General theory of Greenβs theorem wave separation Sβ π π π = π $ + π ) π β ' + π ) π» & ' = π $ π» & P 2 π π ' β π» & ' πΌβ²π 6 π π ) π β, π = 2 ππΌβ²π» & 8ππβ² ; < ' : Causal Greenβs function in the homogeneous whole world π» & 30
Marine experiment Air F.S. Water M.S.(Cable) O.B. Earth F.S. free surface, or air/water boundary O.B. ocean bottom M.S. measurement surface 31
Homogeneous whole-space of water Water 32
π π : Air gun Water π π 33
π π : Air perturbation π π Air F.S. Water π π 34
π π : Earth perturbation π π Air F.S. Water π π O.B. π π Earth 35
Predict a portion of total wave at π β contributed by π π Sβ π π Air F.S. Water π π π β O.B. π π Earth 36
Predict a portion of total wave at π β contributed by π π Sβ π π Air F.S. Water π π π β Up O.B. π π Earth 37
Predict a portion of total wave at π β contributed by π π Sβ π π Air F.S. Down Water π π π β Down Up O.B. π π Earth 38
Predict deghosted data at π β contributed by π π Sβ π π Air F.S. Water π π π β Deghosted O.B. π π Earth 39
Predict deghosted data at π β contributed by π π π π Air F.S. Water π π π β Sβ Deghosted O.B. π π Earth 40
Predict deghosted data at π β contributed by π π π π Air F.S. Water π π π β Sβ M.S.(Cable) Deghosted O.B. π π Earth π >? π ' β π» & ' πΌβ²π 6 π β, π = @ ππΌβ²π» & 8ππβ² A.C. 41
Advantage 1 π >? π ' β π» & ' πΌβ²π 6 π β, π = @ ππΌβ²π» & 8ππβ² A.C. π π Air β’ Can accommodate cable with an arbitrary shape F.S. π π ( Z. Zhang & Weglein, 2016 ) Water π β M.S.(Cable) Deghosted O.B. Earth π π 42
Advantage 1 π >? π ' β π» & ' πΌβ²π 6 π β, π = @ ππΌβ²π» & 8ππβ² A.C. π π Air β’ Can accommodate cable with an arbitrary shape F.S. π π ( Z. Zhang & Weglein, 2016 ) Water π β M.S.(Cable) Deghosted O.B. Earth π π 43
Advantage 1 π >? π ' β π» & ' πΌβ²π 6 π β, π = @ ππΌβ²π» & 8ππβ² A.C. π π Air β’ Can accommodate cable with an arbitrary shape F.S. π π ( Z. Zhang & Weglein, 2016 ) Water π β M.S.(Cable) Deghosted O.B. Earth π π 44
Advantage 1 π >? π ' β π» & ' πΌβ²π 6 π β, π = @ ππΌβ²π» & 8ππβ² A.C. π π Air β’ Can accommodate cable with an arbitrary shape F.S. π π ( Z. Zhang & Weglein, 2016 ) Water π β M.S.(Cable) Deghosted β’ Supersedes π β π E method that assumes cable O.B. Earth π π being horizontal 45
Advantage 2 π >? π ' β π» & ' πΌβ²π 6 π β, π = @ ππΌβ²π» & 8ππβ² A.C. π π Air β’ Needs no information about the βsourcesβ: F.S. π π ΓΌ wavelet, radiation pattern Water ΓΌ air property (shape and reflection at sea surface) π β ΓΌ earth property M.S.(Cable) Deghosted O.B. Earth π π 46
Advantage 2 π >? π ' β π» & ' πΌβ²π 6 π β, π = @ ππΌβ²π» & 8ππβ² A.C. π π Air β’ Needs no information about the βsourcesβ: F.S. π π ΓΌ wavelet, radiation pattern Water ΓΌ air property (shape and reflection at sea surface) π β ΓΌ earth property M.S.(Cable) Deghosted O.B. Earth π π 47
Progress v Deghosting data on a depth variable cable v Wave separation onshore for the removal of ground roll and ghosts v Deghosting data at the ocean bottom 48
ΓΌ Deghosting on a depth variable cable Removal of ground roll & ghost onshore Deghosting ocean bottom data Issue of current Greenβs theorem deghosting method π π Air F.S. π π Water π β M.S.(Cable) O.B. Earth π π 49
ΓΌ Deghosting on a depth variable cable Removal of ground roll & ghost onshore Deghosting ocean bottom data Issue of current Greenβs theorem deghosting method π π Air F.S. π π Water π β M.S.(Cable) O.B. Earth π π 50
ΓΌ Deghosting on a depth variable cable Removal of ground roll & ghost onshore Deghosting ocean bottom data Issue of current Greenβs theorem deghosting method π π Air β’ Can only output deghosted data at a depth F.S. π π above cable ( Weglein et al., 2013 ) Water π β π >? π ' β π» & M.S.(Cable) ' πΌβ²π 6 π β, π = @ ππΌβ²π» & 8ππβ² O.B. A.C. Earth π π 51
ΓΌ Deghosting on a depth variable cable Removal of ground roll & ghost onshore Deghosting ocean bottom data How to deghost the actual acquired data? π π Air β’ π β π E if cable being horizontal F.S. π π Water π β M.S.(Cable) O.B. Earth π π 52
ΓΌ Deghosting on a depth variable cable Removal of ground roll & ghost onshore Deghosting ocean bottom data How to deghost the actual acquired data? π π Air β’ π β π E if cable being horizontal F.S. π π Water β’ A new method without assuming cable being π β horizontal ( J. Wu & Weglein, 2016 ) M.S.(Cable) O.B. Earth π π 53
ΓΌ Deghosting on a depth variable cable Removal of ground roll & ghost onshore Deghosting ocean bottom data The new method π π Air Step 1: F.S. π π Water Deghost data at a depth (at E.S.) above cable πΈ ππ (output) E.S. Use current Greenβs theorem deghosting πΈ , πΈ π (input) M.S.(Cable) ( Weglein 2002, J. Zhang & Weglein 2007 ) O.B. Earth π π E.S. evaluation surface, shallower than cable 54
ΓΌ Deghosting on a depth variable cable Removal of ground roll & ghost onshore Deghosting ocean bottom data The new method π π Air Step 2: F.S. π π Water Predict deghosted data right on the cable πΈ ππ (input) E.S. Use Greenβs theorem one-way wave prediction πΈ ππ (output) M.S.(Cable) ( Weglein et al., 2011a,b, 2016 ) O.B. Earth π π 55
ΓΌ Deghosting on a depth variable cable Removal of ground roll & ghost onshore Deghosting ocean bottom data Numerical test: Model 0m F.S. 10m 40m M.S. Water 300m O.B. Earth Layer Velocity (m/s) Density (kg/m 3 ) Water 1500 1000 Earth 2500 1000 56
ΓΌ Deghosting on a depth variable cable Removal of ground roll & ghost onshore Deghosting ocean bottom data Numerical test: Model 0m F.S. 10m E.S. 25m 40m M.S. Water 300m O.B. Earth Layer Velocity (m/s) Density (kg/m 3 ) Water 1500 1000 Earth 2500 1000 57
ΓΌ Deghosting on a depth variable cable Removal of ground roll & ghost onshore Deghosting ocean bottom data E.S. Input total π at cable cable Offset/m -1000 -500 0 500 1000 0 0.2 1.0 0.5 Time/s 0.4 0 -0.5 -1.0 0.6 x105 0.8 58
ΓΌ Deghosting on a depth variable cable Removal of ground roll & ghost onshore Deghosting ocean bottom data E.S. Input total π at cable cable Offset/m -1000 -500 0 500 1000 0 0.2 1.0 0.5 Primary Time/s 0.4 0 -0.5 -1.0 0.6 x105 0.8 59
ΓΌ Deghosting on a depth variable cable Removal of ground roll & ghost onshore Deghosting ocean bottom data E.S. Input total π at cable cable Offset/m -1000 -500 0 500 1000 0 Direct 0.2 1.0 0.5 Primary Time/s 0.4 0 -0.5 -1.0 0.6 Receiver-ghost x105 0.8 60
ΓΌ Deghosting on a depth variable cable Removal of ground roll & ghost onshore Deghosting ocean bottom data E.S. Deghosted π at E.S. (Step 1) cable Offset/m -1000 -500 0 500 1000 0 0.2 1.0 0.5 Primary Time/s 0.4 0 -0.5 -1.0 0.6 x105 0.8 61
ΓΌ Deghosting on a depth variable cable Removal of ground roll & ghost onshore Deghosting ocean bottom data E.S. Input total π at cable cable Offset/m -1000 -500 0 500 1000 0 Direct 0.2 1.0 0.5 Time/s 0.4 0 -0.5 -1.0 0.6 Receiver-ghost x105 0.8 62
ΓΌ Deghosting on a depth variable cable Removal of ground roll & ghost onshore Deghosting ocean bottom data E.S. Deghosted π at E.S. (Step 1) cable Offset/m -1000 -500 0 500 1000 0 Direct 0.2 1.0 0.5 Time/s 0.4 0 -0.5 -1.0 0.6 Receiver-ghost x105 0.8 63
ΓΌ Deghosting on a depth variable cable Removal of ground roll & ghost onshore Deghosting ocean bottom data E.S. Input total π at cable cable Offset/m -1000 -500 0 500 1000 0 0.2 1.0 0.5 Primary Time/s 0.4 0 -0.5 -1.0 0.6 x105 0.8 64
ΓΌ Deghosting on a depth variable cable Removal of ground roll & ghost onshore Deghosting ocean bottom data E.S. Deghosted π at E.S. (Step 1) cable Offset/m -1000 -500 0 500 1000 0 0.2 1.0 0.5 Primary Time/s 0.4 0 -0.5 -1.0 0.6 x105 0.8 65
ΓΌ Deghosting on a depth variable cable Removal of ground roll & ghost onshore Deghosting ocean bottom data E.S. Deghosted π at cable (Step 2) cable Offset/m -1000 -500 0 500 1000 0 0.2 1.0 0.5 Primary Time/s 0.4 0 -0.5 -1.0 0.6 x105 0.8 66
ΓΌ Deghosting on a depth variable cable Removal of ground roll & ghost onshore Deghosting ocean bottom data E.S. Input total π on cable cable Offset/m -1000 -500 0 500 1000 0 Direct 0.2 1.0 0.5 Primary Time/s 0.4 0 -0.5 -1.0 0.6 Receiver-ghost x105 0.8 67
ΓΌ Deghosting on a depth variable cable Removal of ground roll & ghost onshore Deghosting ocean bottom data E.S. Deghosted π at cable (Step 2) cable Offset/m -1000 -500 0 500 1000 0 0.2 1.0 0.5 Primary Time/s 0.4 0 -0.5 -1.0 0.6 x105 0.8 68
ΓΌ Deghosting on a depth variable cable Removal of ground roll & ghost onshore Deghosting ocean bottom data Summary Step 1 Step 2 One-way wave prediction Current deghosting method output deghosted data on cable output deghosted data above cable v Provide a two-step new method to deghost actual data on the cable, and without limitation of cableβs shape. 69
Progress v Deghosting data on a depth variable cable v Wave separation onshore for the removal of ground roll and ghosts v Deghosting data at the ocean bottom 70
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data Ground roll (Rayleigh wave) ( Boustani et al., Journal of Geophysics and Engineering, 2013 ) 71
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data Deliverable ( J. Wu & A. Weglein, SEG 2016 ) v A new and simplified Greenβs theorem wave separation algorithm ΓΌ for onshore ground roll and ghost removal ΓΌ with a reduced data requirement ΓΌ retains effectiveness of original algorithm ( J. Wu & A. Weglein, SEG 2015 ) 72
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data Marine -- Wave separation ' β π» & ' π K π ππβ² π & = @ ππ K π» & A.C. ( Weglein & Secrest 1990 ) Air Water π & : Reference wave Earth 73
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data Marine -- Data requirement ' β π» & ' π K π ππβ² π & = @ ππ K π» & A.C. ( Weglein & Secrest 1990 ) Air Water π & : Reference wave Earth 74
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data Marine -- Data requirement ' β π» & ' π K π ππβ² π & = @ ππ K π» & A.C. ( Weglein & Secrest 1990 ) β’ π K π was rarely measured in 1990s 75
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data Marine -- Reduce data requirement ' β π» & ' π K π ππβ² π & = @ ππ K π» & A.C. ( Weglein & Secrest 1990 ) β’ π K π was rarely measured in 1990s ' = 0 along the cable β’ Reduce requirement of π K π with π» & ( Osen et al. 1998, Tan 1999 ) 76
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data Onshore -- Wave separation π£ >? = @ β 6 π― & ππβ² π£ 6 π 8 6 π― & β π’ A.C. ( Weglein & Secrest 1990, J. Wu & Weglein, SEG, 2015 ) π£ >? Reflection without ground roll & receiver ghost π― & Greenβs tensor in homogenous whole-space elastic reference π― & Stress tensor of Greenβs function π£ Displacement β π’ Traction 77
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data Onshore -- Wave separation ' β π» & ' πΌβ²π 6 π @ ππΌβ²π» & 8ππβ² A.C. π£ >? = @ β 6 π― & ππβ² π£ 6 π 8 6 π― & β π’ A.C. ( Weglein & Secrest 1990, J. Wu & Weglein, SEG, 2015 ) π£ >? Reflection without ground roll & receiver ghost π― & Greenβs tensor in homogenous whole-space elastic reference π― & Stress tensor of Greenβs function π£ Displacement β π’ Traction 78
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data Onshore -- Wave separation π£ >? = @ β 6 π― & ππβ² π£ 6 π 8 6 π― & β π’ A.C. ( Weglein & Secrest 1990, J. Wu & Weglein, SEG, 2015 ) Not a filtering method of ground roll removal, but a wave theory based wave prediction and no damage on π£ >? 79
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data Onshore -- Wave separation π£ >? = @ β 6 π― & ππβ² π£ 6 π 8 6 π― & β π’ A.C. ( Weglein & Secrest 1990, J. Wu & Weglein, SEG, 2015 ) π£ >? Reflection without ground roll & receiver ghost π― & Greenβs tensor in homogenous whole-space elastic reference π― & Stress tensor of Greenβs function π£ Displacement β π’ Traction 80
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data Onshore -- Data requirement π£ >? = @ β 6 π― & ππβ² π£ 6 π 8 6 π― & β π’ A.C. ( Weglein & Secrest 1990, J. Wu & Weglein, SEG, 2015 ) π£ >? Reflection without ground roll & receiver ghost π― & Greenβs tensor in homogenous whole-space elastic reference π― & Stress tensor of Greenβs function π£ Displacement β π’ Traction 81
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data Onshore -- Data requirement π£ >? = @ β 6 π― & ππβ² π£ 6 π 8 6 π― & β π’ A.C. ( Weglein & Secrest 1990, J. Wu & Weglein, SEG, 2015 ) ππ + 2ππ U π£ U π(π U π£ W + π W π£ U ) π(π U π£ E + π E π£ U ) π U π£ >? Reflection without ground roll & receiver ghost π W β = π(π U π£ W + π W π£ U ) ππ + 2ππ W π£ W π(π W π£ E + π E π£ W ) π’ π E π― & Greenβs tensor in homogenous whole-space elastic reference π(π U π£ E + π E π£ U ) π(π W π£ E + π E π£ W ) ππ + 2ππ E π£ E π― & Stress tensor of Greenβs function π=π U π£ U + π W π£ W + π E π£ E π£ Displacement β π’ Traction 82
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data The data requirement is satisfied ( J. Wu and A. Weglein, SEG 2015 ) β β’ Measure both π and π 83
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data β ) from air/earth Model -- Generate data ( π & π Air A/E boundary M.S. 100m (π£ U , π£ E ) , (π’ U , π’ E ) Earth 400m Layer P Velocity (m/s) S Velocity (m/s) Density (kg/m 3 ) 1(Air) 340 0 3 2(Top earth) 1800 1200 1500 3(Bottom earth) 4000 2500 1800 84
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data Total π£ U input to the separation algorithm Offset/m -3000 -2000 -1000 0 1000 2000 3000 0 0.5 5 4 3 2 1.0 1 Time/s 0 -1 -2 -3 1.5 -4 -5 x10-12 2.0 85
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data Total π£ U input to the separation algorithm Offset/m -3000 -2000 -1000 0 1000 2000 3000 Direct Direct 0 0.5 5 4 3 2 1.0 1 Time/s 0 -1 -2 -3 1.5 -4 -5 x10-12 2.0 86
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data Total π£ U input to the separation algorithm Offset/m -3000 -2000 -1000 0 1000 2000 3000 0 Rayleigh Rayleigh 0.5 5 4 3 2 1.0 1 Time/s 0 -1 -2 -3 1.5 -4 -5 x10-12 2.0 87
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data Total π£ U input to the separation algorithm Offset/m -3000 -2000 -1000 0 1000 2000 3000 0 Primary Primary 0.5 5 4 3 2 1.0 1 Time/s 0 -1 -2 -3 1.5 -4 -5 x10-12 2.0 88
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data Total π£ U input to the separation algorithm Offset/m -3000 -2000 -1000 0 1000 2000 3000 0 Ghost Ghost 0.5 5 4 3 2 1.0 1 Time/s 0 -1 -2 -3 1.5 -4 -5 x10-12 2.0 89
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data Total π£ U input to the separation algorithm Offset/m -3000 -2000 -1000 0 1000 Direct 2000 3000 0 Rayleigh Primary Ghost 0.5 5 4 3 2 1.0 1 Time/s 0 -1 -2 -3 1.5 -4 -5 x10-12 2.0 90
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data Deghosted π£ U from the separation algorithm Offset/m -3000 -2000 -1000 0 1000 2000 3000 0 Primary 0.5 5 4 3 2 1.0 1 Time/s 0 -1 -2 -3 1.5 -4 -5 x10-12 2.0 91
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data The data requirement is satisfied ( J. Wu and A. Weglein, SEG 2015 ) β β’ Measure both π and π β’ Effectively separate ground roll and ghosts out, without damaging reflection data 92
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data The data requirement is satisfied ( J. Wu and A. Weglein, SEG 2015 ) β β’ Measure both π and π β’ Effectively separate ground roll and ghosts out, without damaging reflection data The data requirement is not satisfied ( J. Wu and A. Weglein, SEG 2016 ) β β’ Measure only π , but not π 93
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data 2D onshore experiment Air (πΊ U , πΊ E ) A/E boundary (π£ U , π£ E ) Earth 94
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data 2D onshore experiment -- approximation ΓΌ Vacuum/earth boundary Vacuum (πΊ U , πΊ E ) V/E boundary (π£ U , π£ E ) Earth 95
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data 2D onshore experiment -- approximation ΓΌ Vacuum/earth boundary β(π’)π(π¦ β π¦ C ) , πΊ β π’ : source wavelet ΓΌ A localized force on surface πΊ Vacuum (πΊ U , πΊ E ) V/E boundary (π£ U , π£ E ) Earth 96
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data Boundary condition β(π) : source wavelet β π π π¦ β π¦ C β π¦, π¨ = 0, π = βπΊ πΊ π’ Vacuum (πΊ U , πΊ E ) V/E boundary (π£ U , π£ E ) Earth 97
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data Formula simplification β(π) : source wavelet β π π π¦ β π¦ C β π¦, π¨ = 0, π = βπΊ πΊ π’ Original π£ >? (π β(π β, π C , π) = @ π£ 6 π 8 6 π― & β π’ ββ², π C , π) 6 π― & (π ββ², π β, π) ππβ² A.C. 98
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data Formula simplification β(π) : source wavelet β π π π¦ β π¦ C β π¦, π¨ = 0, π = βπΊ πΊ π’ Original π£ >? (π β(π β, π C , π) = @ π£ 6 π 8 6 π― & β π’ ββ², π C , π) 6 π― & (π ββ², π β, π) ππβ² A.C. Simplified β(π) 6 π― π (π π£ >? (π 8 6 π― & ππ ^ β, π C , π) = @ π£ 6 π + πΊ C , π β, π) A.C. 99
Deghosting on a depth variable cable ΓΌ Removal of ground roll & ghost onshore Deghosting ocean bottom data Formula simplification β(π) : source wavelet β π π π¦ β π¦ C β π¦, π¨ = 0, π = βπΊ πΊ π’ Original π£ >? (π β(π β, π C , π) = @ π£ 6 π 8 6 π― & β π’ ββ², π C , π) 6 π― & (π ββ², π β, π) ππβ² A.C. Require traction everywhere along acquisition surface Simplified β(π) 6 π― π (π π£ >? (π 8 6 π― & ππ ^ β, π C , π) = @ π£ 6 π + πΊ C , π β, π) A.C. Require source wavelet 100
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