Reverse Time Migration of Prism Waves for Salt Flank Delineation

Wei Dai, WesternGecoGerard T. Schuster, King Abdullah University

of Science and Technology

Sep 25, 2013

Outline• Introduction and motivation

• Theory

• Numerical Results

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

Time (s) 20

Horizontal Reflector Embedded in the Velocity0

Z (k

m)

3

0 X (km) 6

0Z

(km

)3

Conventional RTM Image

Reverse Time Migration Formula

𝒎𝒎𝒊𝒈(𝒙)=∑𝝎𝝎𝟐𝑾 ∗(𝝎 )𝑮∗ (𝒙|𝒔 )𝑮∗ (𝒙|𝒈 )𝒅 (𝒈|𝒔 )

Angular Freq. Source SpectrumGreen’s functions

Input Data

0Z

(km

)3

𝒙

𝑮 (𝒙|𝒔 )=𝑮𝒐 (𝒙|𝒔 )+𝑮𝟏(𝒙∨𝒔)𝑮 (𝒙|𝒈 )=𝑮𝒐 (𝒙|𝒈 )+𝑮𝟏(𝒙∨𝒈 )

+ 𝑮𝒐∗ (𝒙|𝒔 )𝑮𝒐

∗ (𝒙|𝒈 ) 𝒅𝟐 (𝒈|𝒔 )

+ +

+ + + Other terms.]

0 X (km) 6

0Z

(km

)3

Ellipses

Rabbit Ears

Prism Wave Kernels

𝒎𝒎𝒊𝒈=∑𝝎𝝎𝟐𝑾 ∗ (𝝎 )𝑮𝟏

∗ (𝒙|𝒔 )𝑮𝒐∗ (𝒙|𝒈 )𝒅𝟐 (𝒈|𝒔 )

𝑮𝟏❑ (𝒙|𝒔 )=∫𝝎𝟐𝒎 (𝒙 ′)𝑮𝒐 (𝒙 ′|𝒔 )𝑮𝒐 ( 𝒙′|𝒙 )𝒅𝒙 ′

Born Modeling

0Z

(km

)3

0 X (km) 6

Migration of Prism Waves

Migration of Prism Waves0

Z (k

m)

30

Z (k

m)

3 0 X (km) 6

𝑥𝑠𝑥𝑔

𝑥1

𝑥2

|𝑥𝑠−𝑥2 ′|𝑐 +

|𝑥2−𝑥𝑔|𝑐 =𝜏𝑠𝑔

𝑥2❑ ′

𝒎𝒎𝒊𝒈=∑𝝎𝝎𝟐𝑾 ∗ (𝝎 )𝑮𝒐

∗ (𝒙|𝒔 ) 𝑮𝟏∗ (𝒙|𝒈 )𝒅𝟐 (𝒈|𝒔 )

𝑮𝟏❑ (𝒙|𝒈 )=∫𝝎𝟐𝒎 ( 𝒙′ )𝑮𝒐 (𝒙 ′|𝒔 )𝑮𝒐 (𝒙 ′|𝒙 )𝒅𝒙 ′

Born Modeling

0Z

(km

)3

0 X (km) 6

Migration of Prism Waves

Migration of Prism Waves0

Z (k

m)

30

Z (k

m)

3 0 X (km) 6

Outline• Introduction and motivation

• Theory

• Numerical Results

L model

Salt model

• Summary

0Z

(km

)3 0 X (km) 6

The L Model• Model size: 301 x 601 • Source freq: 20 hz• shots: 32 • geophones: 601

0Ti

me

(s)

6.4

0 X (km) 6

A Shot Gather of the L Model

0Z

(km

)3

0 X (km) 6

0Z

(km

)3

Prism Wavepath

0Z

(km

)3

0 X (km) 6

0Z

(km

)3

Migration Image of Prism Waves

RTM Image /w Smooth Velocity

The Salt Model

• Model size: 601 x 601

• Source freq: 20 hz

• shots: 601

• geophones: 601

0 X (km) 6

0Z

(km

)6

0Ti

me

(s)

10

0 X (km) 6

A Shot Gather of the Salt Model

0 X (km) 6

0Z

(km

)6

0 X (km) 6

Migration Velocity

RTM with Smooth VelocityRTM Image

RTM Image

0 X (km) 6

0Z

(km

)6

0 X (km) 6

Migration Velocity

If the Horizontal Reflectors are embedded in the velocity

0 X (km) 60 X (km) 6

0Z

(km

)6

Migration Velocity

New Method

RTM Image

0Z

(km

)6

Prism Wave ImageConv. RTM Image

0 X (km) 6

0Z

(km

)6

Migration Velocity

New Method

Filtered RTM Image

0Z

(km

)6 0 X (km) 6

Prism Wave Image

Final Image

0 X (km) 6

Final Image

0 X (km) 6

0Z

(km

)6

Horizontal

Vertical

0Z

(km

)6

RTM Image

Summary• I propose a new method to migrate prism waves

separately.

Limitations• Computational cost is doubled.

Avoid the modification of migration velocity.

Reduce cross interference between different waves by

migrating different waves in separated steps.

Thanks