Least-squares Reverse Time Migration with Frequency-selection

Encoding for Marine Data

Wei Dai, WesternGecoYunsong Huang and Gerard T. Schuster,

King Abdulla University of Science and Technology

Sep 26, 2013

Outline• Introduction and Overview

• Theory

Single frequency modeling

Least-squares migration

• Numerical Results

Marmousi2

Marine field data

• Summary

• Random encoding is not applicable to marine streamer data.

Fixed spread geometry (synthetic)

6 traces

Marine streamer geometry (observed)

4 traces

Mismatch between acquisition geometries will dominate the misfit.

Motivation of Freq. Select. Encoding

4

observeddata

simulateddata

misfit = erroneous

misfit

Marine Data

5

Solution• Every source is encoded with a unique

signature.

observed simulated

• Every receiver acknowledge the contribution from the ‘correct’ sources.

4 shots/group

R(w)

Group 1

Nw frequency bands of source spectrum:

Frequency Selection

2 km

wAccommodate up to Nw shots

Outline• Introduction and Overview

• Theory

Single frequency modeling

Least-squares migration

• Numerical Results

Marmousi2

Marine field data

• Summary

Single Frequency Modeling

(𝜵𝟐+𝝎𝟐

𝒗𝟐 )~𝑷=−𝐖 (𝝎 )𝛅(𝒙 −𝒔)

Helmholtz Equation

(𝜵𝟐− 𝟏𝒗𝟐

𝝏𝟐𝝏𝟐 𝒕 )𝐏=−𝐑𝐞 {𝐖 (𝝎 )𝐞𝐱𝐩 (−𝒊𝝎𝒕 )}𝛅(𝒙−𝒔)

Acoustic Wave Equation

• Advantages: Lower complexity in 3D case. Applicable with multisource technique.

Harmonic wave source

Single Frequency Modeling

(𝜵𝟐− 𝟏𝒗𝟐

𝝏𝟐𝝏𝟐 𝒕 )𝐏=−𝐑𝐞 {𝐖 (𝝎 )𝐞𝐱𝐩 (−𝒊𝝎𝒕 )}𝛅(𝒙−𝒔)

Am

plitu

de

T T

Single Frequency ModelingA

mpl

itude

0 Frequency (Hz) 50

Am

plitu

de

20 Frequency (Hz) 30

11

Single Frequency Modeling

Where do the savings come from?

T

T

∆ 𝑓 = 1𝑇

Frequency sampling rate:

Smaller T larger less samples in frequency

domain

Estimated Frequency Sampling

𝑇𝑚𝑖𝑛

𝑇𝑚𝑎𝑥

∆ 𝑓 𝑚𝑎𝑥=1

𝑇𝑚𝑎𝑥−𝑇𝑚𝑖𝑛

(Mulder and Plessix, 2004)

Theory: Least-squares Migration𝒇 (𝒎 )=𝟏

𝟐‖𝑳𝒎−~𝒅‖𝟐

• Misfit:

~𝒅• Encoded Supergather:

only contains one frequency component for each shot, with

frequency-selection encoding.• Encoding functions are changed at every iteration.

Misfit function is redefined at every iteration.

• N frequency components N iterations.

𝑑𝑖𝑡 ,𝑖𝑔 ,𝑖𝑠 𝑑𝑖 𝜔 ,𝑖𝑔 ,𝑖𝑠𝑁 𝑑𝑖 𝜔𝑠 , 𝑖𝑔

Outline• Introduction and Overview

• Theory

Single frequency modeling

Least-squares migration

• Numerical Results

Marmousi2

Marine field data

• Summary

Marmousi2

0 X (km) 8

0Z

(km

)3.

5

4.5

1.5

km/s

• Model size: 8 x 3.5 km• Shots: 301

• Cable: 2km

• Receivers: 201

• Freq.: 400 (0~50 hz)

Marmousi2• Trace length: 8 sec = 0.125 Hz

∆ 𝑓 𝑚𝑎𝑥=1

𝑇𝑚𝑎𝑥−𝑇𝑚𝑖𝑛≈0.7𝐻𝑧

• According to the formula:

For the frequency bank 0~50 Hz, there are 400

frequency channels accommodate up to 400 shots

• We choose = 0.625 Hz. Each shot contains 80 frequency

components 80 iterations needed.

• At the first iteration, all 400 shots are randomly assigned with a

unique frequency . For next iteration,

0 X (km) 8

0Z

(km

)3.

5

0 X (km) 8

Z (k

m)

3.5

Conventional RTM0

LSRTM Image (iteration=1)LSRTM Image (iteration=20)LSRTM Image (iteration=80) Cost: 3.2

Frequency-selection LSRTM of 2D Marine Field Data

0 X (km) 18.7

0Z

(km

)2.

5

2.1

1.5

km/s

• Model size: 18.7 x 2.5 km • Freq: 625 (0-62.5 Hz) • Shots: 496 • Cable: 6km• Receivers: 480

Marine Field Data• Trace length: 10 sec = 0.1 Hz

∆ 𝑓 𝑚𝑎𝑥=1

𝑇𝑚𝑎𝑥−𝑇𝑚𝑖𝑛≈1𝐻𝑧

• According to the formula:

For the frequency bank 0~62.5 Hz, there are 625

frequency channels accommodate up to 625 shots

• Empirical tests suggest = 0.3 Hz 208 iterations.

One possible reason is the large shot spacing 37.5 m.

Conventional RTM

Frequency-selection LSRTM

Z (k

m)

2.5

0Z

(km

)2.

50

0 X (km) 18.7

Cost: 5

Freq. Select LSRTM

Conventional RTM Conventional RTM

Freq. Select LSRTM

Zoom Views

Summary• MLSM can produce high quality images efficiently.

LSM produces high quality image.

Frequency-selection encoding applicable to marine

data.

• Limitation:

High frequency noises are present. Sensitive to velocity error (5% errors in velocity led to

failure).

Thanks