earthquakes: detection, location & focal geometry · 2011. 4. 20. · normal to the fault...

LECTURE 3

EARTHQ UAKES: DETECTION,

LOCATION & FOCAL GEOMETRY

© Northwestern University, 2007

EARTHQ UAKE LOCA TION

Retrieved (Inverted)from arri val t imesof BODY (principally P) wav es

The problem consists of determining

4 parameters

* L atitude

* L ongitude

[* Depth ]

* Origin Time

{ }EpicenterHypocenter

P times are usually easiest to pick

EXAMPLE: JAPAN SEA Event, 14 NOV 2005

Station AAK (Ala Archa, Kyrgyzstan); ∆ = 52. 4°

→ Gather such data for many stations

Obtain dataset ofobservedarrivals { oi }.

IN THE NEAR FIELD

• In the presence of a dense array, the

Easiest, Simplest, CrudestAlgorithm

consists of identifying

Epicenter ≈ Station withEarliest Arrival

Southern California Seismic Network[www.data.scec.org]

IN THE FAR FIELD

• Position of problem:

Retrieve

LatitudeλLongitudeφDepthhOrigin timet0

from dataset{ oi }.

• P −wave arrival times can be computed as functions ofsource and station parameters (Latitude and LongitudeΛi , Φi ) using amodel of Earth structure.

ci = f ( λ ,φ, h, t0; Λi , Φi )

• DIFFICULTY:

Function f is NON − LINEARNON − LINEAR.

LINEARIZING the PR OBLEM

• AssumeTrial Solution

λ 0 , φ0 , h0 , t00

and compute a set ofpredicted arrival times{ ci } forthat solution, based on a chosen Earth model (Jeffre ys-Bullen, PREM, iaspei91,etc.).

→ If all the data were perfect (no noise), as well as themodel, and we had guessed the right solution, then forall i , we should have ci = oi .

• Define RESIDUALS

δ ti = oi − ci = (o − c)i

Hopefully, theδ ti are small compared with the propagationtimesci − t0

0.

LINEARIZING the PROBLEM (2)

• Then try improving the solution from {λ 0 , φ0 , h0 , t00}

to {λ 1 , φ1 , h1 , t10}

λ 1

φ1

h1

t10

=

λ 0

φ0

h0

t00

+

δ λδ φδ h

δ t0

Again, we expect the termsδ . . . to be small, so that thechange in eachci is simply

δ ci =∂ f

∂λ⋅ δ λ +

∂ f

∂φ⋅ δ φ +

∂ f

∂h⋅ δ h +

∂ f

∂t0⋅ δ t0

• If we know the function f ("direct problem"), weshould be able to compute the partial derivatives such

as∂ f

∂λ.

• Ideally, we would like, for each station,δ ci to beexactlyδ ti = (o − c)i , so we seek to solve

. . .

. . .

. . .

. . .

δ ci

. . .

. . .

. . .

. . .

=

A

⋅

δ λδ φδ h

δ t0

=

. . .

. . .

. . .

. . .

δ ti

. . .

. . .

. . .

. . .


→→ The matrix A is simply made up of the partial

derivatives of f with respect to the source parameters.


The problem has beenLINEARIZEDbut it is still OVER-DETERMINEDas AA is a very tall matrix (4 columns (4unknowns) and tens or hundreds of rows (data points)).

→→ It can be solved by theclassical LEAST−SQUARESalgorithm:

δ λδ φδ h

δ t0

= (AT A)−1 ⋅ AT

. . .

. . .

. . .

. . .

δ ti

. . .

. . .

. . .

. . .

• Note that (AT A) is a 4× 4 matrix, and AT ⋅ δ t is a4−dimensional vector.

DETAILED LOOK at the MA TRIX A

The elements ofA are the partial derivatives of the arrivaltimes ci at stationi with respect to a change in a sourceparameter

• An easy case

∂ ci

∂ t0= 1 for all i

Otherwise, for a spherical Earth, the travel-time tP isfunction of the angular distance∆ to the station, and ofthe source depth,h.

• Source depth

∂ ci

∂ h= −

cos j i

VP(h)j is itself a function

of ∆ andh

NOTE that, at teleseismic distances,j is always a small angle...

DETAILED LOOK at the MA TRIX A (2)

• Latitude and Longitude

If we change the latitude byδ λ, we move the epicenterNorth by an amountδ λ ⋅ l deg. where l deg. = 111. 195km is the length of one degree at the Earth’s surface.

Thus, we change the distance to the stationi byδ∆i = −δ λ ⋅ cosα i , and

∂ ci

∂λ= − cosα i ⋅

∂ TP

∂ ∆

If we change the longitude byδ φ, we move Eastwards,but only by δ φ ⋅ cosλ ⋅ l deg., so that

∂ ci

∂φ= − sinα i ⋅ cosλ ⋅

∂ TP

∂ ∆

α i

δ λ

δ φ ⋅ cosλ

IN SUMMAR Y

• We can compute all the partial derivatives

• We can compute the matrixA

• We can compute(AT A) and invert it

• We can find the"best" change in earthquake sourceparameters to minimize the new residuals

• We can iterate the process until the solution stabilizes

LIMIT ATIONS of THIS ALGORITHM

• The matrix can be inverted only if it is

NON − SINGULARNON − SINGULAR

[ in practiceNOT APPROA CHING SINGULARITY]

• The matrix is singular if 2 rows (or columns) are identi-cal.

• Recall∂ ci

∂ t0= 1 and

∂ ci

∂ h= −

cos j i

VP(h)

If all the stations are at the same distance, then allj i arethe same and the two partials are proportional.

SINGULARITY !

• In practice, if all stations arefar away, then all j i aresmall ( < 10°; rays all take off nearly vertically at thesource), all cosj i ≈ 1 , and one has

PERFECT TRADE-OFF BETWEEN O.T. and DEPTH

In general, the inversion becomes unstable. The onlyway out is to

CONSTRAIN the DEPTH...

SIMILARLY

• Recall

∂ ci

∂λ= − cosα i ⋅

∂ TP

∂ ∆

and

∂ ci

∂φ= − sinα i ⋅ cosλ ⋅

∂ TP

∂ ∆

If all stations are in [approximately] the same azimuth. thetwo columns of partials are proportional, and the matrixfeatures [or approaches]singularity.

→→ STABLE LOCATIONS REQUIRE

A GOOD AZIMUTHAL COVERAGE

[This is usually not an issue for large events ]

α i

INFLUENCE of TRIAL SOLUTION

• If dataset is global, any trial solution (even the antipodes of the true epicenter)will lead to a converging algorithm [Okal and Reymond,2003].

• In practice, one can always use the station with earliest arrival as a trial epicenter.

USGS Earthquake Hazards Program: Phase Data: EASTERN SICHUA... http://neic.usgs.gov/neis/eq_depot/2008/eq_080512_ryan/neic_ryan_p.html

1 of 20 06/07/2008 10:33 AM

USGS HomeContact USGSSearch USGS

Earthquake Hazards Program

Phase Data

Explanation of Parameters

12 MAY 2008 (133)

ot = 06:28:01.49 +/- 0.11 EASTERN SICHUAN, CHINA lat = 30.986 +/- 2.9 lon = 103.329 +/- 2.2 MAGNITUDE 7.9 (GCMT) dep = 19.0 (geophysicist)

80 km (50 miles) WNW of Chengdu, Sichuan, China (pop 2,146,000) 145 km (90 miles) WSW of Mianyang, Sichuan, China (pop 396,000) 350 km (215 miles) WNW of Chongqing, Chongqing, China 1545 km (960 miles) SW of BEIJING, Beijing, China At least 68,800 people killed and severe damage in the Dujiangyan- Mianzhu-Mianyang area. Landslides blocked roads and buried buildings in the Beichuan-Wenchuan area. Felt (VII) at Chengdu, (VI) at Xi'an and (V) at Chongqing and Lanzhou. Felt in much of central, eastern and southern China, including Beijing, Guangzhou, Nanjing, Shanghai, Tianjin, Wuhan and in Hong Kong. Also felt in parts of Bangladesh, Taiwan, Thailand and Vietnam.

nph = 641 of1104 se = 1.23 FE=307 A

error ellipse = ( 21.8, 0.0, 3.9;111.8, 0.0, 2.7; 0.0, 0.0, 0.0)

mb = 6.9 (185) ML = 7.2 ( 1) mblg = 6.2 ( 8) md = 0.0 ( 0) MS = 8.0 (192)

sta phase arrival res dist azm amp per mag amp per mag sta ENH iPnd 06:29:18.09 -2.3 5.4 96 L:1.7+2 .62 7.2 ENH LSA ePnc 06:30:33.98 1.2 10.6 266 LSA CHTO ePn 06:31:04.15 2.3 12.8 199 CHTO eSn 06:33:26.95 S res = 3.5X QIZ ePnd 06:31:06.59 -2.5 13.3 152 QIZ CMAR Pn 06:31:06.73 0.2 13.1 199 CMAR Lg 06:34:53.58 LR 06:37:10.35 CM31 ePn 06:31:08.19 1.7 13.1 199 g:6.2+0 .93 6.0 CM31 BJT ePn 06:31:15.63 -0.3 13.8 46 BJT SONM Pn 06:31:56.91 -1.2 17.0 7 SONM Lg 06:37:07.97 TATO ePn 06:31:57.49 -2.2 17.1 106 b:1.8+4 1.5 7.0 TATO P 06:32:03.30 2.0X ULN ePnc 06:31:58.23 -1.3 17.1 9 b:4.2+4 2.4 7.1 ULN TPUB ePnd 06:31:59.08 -1.5 17.2 112 g:1.4+2 1.9 7.2X b:5.2+3 .95 6.6 TPUB SSLB ePnd 06:31:59.43 -1.4 17.2 110 g:7.7+0 .90 6.3 b:1.1+2 3.3 4.4X SSLB YHNB ePn 06:31:59.58 -0.6 17.1 107 b:6.8+3 1.0 6.7 YHNB NACB ePd 06:32:03.73 -1.3 17.5 108 g:1.3+2 1.9 7.2X b:3.1+4 1.6 7.2 NACB YULB ePnd 06:32:04.55 -2.1 17.7 111 g:3.7+1 1.5 6.8X b:3.3+3 .90 6.5 YULB

EXAMPLE of A UTOMATIC LOCA TION

Sichuan, China, 12 MAY 2008

(based on641arrival times)

ONE−STATION ALGORITHMSDetection and Location

→→ Enhance performance of single station/observatory

• Detection Algorithms

Generally based on the monitoring of energy in theground (or velocity) motion at the station.

* Define aSHORT-TERM AVERAGEover a short slidingwindow

* Compare it with a delayed LONG-TERM AVERAGE.

• WhenE in STA

E in LTAexceeds a given threshold,

TRIGGER DETECTION

* Earthquake spectrum is generallyWHITE, so do this in

SEVERAL FREQUENCY BANDS.Coherence across spectrum is required to trigger detection[or across several stations of a local network, when available

to prevent triggering on human noise.]

→ Arrival timesoi can be defined by evolution of ESTA/ELTA.

SINGLE-STATION LONG-PERIOD LOCATION

• IDEA ONE

" S − P " interval betweenP andS waves can giveDISTANCE


• IDEA TWO

Polarization ofP wave can giveAZIMUTH of ARRIVAL,β

P S

14 NOV 2005

100˚

100˚

110˚

110˚

120˚

120˚

130˚

130˚

140˚

140˚

150˚

150˚

160˚

160˚

170˚

170˚

180˚

180˚

-40˚ -40˚

-30˚ -30˚

-20˚ -20˚

-10˚ -10˚

0˚ 0˚

10˚ 10˚

20˚ 20˚

30˚ 30˚

40˚ 40˚

50˚ 50˚

True

Estimated

NWAO


• CombineDISTANCE and BACK AZIMUTH to obtain

Estimate of Epicenter

EXAMPLE of SINGLE-ST ATION LONG-PERIOD LOCATIONTREMORS — Reymond et al.[1991]

Earthquake located about 300 km from true epicenter

EARTHQ UAKE SOURCE GEOMETRY

Fr om Single Force to Double-Couple

The physical representation of an earthquake source is a system of forces known as aDouble-Couple, the direction of the forces in each couple being the direction of slipon the fault and the direction of the normal to the fault plane.

Mathematically, the system of forces is described by aSecond-Order Symmetric Deviatoric TENSOR

(3 angles and a scalar).

Single Force

Double Couple

Direction of Slip

Normal to the Fault

[Stein and Wysession,2002]


The focal geometry of earthquakes can vary depending onthe orientation of the double-couple representing the source.Here are some basic examples:


HOW CAN WE

• Best describe this gemoetry ?

• Determine it from seismological data ?

• Represent it graphically in simple terms ?


THREE ANGLESare necessary to describe the focal mechanism of an earthquake:


• Thestrike angleφ identifies the azimuth of the trace ofthe fault on the horizontal Earth surface;

• The dip angle δ indicates how steeply the fault pene-trates the Earth;

• The slip angle λ describes the relative motion of thetwo blocks on the fault plane determined byφ andδ .

→→ The physical description of an earthquake source isthus more complex than a vector since it requiresthreeangles as opposed to two.

The strike angleφφ

(between 0° and 360°)

defines the azimuth of the trace of the fault on the Earth’s surface

(theorientation of the knife)

The dip angleδδ


defines the slope (dip) of the fault to be cut through the material

(the inclination of the bladeon the horizontal)

Vertical dip ( δ = 90°) Shallow dip (δ = 30°)

The slip (or rake) angleλλ


defines the direction of motion of the blocks on the fault plane (cut) defined byφ andδ .

Strike-slip (λ = 180°) Dip-slip (λ = 270°)

No vertical motion Motion along line of steepest descent

Varying the slip (or rake) angleλλ (ctd.)

Thrust Faulting ( λ = 90°) Normal Faulting (λ = 270°)

(Typical of subduction zones) (Typical of tensional environments)

Varying the slip (or rake) angleλλ (ctd.)

HYBRID MECHANISMS

Thrust and Strik e-slip (λ = 120°) Normal Faulting and Strike-Slip (λ = 315°)


Double-Couple mechanisms give rise toP waves which canhave positive (first-motion "up") or negative (first-motion"down") initial motions.

The repartition of such motions on a small sphere surround-ing the source involves four alternatingquadrantsin space.

We represent focal mechanisms by giving a stereographicview of a small focal [hemi]sphere with positive quadrantsshaded and negative ones left open.

These are called"focal beachballs".

General case


EXAMPLES of EARTHQ UAKE SOURCE GEOMETRIES


ALL FOCAL MECHANISMS ARE CREATED EQUAL...

They are just ONESOLID ROTATION away from Each Other

Strike-Slip Vertical Dip-Slip

Thrust Normal

Hybrid

DETERMIN ATION of FOCAL MECHANISMS

• Historically

Examine first motion ofP waves and plot them on abeach ball.


DETERMIN ATION of FOCAL MECHANISMS

• Modern Technique

Directly invert wav eforms at many stations for thecom-ponents of the moment tensorrepresenting the double-couple.

"Centroid Moment Tensor Inversion"

→→ This is possible because that seismic ground displace-ment is a linear combination of these components.

Body Wav es

un(x; t) = M pq * Gnp,q =γnγpγq

4π ρα 3 rM pq

t −

r

α

−

γγp−δnp

4π ρ β3 r

γq M pqt −

r

β

P wav es

S wav es

Normal modes

NOTE LINEARITY of all Equations with respect to Mpq .

07/08/02 03:21:46.54 ANDREANOF ISLANDS, ALEUTIAN IS. Epicenter: 51.358 -179.929 MW 6.6

USGS MOMENT TENSOR SOLUTION Depth 21 No. of sta: 54 Moment Tensor; Scale 10**19 Nm Mrr= 0.73 Mtt=-0.59 Mpp=-0.14 Mrt= 0.53 Mrp= 0.41 Mtp=-0.40 Principal axes: T Val= 0.99 Plg=69 Azm=316 N 0.09 4 57 P -1.07 20 148

Best Double Couple:Mo=1.0*10**19 NP1:Strike=246 Dip=25 Slip= 100 NP2: 55 65 85 ------- ----------------- --------------------- -------################-- ------####################### -----#######################--- ---######## #############---- ---######### T ############------ --########## ##########-------- --######################--------- -#####################----------- -##################-------------- ###############---------------- ############------------------- #####--------------- ------ ------------------ P ---- ---------------- -- ----------------- -------

August 2, 2007, ANDREANOF ISLANDS, ALEUTIAN IS., MW=6.7

Goran Ekstrom

CENTROID-MOMENT-TENSOR SOLUTIONGCMT EVENT: C200708020321A DATA: II IU CU IC GE L.P.BODY WAVES: 64S, 165C, T= 50MANTLE WAVES: 62S, 138C, T=125SURFACE WAVES: 64S, 172C, T= 50TIMESTAMP: Q-20070802104412CENTROID LOCATION:ORIGIN TIME: 03:21:51.2 0.1LAT:51.11N 0.00;LON:179.66W 0.01DEP: 32.2 0.2;TRIANG HDUR: 5.6MOMENT TENSOR: SCALE 10**26 D-CMRR= 1.010 0.006; TT=-1.050 0.005PP= 0.031 0.005; RT= 0.740 0.010RP= 0.716 0.010; TP=-0.403 0.004PRINCIPAL AXES:1.(T) VAL= 1.484;PLG=64;AZM=2972.(N) 0.045; 15; 603.(P) -1.538; 21; 156BEST DBLE.COUPLE:M0= 1.51*10**26NP1: STRIKE=271;DIP=27;SLIP= 123NP2: STRIKE= 54;DIP=67;SLIP= 74

----------- ------------------- ---------#####--------- -----#################----- ---#######################-## --############################# -######## ##############----# -######### T #############------# ########## ###########--------- #######################---------- ####################------------- #################-------------- ##############----------------- #########-------------------- ----------------- ------- --------------- P ----- ------------- --- -----------

NEAR-REAL TIME CMT SOLUTIONS

Computed routinely by the NEIC (body wav es) and the Global CMT (ex-Harvard)project (bodyand surfacewaves)

Example:08 JUL 2007, ALEUTIAN ISLANDS

NEIC Global CMT

M0 = 1. 0× 1026 dyn*cm

M0 = 1. 5× 1026 dyn*cm

Note difference in moment

The two focal solutions are separated by a solid rotation of11°.

earthquakes: detection, location & focal geometry · 2011. 4. 20. · normal to the fault...

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