Seismic wave propagation in heterogeneous media: Modeling, signal processing, and inversion...
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Transcript of Seismic wave propagation in heterogeneous media: Modeling, signal processing, and inversion...
Seismic wave propagation in heterogeneous media:
Modeling, signal processing, and inversion
Christian PoppeliersAngelo State University Physics Dept.
21 April, 2006
Seismic wave propagation
• Why?– Listening to waves tells us about the Earth
• 1) seismic wave velocity tomography• 2) seismic imaging discrete structure
– All involve scattering of mechanical waves
Which types of waves?
• Acoustic (sound waves)– Travel through all materials
• Elastic (mechanical waves)– Travel only through solids
--> Most wave modeling in exploration seismology is acoustic
--> Wave propagation in most solids is actually elastic
Some trivia:
1) Main exploration tool of the petroleum industry2) Typical industry scale survey contains 1000’s shots3) 100’s geophones per shot4) datasets can get up to a terrabyte in size!
petroleum industry is the single biggest user of electronic data storage medium
5) $4+ billion a year industry
Assumptions in reflection seismology
• 1) layered structure– Deterministic (i.e. large-scale structures that can be
“mapped”, or delineated)
• 2) “single” scattering• Linear scattering
– i.e. no internal reverberations, etc.
• 3) Acoustic waves ONLY• 4) seismic velocity is smoothly varying
Reality1) Lot’s of internal reflections, and mode conversions
com
pres
siona
l wav
e
shea
r wav
e
Reality, continued2) The Earth might not be perfectly, or vaguely, layered!!
most of the Earth’s crust is crystalline!!!
The Point:Seismic scattering in the earth can be
• the result of STOCHASTIC (“random”) impedance• very STRONG• lots of mode conversions
So what ?…quantifying scattering in stochastic medium can help us “map”this portion of the Earth
This is called statistical seismic imaging
In complex geologic material this can be used as in interpretation tool.
a different paradigm in seismic data analysis:assumptions of horizontal layering, and deterministic structure are removed
- crystalline crust - near-surface, unconsolidated sediments/gravels)- mantle
why is this useful?• a useful interpretational tool...
Inversion:
• Given the data, can we estimate the stochastic fabric?
• First step: develop forward model
• 1) Linear scattering:– reflections are from small velocity perturbations:
– assume impedance is proportional to velocity:
– which means:
Current inversion techniques for statistical seismic imaging assume:
v(x) : Earth’s 3D velocity fieldvo(x) : Earth’s background velocity field
δv(x) : perturbations of velocity about the background
)()()( xxx vvv o
xpxv
xpxpxp 0
linear scattering assumptions, con’t
• Stochastic field is defined as:
1
x
x
x
x
p
p
v
v
reflections occur where δp(x) ≠0
Linear scattering, in 1-D:
dtetwts )()()(
dtetwts )()()(
dtptwts )()()(
)()()( tptwts convolution
In the frequency domain
)()()( fpfwfs
if δp(x)/p(x) <<1, then scattering is “weak”, or approximately linear
Problem: mode conversions and multiple scattering aren’t accounted forSolution: scattering is weak/linear, so they aren’t a problem
“single” or “weak” scattering Born Approximation
s(t) : recorded seismogramw(t) : input seismic pulseδp(t) : acoustic impedance contrasts of the Earth
For now, we simply avoid P-to-S mode conversions:
• P to S mode conversions are avoided if offset is small:
shot
receiver
Ref
lect
ed P
-wav
e
Goal: θ < 200
…and…2) The earth is semi-fractal
• Earth texture of δp is fractal– Model = Von Kármán
2
1
2
122222 )1)((2
)2()(
ak
akp
k = vector wavenumber Γ = Gamma functiona = characteristic lengthυ = Hurst number
)0(
))(()(
2
G
rGHChh
]1,0[,0,)()(
rrKrrG
power spectrum autocorrelation
τ = lagH = rms impedance variationG = Bessel functionr = radius
What does this mean?
• 1) Earth texture is self-similar on all scales– up to the “characteristic length”
• 2) Parameterized by – power spectrum “rollover” size parameter, a– “roughness” --> fractal dimension => D=E+1-v
log wavenumber
log
ampl
itude
1a
“slope” is a functionof fractal dimension
lag
lag, τ
power spectrum auto correlation
steepness controlled by fractal dimension
half-width controlled by a
example: numerical Earth:
a=100mD=2.9
a=100mD=2.5
color corresponds to seismic velocity
4.9 m/s
5.1 m/s
a=500mD=2.5
Is Earth continuously varying?
not really
most crystalline materials contain an n-modal distribution of velocities (n=2,3)
adjust our fractal model
• Goff et. al., 1994; transform continuous fields into binary fields
effects on statistical parameters?
continuous binary
1/a = kc 1/a ≈ 1.15kd
D=E+1-vc D ≈ E+1-(vc/2)
velo
city
, km
/s
P-w
ave velocity, km
/sec
The point, forward model:
Source
Receiver
explosive source
“fine” model “coarse” model
geophones basic observation: the size-scale of the heterogeneities have an affect on the wavefield
Linear inversion:
Color corresponds to estimated “textural size”
Given seismic data that has propagated through a stochastic earth, can we estimate the equivalent fractal “textural parameters”?
Work on this so far
• 1) inversion assuming “white” reflectivity
• 2) iterative inversion with non-white reflectivity
• 3) non-linear inversion
First Attempt; binary model, simple scattering
• recall convolution model:
)()()( tptwts
what this really means:
)(*)()( trtwts
)(2
1)( tp
dt
d
vtr
where
s(t) = seismic reflection dataw(t) = seismic pulseδp(t) = stochastic impedancer(t) = reflectivity
thus, in 1-D:
stochastic velocity:
with corresponding reflectivity
convolved with an illuminating wavelet
will yield this data:
*
But we record data• have to go backwards (inversion)
1) Given data, estimate reflectivity
)()()( krkwks
)(
)()(
kw
kskr known as DEconvolution
since )(2
1)( tp
dt
d
vtr
T
dttrvtp )(2)(ˆ
from δp(t), we can estimate equivalent stochastic model
note that w(k)-1 is known as an “inverse filter”.Its really an inverse operator
The recipe:given data:
deconvolve:
numerically integrate
“clean up” and normalize via a thresholding operator
R
RR
R
T
Tzr
TzrT
Tzr
zr
)(ˆ,1
)(ˆ,0
)(ˆ,1
)(ˆ
from estimate of δp, obtain equivalent fractal model:
grid search through autocorrelations of fractal models
N
zhhvzi aCCN
axE
1
2),:(),(1
),:(
x
where Ei is a minimum, record D, υ
autocorr. of theoretical model
autocorr. of estimated δp,
least-squares
Does it work? • kind of...
known wavelet
UNknown wavelet
Poppeliers and Levander, 2004
Limitations:1) decon requires the wavelet2) wavelet is usually unknown3) deconvolution assumes white reflectivity4) fractal reflectivity is not white5) very sensitive to uncorrelated noise6) not good at estimating fractal dimension
major problem: deconvolution
• reflectivity is NOT white– in fact, reflectivity in the Earth is fractal
• “solution”? – modify the inverse operator in deconvolution
by the non-white, fractal model
The recipe:assume that data is given by
)()()( kWkRkS
with a power spectrum of222
)()()( kRkWkS
initially assume that reflectivity is white, so |R(k)|2 = 1:
2
0
2)()( kWkS
))(()(ˆ 10
11 kWFzw
form inverse filter:
inverse filter
F-1(.) = inverse Fourier transform
deconvolve data to obtain initial estimate of reflectivity:
)(*)(ˆ)(ˆ 10 zszwzr
from r0, threshold, integrate and find a and D as before
From a and D, form reflectivity power spectrum model:
),,(4
)(20
222
1 Dakpv
kkR
and use to modify the inverse operator:
2
1
02
1)(
)()(
kR
kWkW
and form new inverse filter:
)()( 11
11 kWFzw
von Karman model
itera
te
does it work better?
autocorr oforiginal model
autocorr fromspiking decon
autocorr fromiterative decon
Yes! ...1-D simulation observations:A) spiking decon method underestimates char. length, cannot obtain fractal dimension
B) iterative decon method comes closer to original char. length, can estimate fractal dimension
2-D simulationspiking decon
iterative decon
iterative method more accurate and more precise
a (m)
a Da
Synthetic model:a=300mD=2.7
field data: 1986 PASSCAL Basin and Range seismic experiment
Results agree with drill cores
Finally, what if velocities are not discrete?
• Previous methods limited to bi-modal velocity distribution
• what about tri-modal, n-modal, or continuous velocity distributions?
new approach:• Like before, assume that data is from
convolution (linear scattering):
x =
time/space domain
power spectal domain
* =
222
,,ˆ,, kDSkDRkW
“make up” pulse and fractal model who’s product is equal to the data!
given
222,,,,ˆ kaRkWkaS
details:
1) data, wavelet, and reflectivity are discretely sampled in the wavenumber domain2) wavenumber sample interval for the wavelet is much greater than than that of the data3) i.e. more data samples than wavelet samples4) the problem is grossly over-determined5) no information about wavelet’s phase (must assume)
forward model:
222
,,,,ˆsws kaRkWkaS
2
0
2222 4
)(,,ˆV
kkPkWkaS s
sws
data S is a product of the wavelet W and reflectivity R
The reflectivity is a function of the impedance contrasts, δP(k)
2
1
2
122222 )1)((2
)2()(
ak
akp
von
Karm
an
impe
danc
e m
odel
von Karman reflectivity
inverse method:
KkkSkaSE Nn ,0,,ˆ
K
NNRnNn dkkSkaPkWSkSkaSO0
22)(),,()(ˆ;,,ˆ
ig
ii dJm
TnkWkWkWam ,,,,, 21
TniiiT
Ni kaDkaDkaDkDkDkDd ,,ˆ,,,,ˆ,,,ˆ,,, 2121
nNkaSm
J Nn
,,,ˆ2,1
nnn
nnN m
SJ
,...4,3...4,3,
ˆ
inversion must minimize difference between model and data:
K=nyquist wavenumber
objective function:
Newton’s Method:
partial derivatives of von Karman reflectivity
partial derivatives of interpolated pulse power spectrum, or individual spline functions
inversion estimates:
• 1) characteristic length
• 2) fractal dimension
• 3) power spectrum of illuminating wavelet!– but not the phase...
Preliminary tests/results:
a=30m
a=120msimulated shot data:
single shot, at surface fc=10Hz, zero phase illum. wavelet
Poppeliers, 2006
test true parameters recovered az recovered Daz (m), D
noise free 30, 2.7 43.0 ± 5.7 2.61 ± 0.05noise free 120, 2.7 142.1 ± 1.8 2.60 ± 0.15
15% colored noise 30, 2.7 55.7 ± 7.2 2.60 ± .08 15% colored noise 120, 2.7 151.9 ± 12.2 2.59 ± .30
Poppeliers, 2006
Po
pp
elie
rs,
20
06
Poppeliers, 2006
Future work: Question
• Why can’t we use scattering principles from optics?
• Because, of mode conversion issues!– analysis indicates this approach is inappropriate for
large offsets
• What about strongly scattered seismic data?– For large offsets, S converted modes will dominate
the data
If the seismic wavefield that propagates though a stochastic earth is 1) NOT acoustic and 2) NOT singly scattered.
• Then,
• More likely
)()()( krkwks
)()(exp],,,[:)( kwkretcxakFks nPath length
The linear model
x =
but strong scattering looks like
=Some function (with parameters)
x F(f:[a,v,x,etc])expn x
The point:
• 1) Current model for inversions are only good for weakly scattered wavefields– Scattering is not linear
• 2) Forward modeling of seismic waves in a stochastic medium is necessary– First step towards developing inversion– We need a model (based on lots of statistics)
of the function F(f[a,v,x,ect])expn
approach?
Build a family of curves, based on the results of numerical tests, a “find” a function that describes these curves in a general sense.
Results of multiple tests
“test” function
Why?
• Finding the correct function F(f[a,v,x,ect])expn is key to the inverse problem– Need a “forward” model before we can invert
data.• Invert data: given the data, can we determine the
stochastic parameters?
T=30sec
T=30sec
T=100sec
T=100sec
Lightly scattering dimple
Heavily scattering dimple
Notice the difference in the wavefield here. Can we say something about the texture of the dimple?
concluding remarks:
• statistical seismic imaging is a new paradigm of seismic data analysis
• rich in mathematical ideas
• will form a useful tool in interpreting existent data (cheap!)