Kinematics of 2 D specularlyreflected and diffracted multiples

Kinematics of 2 D specularly-reflected and diffracted multiples in data space and image space Gabriel Alvarez Stanford University 1

Goal Understand how specularly-reflected and diffracted 2 D multiples map to subsurface image gathers when migrated with wave equation migration. 2

The Problem Moveout-based multiple attenuation algorithms in image space benefit from the power of migration to handle the complex wave propagation of the primaries. The question remains: What is the moveout of the multiples in image space? 3

Outline • Moveout of 2 D diffracted and specularly-reflected multiples in data space • Mapping of multiples from CMPs to Subsurface Offset Domain Common Image Gathers (SODCIGs) • Mapping of multiples from SODCIGs to Angle Domain Common Image Gathers (ADCIGs) • Discussion and conclusions 4

Surface vs. Subsurface Offset h. D m. D h. D 5

Data Space ξ SODCIGs Image Space m hξ Depth CMPs m h. D Depth Time m D ξ Data Space vs. Image Space ADCIGs Image Space 6

Moveout of speculary-reflected multiples Flat water-bottom: h. D V ts 1 m. D ts 2 h. D tr 2 tr 1 ZD ZD 7

Moveout of speculary-reflected multiples Dipping water-bottom: h. D m. D h. D V ZD φ 2φ 8

Moveout of Diffracted Multiples Flat water-bottom: V h. D ts 1 ts 2 m. D h. D tr 2 tr 1 ZD Xd ts 2 tr 2 ZD 9

Moveout of Diffracted Multiple Dipping water-bottom: h. D m. D h. D ZD V Xd φ 2φ αs: takeoff angle of the source ray 10

Moveout Comparison Dipping water-bottom 11

Moveout of Multiples in Subsurface Offset Domain Common Image Gathers (SODCIGs) 12

Image Coordinates of Non-diffracted Multiple Flat water-bottom: h. D V 1 V 2=ρV 1 m. D h. D αs αr βr βs hξ mξ hξ 13

SODCIG Half-subsurface offset (m) 1000 -400 -200 0 200 400 Depth (m) 1200 1400 Specularly-reflected multiple. Flat water-bottom 14

SODCIG Half-subsurface offset (m) 1000 -400 -200 0 200 400 Depth (m) 1200 1400 Specularly-reflected multiple. Flat water-bottom 15

Image Coordinates of Non-diffracted Multiple h. D m. D ts 1 tr 1 αs+φ βs+φ αr-φ ~ ts 2 ~t r 2 hξ αr βr-φ V 1 φ mξ hξ ~ V 2 ~ ~ 16

Constant subsurface-offset section Horizontal position (m) 1200 800 1400 1600 1800 2000 Depth (m) 1200 1600 Specularly-reflected multiple from a dipping water-bottom 17

SODCIG Half-subsurface offset (m) 600 -800 -400 0 400 800 Depth (m) 1000 1400 Specularly-reflected multiple from a dipping water-bottom 18

Image Coordinates of Diffracted Multiple h. D V 1 m. D h. D ts 1 tr 1 αr αs βr βs V 2 ~ tr 2 Zdiff Xdiff ~t s 2 mξ hξ ~ ~ ~ ~ 19

Constant subsurface-offset sections Half-subsurface offset 0 m Half-subsurface offset -200 m Horizontal position (m) 2000 800 1600 2800 3000 2000 800 Depth (m) 1200 2400 2600 2800 3000 1200 1600 Diffracted multiple from a flat water-bottom 20

SODCIGs Half-subsurface offset (m) -400 1000 0 400 Depth (m) 1200 1400 1600 Half-subsurface offset (m) -400 1000 1400 1600 400 -400 1000 Depth (m) 1200 0 Half-subsurface offset (m) 0 400 1200 1400 1600 Diffracted multiple from a flat water-bottom 21

Image Coordinates of Diffracted Multiple h. D V 1 ts 1 m. D h. D αr αs+φ Zdiff tr 1 V 2 βs+φ αr-φ ~ ts 2 ~t r 2 hξ ~ βr-φ mξ hξ φ ~ ~ 22

Constant subsurface offset sections Half-subsurface offset 0 m Half-subsurface offset -200 m Horizontal position (m) 1600 1200 1600 1800 2000 2200 2400 1600 1200 Depth (m) 1400 1800 2000 2200 2400 1600 1800 Diffracted multiple from a flat water-bottom 23

SODCIGs Half-subsurface offset (m) -800 1000 0 800 Depth (m) 1200 1400 1600 1800 Half-subsurface offset (m) -800 1000 1400 800 -800 1000 Depth (m) 1200 0 Half-subsurface offset (m) 0 800 1200 1400 1600 1800 Diffracted multiple from a dipping water-bottom 24

Moveout of Multiples in Angle-Domain Common-Image-Gathers (ADCIGs) 25

ADCIG for specularly-reflected multiple βr βs ~ tr 2 (xrξ, zrξ) hξ 2γ mξ ~ ts 2 hξ (xsξ, zsξ) (xγξ, zγξ) 26

ADCIG Half-aperture angle (degrees) 1200 0 10 20 30 40 Depth (m) 1400 1600 Specularly-reflected multiple from a flat water-bottom 27

ADDCIG Half-aperture angle (degrees) 1000 -40 -20 0 20 40 Depth (m) 1400 1800 Specularly-reflected multiple from a dipping water-bottom 28

ADCIGs Half-aperture angle (degrees) -40 1200 0 40 Depth (m) 1400 1600 Half-aperture angle (degrees) -40 1200 1600 40 -40 1200 0 40 Diffracted multiple from a flat water-bottom 29 Depth (m) 1400 0 Half-aperture angle (degrees) 1400 1600

ADCIGs Half-aperture angle (degrees) 1400 -40 0 40 Depth (m) 1600 2000 Half-aperture angle (degrees) 1400 2000 0 40 1400 Depth (m) 1600 -40 Half-aperture angle (degrees) -40 0 40 1600 2000 Diffracted multiple from a dipping water-bottom 30

Discussion and Conclusions 31

Discussion Water-bottom, specularly-reflected multiples, migrated with sediment velocity migrate to negative subsurface offsets. h. D m. D V h. D ZD ZD 2 hξ<0 32

Discussion On the other hand, primaries, migrated with slower velocities map to positive subsurface offsets. h. D m. D V h. D ZD 2 hξ>0 ZD 33

Discussion Diffracted multiples may map to positive subsurface offsets in SODCIGs even if migrated with faster velocities. h. D V m. D h. D ZD m. D h. D V ZD ZD 2 hξ<0 2 hξ>0 34

Conclusions Specularly-reflected water-bottom multiples migrate as primaries at twice the water depth and with twice the dip. Diffracted multiples do not migrate like primaries but their moveout in both SODCIGs and ADCIGs can be computed if the location of the diffractor is known. 35

Conclusions Better understanding of the moveout of the multiples in SODCIGs and ADCIGs will help in designing more accurate Radon transforms to attenuate the multiples in image space. 36

Thank you for your attention. I will be happy to entertain your questions. 37

From SODCIGs to ADCIGs βs βr 2γ (xrξ, zrξ) A E hξ mξ B hξ (xsξ, zsξ) C F (mξγ, zξγ) D 38