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Drift time estimation by dynamic time warping
Tianci Cui and Gary F. Margrave
1
Outline
• Drift time
• Well‐based 1D seismogram models
• Matching stationary and nonstationary seismograms
• Dynamic time warping
• Inclusion of internal multiples
• Conclusions and future work
2
Drift time estimation by DTW
• Drift time
• Well‐based 1D seismogram models
• Matching stationary and nonstationary seismograms
• Dynamic time warping
• Inclusion of internal multiples
• Conclusions and future work
3
Drift time: well logs
4
Drift time: fake Q log
: Well logging frequency : Seismic frequency
Kjartanssen (1979)
5
Drift time: frequency‐dependent velocity
: Well logging frequency : Seismic frequency
Kjartanssen (1979)
6
Drift time
7
Drift time estimation by DTW
• Drift time
• Well‐based 1D seismogram models
• Matching stationary and nonstationary seismograms
• Dynamic time warping
• Inclusion of internal multiples
• Conclusions and future work
8
Stationary seismogram: s(t)
=
minimum‐phasedominant frequency: 30 Hz
9
Nonstationary seismogram: sq(t)Synthetic zero‐offset VSP model with Q effects
(Margrave and Daley, 2014)
10
Nonstationary seismogram: sq(t)Synthetic zero‐offset VSP model with Q effects
(Margrave and Daley, 2014)Surface receiver
11
1 D seismogram models
Q effects: 1 Diminishing amplitude2 Widening wavelets3 Delaying events Drift time
12
1 D seismogram models
13
Drift time estimation by DTW
• Drift time
• Well‐based 1D seismogram models
• Matching stationary and nonstationary seismograms
• Dynamic time warping
• Inclusion of internal multiples
• Conclusions and future work
14
Matching without drift time correctionTime‐variant balancing and
time‐variant constant‐phase rotation
15
Matching with theoretical drift time correction
: drift time: stationary seismogram after drift time correction
Drift time correction
16
Matching perfection: time‐variant balancing and time‐variant constant‐phase rotation
Matching with theoretical drift time correction
17
Matching perfection: time‐variant balancing and time‐variant constant‐phase rotation
Matching with theoretical drift time correction
Drift time correction is necessary to match the stationaryto nonstationary seismograms.
Calculation of drift time in industrial practice needs oneof these:
• Knowledge of Q or
• A check‐shot survey or
• Manually stretching and squeezing the syntheticseismogram
18
Drift time estimation by DTW
• Drift time
• Well‐based 1D seismogram models
• Matching stationary and nonstationary seismograms
• Dynamic time warping
• Inclusion of internal multiples
• Conclusions and future work
19
Dynamic Time Warping
Dynamic time warping (Hale, 2012):• Estimates the time shift between two seismograms• Based on constrained optimization algorithm• Realized by dynamic programming• Similar to time‐variant crosscorrelation but more
sensitive to the rapid‐varying time shift
We use dynamic time warping (DTW) to estimate thedrift time between the stationary and nonstationaryseismograms caused by anelastic attenuation
20
DTW: drift time estimation
21
DTW: drift time estimation
22
DTW: drift time estimation
,
23
DTW: drift time estimation
,
24
DTW: drift time estimation
, ,
25
DTW: drift time estimation
, ,
: sample number, : time sample rate: drift time, : drift lag
theoretical drift lag
26
DTW: drift time estimationtheoretical drift lag
• The alignment error is nearly zero along the theoretical drift lag.
• Choose a path traveling from to , sum alignmenterrors along this path. The estimated drift lag sequence is thepath of the minimal cumulative alignment error.
27
DTW: drift time estimationtheoretical drift lag
infinite
Constraint: | |≤1
28
Dynamic Programming
4 7 1
3 5 8
6 2 9
Alignment error array
1
0
‐1
1 2 3n
m
27 possible paths
29
Dynamic Programming: accumulation
4 7 1
3 5 8
6 2 9
Alignment error array
1
0
‐1
1 2 3n
m
Cumulative alignment error array
1
0
‐1
m
n1 2 3
27 possible paths
30
Dynamic Programming: accumulation
Constraint: | |≤1
4 7 1
3 5 8
6 2 9
4
3
6
Alignment error array
1
0
‐1
1 2 3n
m
Cumulative alignment error array
1
0
‐1
m
n1 2 3
27 possible paths
?
31
Dynamic Programming: accumulation
Constraint: | |≤1
4 7 1
3 5 8
6 2 9
4
3
6
Alignment error array
1
0
‐1
1 2 3n
m
Cumulative alignment error array
1
0
‐1
m
n1 2 3
27 possible paths
?
32
Dynamic Programming: accumulation
Constraint: | |≤1
4 7 1
3 5 8
6 2 9
4
3
6
Alignment error array
1
0
‐1
1 2 3n
m
Cumulative alignment error array
1
0
‐1
m
n1 2 3
27 possible paths
?
33
Dynamic Programming: accumulation
Constraint: | |≤1
4 7 1
3 5 8
6 2 9
4 10
3
6
Alignment error array
1
0
‐1
1 2 3n
m
Cumulative alignment error array
1
0
‐1
m
n1 2 3
27 possible paths
34
Dynamic Programming: accumulation
Constraint: | |≤1
4 7 1
3 5 8
6 2 9
4 10
3
6
Alignment error array
1
0
‐1
1 2 3n
m
Cumulative alignment error array
1
0
‐1
m
n1 2 3
27 possible paths
?
35
Dynamic Programming: accumulation
Constraint: | |≤1
4 7 1
3 5 8
6 2 9
4 10
3
6
Alignment error array
1
0
‐1
1 2 3n
m
Cumulative alignment error array
1
0
‐1
m
n1 2 3
27 possible paths
?
36
Dynamic Programming: accumulation
Constraint: | |≤1
4 7 1
3 5 8
6 2 9
4 10
3 8
6
Alignment error array
1
0
‐1
1 2 3n
m
Cumulative alignment error array
1
0
‐1
m
n1 2 3
27 possible paths
37
Dynamic Programming: accumulation
Constraint: | |≤1
4 7 1
3 5 8
6 2 9
4 10
3 8
6 5
Alignment error array
1
0
‐1
1 2 3n
m
Cumulative alignment error array
1
0
‐1
m
n1 2 3
27 possible paths
38
Dynamic Programming: accumulation
Constraint: | |≤1
4 7 1
3 5 8
6 2 9
4 10 9
3 8 13
6 5 14
Alignment error array
1
0
‐1
1 2 3n
m
Cumulative alignment error array
1
0
‐1
m
n1 2 3
27 possible paths 3 possible paths
39
Dynamic Programming: backtracking
Constraint: | |≤1
4 7 1
3 5 8
6 2 9
4 10 9
3 8 13
6 5 14
Alignment error array
1
0
‐1
1 2 3n
m
Cumulative alignment error array
1
0
‐1
m
n1 2 3
27 possible paths 3 possible paths
40
Dynamic Programming: backtracking
Constraint: | |≤1
4 7 1
3 5 8
6 2 9
4 10 9
3 8 13
6 5 14
Alignment error array
1
0
‐1
1 2 3n
m
Cumulative alignment error array
1
0
‐1
m
n1 2 3
27 possible paths 3 possible paths
Estimated drift lag:
41
Dynamic Programming
4 10 9
3 8 13
6 5 14
1
0
‐1
1 2 3n
m
Cumulative alignment error array
1
0
‐1
m
n1 2 3
, ,
Cumulative alignment error array
4 10
3 8
6 5
,
Dynamic: optimal solution varies at different stageWarping path: drift lag
42
DTW: drift time estimationestimated drift lag
infinite
Constraint: | |≤1Further Constraint: ∑ ≤1
The drift lag sequence is constrained to change in blocks of samples
( )
43
DTW: drift time estimationestimated drift lag
infinite
Constraint: | |≤1Further Constraint: ∑ ≤1
The drift lag sequence is constrained to change in blocks of samples
( )
44
DTW: drift time estimation∗
: estimated drift lag: estimated drift time
45
DTW: matching seismograms
46
Drift time estimation by DTW
• Drift time
• Well‐based 1D seismogram models
• Matching stationary and nonstationary seismograms
• Dynamic time warping
• Inclusion of internal multiples
• Conclusions and future work
47
Inclusion of internal multiplessq(t)
sqi(t)
48
Inclusion of internal multiples: sqi(t)
=(Richards and Menke, 1983)
49
Inclusion of internal multiples
50
Conclusions
• DTW succeeds in estimating drift time automaticallywithout knowledge of Q or a check‐shot survey.
• Application of drift time correction results in a muchsimpler residual phase.
• DTW estimates drift time associated with apparent Qincluding both intrinsic and stratigraphic effects.
51
Future work
• Conduct stationary and nonstationary deconvolution onthe seismic trace and tie the deconvolved seismic traceto well log reflectivity by DTW
• Estimate Q value from the drift time estimated by DTW
52
Acknowledgements
• CREWES sponsors
• NSERC: grant CRDPJ 379744‐08
• CREWES staff and students
53
Choosing values
54
Choosing values
55
Applications of DTW
• Tying synthetic to recorded seismograms
• Registration of P– and S–wave images
• Residual normal moveout correction
• Alignment of images computed for differentsource‐receiver offsets or propagation angles.
56
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