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 Gamma Remote Sensing AG, Gümligen 3 Microwaves and Radar Institute, German Areospace Center (DLR)
Dataset: Bisgletscher ● 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 Adapted from: M. Funk, Inventar gefährlicher Gletscher der Schweiz KAPRI during a test
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 • one displacement component • more processing Source: ETH WAV
Differential Interferometry: Signal Model Atmospheric SLC Phase Slant range delay Scattering phase ϕ = 4 π λ R + ϕ atm + ϕ scat Interferogram (difference of SLC phases) Δϕ= 4 π λ tv +Δϕ atm +ϕ noise + k π (differential) Displacement APS
Why Care About Atmospheric Phase Screens (APS)? Where is the displacement? How to separate it from APS? Multiple interferograms can help us!
Signal Model: Multiple Interferograms Building a stack: z = A ( y + ϵ y ) Noise : Interferogram stack APS ● P pixels at M times Decorrelation ● (Previously ΔΦ) SLC Phases : Δ ϕ atm + ϕ noise Incidence matrix : range ● Relate SLC and scattering ● ifgrams Relating it with displacement: Differential Noise APS ● Decorrelation ● z 1,1 z = T v + ϵ z [ z P,M ] time ⋮ Velocity model: z 1 ,M Here « stacking »: 1 velocity space ● every M/S interferograms for ⋮ every pixel : z P, 1 T 1 T M − S T = [ T M ] v 1 ⋮ , v = [ v s ] T 2 T M − S + 1 ⋯ ⋮ ⋮ ⋮ T M / S
Solving the Stack GLS solution for the velocity parameters, given a stack: Covariance of APS + Noise v =( T T Σ − 1 T ) − 1 T T Σ − 1 z ^ 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
Splitting the Absolute APS Idea: model APS as deterministic + stochastic strat + ϵ y turb ϵ y = ϵ y Stratification : Turbulent Mixing : (hydrostatic delay) (wet delay) For each interferogram at Stochastic, assume time i (regressors are the Gaussian: same): turb ∝ N ( 0 , Σ y ) ϵ y strat = X b i ϵ y ,i
From „Absolute APS“ to „Differential APS“ y z z = A y “SLC” “Interferogram” Covariance Turbulence Σ= A Σ y A Σ y transforms linearly! strat =( X ⊗ I M ) b strat =( X ⊗ A ) b Same regression ϵ y ϵ z Stratification model as SLC (with different parameters)! Same regressors for all times!
APS Stratification: Model Performance R 2 for stratification models (600 model runs) Variance of estimates [m 2 /day 2 ]
APS Stratification: Model Performance R 2 for stratification models ● Stratification does not explain most phase (600 model runs) variance! ● Cannot capture local trends Chosen model ● Is turbulence more important?
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 t A locations»[5] s ⊗( A Σ y s ⊗Σ y Σ= A Σ y A T ) t Σ=Σ y Σ y =Σ y Temporal covariance Spatial covariance Same spatial covariance as “absolute” APS
Turbulent APS: Spatial Covariance Model Estimate covariances via variograms: spatial variogram at different timelags Separable variogram: γ st ( s ,t )= C t ( 0 ) γ s ( s )+ C s ( 0 ) γ t ( t )− k γ s ( s ) γ t ( t ) Blue line: Exponential model fit Ribbon: +/- standard deviation For fixed s, increasing t only scales and Dashed line: spatial marginal offesets variogram ! variogram
Turbulent APS: Spatial Covariance Model Change in variogram shape with baseline and time ● Non-stationary ● Non-separable → Different atmospheric turbuelence regimes? → Separability and stationarity only approximate Spatial Variograms by hour
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