Seeing through rock: the mathematics of Reﬂection...

Seeing through rock: the mathematics of

Reflection Seismology

William W. Symes

Reflection seismology is...

• the main prospecting tool of the oil and gas industry

• also useful in civil/environmental engineering and academic earth science

• a $4 billion / year industry

• a big deal in Houston

• rich in mathematical ideas and questions

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The basic idea: collect echoes...

and - through various mathematical operations - convert them to maps of the Earth’sstructure. Infer location of oil and/or gas from structuralmaps, sometimes fromcharacter of echoes also.

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Marine Operations90% of all datacollected worldwide

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Data also acquired on land, but it’s harder...

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Data, data everywhere

Seismic surveys producevastquantities of data!

• each geo/hydro phone collects perhaps 1500 digitized samples per experiment(“shot”).

• each experiment deploys 100’s of geo/hydro phones.

• each survey consists of 100’s (1980’s) to 100,000’s (today)of shots.

• upshot: 750 MB = small, old; 150 GB = modest, contemporary

• equivalent: 1 - 300 CD’s. Immediate future: 100’s of DVD’s!

This data gets turned into...

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An Image of the Earth’s Interior

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(structural image, Mississippi Canyon, offshore Louisiana)

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How do you do that?

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Outline

• Acoustic fundamentals

• Zero offset imaging

• Multioffset imaging

• Velocity via redundancy

• Optimize this!

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Acoustic fundamentals

Basic facts:

• sound ispressurefluctuations about equilibrium - quantify in lb/ft2 or N/m2

• sound isoscillatory- rapid sequence of positive, negative excursions

– A below middle C = 440 cycles/sec (“Hertz”, Hz)

– typical frequency range of significant energy in seismic exploration: 5 - 70Hz

• ⇒ represent pictorially as grey - dark = higher pressure, light = low pressure,plotted vs. time.

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Recording of single hydrophoneNorth Sea survey (thanks: Shell)

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Shot record = recording ofsingle experiment, 64 hydrophonesNorth Sea survey (thanks: Shell)

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CDP 8501

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• sound (in rock or in air) is awave phenomenon

• sound travels at a definite speed, characteristic of material

– sound speed in air∼ 330 m/s∼ 1000 ft/s; so count seconds after lightningflash until thunder arrives, multiply by 1000 to get distancein feet to lightningstrike

• reflection seismographs recordechoes⇒ sound must travel fromsound source(dynamite, air gun) to subsurface structureand backto geo/hydro phone.

• echoes (orreflections) occur when sound reaches a change in mechanical char-acteristics (eg. a change in sound speed!) - air to marble, sandstone to limestone,water-saturated sandstone to gassy sandstone,...

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Offset= distance from source to receiver. Picture shows “near” and“far” offsets.

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Zero Offset

• source and receiver in same place

• can’t really acquire (dynamite!)

• but can fake it by standard techniques

• easiest to understand how to build image...

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Out and back...

/2vt

v = sound speed (orvelocity) - 330 m/s in air, 1500 m/s in water, 2000 - 5000 m/sin sedimentary rock.

If you hear an echo at timet after seismic wave initiation at zero offset, then thesource of the echo must lie somewhere on a circle of radiusvt/2.

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Finding and old shoe...

here!old shoe lies

Time the echo from the old shoe... “hello!” from each source/receiver point, listenat same place, record arrival time of echo.

Shoe must lie on circles of various radii, computed as in previous slide - thereforeon their intersection!

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However...

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Earth structure does not resemble an old shoe -extended, distributed

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Physics of sound reflection - Zero Offset

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vt

Reflection occurs at the place and time at which the wavefrontis tangentto thereflecting surface - thespecular(mirrorlike) reflection principle. Radius fromsource/receiver point isnormal(perpindicular) to surface.

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Locating a reflecting surface - zero offset

/2vt

Reflecting surface must be tangent to each wavefront circle -envelopeof circles.

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Another issue

Do you really want to find thetimes of all the echoes?

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Direct construction of images -ZO Migration

Concept:

(1) for each source/receiver point each sample time t (usually multiples of 2 ms or4 ms),placethe sample value at all possible image points at distancevt/2 (circle).

(2) then sum the resulting images, one for each source/receiver point.

Each image will exhibit circular wavefront.

Principle of Stationary Phase:Sum of oscillatory signalscancels, exceptalongenvelope of wavefronts.

Consequence: after the summation,only the envelope is visible!!!.

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ZO Migration Example (1/4)

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Zero offset simulated data for reflecting surface at depthz = 0.3 km at left edge(x = 0 km), sloping upwards to the right at 5 deg. 101 source/receiver points,spaced 10 m apart, from left to right.

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(1) for each source/receiver point each sample time t (usually multiples of 2 ms or4 ms),placethe sample value at all possible image points at distancevt/2 (circle).

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(2) then sum the resulting images, one for each source/receiver point (this figureshows 5 of the 101 source/receiver images added up, at 0.3, 0.4, 0.5, 0.6, 0.7 kmfrom left edge).

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(2) then sum the resulting images, one for each source/receiver point (this figureshows all 101 source/receiver images added up - the wavefront envelope is recov-ered, sitting exactly on the reflecting surface!)

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Multioffset Migration

Multiple Offsets= data that’s actually measured. Can eitherfake ZO data frommultioffset data (1970’s technology) or directly use MO data - routine today.

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Physics of sound reflection - Nonzero Offset

Recvr

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� � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

Source

Reflection occurs at the place and timet at which ellipse with focii at source, recvr,diam =vt is tangentto the reflecting surface - thespecular(mirrorlike) reflectionprinciple. Bisector of lines from source and receiver points isnormal (perpindicu-lar) to surface (Snell’s law) ⇒ surface isenvelopeof ellipses.

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Direct construction of images -MO Migration

Concept:

(1) for each source/receiver point each sample time t (usually multiples of 2 ms or4 ms),placethe sample value at all possible image points on ellipse of diametervt.

(2) then sum the resulting images, one for each source/receiver pair.

Each image will exhibit elliptical wavefronts.

Principle of Stationary Phase:Sum of oscillatory signalscancels, exceptalongenvelope of wavefronts.

Consequence: after the summation,only the envelope is visible!!!.

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And that’s how we turned lots of this...

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(one of 500 data panels from Exxon GofM data = 750 MB.)

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into this!

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(structural image, Mississippi Canyon, offshore Louisiana)

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Two major details

(1) Q. how do you knowv? A. You don’t!

(2) Spatial variation of velocity and refraction of waves (some other time...)

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Velocity via redundancy (1/6)

• Typical survey contains source-receiver pairs with a variety of offsets.

• Any one set of ellipses with fixed offset (source-receiver distance) is enough todetermine reflecting surfaces by forming envelopes -common offsetconstruction

• the envelopes constructed using different offsets will only agree if the velocityis correct!

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Black circle, ellipse: zero, nonzero offset, correct velocity - both have dashed blackline as possible envelope.

Blue circle, red ellipse: same offsets, velocity 50% high. Envelopes aredifferent,so predicted depths of reflecting surface atsameposition will disagree.

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You can see this with the data directly:

• sort the data recordings into piles, each pile (“gather”) sharing common offset;

• carry out migration for each common offset gather separately: place data valueat timet on ellipse of diametervt, sum together to getcommon offset image(justlike we did for zero offset - a special case!);

• from each image, extract the part of the image that lies belowa specific point onthe surface - a “virtual well”;

• if the velocity is correct, then all “virtual wells” will look same! To compare,put them together, one for each offset, to form acommon image gather. Look atit!

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Synthetic experiments - data from different offsets

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“Virtual wells” displayed next to each other: Left, correctv; Right, 10% low.

“Correct velocity flattens common image gathers” - the practical principle for find-ing sound velocity from seismic data.

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Works with actual data!

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Optimize This!

If you can measure “flatness” somehow, then you can turn image-gather-flatteningtask into optimization problem:minimize a measure of un-flatness of image gathers

An obvious possibility, apr̀es calculus: measure un-flatness by mean square ofderivative in offset of image gathers≡ thedifferential semblanceobjective (func-tion of assumed velocity, data).

Stolk & WWS Inverse Problems03: differential semblance issmooth in velocityand data jointlyandis essentially the only quadratic form in the image gather withthis property - so the only one for which you can use Newton’s method! [theory ofoscillatory integral operators]

Generally: provided a strong enough dichotomy between length scale in velocitymodel and wavelengths in data, can show local uniqueness of global minimizer. Insome special cases,all local minimizers are global![multiscale asymptotics]

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Some Obvious Questions

• what does all of this have to do with physics?

• waves obey the wave equation (acoustic, elastic, whatever)- where is the waveequation?

• what the heck is an image, really? what does it have to do with the mechanics ofrocks? why is it separate, apparently, from “the velocity”?

• I thought you did inverse problems - is there an inverse problem somewhere inhere?

• why not “image” the earth’s structure by modeling the data and fitting the modelpredictions to the data, say by least squares?

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Where to Find Some Answers

• CAAM Colloquium 13 September

• CIME Short Course,www.trip.caam.rice.edu > Downloadable Materi-als

• Mathematical Foundations of Reflection Seismology, ditto

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Credit where credit is due

Photos, drawings: Landmark Graphics, Schlumberger, Western Geco, Veritas DGC,Seismic Consultants, and WWS

Seismic data: Exxon Production Research Co., Shell International Research

Software used to produce data-derived graphics: Seismic Unix (Colorado Schoolof Mines), The Rice Inversion Project (Rice University)

Money: sponsors of The Rice Inversion Project, NSF, DoE

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