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Array Methods for Image Formation

Outline1. Fine scale heterogeneity in the mantle2. CCP stacking versus scattered wave imaging3. Scattered wave imaging systems4. Born Scattering = Single scattering5. Diffraction & Kirchhoff Depth Migration6. Limitations 7. Examples8. Tutorial: I use Matlab R2007b

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SummaryData redundancy suppresses noise.Noise is whatever doesn’t fit the scattering

model.The image is only as good as the 1. The velocity model 2. The completeness of the dataset3. The assumptions the imaging system is

based on.1. We use a single-scattering assumption

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SNORCLE

Subduction forming the Proterozoic continental mantle lithosphere:

Lateral advection of mass

Cook et al., (1999)Bostock (1998)

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5

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Geometry: Fine scales in the mantle

Allegre and Turcotte, 1986Kellogg and Turcotte, 1994

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Coherent Subduction SUMA: Statistical UpperMantle AssemblageReservoirs are distributed

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Structures from basalt extraction

Oman ophiolite

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Regional Seismology: PNE DataHighly heterogeneous models of the CL

Nielsen et al, 2003, GJI

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Two classes of scattered wave imaging systems1. Incoherent imaging systems which are frequency and amplitude

sensitive. Examples are1. Photography2. Military Sonar3. Some deep crustal reflection seismology

2. Coherent imaging systems use phase coherence and are therefore sensitive to frequency, amplitude, and phase. Examples are 1. Medical ultrasound imaging2. Military sonar 3. Exploration seismology4. Receiver function imaging5. Scattered wave imaging in teleseismic studies

3. Mixed systems1. Sumatra earthquake source by Ishii et al. 2005

See Blackledge, 1989, Quantitative Coherent Imaging: Theory and Applications

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Direct imaging with seismic waves

Images based on this model of 1 D scattering are referred to as common conversion point stacks or

CCP images

Converted Wave Imaging from a continuous, specular, interface

From Niu and James, 2002, EPSL

13CCP Stacking blurs the subsurface

The lateral focus is weak

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Fresnel Zones and other measures of wave sampling

• Fresnel zones• Banana donuts • WavepathsAre all about the same thingThey are means to estimate

the sampling volume for waves of finite-frequency, i.e. not a ray, but a wave.

Flatté et al., Sound Transmission Through a Fluctuating Ocean

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Moho

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Moho 410

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Moho

1p2s

2p1s 1p2s

Moho

T0=3s

1. Lateral heterogeneity is blurred

2. Mispositioning in depth from average V(z) profile

3. Pulses broaden due to increase in V with z

4. Crustal multiples obscure upper mantle

5. Diffractions stacked out

VE=1:1

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Scattered wave imaging

Based on diffraction theory: Every point in the subsurface causes scattering

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Scattered wave imaging improves lateral resolutionIt also restores dips€

R = zλ + λ2

4 ⎛ ⎝ ⎜ ⎞

⎠ ⎟

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USArray: Bigfoot

x = 70 km, 7s < T < 30s

21S. Ham, unpublished

SlabMultiples

SlabMultiples

Mismatch

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Resolution Considerations

• Earthquakes are the only inexpensive energy source able to investigate the mantle

• Coherent Scattered Wave Imaging provides about an order of magnitude better resolution than travel time tomography. For wavelength

• Rscat~/2 versus Rtomo ~ (L)1/2

• For a normalized wavelength and path of 100• Rscat~ 0.5 versus Rtomo~ 10• The tomography model provides a smooth

background for scattered wave imaging

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• The receiver function:

– Approximately isolates the SV wave from the P wave

– Reduces a vector system to a scalar system– Allows use of a scalar imaging equation with

P and S calculated separately for single scattering: From Barbara’s lecture:

€

ru P =∇φr u SV =∇ ×

r Ψ :

r Ψ = 0,ψ ,0( )

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Three elements of an imaging algorithm

1. A scattering model 2. Wave (de)propagator

• Diffraction integrals (Wilson et al., 2005)• Kirchhoff Integrals (Bostock, 2002; Levander et al.,

2005)• One-way and two-way finite-difference operators

(Stanford SEP)• Generalized Radon Transforms (Bostock et al., 2001)• Fourier transforms in space and/or time with phase

shifting (Stolt, 1977, Geophysics)3. A focusing criteria known as an imaging condition

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LG(x,t; xs )=δ(t)δ(x−xs)LG(x,ω ) =1

G=L−1

Lu=s(ω)

u=L−1s(ω)u=Gs(ω)

Depropagator: assume smooth coefficientsi.e., c(x,y,z) varies smoothly. Let L be the wave equationOperator for the smooth medium

The Green’s function is the inverse operator to the wave equationFor a general source:

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The Born ApproximationPerturbing the velocity field perturbs the wavefieldc= c+δc -> δL contains δc

[L +δL](u+δu) =s(ω)

Lu+ Lδu+δLu=s(ω)+O(δ )Lu≅sLδu≅−δLu

δu=−L−1δLuδu=−GδLu

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Born Scattering

€

1

c 2 (x)

∂ 2U (x, t)

∂t 2 −∇2U (x, t) = f (t)δ (x − xs)

c(x) = κ (x) /ρ o

x ∈ R n : n = 1,2,3

€

c(x) = co (x) +δc(x)

U (x, t) = Uo (x, t) +δU (x, t)

Perturb the velocity field

Assume a constant density scalar wave equation with a smoothly varying velocity field c(x)

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€

1

c 2

∂ 2δU

∂t 2 −∇2δU =2δc

c 3

∂ 2U (x, t)

∂t 2

Born Scattering: Solve for the perturbed field to first order

€

1

c 2 (x)

∂ 2U (x, t)

∂t 2 −∇2U (x, t) = f (t)δ (x − xs)

The total field consists of two parts, the response to the smooth medium

Plus the response to the perturbed medium δU

The approximate solution satisfies two inhomogenous wave equations. Note that energy is not conserved.It is the scattered field that we use for imaging, noise is anything that doesn’t fit the scattering model

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€

U PS =−V

4π

ω 2

α 2

α

β

⎛

⎝ ⎜

⎞

⎠ ⎟

2exp(−iω(t − r /β ))

r

×δZS

ZS

(sinϑ −β

αsin 2ϑ ) −

δβ

β(sinϑ +

β

αsin 2ϑ )

⎡

⎣ ⎢

⎤

⎦ ⎥

The scattering function for P to S conversion from a heterogeneityWu and Aki, 1985, Geophysics

€

ZS = ρβ

δZS

ZS

=δρ

ρ+

δβ

β

−ω 2 ⇔∂ 2

∂t 2

exp(−iω(t − r /β ))

r⇔

δ (t − r /β )

r

where

Shear wave impedance

Fourier Transform relations

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Regional Seismology: PNE DataHighly heterogeneous models of the CL

Nielsen et al, 2003, GJI

Background model is very smooth

U senses the smooth fieldδU senses the rough field

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Signal Detection:

Elastic P to S Scattering from a discrete heterogeneityWu and Aki, 1985, Geophysics

Specular P to S ConversionAki and Richards, 1980

Levander et al., 2006, Tectonophysics

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€

I(x) =r P

r S (x) = dω F1(ω)

uScat (x,ω)

uInc(x,ω)

⎡

⎣ ⎢

⎤

⎦ ⎥∫

Definition of an image, from Scales (1995), Seismic Imaging

€

RF (x r, t) = dω∫ F2(ω)uSV (xr ,ω)

uP (xr ,ω)

⎡

⎣ ⎢

⎤

⎦ ⎥exp(−iωt)

Recall from the RF lecture the definition of a receiver function

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A specific depropagator: Diffraction Integrals Start with two scalar wave equations, one homogeneous:

€

∇2U(r, t) −1

c2(r)

∂2

∂t2U(r, t) = 0

€

∇2G(r,ro, t) −1

c2(r)

∂2

∂t2G(r,ro, t) = −4πδ(t)δ(r − ro)

and the other the equation for the Green’s function: the response to a singularity in space and time:

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Fourier Transform with respect to timegiving two Helmholtz equations

€

∇2U(r,ω ) + k2U(r,ω ) = 0

€

∇2G (r,rs ,ω) + k 2G (r,rs ,ω) = −4πδ (r − rs)

where k = ω/c

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Apply Green’s Theorem, a form of representation theorem:

€

dV (r) U (r,ω)∇2G (r,rs ,ω) − G (r,rs ,ω)∇2U (r,ω)[ ]∫∫∫

€

= dS0 (r0)∫∫ U (r0,ω)∂G (r0 ,rs ,ω)

∂n− G (r0,rs ,ω)

∂U (r0 ,ω)

∂n

⎡

⎣ ⎢ ⎤

⎦ ⎥

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Given measurements or estimates of two fields U and G, and their normal derivatives on a closed surface S defined by r0, we can predict the value of the field U anywhere within the volume, rs.

The integral has been widely used in diffraction theory and forms the basis of a class of seismic migration operators.

Use the definition of the 3D delta function to sift U(rs,ω)

€

U (rs ,ω) =−1

4πdS0 (r0)∫∫ [U (r0,ω)

∂G (r0,rs ,ω)

∂n− G (r0,rs ,ω)

∂U (r0 ,ω)

∂n]

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We need a Green’s function

The free space asymptotic Green’s function for variable velocity can be written as

€

GFS (r,rs , t) = A(r,r' )δ t −τ (r,rs)( )

GFS (r,rs ,ω) = A(r,r' ) exp(iωτ (r,rs))

€

GFS (x, t;xt ) =δ t − r − rs /c( )

4π r − rs

The constant velocity free space Green’s function is given by

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Here (r,rs) is the solution to the eikonal equation:

And A(r,rs) is the solution to the transport equation:

The solution is referred to as a high-frequency or asymptotic solution. Both equations are solved numerically

€

∇ (r,rs) =1

c(r)

€

2∇τ ⋅∇A − ∇2τ( )A = 0

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€

G (ro ,rs ,ω) = A(r0 ,rs) exp(iωτ (r0 ,rs)) − A(r0 ,rs ' ) exp(iωτ (r0 ,rs ' )

We want a Green’s function that vanishes on So, or

one whose derivative vanishes on So

Recenter the coordinate system so that z=0 lies in SA Green’s function which vanishes at z=0 is given by:

Where r’s is the image around x of rs. At z=0

€

∂G

∂n= 2

∂GFS

∂n z=0

This is also approximately true if So has topography with wavelength long compared to the incident field. This is called Kirchhoff’s approximation.

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The diffraction integral becomes

€

U(r,ω ) =−1

2πdS0∫ U(r0 ,ω )[

∂A

∂n+ iωA

∂τ

∂n] exp(iωτ )

Assume that the phase fluctuations are greater than the amplitude fluctuations we can throw away the first term. This is a far field approximation:

€

U (r,ω) =iω

2πdS0∫ U (r0 ,ω)

A(r0 ,r) cosθ

c(r0)exp(iωτ )

This is the Rayleigh-Sommerfeld diffraction integral(Sommerfeld, 1932, Optics)

If we use it to backward propagate the wavefield, rather than forward propagate it, it is an imaging integral.

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Imaging ConditionThe imaging condition is given by the time it took the source

to arrive at the scatterer plus the time it took the scattered waves to arrive at the receiver. If we have an accurate description of c( r) then the waves are focused at the scattering point. (attributed to Jon Claerbout)

The image is as good as the velocity model.

For an incident P-wave and a scattered S-wave, and an observation at ro, then the image would focus at point r, when

€

0 = τ p (r,re ) +τ s (ro ,r) − te (ro ,re )

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€

I (r) =1

4π 2 dω−∞

∞

∫ (iω) dS0− L

L

∫ exp(−iω(t − (τ p +τ s )))U (r0 ,ω)A(r,r0) cosθ

c(r0)t=τ p+τ s

Because the phase is zero at the imaging point, the inverse Fourier transform becomes a frequency sum

Recall that U is a receiver function and the desired image is

€

I (x) =r P

r S (x) = dω F1(ω )

u SV (x,ω )

uP (x,ω )

⎡

⎣ ⎢

⎤

⎦ ⎥∫

€

SPS (r) = AS (r) /AP (r)

Define

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€

I (r) =r P

r S (r) =

1

4π 2 dω−∞

∞

∫ (iω) dS0− L

L

∫ exp(−iω(t − (τ P +τ S ))RF (r0,ω)AS (r,r0) cosθ (r0)

AP (r,re )β (r0)t= te

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S(x,z)

P(x,z)

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Sensitivities• Spatial aliasing and Aperture

– First Fresnel zone has to be well sampled in a Fourier sense

– Generally 2L > ztarget We need lots of receivers

€

RF ≈ zt argetλ

Δx =βmin

2 fmax sinθ

€

Ninstruments = M *2R f /Δx

Ninstruments = M *4 fmax sinθ f zt arg etλ

βmin

Ex: =3.6, f=0.5, 45 degx = 5 km

L~1000 km

400 stations

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• Data redundancy– Redundancy builds Signal/(Random Noise) as

N0.5, where N is the number of data contributing at a given image point: N=Nevent*Nreceiver

– Events from different incidence angles, i.e. source distances, suppresses reverberations

– Events from different incidence angles reconstruct a larger angular spectrum of the target

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Other considerations

1. A good starting velocity model is required– Use the travel-time tomography model

2. Data preconditioning– DC removal – Frequency filter– Remove the direct wave and other signals that

don’t fit the scattering model– Wavenumber filters (ouch!)

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Vp~ 7.75 km/s1% Partial Melt

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Jemez Lineament: Mazatzal-Yavapai BoundaryJemez Lineament: Mazatzal-Yavapai

Zone of melt production

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Structures from basalt extraction

Oman ophiolite

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RISTRA Receiver FunctionsDeepProbe

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Example: Kaapvaal Craton

x ~ 35 kmAL ~ 20o

9 Eqs

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Levander et al., 2005,AGU Monograph

58RF from Niu et al., 2004, EPSL

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Bostock et al.,

2001, 2002

Generalized Radon Transform

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Alaska Subduction Zone

dVp/Vo (Ppdp)dVs/Vo (Pds, Ppds, Psds)

dV/V0 (%)

0 10-10

Rondenay et al., 2008

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CMB Imaging using ScS precursors and the Generalized Radon Transform

Wang et al., 2006Ma et al., 2008

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D’’ Wang et al., 2006; Ma et al, 2008

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SummaryData redundancy suppresses noise.Noise is whatever doesn’t fit the scattering model.

The image is only as good as the 1. The velocity model 2. The completeness of the dataset3. The assumptions the imaging system is based on.4. We use a single-scattering assumption