Research note: derivations of gradients in angle‐independent joint migration inversion

Sun, Yimin; Verschuur, Eric; Qu, Shaobo

doi:10.1111/1365-2478.12738

Cited by 12 publications

(16 citation statements)

References 14 publications

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“…; Sun et al . ). The redefined objective function in the whole solution space is as follows:

\begin{matrix} true \\ right J_{1} ((), z_{m}) & = & left \frac{1}{2} | | Δ p^{+} ((), z_{m}) {false| false|}_{2}^{2}, \\ right J_{2} ((), z_{m}) & = & left \frac{1}{2} | | Δ p^{-} ((), z_{m}) {false| false|}_{2}^{2}, \\ right J_{3} ((), z_{m}) & = & left \frac{1}{2} | | Δ q^{+} ((), z_{m}) {false| false|}_{2}^{2}, \\ right J_{4} ((), z_{m}) & = & left \frac{1}{2} | | Δ q^{-} ((), z_{m}) {false| false|}_{2}^{2}, \\ right J & = & left \sum_{m = 0}^{N_{z} - 1} {J_{1} (z_{m}) + J_{2} (z_{m}) + J_{3} (z_{m}) + J_{4} (z_{m})}, \end{matrix}

where four redatumed residual wavefields are as follows: the down‐going incoming residual wavefield

Δ {boldp}^{+} false(z_{m} false)

, the up‐going incoming residual wavefield

Δ {boldp}^{-} false(z_{m} false)

, the down‐going outgoing residual wavefield

Δ {boldq}^{+} false(z_{m} false)

…”

Section: Theory Of Joint Migration Inversionmentioning

confidence: 97%

“…For one depth level, the reflectivity/transmission operators (

{boldR}^{\cup / \cap} false(z_{m} false)

{boldT}^{\pm} false(z_{m} false)

) are matrices with the dimension of

(N_{x} \times N_{x})

. The wavefields are connected by reflectivity and transmission effects (see also Berkhout ; Davydenko and Verschuur ; Sun, Verschuur and Qu ):

\begin{matrix} true \\ right q^{+} ((), z_{m}) & = & left s^{+} ((), z_{m}) + T^{+} ((), z_{m}) p^{+} ((), z_{m}) + R^{\cup} ((), z_{m}) p^{-} ((), z_{m}), \\ right q^{-} ((), z_{m}) & = & left s^{-} ((), z_{m}) + T^{-} ((), z_{m}) p^{-} ((), z_{m}) + R^{\cap} ((), z_{m}) p^{+} ((), z_{m}) . \end{matrix}

The transmission operator

{boldT}^{\pm} false(z_{m} false)

can be further written as follows:

{boldT}^{\pm} (z_{m}) = I + δ {boldT}^{\pm} (z_{m}),

where

boldI

is a unit matrix sharing the same size as the other operators and

δ {boldT}^{\pm} false(z_{m} false)

is the differential transmission in the down/up direction (Berkhout ).…”

Section: Theory Of Joint Migration Inversionmentioning

confidence: 99%

“…The four wavefields of two consecutive levels are connected by propagation operators (see also Berkhout ; Davydenko and Verschuur ; Sun et al . ):

\begin{matrix} true \\ left p^{+} ((), z_{m}) = boldW (() z_{m}, z_{m - 1}) q^{+} ((), z_{m - 1}), \\ left p^{-} ((), z_{m - 1}) = boldW (() z_{m - 1}, z_{m}) q^{-} ((), z_{m}) . \end{matrix}

We build the propagation operator in the frequency‐wavenumber domain first and then transform it back to the frequency–space domain. Within one layer, the dimension of the propagation operator is

(N_{x} \times N_{x})

.…”

Section: Theory Of Joint Migration Inversionmentioning

confidence: 99%

See 2 more Smart Citations

Simultaneous joint migration inversion for high‐resolution imaging/inversion of time‐lapse seismic datasets

Verschuur

2020

Geophysical Prospecting

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A B S T R A C TThe current time-lapse practice is to exactly repeat well-sampled acquisition geometries to mitigate acquisition effects on the time-lapse differences. In order to relax the rigid requirements on acquisition effects, we propose simultaneous joint migration inversion as an effective time-lapse tool for reservoir monitoring, which combines a joint time-lapse data processing strategy with the joint migration inversion method. Joint migration inversion is a full-wavefield inversion method that explains the measured reflection data using a parameterization in terms of reflectivity and propagation velocity. Both the inversion process inside the imaging/inversion scheme and the extra illumination provided by including multiples in joint migration inversion makes the obtained velocity and reflectivity operator largely independent of the utilized acquisition geometry and, thereby, relaxes the strong requirement of non-repeatability during the monitoring. Because simultaneous joint migration inversion inverts for all datasets simultaneously and utilizes various constraints on the estimated reflectivities and velocity, the obtained time-lapse differences have much higher accuracy compared to inverting each dataset separately. It allows the baseline and monitor parameters to communicate with each other dynamically during inversion via a userdefined spatial weighting operator. In order to get more localized time-lapse velocity differences, we further extend the regular simultaneous joint migration inversion to a robust high-resolution simultaneous joint migration inversion process using the timelapse reflectivity difference as an extra constraint for the velocity estimation during inversion. This constraint makes a link between the reflectivity-and the velocity difference by exploiting the relationship between them. We demonstrate the feasibility of the proposed method with a highly realistic synthetic model based on the Grane field offshore Norway and a time-lapse field dataset from the Troll Field.

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“…; Sun et al . ). The redefined objective function in the whole solution space is as follows:

\begin{matrix} true \\ right J_{1} ((), z_{m}) & = & left \frac{1}{2} | | Δ p^{+} ((), z_{m}) {false| false|}_{2}^{2}, \\ right J_{2} ((), z_{m}) & = & left \frac{1}{2} | | Δ p^{-} ((), z_{m}) {false| false|}_{2}^{2}, \\ right J_{3} ((), z_{m}) & = & left \frac{1}{2} | | Δ q^{+} ((), z_{m}) {false| false|}_{2}^{2}, \\ right J_{4} ((), z_{m}) & = & left \frac{1}{2} | | Δ q^{-} ((), z_{m}) {false| false|}_{2}^{2}, \\ right J & = & left \sum_{m = 0}^{N_{z} - 1} {J_{1} (z_{m}) + J_{2} (z_{m}) + J_{3} (z_{m}) + J_{4} (z_{m})}, \end{matrix}

where four redatumed residual wavefields are as follows: the down‐going incoming residual wavefield

Δ {boldp}^{+} false(z_{m} false)

, the up‐going incoming residual wavefield

Δ {boldp}^{-} false(z_{m} false)

, the down‐going outgoing residual wavefield

Δ {boldq}^{+} false(z_{m} false)

…”

Section: Theory Of Joint Migration Inversionmentioning

confidence: 97%

“…For one depth level, the reflectivity/transmission operators (

{boldR}^{\cup / \cap} false(z_{m} false)

{boldT}^{\pm} false(z_{m} false)

) are matrices with the dimension of

(N_{x} \times N_{x})

. The wavefields are connected by reflectivity and transmission effects (see also Berkhout ; Davydenko and Verschuur ; Sun, Verschuur and Qu ):

\begin{matrix} true \\ right q^{+} ((), z_{m}) & = & left s^{+} ((), z_{m}) + T^{+} ((), z_{m}) p^{+} ((), z_{m}) + R^{\cup} ((), z_{m}) p^{-} ((), z_{m}), \\ right q^{-} ((), z_{m}) & = & left s^{-} ((), z_{m}) + T^{-} ((), z_{m}) p^{-} ((), z_{m}) + R^{\cap} ((), z_{m}) p^{+} ((), z_{m}) . \end{matrix}

The transmission operator

{boldT}^{\pm} false(z_{m} false)

can be further written as follows:

{boldT}^{\pm} (z_{m}) = I + δ {boldT}^{\pm} (z_{m}),

where

boldI

is a unit matrix sharing the same size as the other operators and

δ {boldT}^{\pm} false(z_{m} false)

is the differential transmission in the down/up direction (Berkhout ).…”

Section: Theory Of Joint Migration Inversionmentioning

confidence: 99%

“…The four wavefields of two consecutive levels are connected by propagation operators (see also Berkhout ; Davydenko and Verschuur ; Sun et al . ):

\begin{matrix} true \\ left p^{+} ((), z_{m}) = boldW (() z_{m}, z_{m - 1}) q^{+} ((), z_{m - 1}), \\ left p^{-} ((), z_{m - 1}) = boldW (() z_{m - 1}, z_{m}) q^{-} ((), z_{m}) . \end{matrix}

We build the propagation operator in the frequency‐wavenumber domain first and then transform it back to the frequency–space domain. Within one layer, the dimension of the propagation operator is

(N_{x} \times N_{x})

.…”

Section: Theory Of Joint Migration Inversionmentioning

confidence: 99%

See 1 more Smart Citation

Simultaneous joint migration inversion for high‐resolution imaging/inversion of time‐lapse seismic datasets

Verschuur

2020

Geophysical Prospecting

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“…Starting with the surface-recorded wavefield p − 0 (z 0 ), the backward propagation scheme (Staal, 2015;Sun et al, 2019) reconstructs the wavefields q − 0 (z 1 ) and p − 0 (z 1 ) as follows:…”

Section: N V E R S E P Ro Pag At I O N I N 1 5 -D I M E N S I O Na L a M P L I T U D E -V E R S U S -O F F S E T J O I N T M I G R At I mentioning

confidence: 99%

“…Note the traditional JMI (Berkhout, 2014b;Verschuur et al, 2016;Sun et al, 2019 adopts conditions similar to equations ( 14) and ( 15).…”

Section: 5 -D I M E N S I O Na L a M P L I T U D E -V E R S U S -O F ...mentioning

confidence: 99%

Research note: Inverse propagation in 1.5‐dimensional amplitude‐versus‐offset joint migration inversion

Sun

Verschuur

2021

Geophysical Prospecting

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The state-of-the-art joint migration inversion faces the so-called amplitude-versusoffset challenge, due to adopting over-simplified one-way propagation, reflection and transmission operators to avoid over-parameterization in the inversion process. To overcome this challenge, we apply joint migration inversion to horizontally layered media (or 1.5-dimensional media) and parameterize the solution space via density and velocity models. In this scenario, one-way propagation, reflection and transmission operators required by the joint migration inversion process can be analytically and correctly derived from the subsurface models, so the amplitude-versus-offset challenge is successfully overcome. We introduce a new concept, which is named 'inverse propagation', into our 1.5-dimensional amplitude-versus-offset joint migration inversion. It can correctly reconstruct subsurface wavefields by using a surface-recorded receiver wavefield with all the influence of transmission, reflection and multiples accounted for. A synthetic example is used to demonstrate the correctness of the inverse propagation. This work is the foundation to further develop the 1.5-dimensional amplitude-versus-offset joint migration inversion technology.

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One-way reflection waveform inversion with depth-dependent gradient pre-conditioning

Abolhassani,

Hoogerbrugge,

Verschuur

2024

Geophysical Journal International

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Summary Reflection Waveform Inversion (RWI) is a technique that uses pure reflection data to estimate subsurface background velocity, relying on evolving seismic images. Conventional RWI operates in a cyclic workflow, with two key components in each cycleâmigration and reflection tomography. Conventional RWI may result in suboptimal background velocity estimation, partly due to limited or unresolved resolution within each component in each cycle. While gradient preconditioning with the reciprocal of Hessian information helps resolve this issue in both components of RWI, it becomes impractical for a large number of model parameters. One-way reflection waveform inversion (ORWI) is a reflection waveform inversion technique in which the forward modeling scheme operates in one direction (downward and then upward) via virtual parallel depth levels within the medium. Leveraging the ORWI framework, we decompose and reduce the linear Hessian operator (also known as the approximate Hessian or Gauss-Newton Hessian) into multiple smaller sub-operators. In particular, the diagonal blocks of the mono-frequency approximate Hessian operators, each corresponding to a single depth level within the medium, are extracted and inverted to precondition the corresponding mono-frequency gradients in both the migration and reflection tomography components of ORWI. This depth-dependent gradient preconditioning transforms standard ORWI into a high-resolution, yet computationally feasible version aimed at addressing suboptimal velocity estimation, referred to as high-resolution ORWI (HR-ORWI). The effectiveness of the proposed approach is demonstrated through successful applications to synthetic data examples.

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Research note: derivations of gradients in angle‐independent joint migration inversion

Cited by 12 publications

References 14 publications

Simultaneous joint migration inversion for high‐resolution imaging/inversion of time‐lapse seismic datasets

Simultaneous joint migration inversion for high‐resolution imaging/inversion of time‐lapse seismic datasets

Research note: Inverse propagation in 1.5‐dimensional amplitude‐versus‐offset joint migration inversion

One-way reflection waveform inversion with depth-dependent gradient pre-conditioning

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