Reverse Time Migration of Prism Waves for Salt Flank Delineation Wei Dai, WesternGeco Gerard T. Schuster, King Abdullah University of Science and Technology Sep 25, 2013

Outline Introduction and motivation Theory Numerical Results Summary L model Salt model Summary

Introduction Problem: Vertical boundaries (salt flanks) are difficult to image because they are usually not illuminated by primary reflections. Solution: Prism waves contain valuable information.

Conventional Method When the known boundaries are embedded in the velocity model, conventional RTM can migrate prism waves correctly.

Recorded Trace 𝒅 𝒈 𝒔 = 𝒅 𝟏 𝒈 𝒔 + 𝒅 𝟐 (𝒈|𝒔) Time (s) 2

Horizontal Reflector Embedded in the Velocity Z (km) 3 X (km) 6 Z (km) 3 Conventional RTM Image

𝒎 𝒎𝒊𝒈 (𝒙)= 𝝎 𝝎 𝟐 𝑾 ∗ (𝝎)𝑮 ∗ 𝒙 𝒔 𝑮 ∗ 𝒙 𝒈 𝒅 𝒈 𝒔 Reverse Time Migration Formula 𝒎 𝒎𝒊𝒈 (𝒙)= 𝝎 𝝎 𝟐 𝑾 ∗ (𝝎)𝑮 ∗ 𝒙 𝒔 𝑮 ∗ 𝒙 𝒈 𝒅 𝒈 𝒔 Angular Freq. Source Spectrum Green’s functions Input Data Z (km) 3 𝒙 𝑮 𝒙 𝒔 = 𝑮 𝒐 𝒙 𝒔 + 𝑮 𝟏 (𝒙|𝒔) 𝑮 𝒙 𝒈 = 𝑮 𝒐 𝒙 𝒈 + 𝑮 𝟏 (𝒙|𝒈)

𝒎 𝒎𝒊𝒈 𝒙 = 𝝎 𝝎 𝟐 𝑾 ∗ 𝝎 𝑮 𝒐 ∗ 𝒙 𝒔 + 𝑮 𝟏 ∗ 𝒙 𝒔 𝑮 𝒐 ∗ 𝒙 𝒈 + 𝑮 𝟏 ∗ 𝒙 𝒈 𝒎 𝒎𝒊𝒈 𝒙 = 𝝎 𝝎 𝟐 𝑾 ∗ 𝝎 𝑮 𝒐 ∗ 𝒙 𝒔 + 𝑮 𝟏 ∗ 𝒙 𝒔 𝑮 𝒐 ∗ 𝒙 𝒈 + 𝑮 𝟏 ∗ 𝒙 𝒈 [ 𝒅 𝟏 𝒈 𝒔 + 𝒅 𝟐 𝒈 𝒔 ] = 𝝎 𝝎 𝟐 𝑾 ∗ 𝝎 [ 𝑮 𝒐 ∗ 𝒙 𝒔 𝑮 𝒐 ∗ 𝒙 𝒈 𝒅 𝟏 𝒈 𝒔 + 𝑮 𝒐 ∗ 𝒙 𝒔 𝑮 𝒐 ∗ 𝒙 𝒈 𝒅 𝟐 𝒈 𝒔 Ellipses + 𝑮 𝟏 ∗ 𝒙 𝒔 𝑮 𝒐 ∗ 𝒙 𝒈 𝒅 𝟏 𝒈 𝒔 + 𝑮 𝒐 ∗ 𝒙 𝒔 𝑮 𝟏 ∗ 𝒙 𝒈 𝒅 𝟏 𝒈 𝒔 Rabbit Ears + 𝑮 𝟏 ∗ 𝒙 𝒔 𝑮 𝒐 ∗ 𝒙 𝒈 𝒅 𝟐 𝒈 𝒔 + 𝑮 𝒐 ∗ 𝒙 𝒔 𝑮 𝟏 ∗ 𝒙 𝒈 𝒅 𝟐 𝒈 𝒔 Prism Wave Kernels + Other terms.] Z (km) 3 X (km) 6

Migration of Prism Waves 𝒎 𝒎𝒊𝒈 = 𝝎 𝝎 𝟐 𝑾 ∗ 𝝎 𝑮 𝟏 ∗ 𝒙 𝒔 𝑮 𝒐 ∗ 𝒙 𝒈 𝒅 𝟐 𝒈 𝒔 𝑮 𝟏 𝒙 𝒔 = 𝝎 𝟐 𝒎 𝒙 ′ 𝑮 𝒐 𝒙 ′ 𝒔 𝑮 𝒐 𝒙 ′ 𝒙 𝒅𝒙′ Born Modeling Z (km) 3 X (km) 6

Migration of Prism Waves 𝑥 𝑔 𝑥 𝑠 𝑥 2 Z (km) 𝑥 1 3 𝑥 2 ′ 𝑥 𝑠 − 𝑥 2 ′ 𝑐 + 𝑥 2 − 𝑥 𝑔 𝑐 = 𝜏 𝑠𝑔 Z (km) 3 X (km) 6

Migration of Prism Waves 𝒎 𝒎𝒊𝒈 = 𝝎 𝝎 𝟐 𝑾 ∗ 𝝎 𝑮 𝒐 ∗ 𝒙 𝒔 𝑮 𝟏 ∗ 𝒙 𝒈 𝒅 𝟐 𝒈 𝒔 𝑮 𝟏 𝒙 𝒈 = 𝝎 𝟐 𝒎 𝒙 ′ 𝑮 𝒐 𝒙 ′ 𝒔 𝑮 𝒐 𝒙 ′ 𝒙 𝒅𝒙′ Born Modeling Z (km) 3 X (km) 6

Migration of Prism Waves Z (km) 3 Z (km) 3 X (km) 6

Outline Introduction and motivation Theory Numerical Results Summary L model Salt model Summary

The L Model Model size: 301 x 601 Source freq: 20 hz shots: 32 geophones: 601 Z (km) 3 X (km) 6

A Shot Gather of the L Model Time (s) 6.4 X (km) 6

Prism Wavepath Z (km) 3 Z (km) 3 X (km) 6

RTM Image /w Smooth Velocity Z (km) 3 X (km) 6 Z (km) 3 Migration Image of Prism Waves

The Salt Model Model size: 601 x 601 Source freq: 20 hz shots: 601 Model size: 601 x 601 Source freq: 20 hz shots: 601 Z (km) geophones: 601 6 X (km) 6

A Shot Gather of the Salt Model Time (s) 10 X (km) 6

RTM with Smooth Velocity Migration Velocity RTM Image Z (km) 6 X (km) 6 X (km) 6

If the Horizontal Reflectors are embedded in the velocity Migration Velocity RTM Image Z (km) 6 X (km) 6 X (km) 6

New Method Migration Velocity Conv. RTM Image Prism Wave Image Z (km) 6 Migration Velocity Conv. RTM Image Prism Wave Image RTM Image Z (km) X (km) 6 6 X (km) 6

New Method Migration Velocity Prism Wave Image Filtered RTM Image Z (km) 6 Migration Velocity X (km) 6 Prism Wave Image Filtered RTM Image Z (km) 6 X (km) 6

Final Image Horizontal Final Image RTM Image Vertical Z (km) 6 Z (km) Z (km) 6 Horizontal Final Image RTM Image Vertical Z (km) X (km) 6 6 X (km) 6

Summary I propose a new method to migrate prism waves separately. Avoid the modification of migration velocity. Reduce cross interference between different waves by migrating different waves in separated steps. Limitations Computational cost is doubled.

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