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Feb 03, 2023 264 likes •499 views

Geostatistical Analysis and Mitigation Of Atmosphere Induced Phase In Terrestrial Radar Interferometric Observations Of An Alpine Glacier Simone Baffelli 1 , Othmar Frey 1,2 , Irena Hajnsek 1,3 1 Earth Observation and Remote Sensing, ETH Zurich 2

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SLIDE 1

Geostatistical Analysis and Mitigation Of Atmosphere Induced Phase In Terrestrial Radar Interferometric Observations Of An Alpine Glacier

Simone Baffelli1, Othmar Frey1,2, Irena Hajnsek1,3

1Earth Observation and Remote Sensing, ETH Zurich 2 Gamma Remote Sensing AG, Gümligen 3 Microwaves and Radar Institute, German Areospace Center (DLR)

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SLIDE 2

Dataset: Bisgletscher

Adapted from: M. Funk, Inventar gefährlicher Gletscher der Schweiz

  • The Bisgletscher, above Randa (VS)

very fast (up to 2 m/day)

  • Several icefalls in the past [1]
  • In similar glaciers: acceleration phase before rupture [2]
  • Early warning could be possible

KAPRI during a test

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SLIDE 3

Bisgletscher Monitoring History

Starting 2012, automatic camera:

  • no measurements at night + fog
  • 2D displacement maps

2014-2015 radar, differential interferometry:

  • 24/7, with fog + clouds
  • ne displacement component
  • more processing

Source: ETH WAV

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SLIDE 4

Differential Interferometry: Signal Model

SLC Phase Interferogram (difference of SLC phases)

Δϕ= 4 π λ tv+Δϕatm+ϕnoise+k π ϕ=4 π λ R+ϕatm+ϕscat

Scattering phase Atmospheric delay Slant range Displacement (differential) APS

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SLIDE 5

Why Care About Atmospheric Phase Screens (APS)?

Where is the displacement? How to separate it from APS? Multiple interferograms can help us!

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SLIDE 6

Signal Model: Multiple Interferograms

Building a stack: Relating it with displacement:

z=T v+ϵ z

Velocity model:

  • Here «stacking»: 1 velocity

every M/S interferograms for every pixel:

[

z1,1 ⋮ z1,M ⋮ z P,1 ⋮ z P,M]

time space

T=[ T 1 T 2 ⋮ T M /S ⋯ T M−S T M−S+1 ⋮ T M ] , v=[ v1 ⋮ vs]

Differential Noise

  • APS
  • Decorrelation

Interferogram stack P pixels at M times (Previously ΔΦ) Noise:

  • APS
  • Decorrelation

z=A( y+ϵ y)

SLC Phases:

  • range
  • scattering

Incidence matrix: Relate SLC and ifgrams Δ ϕatm+ϕnoise

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SLIDE 7

Solving the Stack

^

v=(TT Σ−1T)−1TT Σ−1z

Is this possible?

  • We don’t know the (large) covariance matrix!
  • The matrix is too large to invert at once

We should study the APS statistics first GLS solution for the velocity parameters, given a stack: Covariance of APS + Noise

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SLIDE 8

Splitting the Absolute APS

ϵ y=ϵ y

strat+ϵ y turb

Stratification: (hydrostatic delay) For each interferogram at time i (regressors are the same):

ϵ y ,i

strat=X bi

Turbulent Mixing: (wet delay) Stochastic, assume Gaussian:

ϵ y

turb∝N (0 ,Σ y)

Idea: model APS as deterministic + stochastic

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SLIDE 9

From „Absolute APS“ to „Differential APS“

z=A y Σ=A Σ y A Σy

y “SLC” z “Interferogram” Turbulence Stratification

ϵy

strat=(X⊗I M)b

ϵz

strat=(X⊗A)b

Same regressors for all times! Same regression model as SLC (with different parameters)! Covariance transforms linearly!

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SLIDE 10

APS Stratification: Model Performance

R2 for stratification models (600 model runs) Variance of estimates [m2/day2]

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SLIDE 11

APS Stratification: Model Performance

  • Stratification does not explain most phase

variance!

  • Cannot capture local trends
  • Is turbulence more important?

R2 for stratification models (600 model runs) Chosen model

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SLIDE 12

Turbulent APS: Covariance Model

  • Stationary: statistics depends only on

spatial and temporal separations

  • Separable: no space-time interactions
  • Spatial covariances at different time

lags are proportional

  • «all the locations in space have the

same temporal covariance structure»[6]

  • «… temporal evolution of the process

at a given spatial location does not depend directly on the process’ temporal evolution at other locations»[5]

Σ=A Σ y A Σy=Σy

s ⊗Σ y t

Σ=Σy

s ⊗( A Σy t A T)

Spatial covariance Temporal covariance Same spatial covariance as “absolute” APS

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Turbulent APS: Spatial Covariance Model

Estimate covariances via variograms: spatial variogram at different timelags Separable variogram: For fixed s, increasing t only scales and

  • ffesets variogram!

γst(s ,t)=Ct(0)γs(s)+Cs(0) γt(t)−k γs(s) γt(t)

Blue line: Exponential model fit Ribbon: +/- standard deviation Dashed line: spatial marginal variogram

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SLIDE 14

Turbulent APS: Spatial Covariance Model

Spatial Variograms by hour

Change in variogram shape with baseline and time

  • Non-stationary
  • Non-separable

→ Different atmospheric turbuelence regimes? → Separability and stationarity only approximate

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SLIDE 15

Turbulent APS: Covariance Model

Temporal variogram

  • Temporal variance grows quickly
  • Cannot use long-baseline

interferograms:

  • Decorrelation
  • Phase wrapping on moving parts

Phase wrapping for 2 m/day

  • 5

10 1000 2000 3000

Temporal lag [s] Semivariance[ rad2]

Jul 13 Jul 20 Jul 27

master_id

Blue line: Exponential model fit Dots: temporal marginal variogram

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SLIDE 16

Putting it Together: APS Prediction + Inversion

Recall: GLS Now we have an approximate Σ. But: the matrix is large! (spatial + temporal correlation)

  • Predict APS with Regression-
  • Kriging. Should remove spatial

correlation

  • Compute GLS inversion pixelwise

^

v=(TT Σ−1T)−1TT Σ−1z

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SLIDE 17

Regression-Kriging Performance

  • Kriging removes low spatial-

frequency APS

  • Deformation pattern better

visible

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SLIDE 18

Regression-Kriging Performance

Residual velocity estimated on PS where v=0

  • Linear model (LM) does not reduce

variance → Explains observed high R2

  • Kriging → Significant reduction
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SLIDE 19

Regression-Kriging Performance

Residual velocity estimated on PS where v=0

  • Solve GLS after Kriging
  • Assume no spatial correlation
  • Remove high-frequency APS
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SLIDE 20

Example: Timeseries

Stack size 30 minutes

LOS Velocity [m/day]

Only OLS RK + GLS

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SLIDE 21

Example: Animations

Same stack size (~30 minutes)! LOS Velocity [m/day] Only OLS (no covariance) Kriging + GLS

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References

[1] Raymond, M., Wegmann, M., & Funk, M. (2003). Inventar gefahrlicher Gletscher in der

  • Schweiz. Mitteilungen Nr 182 der Versuchsanstalt fur Wasserbau, Hydrologie und Glaziologie

an der Eidgenossischen Technischen Hochschule Zurich. Retrieved from http://cat.inist.fr/?aModele=afficheN&cpsidt=15381742 [2] Faillettaz, J., Pralong, A., Funk, M., & Deichmann, N. (2008). Evidence of log-periodic

  • scillations and increasing icequake activity during the breaking-off of large ice masses.

Journal of Glaciology, 54(187), 725–737. https://doi.org/10.3189/002214308786570845 [3] Strozzi, T., Wegmüller, U., Tosi, L., Bitelli, G., & Spreckels, V. (2001). Land subsidence monitoring with differential SAR interferometry. Photogrammetric Engineering & Remote Sensing, 67(11), 1261–1270. [4] www.nextflow.io [5] http://faculty.missouri.edu/~wiklec/ST_3_wikle_Boston_Descriptive.pdf [6]IBozza, S. & O'Hagan, A. A Bayesian Approach for the Estimation of the Covariance Structure of Separable Spatio- Temporal Stochastic Processes Between Data Science and Applied Data Analysis, Springer Berlin Heidelberg, 2003 , 165-172