Least Squares Natural Migration (LSNM) of Surface Waves

Zhaolun Liu2015.12.9

Outline

• Introduction

• Theory

•Numerical tests

• Conclusion

•Next works

Outline

• Introduction

• Theory

•Numerical tests

• Conclusion

•Next works

Introduction

• suffers migration artifacts due to the recording geometry and the non-uniform illumination (AlTheyab et.al, 2015).

• Least squares method to solve Lm=d

, where

𝐦=(𝐋†𝐋 )− 1𝐋†𝐝

Outline

• Introduction

• Theory

•Numerical tests

• Conclusion

•Next works

Theory

• Iterative method: • Inverse of Hessian method

1. SVD• • pinv, where is pseudoinverse of W • =pinvH

2. CG method to solve

1st : modeling

3rd : migration

2nd : modeling

1 migration

Outline

• Introduction

• Theory

•Numerical tests

• Conclusion

•Next works

Numerical tests

• 301 receivers • 301 sources• 1 m spacing• Dominant fre: 10 Hz

• Pre-processing1. Mute the scattered waves to obtain the 2. Mute the transmitted waves to obtain the

• Natural migration

1. Calculate the kernel:

2. Calculate the migration image for each

shot:

3. Stack the image of all shots:

• Buried corners locations: • x1=32 m, x2=46 m

• x3=122 m, x4=151 m • x5=227 m, x6=271 m• All depth is 10 m

x1 x2x3

x4 x6x5

• Six corners model

Nature Migration image

• Problem 2: the weak energy at the middle.

• Problem 1: the polarity for the fault is different.

Analysis for two-corner model

• Buried corners locations: • x1=122 m, x2=151 m

x1 x2

Analysis for two-corner model

Polarity analysis

• One corner model

• Two source-receiver pairs:

𝑥𝑠1𝑥𝑔1 𝑥𝑠2𝑥𝑔2

• Calculate the kernel𝑥𝑠1 𝑥𝑔1 𝑥𝑟

𝐺𝑡𝑟𝑎𝑛𝑠 (𝑥𝑟∨𝑥𝑠1 ,𝑡)𝐺𝑡𝑟𝑎𝑛𝑠 (𝑥𝑟∨𝑥𝑔1 , 𝑡)

∗

L(𝑥𝑟 , 𝑡)¿

,

, • Buried corners locations:=122 m

Polarity analysis

• Obtain the data (only backscattered waves)

𝑚 (𝑥𝑟 )=∫L (𝑥𝑔1|𝑥𝑟|𝑥𝑠1 , 𝑡 )𝐝 (𝑥𝑔1 ¿𝑥 𝑠1 ,𝒕 )𝒅𝒕• Calculate the image

∙

∙

¿

¿

𝑥𝑠1 𝑥𝑔1 𝑥𝑟𝐝 (𝑥𝑔1¿ 𝑥𝑠1 ,𝒕 )  

Velocity discontinuity polarity change?

Energy analysis

• For one source and all the receivers𝑥𝑟 1

𝑥𝑟

𝑥𝑟 1

𝑥𝑠1 𝑥𝑔1 𝑥𝑟

𝐝 (𝑥𝑔1¿ 𝑥𝑠1 ,𝒕)  

Migration image of six-corner model

• Change polarity at one side of the source• Then stack the images of all the shots Still too weak at the middle, and how to improve?

The inverse of Hessian

The energy is balanced!

• Six-corner model

The inverse of Hessian

• The stacked images with different number of singular values (SV)

The inverse of Hessian

• One corner model

The inverse of Hessian

• The stacked images with different number of SVs

Outline

• Introduction

• Theory

•Numerical tests

• Conclusion

•Next works

Conclusion

• For a one-corner fault at near surface, the polarities of the corresponding image of NM with source-receiver at two sides of the fault are opposite. • The energy of one-shot NM images for different buried one-corner

faults at the same depth depends on the distance between the position of source and fault.• The inverse of Hessian can balance the energy of NM image to some

extent.

Outline

• Introduction

• Theory

•Numerical tests

• Conclusion

•Next works

Next works

• Apply LSNM to the Aqaba seismic data.

• Apply LSNM to 3D problem and analyze the polarity and image energy.

• Applying this method to USArray data.

(Han Yu et.al, 2014)