![Page 1: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/1.jpg)
Seismic Tomography and Double-Difference Seismic Tomography
Clifford ThurberUniversity of Wisconsin-Madison
Haijiang ZhangUniversity of Science and Technology of China
![Page 2: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/2.jpg)
Acknowledgements
• Felix Waldhauser, for hypoDD, sharing data, and providing many constructive comments
• Bill Ellsworth, for suggesting the name "tomoDD"
• Charlotte Rowe for assistance• Defense Threat Reduction Agency, NSF, and
USGS for financial support
![Page 3: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/3.jpg)
Outline
• Seismic tomography basics – conventional and double-difference
• Synthetic tests and example applications• Usage of tomoDD
![Page 4: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/4.jpg)
Consider residuals from one earthquake
*
*
*
* *
0 90 180 270STATION AZIMUTH
LATE
EARLY
Arrival Time Misfit
Trial Location
MapView
![Page 5: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/5.jpg)
Interpretation #1 - earthquake is farther north
*
*
*
* *
0 90 180 270STATION AZIMUTH
LATE
EARLY
*
* * * *
Arrival Time Misfit
MapView
True Location
![Page 6: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/6.jpg)
Is mislocation the only explanation?
*
*
*
* *
0 90 180 270STATION AZIMUTH
LATE
EARLY
Arrival Time Misfit
Trial Location
MapView
![Page 7: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/7.jpg)
Alternative interpretation - velocity structure is slower near event and to the south and
faster near the northern station!
*
*
* *
0 90 180 270STATION AZIMUTH
LATE
EARLY
*
FASTER
SLOWER
MapView
True Location
![Page 8: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/8.jpg)
Alternative interpretation - velocity structure is slower near event and to the south and
faster near the northern station!
*
*
* *
0 90 180 270STATION AZIMUTH
LATE
EARLY
** * * *
FASTER
SLOWER
Compensate for Structure
MapView
True Location
![Page 9: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/9.jpg)
Alternative interpretation - velocity structure is slower near event and to the south and
faster near the northern station!
*
*
* *
0 90 180 270STATION AZIMUTH
LATE
EARLY
** * * *
FASTER
SLOWER
Compensate for Structure
MapView
True Location
How can we determine the heterogeneity?
![Page 10: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/10.jpg)
How does seismic tomography work?"Illuminate" fast velocity anomaly
with waves from earthquake to array
Localizes anomaly to a "cone"
![Page 11: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/11.jpg)
How does seismic tomography work?"Illuminate" fast velocity anomaly
with waves from earthquake to array
Localizes anomaly to a "cone"
"Illuminate" fast anomaly with waves from another earthquake
Localizes anomaly to another "cone"
![Page 12: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/12.jpg)
Combine observations from multiple earthquakes to image anomaly
![Page 13: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/13.jpg)
h
hslowness si = 1/velocity
s1 s2
s3 s4
Simple Seismic Tomography Problem
![Page 14: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/14.jpg)
h
hslowness si = 1/velocity
s1 s2
s3 s4
Simple Seismic Tomography Problem
![Page 15: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/15.jpg)
h
h
d = G m
slowness si = 1/velocity
data model
s1 s2
s3 s4
Simple Seismic Tomography Problem
![Page 16: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/16.jpg)
h
h
d = G m
slowness si = 1/velocity
data model
s1 s2
s3 s4
Simple Seismic Tomography Problem
QUESTIONS SO FAR?
![Page 17: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/17.jpg)
Consider pairs of closely-spaced earthquakes
01
1
1
1 1
0 90 180 270AZIMUTH
LATE
EARLY
Relative Arrival Time
![Page 18: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/18.jpg)
0 2
2
2
2 2
0 90 180 270AZIMUTH
LATE
EARLY
Relative Arrival Time
![Page 19: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/19.jpg)
03
3
3
3 3
0 90 180 270AZIMUTH
LATE
EARLY
Relative Arrival Time
![Page 20: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/20.jpg)
04
4
4
4 4
0 90 180 270AZIMUTH
LATE
EARLY
Relative Arrival Time
![Page 21: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/21.jpg)
04
4
4
4 4
0 90 180 270AZIMUTH
LATE
EARLY
Relative Arrival Time
So relative arrival times tell you relative locations
![Page 22: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/22.jpg)
Consider effect of heterogeneity - linear horizontal velocity gradient
01
0 90 180 270AZIMUTH
LATE
EARLY
SLOWER ====> FASTER
Relative Arrival Time
1
1
1 1
gray = homogeneous case
![Page 23: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/23.jpg)
Consider effect of heterogeneity – linear horizontal velocity gradient
01
1
1
1 1
0 90 180 270AZIMUTH
LATE
EARLY
SLOWER ====> FASTER
Relative Arrival Time
1
1
1 1
gray = homogeneous case
![Page 24: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/24.jpg)
0 2
0 90 180 270AZIMUTH
LATE
EARLY
SLOWER ====> FASTER
Relative Arrival Time
2
2
2 2
gray = homogeneous case
![Page 25: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/25.jpg)
0 2
2
2
2 2
0 90 180 270AZIMUTH
LATE
EARLY
SLOWER ====> FASTER
Relative Arrival Time
2
2
2 2
gray = homogeneous case
![Page 26: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/26.jpg)
03
3
3
3 3
0 90 180 270AZIMUTH
LATE
EARLY
SLOWER ====> FASTER
Relative Arrival Time
3
3
3 3
gray = homogeneous case
![Page 27: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/27.jpg)
04
4
4
4 4
0 90 180 270AZIMUTH
LATE
EARLY
SLOWER ====> FASTER
Relative Arrival Time
4
4
4 4
gray = homogeneous case
![Page 28: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/28.jpg)
Ignore heterogeneity – some locations will be distorted, some residuals will be larger!
04 2
1
3
4 21
3
gray = true white = relocated
![Page 29: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/29.jpg)
Consider effect of different heterogeneity - low velocity fault zone
01
1
1
1 1
0 90 180 270AZIMUTH
LATE
EARLY
Relative Arrival Time
FAST SLOW FAST1
1
1 1
gray = homogeneous case
![Page 30: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/30.jpg)
0 2
2
2
2 2
0 90 180 270AZIMUTH
LATE
EARLY
Relative Arrival Time
FAST SLOW FAST2
2
2 2
gray = homogeneous case
![Page 31: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/31.jpg)
03
3
3
3 3
0 90 180 270AZIMUTH
LATE
EARLY
Relative Arrival Time
FAST SLOW FAST
3
3
3 3
gray = homogeneous case
![Page 32: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/32.jpg)
04
4
4
4 4
0 90 180 270AZIMUTH
LATE
EARLY
Relative Arrival Time
FAST SLOW FAST
gray = homogeneous case
4
4
4 4
![Page 33: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/33.jpg)
Result - locations are very distorted!
04 2
1
3
4 2
1
3
gray = true white = relocated
![Page 34: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/34.jpg)
Implications
• Ignoring heterogeneous earth structure will bias estimated locations from true locations
• Different heterogeneities have different "signatures" in arrival time difference patterns - so there should be a "signal" in the data that can be modeled
![Page 35: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/35.jpg)
Implications
• Ignoring heterogeneous earth structure will bias estimated locations from true locations
• Different heterogeneities have different "signatures" in arrival time difference patterns - so there should be a "signal" in the data that can be modeled
QUESTIONS?
![Page 36: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/36.jpg)
Our DD tomography approach• Determine event locations and the velocity
structure simultaneously to account for the coupling effect between them.
• Use absolute and high-precision relative arrival times to determine both velocity structure and event locations.
• Goal: determine both relative and absolute locations accurately, and characterize the velocity structure "sharply."
![Page 37: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/37.jpg)
Seismic tomography
Arrival-time residuals can be linearly related to perturbations to the hypocenter and the velocity structure:
Nonlinear problem, so solve with iterative algorithm.
![Page 38: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/38.jpg)
Double-difference seismic tomography
For two events i and j observed at the same station k
Subtract one from the other
Note:
![Page 39: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/39.jpg)
Combine conventional and double-difference tomography into one system of equations
involving both absolute and double-difference residuals
absolute
doubledifference
![Page 40: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/40.jpg)
Test on "vertical sandwich" model• Constant velocity (6 km/s) west of "fault"• Sharp lateral gradient to 4 km/s• Few km wide low-velocity "fault zone"• Sharp lateral gradient up to 5 km/s• Gentle lateral gradient up to 6 km/s• Random error added to arrival times but not
differential times (so latter more accurate)• Start inversions with 1D model
![Page 41: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/41.jpg)
Conventional tomography solution
True model, all depths
![Page 42: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/42.jpg)
Double-difference tomography solution
True model, all depths
![Page 43: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/43.jpg)
Difference between solutions and true modelDouble difference Conventional
Marginal resultsnear surface
Poor resultsat model base
DD resultssuperior
throughoutwell resolved
areas
![Page 44: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/44.jpg)
Peacock, 2001
Application to northern Honshu, Japan
![Page 45: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/45.jpg)
Examples of previous results for N. HonshuNakajima et al., 2001
Zhao et al., 1992
Note relative absence of structural variations within the slab
![Page 46: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/46.jpg)
Events, stations, and inversion grid
Y=40 km
Y=-10 km
Y=-60 km
Zhang et al., 2004
![Page 47: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/47.jpg)
Cross section at Y=-60 kmVp
Vp/Vs
Vs
![Page 48: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/48.jpg)
Test 1: with mid-slab anomaly
Vp Vs
Inputmodel
Recoveredmodel
![Page 49: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/49.jpg)
Test 2: without mid-slab anomaly
Vp Vs
Inputmodel
Recoveredmodel
![Page 50: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/50.jpg)
Preliminary study of the southern part of New Zealand subduction zone
![Page 51: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/51.jpg)
Preliminary study of the New Zealand subduction zone - Vp
![Page 52: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/52.jpg)
Preliminary study of the New Zealand subduction zone - Vs
![Page 53: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/53.jpg)
Preliminary study of the New Zealand subduction zone - Vp/Vs
![Page 54: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/54.jpg)
Comparing Northern Honshu (top) to New Zealand (bottom)
![Page 55: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/55.jpg)
Application to ParkfieldFollowing 4 workshops in 2003-2004, a site just north of the rupture zone for the M6
Parkfield earthquake was chosen for SAFOD because:
Surface creep and abundant shallow seismicity allow us to accurately target the subsurface position
of the fault.
Clear geologic contrast across the fault - granites on SW side and Franciscan melange on NE -
should facilitate fault's identification (or so we thought!).
Good drilling conditions on SW side of fault (granites).
Fault segment has been the subject of extensive geological and geophysical studies and is within the most intensively instrumented part of a major plate-
bounding fault anywhere in the world (USGS Parkfield Earthquake Experiment).
![Page 56: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/56.jpg)
San
And
reas
Fau
lt Zo
ne
Phase 1: Rotary Drilling to 2.5 km (summer 2004)
Phase 2: Drilling Through the Fault Zone (summer 2005)
Phase 3: Coring the Multi-Laterals (summer 2007)
Resistivities: Unsworth & Bedrosian, 2004
Earthquake locations: Steve Roecker, Cliff Thurber, and Haijiang Zhang, 2004
SAFOD Drilling Phases
Pilot Hole (summer 2002)
Target Earthquake
1
2 3
![Page 57: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/57.jpg)
PASO-DOS, SUMMER 2001 – FALL 2002
![Page 58: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/58.jpg)
![Page 59: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/59.jpg)
Relationship of Seismicity to 3D Structure – Fault-Normal View
NE SW
Z=7.0 km
Z=-0.5 km
Viewed from the northwest
![Page 60: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/60.jpg)
Relationship of Seismicity to 3D Structure – Fault-Parallel View
SE NW
Z=7.0 km
Z=-0.5 km
Viewed from the northeast
![Page 61: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/61.jpg)
Zoback et al. (2011)
Revised Locations of Target Events and Borehole Features
![Page 62: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/62.jpg)
SUMMARY
• DD tomography provides improved relative event locations and a sharper image of the velocity structure compared to conventional tomography.
• In both Japan and New Zealand, we find evidence for substantial velocity variations within the down-going slab, especially low Vp/Vs zones around the lower plane of seismicity.
• In Parkfield, earthquakes "hug" the edge of the high-velocity zone and repeating earthquakes correlate with structures seen in borehole.
![Page 63: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/63.jpg)
Extensions of tomoDD
• Regional scale tomoDD
• Adaptive tomoDD
• Global scale tomoDD
![Page 64: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/64.jpg)
Regional scale version tomoFDD
• Considers sphericity of the earth.• Finite-difference ray tracing method
[Podvin and Lecomte, 1991; Hole and Zelt, 1995] is used to deal with major velocity discontinuities such as Moho and subducting slab boundary.
• Discontinuities are not explicitly specified.
![Page 65: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/65.jpg)
Insert the Earth into a cubic box.
Use the rectangular box
to cover the region of interest
2D slice
Treating sphericity of the Earth
Flanagan et al., 2000
![Page 66: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/66.jpg)
Adaptive-mesh version tomoADD
• Uneven ray distribution requires irregular inversion mesh.
• Linear and natural-neighbor interpolation based on tetrahedral and Voronoi diagrams.
Zhang and Thurber, 2005, JGR
![Page 67: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/67.jpg)
Uneven ray distribution• Nonuniform station geometry• Noneven distribution of sources• Ray bending• Missing data
Mismatch between ray distribution and cells/or grids causes instability of seismic tomography
Using damping and smoothing → possible artifacts
![Page 68: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/68.jpg)
The advantage of adaptive grid/cells (or why do we bother to use?)
• The distribution of the inversion grid/cells should match with the resolving power of the data.– The inverse problem is better conditioned.– Weaker or no smoothing constraints can be
applied.– Less memory space (less computation time?)
![Page 69: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/69.jpg)
Construct tetrahedral and Voronoi diagrams around irregular mesh
• Represent the model with different scales• Represent interfaces• Place nodes flexibly
![Page 70: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/70.jpg)
Linear interpolation
• Based on tetrahedra in 3D
kk
k f) (φ)v(
4
1
rr
)r(r)r(r)r(r
)r(r)r(r)r(rr13121k
13121
)(φk
![Page 71: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/71.jpg)
Natural neighbor (NN) interpolation
)(f)(φ)v( k
n
kk rrr
1
where is the natural-neighbor “coordinate”
)(φk r
![Page 72: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/72.jpg)
linear interpolation vs. natural neighbor interpolation
• Linear interpolation– Using 4 nodes– Continuity in first derivatives– Easier to calculate
• Natural neighbor interpolation– Using n nodes– Continuity in both first and 2nd derivatives– More difficult to calculate
![Page 73: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/73.jpg)
Automatic construction of the irregular mesh
![Page 74: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/74.jpg)
Application to SAFOD project
~800 earthquakes, ~100 shots, subset of high-resolution
refraction data (Catchings et al.,
2002); 32 "virtual
earthquakes" (receiver gathers from Pilot Hole)
![Page 75: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/75.jpg)
The inversion grids for (a) P and (b) S waves at the final iteration using only the absolute data.
![Page 76: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/76.jpg)
The DWS value distribution (ray sampling density) for P waves
Regular grid Irregular grid
![Page 77: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/77.jpg)
The across-strike cross-section of P-wave velocity structure through Pilot Hole
(absolute and differential data)
Natural neighbor interpolationLinear interpolation
![Page 78: S eismic T omography and Double- D ifference S eismic T omography](https://reader035.vdocuments.us/reader035/viewer/2022062310/568165f7550346895dd91f8c/html5/thumbnails/78.jpg)
Global scale DD tomography