GEOS 657 MICROWAVE REMOTE SENSING - University of Alaska ...
Transcript of GEOS 657 MICROWAVE REMOTE SENSING - University of Alaska ...
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
GEOS 657, Spring 2016 - 3
How InSAR Works
• 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.
Franz J Meyer, UAF
GEOS 657, Spring 2016 - 4
How InSAR Works
• InSAR phase is a function of distance from satellite to ground (range)
Pass 1 Pass 2No phase shift
No deformation
Franz J Meyer, UAF
GEOS 657, Spring 2016 - 5
How InSAR Works
• InSAR phase is a function of distance from satellite to ground (range)
Pass 1 Pass 2
phase shift:
120 degrees
Ground subsidence: 9 mm
Franz J Meyer, UAF
GEOS 657, Spring 2016 - 6
InSAR Processing – Steps 1 – 7:Find and Download Images, Co-Register & Form Interferogram
=∙
• To form interferogram, we calculate: 𝐼 = 𝑢1 ∙ 𝑢2∗ (where ∎∗ is complex
conjugate)
SAR Image 𝑢1: SAR Image 𝑢2:∗
Interferogram 𝐼:
Example: Mt. Peulik volcano, Alaska
Franz J Meyer, UAF
GEOS 657, Spring 2016 - 7
InSAR Processing – Step 8:Subtraction of Flat Earth Phase
• Example:
– ERS-2 Interferogram of Mt. Peulik volcano, Alaska
Before Flat Earth Phase Compensation After Flat Earth Phase Compensation
Flat Earth Phase
Franz J Meyer, UAF
GEOS 657, Spring 2016 - 8
InSAR Processing – Step 9 & 10:Phase Filtering & Topographic Phase Correction using DEM
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InSAR Imaging Geometry & Parameters
Topographic Phase Correction using DEMin
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0 2.83 cm
𝜟𝝓 ≈ 𝝓𝒅𝒆𝒇𝒐 =𝟒𝝅
𝝀∆𝑹𝒅𝒆𝒇𝒐
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InSAR Processing Step 9 - Deformation interferogram in map projection
(including phase unwrapping and geocoding)Interpretation of Geocoded Differential Phase 𝜟𝝓
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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:• Spherical point source (Mogi source)• The model source is located at a depth of 6.5 ± 0.2 km. • The calculated volume change of magma reservoir is 0.043 0.002 km3.
0 2.83 cm
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Franz J Meyer, UAF
GEOS 657, Spring 2016 - 14
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:• Quality measure describing
noise level of InSAR phase
Useful for:• How accurate is a
topography or deformation
estimate from InSAR
• Contributions to Phase Noise:
Deformation Modeling Problem
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Magma intrusion
deformation: what we see (InSAR)
magma dynamics: what we want to know
?
Deformation Modeling Problem
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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
Franz J Meyer, UAF
GEOS 657, Spring 2016 - 19
Solving for Model Parameters using Model Inversion
𝐺 ∙ 𝑥 = 𝑏
• If the covariance matrix for errors in the observation (𝑏) is Σ𝑏, then the weighted least-squares (maximum likelihood) solution for 𝑥 is
ො𝑥 = 𝐺𝑇 ∙ Σ𝑏−1 ∙ 𝐺
−1∙ 𝐺𝑇 ∙ Σ𝑏
−1 ∙ 𝑏
and the covariance matrix for the estimated vector components is
Σ𝑥 = 𝐺𝑇 ∙ Σ𝑏−1 ∙ 𝐺
−1
• In the case where we assume that observation errors are independent and have equal standard deviations, 𝜎, we get
Σ𝑥 = 𝜎2 𝐺𝑇 ∙ 𝐺 −1
– The square roots of the diagonal terms of Σ𝑥 are the standard errors of the estimated parameters
What is the Forward Model in Volcano Deformation?
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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 = ƒ(model parameters)
Franz J Meyer, UAF
GEOS 657, Spring 2016 - 21
What Is the Forward Model?Simple Model: Inflating Point Source Model
• A component of deformation vector (𝑢𝑖) and the displacement at the free surface (𝑥3 = 0) takes the form
𝑢𝑖 𝑥1 − 𝒙𝟏′ , 𝑥2 − 𝒙𝟐
′ , −𝒙𝟑′ = 𝑪
𝑥𝑖 − 𝒙𝒊′
𝑅3
– 𝒙𝒊′ is a source location, 𝑪 is a combination of material properties and source strength, and 𝑅 is
the distance from the source to the surface location
• 𝑪 is defined as follows:
𝑪 = ∆𝑃 1 − 𝜈𝑟𝑠3
𝐺= 𝜟𝑽
1 − 𝜈
𝜋
– ∆𝑃 - change in pressure of magma chamber
– 𝛥𝑉 - change in volume of magma chamber
– 𝜈 - Poisson’s ratio
– 𝑟𝑠 - radius of the sphere
– 𝐺 - shear modulus of country rock
Unknown (target) parameters marked in red
Franz J Meyer, UAF
GEOS 657, Spring 2016 - 22
Forward Model: Inflating Point Source
Courtesy of M. Lisowski
α << d
D. Dzurisin, 2007
Franz J Meyer, UAF
GEOS 657, Spring 2016 - 23
Forward Model: Inflating Point Source
Courtesy of M. Lisowski
α << d
D. Dzurisin, 2007
Horizontal
Vertical
α/d = 0.4
α/d = 0.6
α/d = 0 (point source)
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GEOS 657, Spring 2016 - 24
Forward Model: Sill Model
Horizontal
Vertical
Point source
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
• defines search space for source model location – Source Depth and volume change
is fixed for this experiment
• 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
• 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:
• The model source is located at a depth of 6.5 ± 0.2 km.
• 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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• Superimposition of individual deformation sources
• 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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Quaternary strain localized to ~60 km long zones of fissures, aligned eruptive centers and faults -“magmatic segments“
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
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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• 2.2 km3 magma intruded along dyke (Mt St Helens 1980 1.2 km3)
• 0.5 km3 sourced from Dabbahu and Gabho volcanoes at North.
• Earthquakes can be responsible for < 10 % of moment release.
Wright et al., Nature, 2005