Intrinsic random functions for mitigation ofatmospheric effects in ground based radarinterferometryJ.A. Butt, A. Wieser, S. ConzettInstitute of Geodesy and Photogrammetry,ETH Zurich, Stefano-Franscini-Platz 15, 8093 Zurich, Switzerland

Abstract. The potential benefits of terrestrialradar interferometry (TRI) for deformation mon-itoring are restricted by the influence of changingmeteorological conditions contaminating the po-tentially highly precise measurements with spu-rious deformations. This is the case especiallywhen the measurement setup includes long dis-tances between instrument and objects of inter-est and the topography affecting atmospheric re-fraction is complex. These situations are typi-cally encountered with geo-monitoring in moun-tainous regions, e.g. with glaciers, landslides orvolcanoes.

We propose and explain an approach for themitigation of atmospheric influences based onthe theory of intrinsic random functions of or-der k (IRF-k) generalizing existing approachesbased on ordinary least squares estimation oftrend functions. This class of random functionsretains convenient computational properties al-lowing for rigorous statistical inference while stillpermitting to model stochastic spatial phenom-ena which are non-stationary in mean and vari-ance. We explore the correspondence betweenthe properties of the IRF-k and the propertiesof the measurement process. Drifts in varianceare linked to the additive nature of atmosphericeffects along the propagation path and to the ir-regularity of the terrain.

In an exemplary case study, we find thatour method reduces the time needed to obtainreliable estimates of glacial movements from12 h down to 0.5 h. We relate the estimatederror variance of the results to details of ourmeasurement campaign like roughness of ter-rain, altitude differences and involved distancesthus indicating how this method is expected toperform when applied under different conditions.

Keywords. Atmospheric correction, intrinsicrandom functions, ground based radar inter-ferometry, monitoring in alpine environment,Geostatistics

1 Introduction

Terrestrial radar interferometry (TRI) is a tech-nology providing spatiotemporally dense mea-surements for the quantification of geometric sur-face changes along the line-of-sight over distancesup to a few km. This is not only of immediatepractical interest in the context of applicationslika structural health monitoring, operation andsafeguarding of open pit mines or monitoring ofrockfalls endangering critical infrastructure butcould also facilitate better understanding of dy-namic processes underlying large-scale geologicalnatural hazards through scientific measurements.

TRI is therefore regularly deployed in moun-tainous regions, e.g. to survey glaciers (Voytenkoet al., 2012), assess the likelihood of geologi-cally predisposed areas becoming active land-slides (Caduff et al., 2014) or to observe theflanks of volcanoes for deformation patterns in-dicating an increase in activity (Rodelsperger etal., 2010). But even though TRI as a remotesensing technology has certain advantages overclassical point-based geodetic techniques in formof its high sensitivity to small surface displace-ments, inherently areal sampling and its remoteoperability without need for any in-situ compo-nents, it shares with them some of their limita-tions.

Like for any other type of measurement TRIdata consist of a signal part and a noise part.The latter is further decomposable into a spa-tiotemporally highly autocorrelated componenthaving its origins in the essentially unpredictable

meteorological changes along the propagationpath and a second, spatiotemporally uncor-related, component subsuming thermal noise,crosstalk between electronic circuits and move-ments of the surveyed object on very shortlength- or timescales or in manners otherwise in-accessible to systematic analysis.

This second component becomes directly visi-ble in the interferograms at locations correspond-ing to regions with weak backscattering. There itappears in the form of unsystematic disturbancesseemingly adhering to the restrictive probabil-ity laws of white noise and is therefore easilymodeled stochastically and amenable to statis-tical standard treatments.

Fig. 1: Two successive 2-minute interferograms, inwhich the APS masks the deformations that are ac-tually limited to the area outlined in black.

The noise component corresponding to at-mospheric influences is more problematic. Notonly can this so called atmospheric phase screen(APS) reach magnitudes and spatial extents al-lowing it to completely mask the underlyingsignal (see Fig. 1); its stochastic properties de-pend on the topography and are consequentlynon-constant because the APS associated witha point t ∈ T (e.g., T = R3) is the result of

an integration of differential atmospheric effectsalong the line joining t and the instrument posi-tion. Accordingly, the APS cannot be second or-der stationary, and thus widely known inferencemethods like simple Kriging hold no optimalityproperties. This raises the question of stochasti-cally optimal inference of an instationary APS.

This inference problem has been approachedfrom different perspectives. Most of them em-ploy a blend of (i) deterministic relations be-tween APS, refractive index and meteorologicalquantities and (ii) stochastic relations of the de-terministically unmodelled residuals to deducethe APS from the measurements. A purely de-terministic approach was investigated by Gonget al. (2010), who gathered meteorological dataand predicted the atmospheric phase delay us-ing tools from weather forecasting. Iglesias et al.(2014) mixed a polynomial model for the APSwith a multiple regression model meant to ex-plicitly incorporate the information from altitudeand phase measurements on known stable pointsinto the coefficients of the polynomial APS. Pre-supposing less regularity of the APS and placingmore emphasis on the structure found in the datathemselves lead Caduff et al. (2014) to estimatethe APS over the area potentially containing dis-placements with a spatial low-pass filter.

It is in recognition of the irregular move-ment patterns and highly variable meteorologicalproperties of air in mountainous terrain that weadopt a data-driven viewpoint similar to the oneproposed by Caduff et al. (2014). In section 2 aninterpretation of the APS as an intrinsic randomfunction will be presented and followed in sec-tion 3 by a scheme allowing rigorous statisticalinference for intrinsic random functions in formof what is called the BLIE (Best Linear Intrin-sic Estimator) in the geostatistical literature, seeMatheron (1973). Section 4 contains results val-idating the proposed correction method in thecontext of a monitoring campaign targeting analpine glacier in southern Switzerland.

2 A mathematical model for the APS

The area potentially containing moving objectswill be assumed known. A set of persistent scat-terers (PS) lying outside of it have to be ex-tracted from the data for later use as referencepoints with good backscattering characteristicsand signal-to-noise ratio that provide reliable in-formation regarding the APS over stable regions.

2

To this end the amplitude dispersion index (ADI)described for example in Noferini et al. (2005)may be used although care must be taken to elim-inate any PS detected in unstable areas. Thegoal is then to derive for all locations t ∈ Tthe least squares estimator minimizing the ex-pected quadratic error E[(A(t)−A(t))2] whichrepresents the deviation between predicted andtrue APS, and to quantify the uncertainties as-sociated with the resulting estimates. Acknowl-edging the effect of noise unaccounted for oth-erwise, we will refrain from employing a strictinterpolation procedure and instead perform es-timation and smoothing simultaneously using acoherence-based noise parameter.

The APS itself will be considered a randomfield

A(·) : T 3 t 7→ A(t) ∈ L2(Ω) (1)

where L2(Ω) denotes the Hilbert space of squareintegrable random variables on the probabilityspace Ω, see for example Christakos (2013, p 25).The above definition is to be interpreted as A(·)being an assignment of a random variable to eachlocation t ∈ T . At certain points during thederivation of the BLIE it will also be necessaryto consider the space Ξ of all random fields. Mea-surements M(·) on the PS will be demanded tosatisfy

M(·) = A(·) +N(·) (2)

with Noise N(·) being a zero-mean random fieldwith diagonal covariance matrix ΣN and A(·)and N(·) uncorrelated. To model drifts in vari-ance and expected value of A(·), the usual reg-ularity assumption of second order stationarity(s.o.s.) of A(·) is dropped and replaced by theweaker assumption of A(·) being an intrinsic ran-dom function of order k (IRF-k).

IRF-k’s can be understood as random func-tions, for which a differential operator ∇k+1

of order k + 1 exists, such that ∇k+1A(·)satisfies E[∇k+1A(t)] = 0 ∀t ∈ T andE[∇k+1A(t)∇k+1A(s)] = C(t− s) ∀s, t ∈ T andis thus s.o.s. although A(·) itself, as a sum ofatmospheric phase variations along wave propa-gation paths of different lengths, will exhibit aninstationary variance depending on the distancebetween instrument and targeted area. Whileit can therefore not be s.o.s., the supposition ofs.o.s. derivatives of A(·) seems justifiable (seeFig. 2) and will make rigorous inference possible.Arguments and definitions will be made moreformal in the following section adhering closely

to notations laid out in Chiles and Delfiner (2012,pp. 238-270) and loosely to the theory presentedin Matheron (1973).

Fig. 2: Measured displacement velocities and theirderivative in range-direction. The latter seems to bezero-mean with a homogeneous correlation function.

3 Inference in the space of IRF-k

3.1 Intrinsic random functions

Let δ be the Dirac measure and δt, t ∈ T denoteits translate i.e., δt(T ′) = 1 if t ∈ T ′ ⊂ T andδt(T ′) = 0 otherwise. Then the discrete measureλ =

∑i λiδti , acts on a function X : T → U via

Xλ :=∑i

∫T

λiX(s)δti(ds) =∑i

λiX(ti).

If U is either L2(Ω) or R and X therefore a ran-dom field or a normal, scalar function, define theaction of λ on X around t as

Xλ(t):=∑i

∫T

λiX(s)δt+ti(ds) =∑i

λiX(t+ ti).

(3)The space Λk of discrete measures allowable atorder k consists of those λ, for which all Polyno-

3

mials P k : T → R of degree at most k vanish:

λ ∈ Λk ⇔∑i

λiPk(t+ ti) = 0 ∀ t ∈ T

whereby the number of λi is almost arbitrary.This emulates closely polynomials of degree kbeing in the kernel of the differential operatorof order k + 1. Such Xλ are then called allow-able linear combinations of order k, or ALC-kfor short. If λ ∈ Λk then polynomial drifts ofX(·) up to order k in mean and up to order 2kin variance are mapped to 0 by λ according to

X(t) := Y (t) + P k(t)Xλ(t) = Yλ(t) + P kλ (t)︸ ︷︷ ︸

0 since λ∈Λk

= Yλ(t)

E[X(s)X(t)] := K(s, t)︸ ︷︷ ︸only function of t−s

+∑i6k

P i(s)f1i (t)

+∑j6k

P j(t)f2j (s) +

∑i,j6k

P i(s)P j(t)

E[Xλ(s)Xλ(t)] = Kλλ(s, t) +∑i6k

P iλ(s)f1i λ(t)

+∑j6k

P jλ(t)f2j λ

(s) +∑i,j6k

P iλ(s)P jλ(t)

= Kλλ(s, t)︸ ︷︷ ︸only function of t−s

where Kλλ(s, t) is understood as a notational ex-tension of eq. (3) equivalent to

Kλλ(s, t) =∑i,j

λiλjK(t+ ti, s+ sj).

Xλ(t) is therefore free of the influences of thosepolynomial drifts and would be stationary ifthese were the only terms inciting instationarybehavior of X(·). If indeed this is the case andthe random field Xλ(·) : T 3 t 7→ Xλ(t) ∈ L2(Ω)is zero-mean and s.o.s. for λ ∈ Λk:

E[Xλ(t)] = 0 ∀t ∈ TE[Xλ(s)Xλ(t)] = C(t− s) ∀s, t ∈ T (4)

then X(·) is called an IRF-k. For explanatorypurposes we will showcase the necessary con-structions and the derivation of the BLIE for theAPS A(·) in the 1D case only. The extensionsrequired to handle data in more than one dimen-sion are straightforward (see Matheron, 1973)and any non-obvious adjustments to the proce-dure —necessary for the application to 2D ran-dom fields in sec. 4—will only originate from the

polar geometrical nature of the measurements.To avoid unnecessary clutter of symbols, the lin-ear map Lλ : F 3 f(·) 7→ fλ(·) ∈ F with F somespace of functions will also be denoted by λ. Noconfusion should arise due to the different mean-ings of λ as a measure and the linear map thatmaps a function f to fλ.

We will now define two specific measures ∇and ∫ that are inverse to each other in a cer-tain sense to be made explicit later. In section3.3 they will be the basis for a formalization ofthe idea that differencing in range direction canmake interferograms s.o.s. Let T be a set of pixelindices now, and

∇ : Ξ 3 A(·) 7→ A∇ = A(·)−A(· − 1) ∈ Ξ

∫ : Ξ 3 A(·) 7→ A∫ (·) =·−a∑k=0

A(a+ k) ∈ Ξ

where A(·) : T 3 t 7→ A(t) ∈ L2(Ω), T ⊂ Z is adiscrete (1D) random field and a is an arbitraryinteger constant corresponding to the lower limitof integration. A short calculation shows

∇∫ A(·) = ∇( ·−a∑k=0

A(a+ k))

(5)

=·−a∑k=0

A(a+ k)−·−a−1∑k=0

A(a+ k)

= A(·)∫ ∇A(·) = ∫ (A(·)−A(· − 1))

=·−a∑k=0

A(a+ k)−A(a+ (k − 1))

= A(·)−A(a− 1)

so that ∇ and ∫ are not strictly inverses on Ξ.However, analogous to integration and differen-tiation in real analysis, ∫ ∇ differs from the iden-tity function idΞ only by A(a − 1) which is anelement of ker∇ := X ∈ Ξ : X∇ = 0 and canbe considered as giving rise to the identity on thequotient space Ξ / ker∇. By the universal prop-erty of quotient spaces (Ash, 2013 p. 89) ∃!∇ suchthat

Ξ Im∇

Ξ/ ker∇

...................................................................................................................... ............∇

........................................................................................... ............

π

....................................................................................................

∇π : Ξ 3 X 7→ X ∈ Ξ/ ker∇

∇ = ∇ πX = Y ∈ Ξ : Y−X ∈ ker∇

commutes, that is ∇ πX = ∇X ∀ X ∈ Ξ.4

Here, as in X, the overbar is understood to in-dicate the equivalence class of X under the re-lation X ∼ Y ⇔ X − Y ∈ ker∇ and Ξ/ ker∇is the space of equivalence classes with additionX+ Y = X + Y and scalar multiplication αX =αX,α ∈ R. Since ∇X = ∇X is well defined ac-cording to X ∼ Y ⇒ ∇(X−Y ) = 0⇔ X∇ = Y∇and by the first isomorphism theorem for mod-ules (Ash, 2013 p. 89) Im∇ ∼= Ξ/ ker∇ we sur-mise that ∇ might be an isomorphism betweenIm∇ and Ξ/ ker∇. Indeed it is

surjective by ∇(Ξ/ ker∇) = ∇ π(Ξ) = Im∇injective by X 6= Y ⇒ X − Y /∈ ker∇

⇒ ∇X 6= ∇Y ⇒ ∇X 6= ∇Yand lastly a homomorphism by

∇(X + Y ) = ∇(X + Y ) = ∇X + ∇Y∇(αX) = ∇αX = α∇X .

Furthermore, ∫ = π ∫ : Im∇ → Ξ/ ker∇ servesas a two sided inverse homomorphism. Thismeans that the following diagrams commute.

Im∇ Im∇

Ξ/ ker∇

............................................................................................................ ............

idIm∇

....................................................................................................∫ ........................................................................................... .......

.....

∇ ∇∫X(·) = ∇π ∫ X(·)= X(·)

Ξ/ ker∇ Ξ/ ker∇

Im∇

......................................................................... ............

idΞ/ ker∇

....................................................................................................

∇........................................................................................... .......

.....

∫∫ ∇X(·) = π ∫ ∇X(·)

= X(·)

As of now ∇ : Ξ/ ker∇ → Im∇ is identified as anisomorphism with inverse ∫ : Im∇ → Ξ/ ker∇.This implies in fact the recapturability of theequivalence class of A(·) from A∇(·) since A(·) =∫A∇(·) satisfies ∇A(·) = A∇(·) meaning that Ais the equivalence class of all solutions A(·) to∇A(·) = A∇(·) with A∇(·) given. By exchangingthe space of functions on T ×T for Ξ in the quo-tient space constructions, it is possible to showthat equivalence classes σ of covariance functionsare recoverable from C in the same way equiv-alence classes A of random functions are fromA∇. They turn out to be not only the mini-mum requirement for best linear estimation butalso directly inferable from the data by extend-ing the inversion procedure on the quotient spaceΞ/ ker∇ to F/ ker∇∇ and applying it to a para-metric model of C.

3.2 Derivation of the BLIE

The main impediment complicating statisticalinference for instationary random fields is thefact that the covariance function σ depends notonly on the difference between pixels but alsoon the location, making it impossible to infer itfrom one realization of a random field. But itwill turn out, that for optimal estimation theinstationary covariance is in fact not strictlyrequired; the equivalence class of its station-ary part, termed generalized covariance (GC) byMatheron (1973), will be sufficient. And the GCin turn depends only on the equivalence class ofthe random field enabling reliable estimation inpresence of drift in mean and variance. SinceA∇(·) can be assumed stationary, the covariancefunction of the ALC-0 A∇(·)

E [A∇(s)A∇(t)] = σ∇∇(t, s) = C(t, s)

is stationary and can be inferred from the data.Employing arguments from section 3.1 it can beshown that for ∇1σ = σ∇δ and ∇2σ = σδ∇ theinverse of ∇2∇1 : F/ ker∇2∇1 → ∇2∇1(F ) isπ21 ∫1 ∫2 with F some function space containingσ and other quantities as defined below.

F ∇1(F )

F/ ker∇1

.............................................................................................................. ............

∇1

....................................................................................................

π1...................................................................................... .......

.....

∇1

∇1(F ) ∇2∇1(F )

∇1(F )/ ker∇2

.............................................................................. ............

∇2

..................................................................................................

π2...................................................................................... .......

.....

∇2

π is natural projection onto the quotient space∇σ = ∇σ ; σi equiv. class of σ in F/ ker∇i∇i(F ) is space of functions of type ∇iσ, σ ∈ F

F ∇1(F ) ∇2∇1(F )

F/ ker∇2∇1

........................................................................................................................................................... ............∇1

........................................................................................................................... ............∇2

........................................................................................................................................................................................................................................................................................................................................................... ............

π21

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

...................

............

∇2∇1

Checking that they are inverses is simple:

For C ∈ ∇2∇1(F ), ∇1e = d, ∇2d = C

i) ∇2∇1π21 ∫1∫2C = ∇2∇1π21 ∫

1(d+ q2)

= ∇2∇1π21(e+ q1︸︷︷︸∈ker∇1

+ ∫1q2︸︷︷︸

∈ker∇2

)

= ∇2∇1e21 = ∇2∇1e = C

5

ii) π21 ∫1∫2∇2∇1e

21 = π21 ∫1∫2∇2(∇1e)

= π21 ∫1∇1e+ q2

= π21(e+ q1 + ∫1q2) = e21

This directly translates to the formula

σ21 = π21∫1∫2C (6)

since ∇2∇1σ = σ∇∇ = C is known allowing in-ference not of the instationary covariance func-tion but only of its equivalence class π21σ. Forany λλ : σ 7→ σλλ with kerλλ ⊇ ker∇2∇1λλ : F/ ker∇2∇1 3 σ21 7→ σλλ ∈ ∇2∇1(F ) iswell defined:

σ1 ∼ σ2 ⇔ σ1 − σ2 ∈ ker∇2∇1 ⊆ kerλλ⇒ λλ(σ1 − σ2) = 0⇔ λλσ1 = λλσ2

Thus for any λ ∈ Λ0 σλλ only depends on theequivalence class σ21 which can be calculatedfrom the stationary covariance C of the firstderivatives of A(·). We estimate the atmosphereat t0 ∈ T linearly from n measurements:

A(t0) =n∑j=1

λjM(t0 + sj) = λTM

The error ε(t0) = A(t0)−A(t0) = Aε +Nλ withε =

∑nj=0 λjδsj − δ is an instationary random

field. Forcing the residual term Aε to be drift in-dependent and stationary is equivalent to addingthe constraint ε ∈ Λ0 effectively constraining εto eliminate constant polynomials translating tothe equation

∑ni=1 λi − 1 = 0. Setting K =

π21 ∫1 ∫2 C and C = σ∇∇ the error variance canbe written as

E[ε2] = E[(Aε −Nλ)2]AqN= E[A2ε ] + E[N2

λ ]= σεε(t0, t0) + λTΣNλ

ε∈Λ0

= Kεε(t0, t0) + λTΣNλ= λT (Kij + ΣN )︸ ︷︷ ︸

K

λ− 2λTKt0 +K(t0, t0)

Kijkl = K(t0 + sk, t0 + sl)Kt0k = K(t0 + sk, t0) k, l = 1, ..., n

Minimizing E[ε2] subject to∑nj=1 λj − 1 = 0

leads to the system of equations (Berlinet andThomas-Agnan, 2011 pp. 84-86)

Kλ+ 1−µ = Kt0 (7)

1−Tλ = 1

where µ is a single Lagrange-multiplier and 1− ∈Rn is a vector of ones. Successive substitutiongives as the final result

A(t0) =n∑j=1

λjM(t0 + sj) (8)

λ = K−1(Kt0 − 1−(1−

TK−11−)−1(1−TK−1Kt0 − 1)

)This estimator can be seen to coincide with or-dinary Kriging (see Cressie, 1990 for a similarformulation) but with the GC K instead of thestationary covariance C and an additional termaccounting for the effects of noise. How this for-mula is to be adapted for practical applicationunder consideration of coordinate dependent as-pects will be discussed in the next section.

3.3 Geometrical considerations and practi-cal implementation

Let s ∈ T be a point and let cs = (xs, ys)T andps = (rs, ϕs)T denote Cartesian and polar coor-dinates of this point, respectively. The coordi-nate transform given by

φ : [0,∞)× [0, 2π) 3 ps 7→ cs ∈ R2

φ(ps) =[rs cos(ϕs)rs sin(ϕs)

]φ−1(cs) =

[ √x2s + y2

s

atan(ys/xs)

]maps the polar coordinates of a point s to itsCartesian coordinates. If f(·) is a function fromT to R then f c(·) and fp(·) will be the corre-sponding functions acting on the Cartesian andpolar coordinates satisfying f(s) = f c(cs) andf(s) = fp(ps). The identity f(s) = f c(cs) =f c(φps) implies the transformation laws f c φ =fp and fp φ−1 = f c. An analogous statementholds for functions with multiple inputs for whichthe coordinate transform is applied to each inputseparately, e.g.:

σc (φ, φ) = σp

σc = σp (φ−1, φ−1)

Following the line of argumentation outlined insection 2 the derivative of the APS A(·) in line-of-sight may be regarded as s.o.s.. It will be nu-merically approximated by the discrete deriva-tive of A(·) in range direction. Introducing themeasure ∇ as below, A∇(·) ≈ (∂/∂r)A(·) and alinear map ∫ can be found such that the relationbetween ∇ and ∫ closely resembles that of ∇ and∫ as defined in the 1D case in section 3.1.

6

Note however that apart from the errors intro-duced by discretization, further sources of uncer-tainty exist.

1. The derivative in range direction is used in-stead of the inaccessible derivative in line-of-sight.

2. Height information is not used although itmight in reality have an integral part to playin the determination of atmospheric correla-tions.

3. The covariance of A∇ is treated as thoughit were stationary even if it is likely tobe height dependent and empirically knownonly at ground level.

Given the lack of reliable information regardingthe stochastic properties of differential phase de-lay and the ill-posedness of the estimation prob-lem, none of these simplifying neglections can beproven justified within the framework, in whichthe BLIE is valid. For the time being we willjust assume their validity and not try to find amore faithful stochastic model. However, ∇ and∫ still need to be defined in terms of coordinatesto provide a link between image geometry andstochastic aspects of the measurements:

∇ : A(·) 7→ A∇(·) = 1∆r [Ap(p·)−Ap(p· −∆)]

∫ : A(·) 7→ A∫ (·) =

r·−ra∆r∑k=0

Ap(p· + k∆)∆r

Here ∆ = (∆r, 0)T, ∆r is the range differencebetween two pixels and a ∈ T is a point unde-termined apart from its membership to the linejoining · and the instrument. Closedness of pTunder translation by ∆ is presumed only to avoidproblems of the definitions in the vicinity of theboundary of pT .

Calculations mirroring those in section 3.2show that ∇ and ∫ are isomorphisms betweenΞ/ ker∇ and Im∇ and π21∫1∫2 and∇2∇1 are iso-morphisms between F/ ker∇2∇1 and Im∇2∇1for F 3 σ and ∇2∇1σ = σ∇2∇1 . Thus one of theGC’s K satisfying K−σ ∈ ker∇2∇1 as requiredin the BLIE can be seen to be

K(s, t) = ∫1∫2Cp(ps, pt) (9)

=

rs−ra∆r∑k=0

rt−rb∆r∑j=0

Cp(pa + k∆, pb + j∆)∆r2

Here a = (r0, ϕs)T and b = (r0, ϕt)T are chosento lie on the lower border of the interferogram.Equation 9 can be interpreted as assuring thatthe GC K(s, t) can be computed as the doubleintegral of the stationary covariance function Cc(Cp need not be stationary) along the two linesLs, Lt joining points a, b to s, t (see Fig. 3).

Fig. 3: Geometric relations between A (top layer)and A∇ (lower layer) are inherited by the covariancefunctions.

We therefore recognize equation 9 as a discretiza-tion of the integral representation of K(s, t) inthe continuous case:

K(s, t) =∫Ls

∫Lt

C(u, v)dudv

=∫ rs

r0

∫ rt

r0

Cp([

r1ϕs

],

[r2ϕt

])dr1dr2

=∫ rs

r0

∫ rt

r0

Cc(φ

[r1ϕs

], φ

[r2ϕt

])dr1dr2

This can be understood intuitively. The totalcorrelation between the APS associated to pointss and t consists of the sum of all individual cor-relations between points in the atmosphere alongthe propagation paths Ls and Lt. Even thoughK(s, t) is instationary, it differs from a stationaryGC only by an element of ker ∇2∇1 as proven inGelfand and Vilenkin (1964, pp. 179-186).

To summarize our findings and tie them toa practically feasible algorithm, we propose thefollowing scheme:

1. Choose PS lying outside the deformationarea by thresholding on the ADI and usingprior knowledge on stable areas.

2. Calculate A∇ from a version of A previ-ously smoothed with a median filter to re-

7

duce the impact of noise on the estimationof the derivative.

3. Infer Cc(cs, ct) = E[A∇(s)A∇(t)] by min-imizing the squared error between theempirical covariance and the parametricanisotropic stationary covariance model.

4. Calculate the GC K(s, t) using eq. 9 andestimate ΣN by using formulas linking co-herence and signal-to-noise ratio (see e.g.Hanssen, 2006 p. 98) and a rough prior ofthe variance of the APS.

5. Estimate the APS at the unobserved loca-tions using its noisy observations on the PSusing eq. 8 and subtract it from the measure-ments to estimate the deformations.

We finally propose to perform these steps for aset of overlapping windows to lessen the compu-tational burden and the impact of neglected butpotentially existing instationarity in Cc(cu, cv).

In our tests step 3. proved to be problem-atic. In accordance to Fig. 2 the estimation ofan anisotropic covariance function yielded veryshort correlation lengths in range-direction. Thishowever affected unfavourably the robustness ofthe output and the runtime of the algorithm dueto the demand for smaller steps in the numericalintegration of Cc. To sidestep these challenges,isotropy of Cc was assumed during the produc-tion of the results presented in section 4. Addi-tionally r0 was set to 0 and Cc chosen in a wayto optimize the fit between K and the empiri-cal covariance of A(·) circumventing occasionallyunreliable inference of K from a single dataset.

4 Validation and conclusion

For testing and validation purposes we drawupon the data gathered during a 2014 measure-ment campaign in the alpine regions of southernSwitzerland, see Butt et al. (2016). The datasetfeatures rugged terrain with height differencesexceeding 1 km, a spatiotemporally highly vari-able APS, and stable reference areas surroundinga rapidly moving glacier. Data were collected ata sampling interval of 2 minutes from a locationacross the valley, at a distance of up to 8 km fromthe monitored area. Even though movements lo-cally reach 2 m/day the influence of the APS canmask the signal to the point where atmosphericartifacts and real displacements become indistin-guishable (see Fig. 4).

Fig. 4: From top to bottom: The average of fivetwo-minute interferograms, estimated velocities aftersubtraction of the APS and the error variance.

For the estimation of the APS the phase val-ues of approximately 3500 PS were used, thedistribution of which was very sparse in theright fifth of the interferograms shown in Fig. 4(≈ 25PS/km2) and increasingly dense towardstheir center (≈ 350PS/km2). As can be seenin Fig. 4 the correction scheme performs well inregions with a high amount of PS and producespoor results in regions with few PS (right) or badcoherence (left). This is also supported by theaccompanying estimation of the error variance,which is helpful for quantifying the uncertaintyassociated with the APS correction procedure. Itis worth noting that the left parts of the imagescontain the backscattering of objects in a dis-tance of about 4 km to the instrument while forthe right, topographically highly irregular partsthis distance is about 8 km. Consequently the er-ror variances are significantly higher in the latterarea.

8

Depending on the number of PS the errorvariance for the deformation estimation using 2-minute interferograms is in certain regions of amagnitude that rivals the amplitude of the sig-nal. This can be explained with short correlationlengths induced by turbulent atmospheric behav-ior; averaging in time improves the stochasticproperties of the residuals by mitigating tempo-ral high-frequency components. Fig. 5 shows theaverage total errors derived from cross validationon the stable areas as a function of averagingtime and size of the area without any PS in it.

Fig. 5: Average total errors of estimated deforma-tions for the IRF-k approach and simple stacking.

Two obvious trends are clearly visible: largeraveraging times lead to more reliable estimationsin both cases and the larger the area without re-liable APS-observations, the more uncertain theresults. Less than 0.05m/day average total er-ror are reached within 20 minutes with our ap-proach, whereas simple averaging in time needs12 h and the widely employed fitting of secondorder polynomials (not shown) around 2 h.

We conclude the suitability of IRF-k’s for mit-igating the atmospheric phase screen and facil-itating deformation monitoring based on inter-ferograms heavily affected by instationary au-tocorrelated atmospheric noise as expected inmountaineous terrain. We will further pursuean approach generalizing IRF-k’s to linear ran-dom functionals and reproducing kernel Hilbertspaces of distributions to better include geomet-ric correlations induced by the topography andbest represented via linear operators acting onstationary covariance functions in 3D space.

Acknowledgement. The radar data used inthis paper were collected in cooperation withProf. Martin Funk, ETH Zurich, and Prof. Mar-tin Truffer, University of Alaska, who also pro-vided the instrument. The Swiss Federal Officefor the Environment has supported the data col-lection financially.

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