Class 3: Refraction Traveltime Interpretation
Wed, Sept 16, 2009
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Analytical refraction solutions
Detailed approach in delay-time method
Generalized Linear Inversion (GLI) for layer model inversions
Review of commercial software solutions
Computer exercise – process synthetic and real data
This class introduces the conventional refraction traveltime methods for near-surface
structure interpretation. These approaches produce layered models, appropriate for
relatively simple velocity structures. Two influential approaches include delay-time
method and generalized liner inversion method. We invited two distinguished
geophysicists in these areas to give presentations, Chuck Diggins, Chief Geophysicist
of FusionGeo, and Dan Hampson, President of Hampson and Russell. However, both
of them are traveling this week. They will join us on Sept 28, 2009.
Analytical Refraction Solutions
Assuming 1D velocity structures with multiple layers and velocity increasing with depth,
we can derive an exact analytical solution from refraction traveltimes.
Two-Layer Model (One layer over half a space):
x
t
h1
(x1,t1)
v1
v2>v1
t1= x1/v2 + 2h1(v22-v12)1/2/(v2v1)
shot offset
(1)
Three-Layer Model (Two layers over half a space):
t2= x2/v3 + 2h1(v32-v12)1/2/(v3v1) + 2h2(v32-v22)1/2/(v3v2) (2)
t
shot offset
N-Layer Model
tn-1= xn-1/vn + (2/vn)Σhi(vn2-vi2)1/2/vi
(3)
tn-1= a0 xn-1 + a1h1 + a2h2 + a3h3 + …
(3a)
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Notes:
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Other analytical solutions – multiple dipping interfaces
All analytical solutions _ assume velocity increasing with depth
Hidden layers – refraction methods fail
Any other structures – no analytical solutions, need to be derived otherwise
All approaches are under some assumptions
Delay-Time Method
Map irregular refractor interfaces
E
G
hG
V1
A
B
V2
C
D
Total delay time for source and receiver: TEG = TE + TG
(1) Delay-time at G: TG = CG/v1 – CD/v2 = hG ((v22-v12)1/2 /(v1v2) )
(2) y
x
EF
G
ER
Derive reliable v2 when the interface varies:
tEFG – tERG = TEF – TER + 2x/v2 – y/v2
(3)
Questions: what are the assumptions behind the delay-time method?
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Generalized Linear Inversion (GLI) for Layer Models
By Hampson and Russell (1984)
D0(x) V0
D1(x) V1(x)
D2(x) V2(x)
V3(x)
Objective: given traveltimes observed at the surface, invert layer depths Di(x), and velocity Vi(x) Model:
Data:
M=(m1, m2, m3 …, mm)
T=(t1, t2, t3, …, tn)
(thickness and slowness) (direct-wave and refraction arrivals) Physics: ti=Aj(m1, m2, m3 …, mm), i=1,2,3, … n.
Given a velocity model and geometry, generate traveltimes of rays
Inverse Problem:
BΔM=ΔT Bij=∂ti/∂mj
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Objective Function: ψ = ⎜⎜T – A(m)⎜⎜
T
-1 T
Model Update: ΔM=(B B) B ΔT
M=M+ΔM
A more sophisticated inversion scheme:
Objective Function: ψ = ||T – A(m)||2 + τ ||L(m)||2
Solving an inverse problem:
(BTB +τLTL)ΔM=BT ΔT - τLTL(M)
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Commercial Software Packages:
Delay-Time Solutions:
Fathom:
Seismic Studio:
TomoPlus:
Green Mountain Geophysics (ION), Denver, CO
FusionGeo, Houston, Texas
GeoTomo, Houston, Texas
Generalized Linear Inversion:
GLI3D:
Hampson-Russell, Calgary, Canada
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MIT OpenCourseWare
http://ocw.mit.edu
12.571 Near-Surface Geophysical Imaging�
Fall 2009
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