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Gradient Approach to INSAR Modelling of Glacial Dynamics and Morphology

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Gradient Approach to INSAR Modelling of Glacial Dynamics and Morphology Gradient approach to INSAR modelling of glacial dynamics and morphology A. I. Sharov Institute of Digital Image Processing, Joanneum Research, Wastiangasse 6, A - 8010 Graz, Austria E-mail: [email protected] Keywords: differential interferometry, phase gradient, change detection, glacier velocity, geodetic survey ABSTRACT: This paper describes the development and testing of an original gradient approach (GINSAR) to the reconstruction of glacial morphology and ice motion estimation from the interferometric phase gradient that does not involve the procedure of interferometric phase unwrapping, thus excluding areal error propagation and improving the modelling accuracy. The global and stringent GINSAR algorithm is applicable to unsupervised glacier change detection, ice motion estimation and glacier mass balance measurement at regional scale. Experimental studies and field surveys proved its efficacy and robustness. Some algorithmic singularities and limitations are discussed. 1 INTRODUCTION detect and to measure quite small glacier motions and ice deformations in the centimetre range from There is a close interrelation between glacier motion spaceborne SAR interferograms with a nominal and glacial topography. Irregularities in the glacier ground resolution of several tens of meters. Accurate width, thickness, slope or direction alter the reconstruction of the configuration and external character of ice flow. On the other hand, glacial structure of the glacier surface from INSAR data is topography is dynamically supported by the moving also possible. Differential interferometry (DINSAR) ice (Fatland & Lingle 1998). Both, glacier dynamic based on differencing between two different SAR quantities and topographic variables are essential interferograms of the same glacier allows the parameters for studying the glacier mass balance and impacts of glacial topography and surface monitoring actual and potential glacier changes. The displacement on the interferometric phase to be detailed and equivalent spatial models derived from analysed separately, but involves some technological remote sensing data that reliably describe the glacier complications and restrictions (Gabriel et al. 1993). shape and spatial distribution of ice velocity at This paper reports on the design of a stringent, different instants of time are of fundamental albeit very simple algorithm for the reconstruction of significance for such studies. glacial morphology and ice motion estimation from While the generation of glacier surface models, the interferometric phase gradient that does not e.g. by stereophotogrammetric methods, is well involve complex process artifices and does not established, the generation of glacier kinematic require additional topographic reference models. models representing the ice flow pattern or velocity The following methodical sequence of actions is vector field still remains an area of active research. considered This is mostly because of the typically low velocity • calculation of interferometric phase gradients of glacier ice motion that restricts the application of and their conversion to the glacier slope map; remote sensing data with limited spatial resolution • unsupervised glacier change detection by and short temporal coverage. The common rate of differencing multitemporal INSAR slope maps; flow of the Alpine glaciers is from ten to twenty • determination of glacier dynamic quantities and inches per day in summer, and about half that in estimation of mass balance components such as winter (Webster’s Revised Unabridged Dictionary, longitudinal strain rate, motion field, flux 1998). The ground resolution of available remote divergence, etc.; sensing data is somewhat coarser and a quite long • verification of results by field surveys in the time interval (months to years) between subsequent European Arctic. surveys is usually required in order to detect and to An experimental data set included the ERS-1/2- measure the glacier motion accurately. INSAR pairs obtained over the Svartisen Ice Cap in The advancement of satellite radar interferometry Norway and the Northern Ice Dome in Novaya (INSAR) in the 1990s gave new impetus to the study Zemlya, Russia, which were processed using the of glacier dynamics from space. Apart from large RSG 4.1 software package. Terrestrial surveys of terrestrial coverage of up to 10,000 km² and general glacier velocities were undertaken at 5 outlet advantages of all-weather SAR systems, the INSAR glaciers in Novaya Zemlya in September 2001. method provides an astonishing opportunity to 2 INSAR MODELLING OF THE GLACIER The next relation gives another useful approximation SURFACE: SLOPE MAP VERSUS DEM using the central difference ∂ϕ(x, y) ϕ(x + ∆x, y)(−ϕ x − ∆x, y ) In addition to glaciological maps, two basic types of ≅ and ∂x 2∆x spatial models are presently used for the documentary representation of the glacier surface: ∂ϕ(x, y) ϕ(x, y + ∆y)(−ϕ x, y − ∆y ) ≅ . (2) • topographic models represent the glacier relief ∂y 2∆y by specifying the geodetic location and absolute elevation of glacier points with respect to the Further increase in the shift value coarsens the standard reference surface or co-ordinate system; spatial resolution of the topogram. • morphometric models describe quantitatively the The full two-dimensional derivative of configuration and external structure of the glacier interferometric phase is given as an algebraic sum of surface by specifying relative spatial parameters. partial derivatives A digital elevation model (DEM) and a slope ∂ϕ(x, y) ∂ϕ()x, y dϕ(x, y) = dx + dy ≅ ∇ϕ ⋅ ∆x + ∇ϕ ⋅ ∆y map/model (SM) are typical examples of ∂x ∂y x y topographic and morphometric models, respectively. or ∇ϕ(x, y) = ϕ(x + ∆x, y) +ϕ(x, y + ∆y) − 2ϕ(x, y) . (3) In common practice, slope maps are produced as a secondary product from available DEMs by using The value of the phase derivative is proportional to standard algorithms. An inverse procedure is seldom the difference in grey level between adjacent pixels performed. st of the interferogram and can take both positive and At the 21 EARSeL Symposium in Paris it was negative values. demonstrated, however, that the direct production of The biggest advantage of such an approach to glacier slope maps from INSAR data could be INSAR data processing is probably that the resultant performed in a more straightforward and a much less picture can be scaled with any real, not necessarily complicated manner than by interferometric DEM integer factor, that makes further processing flexible generation. An original gradient approach to and allows for any linear combination between the geometric processing of SAR interferometric data derivatives obtained from different interferograms. (GINSAR) has been shown to be capable of Considering the discrete character of the providing quite accurate glacier slope maps (Sharov interferometric phase, partial derivatives ∇ϕ and et al. 2002). The underlying concept of the GINSAR x ∇ϕ can be directly converted to the relative height technique is based on the transition from performing y operations on original SAR interferograms to the increments ∆hx and ∆hy by using the next equation analysis of their derivatives. Differentiation of original interferometric phase results in the gradient ∆hx, y ≅ C(x, y)⋅ ∇ϕ x, y , (4) picture, which provides a realistic view of the terrain where C(x, y) = 0.25π −1 ⋅ λ ⋅ B −1 (x) ⋅ R(y) ⋅ sinθ (y) (5) and can be unambiguously related to the glacier ⊥ relief (Fig. 1, a, b, c, d). is the conversion factor depending on the imaging The partial derivatives of interferometric phase geometry; λ = 5.66 cm is the wavelength of SAR ϕ()x, y in azimuth (x) and in range (y) direction are signal, B⊥ (x) is the length of the perpendicular well approximated by differences as (Sharov & component of spatial baseline, R(y) - the slant range, Gutjahr 2002) θ(y) - the look angle. ∂ϕ()x, y ϕ()()x + ∆x, y −ϕ x, y The full height increment is defined as follows ≅ ∇ϕ ()x, y = and ∂x x ∆x ∆h(x, y) ≅ C x, y ⋅∇ϕ(x, y) = ∆h ⋅ ∆x + ∆h ⋅ ∆y . (6) ∂ϕ(x, y) ϕ()()x, y + ∆y −ϕ x, y ( ) x y ≅ ∇ϕ ()x, y = . (1) ∂y y ∆y The INSAR image product specifying continuously Thus, in our practice, the phase derivatives are the glacier height increments is called topogram. calculated by subtracting an original interferential Our topograms are represented in the form of an picture from a translated version of the same RGB image. The first two layers give the partial interferogram. In mathematics, such an approach to height increments ∆hx and ∆hy . The third layer is image differentiation is known as method of finite occupied with the full height increment representing differences (Piskunov 1978). The shift values ∆x the rate of increase of h()x, y per unit distance. One and ∆y are equal 1 pixel usually, but, in general, can typical example of an INSAR topogram showing be manipulated separately in azimuth and range Fingerbreen Outlet Glacier at the Eastern Svartisen direction within the interval from 0 to several pixels. Ice Cap, Northern Norway is given in Figure 1, d. The terrestrial slope value ε can be calculated on GINSAR SM generation involves less complicated a pixel-by-pixel basis as follows operations than that of producing the INSAR DEM, yet without compromising on accuracy. All 2 2 2 − 0.5 ε = arccos []∆p ⋅ ()∆hx + ∆hy + ∆p , (7) technological operations involved act globally, i.e. over the whole image scene without gaps. For the where ∆p is the pixel diagonal size on the ground great majority of points in the interferential picture depending on the kind of product, e.g. ∆p ≈ 28 m in partial derivatives of the wrapped phase are equal to 5-look ERS-1/2-SAR interferometric products and, partial derivatives of the unwrapped phase. in 20-look interferograms, we have ∆p ≈ 56 m. The Therefore, the interferometric phase unwrapping, formula (7) provides a basic equation for which is reputed to be the most sophisticated and transforming the INSAR topogram in the slope map. problematic calculus in interferometric signal INSAR slope maps are also generated in the form processing, becomes redundant, thus excluding areal of 3-layer images.
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