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Spring 2017 – UAF Class GEOS 657

GEOS 657 – MICROWAVE REMOTE SENSINGSPRING 2017

Lecturer: F.J. Meyer, Geophysical Institute, University of Alaska Fairbanks; [email protected]

Lecture 14: On the Use of InSAR in Geophysics

Franz J Meyer, UAF

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REVIEW OF INSAR WORKFLOW FOR A GEOPHYSICAL APPLICATION

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How InSAR Works

β€’ InSAR phase is a function of distance from satellite to ground (range)

Path difference results in phase shift

780 k

m

By precisely measuring the phase difference 𝝓, the change in distance from satellite to ground can be calculated to an accuracy of sub-wavelength.

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How InSAR Works

β€’ InSAR phase is a function of distance from satellite to ground (range)

Pass 1 Pass 2No phase shift

No deformation

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How InSAR Works

β€’ InSAR phase is a function of distance from satellite to ground (range)

Pass 1 Pass 2

phase shift:

120 degrees

Ground subsidence: 9 mm

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InSAR Processing – Steps 1 – 7:Find and Download Images, Co-Register & Form Interferogram

=βˆ™

β€’ To form interferogram, we calculate: 𝐼 = 𝑒1 βˆ™ 𝑒2βˆ— (where βˆŽβˆ— is complex

conjugate)

SAR Image 𝑒1: SAR Image 𝑒2:βˆ—

Interferogram 𝐼:

Example: Mt. Peulik volcano, Alaska

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InSAR Processing – Step 8:Subtraction of Flat Earth Phase

β€’ Example:

– ERS-2 Interferogram of Mt. Peulik volcano, Alaska

Before Flat Earth Phase Compensation After Flat Earth Phase Compensation

Flat Earth Phase

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InSAR Processing – Step 9 & 10:Phase Filtering & Topographic Phase Correction using DEM

Sim

ula

ted

To

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grap

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on

ly i

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rfe

rogr

am

0 360

DEM

in R

adar

Co

ord

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InSAR Imaging Geometry & Parameters

Topographic Phase Correction using DEMin

terf

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gram

Top

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y-o

nly

in

terf

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gram

Def

orm

atio

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gram

0 2.83 cm

πœŸπ“ β‰ˆ 𝝓𝒅𝒆𝒇𝒐 =πŸ’π…

π€βˆ†π‘Ήπ’…π’†π’‡π’

9

Filtering πœŸπ“ for Noise Suppression

Filt

ere

d d

efo

rmat

ion

inte

rfe

rogr

am

0 2.83 cm

10

InSAR Processing Step 9 - Deformation interferogram in map projection

(including phase unwrapping and geocoding)Interpretation of Geocoded Differential Phase πœŸπ“

11

InSAR Processing Step 9 - Deformation interferogram in map projection

(including phase unwrapping and geocoding)Representation of Interferometric Phase

28 mm0

28.3

84.9

Uplift mm

Courtesy of G. Bawden, USGS12

InSAR Processing Step 9 - Deformation interferogram in map projection

(including phase unwrapping and geocoding)Deformation Modeling – Concept [Mt. Peulik Example]

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RESIDUAL

575

0

574

0

574

5

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OBSERVED

4 Oct. 95 - 9 Oct. 97

5 km

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SYNTHETIC

Best-fit source parameters:β€’ Spherical point source (Mogi source)β€’ The model source is located at a depth of 6.5 Β± 0.2 km. β€’ The calculated volume change of magma reservoir is 0.043 0.002 km3.

0 2.83 cm

13

Franz J Meyer, UAF

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InSAR Coherence: A Phase Quality Descriptor

m5

m25

ERS resolution element

interferogram

InSAR

processor

receiver noise

temporal

changes of

surface

scattering

conditions

propagation

effects

processor errors

Coherence:β€’ Quality measure describing

noise level of InSAR phase

Useful for:β€’ How accurate is a

topography or deformation

estimate from InSAR

β€’ Contributions to Phase Noise:

L-band (𝝀 = πŸπŸ“π’„π’Ž): Higher coherenceC-band (𝝀 = πŸ“. πŸ”π’„π’Ž): Lower coherence

Wavelength Vs. Coherence

15

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AN EXAMPLE OF THE USE OF INSAR IN GEOPHYSICS

Deformation Modeling Problem

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Magma intrusion

deformation: what we see (InSAR)

magma dynamics: what we want to know

?

Deformation Modeling Problem

18

Estimate source characteristics from InSAR deformation data

up

s

forward model

InSAR image

displacement

(vector)source

parameters

G s = d

design matrix

inverse

model

s = G dinv

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Solving for Model Parameters using Model Inversion

𝐺 βˆ™ π‘₯ = 𝑏

β€’ If the covariance matrix for errors in the observation (𝑏) is Σ𝑏, then the weighted least-squares (maximum likelihood) solution for π‘₯ is

ොπ‘₯ = 𝐺𝑇 βˆ™ Ξ£π‘βˆ’1 βˆ™ 𝐺

βˆ’1βˆ™ 𝐺𝑇 βˆ™ Σ𝑏

βˆ’1 βˆ™ 𝑏

and the covariance matrix for the estimated vector components is

Ξ£π‘₯ = 𝐺𝑇 βˆ™ Ξ£π‘βˆ’1 βˆ™ 𝐺

βˆ’1

β€’ In the case where we assume that observation errors are independent and have equal standard deviations, 𝜎, we get

Ξ£π‘₯ = 𝜎2 𝐺𝑇 βˆ™ 𝐺 βˆ’1

– The square roots of the diagonal terms of Ξ£π‘₯ are the standard errors of the estimated parameters

What is the Forward Model in Volcano Deformation?

20

Predicts deformation (u) caused by magma intrusion(relates magma intrusion to deformation)

z

magma

intrusion

displacement

x

2 ui + = – Fi

2uk

(1-2) xi yk

elasto-static behavior

u = Ζ’(model parameters)

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What Is the Forward Model?Simple Model: Inflating Point Source Model

β€’ A component of deformation vector (𝑒𝑖) and the displacement at the free surface (π‘₯3 = 0) takes the form

𝑒𝑖 π‘₯1 βˆ’ π’™πŸβ€² , π‘₯2 βˆ’ π’™πŸ

β€² , βˆ’π’™πŸ‘β€² = π‘ͺ

π‘₯𝑖 βˆ’ π’™π’Šβ€²

𝑅3

– π’™π’Šβ€² is a source location, π‘ͺ is a combination of material properties and source strength, and 𝑅 is

the distance from the source to the surface location

β€’ π‘ͺ is defined as follows:

π‘ͺ = βˆ†π‘ƒ 1 βˆ’ πœˆπ‘Ÿπ‘ 3

𝐺= πœŸπ‘½

1 βˆ’ 𝜈

πœ‹

– βˆ†π‘ƒ - change in pressure of magma chamber

– π›₯𝑉 - change in volume of magma chamber

– 𝜈 - Poisson’s ratio

– π‘Ÿπ‘  - radius of the sphere

– 𝐺 - shear modulus of country rock

Unknown (target) parameters marked in red

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Forward Model: Inflating Point Source

Courtesy of M. Lisowski

Ξ± << d

D. Dzurisin, 2007

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Forward Model: Inflating Point Source

Courtesy of M. Lisowski

Ξ± << d

D. Dzurisin, 2007

Horizontal

Vertical

Ξ±/d = 0.4

Ξ±/d = 0.6

Ξ±/d = 0 (point source)

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Forward Model: Sill Model

Horizontal

Vertical

Point source

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Forward Model: Dike Model

Ultimate Goal of Deformation Modeling:

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Minimize

2)],(),(),([ yxobsyxlosyxu iii

ui is a theoretical calculation of ground surface deformation vector (i=1, 2, 3)losi is the InSAR line-of-sight vectorobsi is the observed deformation (InSAR image)(x, y) is the image coordinate

Non-linear inversion!!!!

Find the best-fitting Model ParametersA Simple Approach

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1. Loop through model parameters

2. calculate the residual (observed – modeled) for each set of model parameters

3. Find the set of model parameters that renders the smallest residual

best-fitting model parameters

A Matlab Lab for Estimating Source Parameters

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xy_gridsearch.m

β€’ defines search space for source model location – Source Depth and volume change

is fixed for this experiment

β€’ For each set of x and y coordinate parameters:

– Run forward model to produce predicted surface deformation result [mogi2InSAR.m]

– Calculate difference (residuals) between predicted and measured deformation

β€’ Best fitting model parameters are those that minimize residuals

Z (depth)

v (volume change)

Mt. Peulik Example

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Best-fit source parameters:

β€’ The model source is located at a depth of 6.5 Β± 0.2 km.

β€’ The calculated volume change of magma reservoir is 0.043 0.002 km3.

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RESIDUAL

575

0

574

0

574

5

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OBSERVED

5 km

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SYNTHETIC

0 2.83 cm

VR

xxvxxxxu ii

i

32211

)1()0,,(

where x1, x2 , and x3 are horizontal locations and depth of the center of the sphere, R is the distance

between the sphere and the location of observation (x1, x2 , and 0), and v is the Poisson’s ratio of host rock.

Spherical point source (Mogi source)

Model Inversion for Multiple Sources

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β€’ Superimposition of individual deformation sources

β€’ Smoothing (spatial + temporal)

– The total displacement on a

given patch…

– …is related to that of patches

adjacent to it, by a finite-

difference Laplacian

approximation

(schematic)

a1

a2

a3

a4

a5

a6

a7

a8

a9

aM

(a2–a5)–(a5–a8)+(a4–a5)–(a5–a6) = 0

a2 + a4 – 4a5 + a6 + a8 = 0

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Example: Earthquake Modeling

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Afar –triple junction

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Smith and Cann, JGR, 2000

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Quaternary strain localized to ~60 km long zones of fissures, aligned eruptive centers and faults -β€œmagmatic segmentsβ€œ

Courtesy of T. Wright

35Courtesy of T. Wright

14/9/2005 to11/05/2005

163 earthquakes (mb<6) detected by NEIC.

Relocated by Anna Stork

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3D displacements measured from radar data

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Deflating

Magma

ChambersDeflating

Magma

Chambers

Collapsed

Zone

along Rift

Collapsed Zone

along Rift

Map view

Cross section

Deformation Modelling

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β€’ 2.2 km3 magma intruded along dyke (Mt St Helens 1980 1.2 km3)

β€’ 0.5 km3 sourced from Dabbahu and Gabho volcanoes at North.

β€’ Earthquakes can be responsible for < 10 % of moment release.

Wright et al., Nature, 2005

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What’s Next?

β€’ This is what awaits next:

– Thursday : Lab on InSAR Processing Using the SNAP Toolbox

– Next week Tuesday: Project Concept Presentations