Reduced-Time Migration of Converted Waves David Sheley University of Utah

Outline Motivation Migration Theory Error Analysis Synthetic Data Results Field Data Result Conclusions & Future Work

PP vs PS Transmission Migration Depth Source Well Receiver Well This crude example is for crosswell and most specifically for orebody delineation. If you are more interested in salt flank imaging with VSP let use a bit of imagination. Z Offset X

PP Reflection Migration P P = ? =? Depth Source Well Receiver Well PP Migration allow us to image sub-horizontal interfaces. The migration algorithm is highly dependent on the velocity information. Z Offset X

Converted Wave Migration Vp,Vs = ? P Depth P PS Source Well Receiver Well PS converted wave migration allows us to image subvertical boundaries giving us information about what could be happening parallel to the boreholes. Unfortunately it is not only dependent on the P-wave velocity model but also on the S-wave velocity model. To add to our grief, for deep holes there are often positioning errors. So you’re probably thinking to yourself “this kid thinks converted wave migration may actually be useful for real geologic problems. Perhaps during the course of this talk I can remove some …. Z Offset X =?

Outline Motivation Migration Theory Error Analysis Synthetic Data Results Field Data Result Conclusions & Future Work

Conventional Migration m(r) = S(zg, tsr + trg) g The green line is an isocron. s g tsr trg r

PS Transmission Migration m(r) = S(zg, dsr/Vp + drg/Vs) g This equtaion and figure model PS converstion SP conversion is also possible to migrate simply by swapping Vp and Vs. Again the green line is an isocron. If we examine this equation a bit we can see that if the image point r is at the geophone the green isochron will collapse about the receiver and the time argument will be equal to the travel time of the direct P-wave from the source to the receiver. s g drg/Vs dsr/Vp r

Problem Receiver Source Well Well 20 40 60 50 km/sec Depth (m) km/sec 6.0 5.0 7.0 Receiver Source Well Well 20 Depth (m) 40 I will prove that mapping the direce P-wave to the receiver will acomplish the goal later. First, I will discuss how, even in the presence of velocity errors the direct-P is mapped to the reciever. 60 Offset (m) 50

Reduced-Time Migration Data time shift S’(g, t) = S(zg, t + tsg ) tsg = Observed direct-P time obs obs

Data Shift Original Data Shifted Muted Data SP PS P S Depth (m) 114 20 114 20 35 Time (ms) Original Data Depth (m) 114 8 2 Time (s) Shifted Muted Data SP PS P S

Reduced-Time Migration Data time shift S’(zg, t) = S(zg, t + tsg ) tsg = Observed direct-P time obs obs Modify the migration equation m(r) = S(zg, tsr + trg - tsg + tsg ) calc obs g calc m(r) = S’(zg, tsr + trg – tsg ) g

Outline Motivation Migration Theory Error Analysis Synthetic Data Results Field Data Result Conclusions & Future Work

m(r) = S(zg, tsr + trg psr) Error Analysis -- CWM Assumptions: Single trace Homogeneous media True velocity = c Migration velocity c’ = c + dc Vp/Vs = psr m(r) = S(zg, tsr + trg psr) g l m(r) = S(zg, (dsr + drg psr)/c’ ) g l

Error Analysis Conventional Migration (dsr + drg psr)/c’ l (dsr + drg psr)/(c + dc) = ~ (dsr + drg psr)(s – s dc) l 2 e = - (dsr + drg psr) s dc 2 l cm

Error Analysis Reduced-Time Migration m(r) = S(g, g obs calc tsr + trg - tsg + tsg )

Error Analysis Reduced-Time Migration obs calc tsr + trg - tsg + tsg = l (dsr + drg psr - dsg)(s – s dc) + dsg s 2 e = - (dsr + drg psr - dsg) s dc rtm 2 l

Error Functions CWM vs. RTM cm e = - (dsr + drg psr) s dc 2 l rtm e = - (dsr + drg psr - dsg) s dc 2 l

e e Imaging-Time Error cm rtm Offset (m) 500 16 Depth (m) 12 Imaging Offset (m) 500 16 cm e Depth (m) 12 Imaging Error (ms) 250 8 rtm e Depth (m) 4 250 Offset (m) 500

Outline Motivation Migration Theory Error Analysis Synthetic Data Results Field Data Result Conclusions & Future Work

Crosswell Model V V V = 5000 m/s V = 5500 m/s Vp /Vs = 1.5 V = 5500 m/s 2 V 2 Vp /Vs = 1.5 Source = 1500 Hz Depth (m) ds = 2 m V 1 dg = 2 m Well Separation = 100 m 114 114 Offset (m)

Synthetic Data Original Data Shifted Muted Data P PS S SP SP PS Depth (m) 114 20 35 Time (ms) Original Data Depth (m) 114 8 2 Time (s) Shifted Muted Data P PS S SP SP PS

PS Transmission Migration True Velocity Depth (m) 114 Offset (m) 114

Conventional PS Migration + 10 % Velocity Depth (m) 114 Offset (m) 114

Reduced-Time PS Migration + 10% Velocity Depth (m) 114 Offset (m) 114

Outline Motivation Migration Theory Error Analysis Synthetic Data Results Field Data Result Conclusions & Future Work

Data Problems Time Delay = 3 ms ? Well location Velocity Model

Kidd Creek Receiver Source Well Well 20 40 60 50 km/sec Depth (m) km/sec 6.0 5.0 7.0 Receiver Source Well Well 20 Depth (m) 40 I will prove that mapping the direce P-wave to the receiver will acomplish the goal later. First, I will discuss how, even in the presence of velocity errors the direct-P is mapped to the reciever. 60 Offset (m) 50

Time Shifted CRG 20 Depth (m) 40 60 Time (ms) 6

Conventional PS Migration 20 Depth (m) 40 I will prove that mapping the direce P-wave to the receiver will acomplish the goal later. First, I will discuss how, even in the presence of velocity errors the direct-P is mapped to the reciever. 60 Offset (m) 50

Reduced-Time PS Migration 20 Depth (m) 40 I will prove that mapping the direce P-wave to the receiver will acomplish the goal later. First, I will discuss how, even in the presence of velocity errors the direct-P is mapped to the reciever. 60 Offset (m) 50

RTM-PS CRG #8 20 40 60 50 Depth (m) Offset (m) 20 40 60 50 Depth (m) Offset (m) I will prove that mapping the direce P-wave to the receiver will acomplish the goal later. First, I will discuss how, even in the presence of velocity errors the direct-P is mapped to the reciever.

Kidd Creek 20 40 60 50 50 Offset (m) Offset (m) 20 40 I will prove that mapping the direce P-wave to the receiver will acomplish the goal later. First, I will discuss how, even in the presence of velocity errors the direct-P is mapped to the reciever. 60 Offset (m) 50 Offset (m) 50

Outline Motivation Migration Theory Error Analysis Synthetic Data Results Field Data Result Conclusions & Future Work

Discussion & Conclusions PS migration can image structure invisible to reflection migration. Reduced-time migraton decreases the error of an incorrect velocity model. Converted wave reduced-time migration can successfully image a transmitting boundary.

Future Work Model and migrate salt proximity VSP data with converted wave RTM. Model and test PP RTM. Search for other applications of RTM. Graduate.