation based angular Hessian for amplitude correction

“…From the compensation result shown in Figure 2, we can see the subsalt areas have higher resolution and the faults in the subsalt area are better imaged. This result can be compared to the previous works (Ren et al, 2009a, Plessix andMulder 2004).…”

An efficient illumination compensation method for reverse time migration

“…Thus, the whole illumination compensation operator (Equation 13) can be efficiently implemented during wavefield extrapolation. It is interesting to observe that our proposed illumination compensation operator shares similar forms with the classical Type 3 migration weights as proposed by Plessix and Mulder (2004). The small differences are the specific inverse trigonometric functions for receiver side's operators and an additional cos 𝛽 s term for source side's operators.…”

Coupling Modeling and Migration for Seismic Imaging

“…These results are consistent with those obtained by Ren et al (2011). From these illustrations, it appears that a good approximation of the Hessian, as for example proposed by Plessix & Mulder (2004), should take into account …”

Frequency domain wave equation based angular Hessian for amplitude correction

“…From the compensation result shown in Figure 2, we can see the subsalt areas have higher resolution and the faults in the subsalt area are better imaged. This result can be compared to the previous works (Ren et al, 2009a, Plessix andMulder 2004).…”

An efficient illumination compensation method for reverse time migration

“…Thus, the whole illumination compensation operator (Equation 13) can be efficiently implemented during wavefield extrapolation. It is interesting to observe that our proposed illumination compensation operator shares similar forms with the classical Type 3 migration weights as proposed by Plessix and Mulder (2004). The small differences are the specific inverse trigonometric functions for receiver side's operators and an additional cos 𝛽 s term for source side's operators.…”