LA-UR-17-30672

Approved for public release; distribution is unlimited.

Title: Introduction to Seismic Tomography

Author(s): Rowe, Charlotte Anne

Intended for: Lecture to graduate students, Univ. de Guadaljara Coastal Campus

Geophysics Program

Issued: 2017-11-21
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 INTRODUCTION TO SEISMIC TOMOGRAPHY

Presented to SisVOc by CHARLOTTE ROWE 22 Noviembre 2017

WHAT IS TOMOGRAPHY?
TOMOGRAPHY IS A METHOD OF
OBTAINING AN IMAGE OF A 3D OBJECT BY
OBSERVING THE BEHAVIOR OF ENERGY
TRANSMISSIONS THROUGH THE OBJECT.

THE IMAGE IS OBTAINED BY

INTERROGATING THE OBJECT WITH
ENERGY SOURCES AT A VARIETY OF
LOCATIONS AND OBSERVING THE
OBJECT’S EFFECTS ON THE ENERGY AT A
VARIETY OF SENSORS.

TOMOGRAPHY WAS FIRST

USED TO BUILD 3-DIMEN-
SIONAL SCANS THROUGH
HUMAN BODIES. THESE
ARE CALLED COMPUTED
TOMOGRAPHIC (CT) SCANS.
WHAT IS SEISMIC
TOMOGRAPHY?
WHEN WE TALK ABOUT SEISMIC
TOMOGRAPHY WE USUALLY MEAN
SEISMIC WAVE SPEED TOMOGRAPHY –
WE ARE TRYING TO BUILD A 3D
REPRESENTATION OF HOW FAST SEISMIC
WAVES PROPAGATE THROUGH THE
EARTH.

WE CAN ALSO PERFORM TOMOGRAPHY

TO FIND SEISMIC ATTENUATION
CHARACTERISTICS OF THE EARTH – BUILD
A THREE-DIMENSIONAL MODEL OF HOW
THE SEISMIC WAVES ARE ATTENUATED AS
THEY PASS THROUGH THE EARTH.

BUT TODAY LET’S JUST TALK ABOUT

SEISMIC WAVE SPEED.
WHY DO WE NEED TO KNOW SEISMIC
WAVE SPEEDS?
1) KNOWING SOMETHING ABOUT SEISMIC WAVE
SPEED TELLS US SOMETHING ABUOT THE
PROPERTIES OF THE ROCKS
2) WE NEED TO USE SEISMIC WAVE SPEEDS
(VELOCITY) IN ORDER TO CALCULATE THE
LOCATION OF AN EARTHQUAKE WHOSE SOURCE
IS OF INTEREST TO US.

HOW WELL MUST WE KNOW THE VELOCITY?

THIS DEPENDS ON WHAT KIND OF PROBLEM WE

ARE TRYING TO SOLVE.

TO INTERPRET GENERAL, BULK PROPERTIES OF THE

EARTH, IF OUR VELOCITY ESTIMATE IS PRETTY
CLOSE TO CORRECT AND IS GENERALIZED OVER A
LARGE VOLUME, IT IS PROBABLY GOOD ENOUGH.
HOW MUCH DO WE NEED TO KNOW
ABOUT SEISMIC WAVE SPEEDS IN THE
EARTH?
TO OBTAIN A ROUGH IDEA OF WHERE AN
EARTHQUAKE HAS OCCURRED, ESPECIALLY AT
TELESEISMIC DISTANCES, A SIMPLE MODEL IS
PROBABLY GOOD ENOUGH.

THE MODEL SHOWN HERE IS WHAT WE

CALL A ONE-DIMENSIONAL MODEL. YES,
IT IS ACTUALLY A 3D OBJECT, BUT THE
SEISMIC VELOCITY ONLY VARIES IN ONE
DIRECTION – RADIALLY. WE CALL THIS A
RADIAL 1-D MODEL.

SUCH MODELS WERE OBTAINED BY

LOOKING AT THE VARIOUS SEISMIC WAVES
FROM LARGE, DISTANT EARTHQUAKES,
AND MEASURING THE TIME IT TOOK THEM
TO TRAVEL FROM THE EARTHQUAKE TO
SEISMOMETERS AT MANY DISTANCES AROUND THE
WORLD.
EXAMPLES OF REFERENCE EARTH MODELS
HERE ARE THREE STANDARD REFERENCE EARTH MODELS (ONE-DIMENSIONAL, RADIALLY
SYMMETRIC EARTH). THE ONES WE KNOW BEST ARE CALLED AK135, iasp91 AND PREM. THEY
ARE ALL VERY SIMILAR, ESPECIALLY AT DEPTH, BUT THEY HAVE SOME VARIATIONS AT
SHALLOWER LEVELS.

ALL THREE REFERENCE MODELS

ARE VERY GOOD MODELS OF
AVERAGE SEISMIC WAVE SPEED
FOR THE WHOLE EARTH.

NOT ONE OF THESE

MODELS IS A VERY GOOD
REPRESENTATION OF THE
EARTH IN ANY PARTICULAR
PLACE.

SO FOR PREDICTING THE TIME

IT TAKES A SEISMIC WAVE TO
TRAVEL OVER A GREAT DISTANCE,
THEY ARE OKAY.

FOR SMALL REGIONAL OR LOCAL STUDIES THEY ARE TERRIBLE.

AS WE WORK AT SMALLER SCALES, THE EARTH BECOMES
MORE COMPLICATED.

INSTEAD OF GLOBALLY UNIFORM MANTLE

DEPTH, WE BEGIN TO SEE DIFFERENCES IN
CRUSTAL THICKNESS AT REGIONAL AND LOCAL
SCALES.

INSTEAD OF LATERALLY HETEROGENEOUS

AND UNIFORM LAYERS OF “CRUST” AND
“MANTLE” AND “CORE,” WE ARE DEALING
WITH MANY GEOLOGIC VARIATIONS IN THE
FABRIC, LITHOLOGY, STRENGTH AND CON-
TINUITY OF ROCKS.

THERE ARE FAULTS AND

BASINS AND INTRUSIONS. ALL OF THESE
THINGS AFFECT THE SEISMIC WAVE SPEEDS
AND MUST BE ACCOUNTED FOR IN OUR
MODELS USED FOR EARTHQUAKE LOCATION.
TO BEGIN WITH, WE NEED DATA FROM EARTHQUAKES. WE MUST HAVE HIGH QUALITY
PICKS OF P OR P AND S WAVES, AND REASONABLE ESTIMATES OF EARTHQUAKE LOCATIONS
SO THAT WE CAN FIND THE TRAVEL TIME OF THE P AND S PHASES FROM THE SOURCE TO
THE STATION.
SO WE JUST TRACE THE WAVES THROUGH THE ROCKS AND WE
CAN GET THE VELOCITIES, RIGHT?
DISTANCE TO STATION DIVIDED BY TRAVEL TIME IS THE
A VELOCITY IN KM PER SECOND.

So V(A) = length(A) / traveltime(A)

And V(B) = length(B) / traveltime(B)
B

BUT WHAT ABOUT A LAYERED EARTH?

V(A) = length(A) / traveltime(A)

For B we can estimate the average velocity:
A
V(B) = [length(pink) + length(white)] / (total traveltime)
Layered V(B) will not be the same as V(A).
Earth B
V(Bw) = V(A)
V(Bp) = length(pink) / traveltime(pink)

How can we determine this if we don’t know length(pink)?

HOW DO WE DETERMINE LENGTH(PINK)?

WE CAN TRY A FORWARD MODEL.

GUESS AT THE LENGTH. CALCULATE THE EXPECTED TRAVEL TIME IF WE ASSIGN A VELOCITY TO
PINK. ADJUST VELOCITY AND LENGTH OF PINK UNTIL WE GET AN ANSWER THAT FITS THE
OBSERVATIONS.

IS THIS THE UNIQUE, CORRECT ANSWER? NO, OF COURSE NOT.

IF WE ASSUME A DIFFERENT VELOCITY FOR PINK, THEN TO FIT THE TRAVEL TIME, WE WOULD
NEED A DIFFERENT SIZE FOR PINK. ALTHOUGH WE KNOW WHAT V(Bw) IS, CHANGING V(Bp)
WILL REQUIRE A CHANGE OF length(W) AS WELL AS length(P) TO FIT THE OBSERVED TRAVEL
TIME

THE ANSWER IS THAT WE CLEARLY NEED MORE A

RAYS . WE NEED ENOUGH SOURCES AND RECEIVERS
(EARTHQUAKES AND SEISMIC STATIONS), IN THE CORRECT
POSITIONS, SO THAT WE CAN CONSTRAIN THE SIZE OF Layered
THE PINK LAYER BY ADJUSTING ITS SIZE AND VELOCITY Earth B
UNTIL WE FIT ALL THE KNOWN TRAVEL TIMES.

WE WILL NEVER FIT THEM ALL PERFECTLY. WE SEEK TO

MINIMIZE THE MISFIT.
RESOLVING THE SIZE AND VELOCITY OF THE PINK LAYER IS AN EXERCISE IN
REGRESSION. A SIMPLE EXAMPLE OF REGRESSION TO MINIMIZE A MISFIT FOR MANY
DATA POINTS IS THE FITTING OF A STRAIGHT LINE THROUGH DATA POINTS.

TO BEST FIT THE DATA WE WANT TO FIND THE SLOPE AND INTERCEPT DEFINING A
LINE THAT MINIMIZES ALL THE
MISFITS (DISTANCE FROM OUR LINE)
FOR ALL DATA POINTS.

NO DATA ARE FIT PERFECTLY

BUT WE HAVE A GOOD GUESS
AT THE FUNCTION THAT FITS
THE DATA.

THE REMAINING
MISFIT MAY COME
FROM ERRORS IN THE
DATA MEASUREMENTS.
IT MAY MEAN THAT A
STRAIGHT LINE IS NOT
THE CORRECT FUNCTION
(MODEL) TO USE.
COULD WE FIT ALL THE DATA PERFECTLY?
SURE. LOOK AT THE YELLOW FUNCTION. THIS FITS THE DATA PERFECTLY.

WHY IS THIS YELLOW SOLUTION PROBABLY INCORRECT?

IT PROBABLY MAKES NO PHYSICAL SENSE.

IT COULD BE FITTING DATA ERROR,

FORCING THAT TO BECOME PART
OF THE MODEL.

WE HAVE TO BRING SIMILAR

CONSIDERATIONS TO OUR
TOMOGRAPHY PROBLEM.

IN THE END WE HAVE TO

BALANCE FITTING THE
DATA AGAINST FINDING
A SENSIBLE MODEL.
 PARAMETERIZING OUR MODEL

HERE WE SHOW TWO SOURCES AND A NETWORK OF

STATIONS ON THE SURFACE. THE EARTH WE ARE
MODELING HAS BEEN DIVIDED INTO SEVERAL BOXES,
EACH OF WHICH WE COULD ASSIGN A SEISMIC VELOCITY.
WE SAY THAT THE MODEL HAS BEEN PARAMETERIZED
AS A VOLUME OF DISCRETE CELLS OR ELEMENTS.

EACH ELEMENT THAT HAS MULTIPLE RAYS CROSSING IT

CAN PROBABLY BE RESOLVED AND A VELOCITY FOR THAT
CELL CAN BE FOUND.

IF OUR CELL SIZES ARE TOO SMALL ,THEN MANY WILL NOT BE SAMPLED AND CANNOT BE
RESOLVED.

IF OUR CELL SIZES ARE TOO LARGE, THEN WE WILL MISS OUT ON SMALL VELOCITY
CHANGES; THEY WILL BE AVERAGED INTO A LARGER BOX.
 SOME OTHER CONSIDERATIONS

WE MAY CHOOSE TO
PARAMETERIZE OUR 3D
MODEL AS A GRID OF
NODES, OR POINTS, EACH OF
WHICH DEFINES VELOCITY AT
A POINT, INSTEAD OF AS A
VOLUME OF BOXES HAVING
DISCRETE VELOCITIES. THIS
APPROACH THEN ASSUMES
THAT THE VELOCITY FROM
ONE NODE TO THE NEXT
VARIES LINEARLY.

WE MAY CHOOSE TO
PARAMETERIZE OUR MODEL WITH VARIABLE NODE DENSITY (OR BOX
SIZES), SO IN PLACES WHERE WE EXPECT TO HAVE MANY CROSSING RAYS, WE CAN
HAVE A MODEL WITH MORE DETAIL.
 WE WORRY A LOT ABOUT RAY COVERAGE

HERE WE SEE A RAY DIAGRAM FOR

A TOMOGRAPHIC EXPERIMENT IN
AUSTRALIA. PURPLE DOTS SHOW
THE SEISMIC STATIONS USED, AND
THE BLACK LINES ARE RAYS FOR
THE EARTHQUAKES USED IN THE
STUDY.

THE CONTINENT AND AREA JUST

EAST OF IT MAY BE WELL-RESOLVED, BUT WE CAN SEE THAT THERE ARE VERY FEW CROSSING
RAYS IN THE OCEAN, ESPECIALLY TO THE SOUTH AND WEST. FOR THIS PART OF THE EARTH IT
WILL NOT BE POSSIBLE TO OBTAIN A RELIABLE VELOCITY MODEL USING THESE DATA.
ANOTHER COMPLICATION….

SEISMIC WAVES ARE AFFECTED BY VELOCITY

CHANGES THROUGH WHICH THEY TRAVEL.

THEY DO NOT KEEP TRAVELING IN THE SAME

DIRECTION. YES – THEY CHANGE DIRECTION
WHEN THEY ENCOUNTER A CHANGE IN VELOCITY.

SEISMIC RAYS BEND WHEN THEY MOVE FROM

ONE VELOCITY MEDIUM INTO ANOTHER. THE
AMOUNT OF BENDING DEPENDS ON THE
VELOCITY CONTRAST AND IS GOVERNED BY
SNELL’S LAW, WHICH DEFINES A RELATIONSHIP

WHERE THETA(1) AND THETA(2) ARE INCIDENCE

AND REFRACTED ANGLES AND v1 AND v2 ARE
THE RESPECTIVE VELOCITIES.
ANOTHER COMPLICATION….

NOT ONLY MUST WE WORRY ABOUT OUR RAY COVERAGE WHEN WE WANT TO SOLVE
A TOMOGRAPHIC PROBLEM, BUT WE HAVE TO KNOW THAT THE RAY PATHS ARE NOT STRAIGHT.

WHEN WE PERFORM OUR TOMOGRAPHY AND WE CHANGE THE MODEL, THEN OUR RAYS WILL
ALSO CHANGE.

SO WE ONLY ALLOW THE MODEL TO CHANGE A LITTLE, THEN WE TRACE THE NEW RAYS
THROUGH THE NEW MODEL AND ADJUST THE MODEL AGAIN, THEN WE TRACE NEW RAYS. AND
THEN WE DO IT AGAIN. AND AGAIN….. THIS ITERATIVE PROCESS DEPENDS ON CAREFUL CHOICE
OF MODEL STABILITY (DAMPING) AND REQUIRING THE MODEL TO BE PHYSICALLY SENSIBLE
(SMOOTHING). IT IS AN ART, AND REQUIRES UNDERSTANDING THE GEOLOGY, THE DATA
AND THE BASIC MATHEMATICS OF THE PROCESS.
OH, AND ANOTHER THING….

IF WE ARE USING LOCAL EARTHQUAKES AND STATIONS TO SOLVE FOR A LOCAL VELOCITY
MODEL, WE HAVE TO UNDERSTAND THAT ONCE WE HAVE CHANGED THE VELOCITY MODEL,
THEN THE TRAVEL TIMES WE HAVE USED, BASED ON OUR INITIAL EARTHQUAKE HYPOCENTER
LOCATIONS, WILL HAVE CHANGED.

IF THE TRAVEL TIME FOR THESE HYPOCENTERS CHANGE, THEN THE LOCATION ESTIMATES NEED
TO CHANGE.

SO AN IMPORTANT STEP IS TO RE-LOCATE ALL THE EARTHQUAKES USED IN THE TOMOGRAPHIC

MODEL, USING THE NEW MODEL.

ONCE WE HAVE RE-LOCATED THE EARTHQUAKES, THEN WE OBTAIN NEW HYPOCENTERS, NEW
TRAVEL TIMES (IN THE NEW MODEL) AND WE MUST PERFORM THE TOMOGRAPHY AGAIN TO
UPDATE THE MODEL BASED ON OUR NEW EARTHQUAKE LOCATIONS.

REPEAT UNTIL THE MODEL STOPS CHANGING AND THE HYPOCENTERS DO NOT MOVE ANY
MORE.
 THINGS TO CONSIDER FOR 3D SEISMIC TOMOGRAPHY
• DECIDE WHAT YOUR TARGET IS

• ASSESS YOUR SOURCES AND RECEIVERS AND THE GEOMETRY.

• DO YOU HAVE ENOUGH DATA? DO YOU HAVE THE RIGHT DATA? CAN YOU GET MORE DATA?

• YOU NEED TO PARAMETERIZE YOUR MODEL AREA. GRID? CELLS? WHAT SIZE?

• YOU NEED A STARTING MODEL. PROBABLY A SIMPLE 1-D LAYERED MODEL WILL DO, UNLESS
YOU KNOW MANY DETAILS ABOUT YOUR AREA ALREADY. THE STARTING MODEL SHOULD BE
AS CLOSE TO CORRECT AS POSSIBLE. REMEMBER, TINY CHANGES.

• DECIDE WHAT TOMOGRAPHY CODE IS APPROPRIATE FOR YOUR NEEDS.

• PERFORM DATA QUALITY CONTROL. IF YOU START OUT WITH POOR EARTHQUAKE
LOCATIONS AND BAD PICKS, YOUR INVERSION WILL FAIL OR YOUR RESULTING MODEL WILL
BE NONSENSE.

• COMPARE YOUR RESULTS WITH OTHER STUDIES IN THE TARGET AREA. COMPARE TO KNOWN
GEOLOGIC AND TECTONIC FEATURES. IS IT CONSISTENT WITH GRAVITY , MT, AND OTHER
INDEPENDENT ANALYSES? HOW DOES IT DIFFER FROM OTHER SEISMIC MODELS?
THANK YOU