Analysis of forest-slab height inversion from multibaseline SAR data A. Liseno Universit` a di Napoli “Federico II” Dip. di Ing. Elettr. e delle Telecomunicazioni via Claudio 21 I-80125 Naples (Italy) K.P. Papathanassiou, A. Moreira German Aerospace Center (DLR) Microwaves and Radar Institute D-82230 Oberpfaffenhofen (Germany) R. Pierri Seconda Universit` a di Napoli Dip. Ing. Informazione via Roma 29 I-81031 Aversa (Italy) Abstract—We address the problem of reconstructing parame- ters regarding the vertical structure of forests from multibaseline SAR data. The attention is focused on the retrieval of the forest height so that the case of vanishing ground scattering is considered. The cross-spectral density of the scattered field is taken as datum of the problem and the forest height is searched for by tackling an optimization problem. An analytical and numerical analysis of the involved functional is performed by taking due care to the discrete nature of the scatterers. I. I NTRODUCTION Microwave remote sensing has a potential role in deter- mining forest parameters, and thus relevant climatologic and hydrologic indicators, since forest biomass could be extracted from remotely sensed forest structure, especially forest height [1]. Traditional Synthetic Aperture Radar (SAR) reconstructs two- dimensional images of the investigated scene and thus the possibility of recovering information about parameters of the forest vertical structure amounts at acquiring data regarding the “height” dimension. The “smallest” possible sensing con- figuration to gain information about the “height” is Interfero- metric SAR (InSAR) [2]. However, the single-baseline/single- polarization coherency coefficient provides only two real ob- servables for each pixel, so that the inversion problem can be faced only under strong simplifying assumptions, or a priori information, and results to be severely ill-conditioned. In order to increase the number of available data, (single- baseline) Polarimetric SAR Interferometry, or Pol-InSAR, exploits the knowledge of the interferometric coherence at different polarizations [3]. In this way, for each pixel, six real observables are provided. Further enlargement of the number of available observables, through dual-baseline Pol-InSAR, has been also recently proposed [4]. By facing the forest height reconstruction problem as an optimization one leads to the minimization of a non-quadratic functional [5]. A such functional might exhibit several local minima, especially if the ratio between the number of (inde- pendent) data and the number of unknowns is low [6], so that gradient-based minimization procedures could be trapped into false solutions. From this point of view, defining strategies to improve the number of data to be exploited during the inversion, thus increasing the ratio between the number of observables and that of the unknowns, is mandatory to provide reliability to the reconstruction procedure and to reduce the ill-conditioning. Furthermore, it should be mentioned that new trends in next satellite missions foresee to combine space-borne radars with sets of passive receivers onboard of a constellation of microsatellites or clusters of fully active sensors in a multistatic configuration [7]. This encourages the investigation towards “multi-baseline” approaches to forest structure parameter retrieval for improvements of the quantity of available data. The aim of this paper is to present and analyze a multibaseline reconstruction approach for determining parameters of the forest vertical structure by exploiting knowledge of the cross- spectral density of the scattered field acquired over different flight tracks. Attention is particularly focused on forest height retrieval and, in order to analyze the reconstruction capabilities of the approach as a function of the parameters of the sensing configuration in a simplified situation, the working assumptions of negligible ground scattering contribution and single-polarization are carried on. II. THE SPECTRAL DENSITY OF THE SCATTERED FIELD Let us consider the one-dimensional geometry depicted in Fig. 1 for which the measurements of the scattered field E s (r, s) are performed at the same quota h within the interval (-b M ,b M ). The relevant link between the scatterers’ distribution χ(s ′ ) and the data of the problem can be written as [5] E s (r, s)=  s ′ 1 s ′ 0 χ(s ′ )e -2σ(s ′ 1 -s ′ ) e j2k0R(r,s;r ′ ,s ′ ) ds ′ , (1) where s ′ 1 = s ′ 0 + h ′ V , h ′ V = h V / cos θ i is the projection of the “forest-slab” height along the s ′ direction, R(r, s; r ′ ,s ′ )=  (r - r ′ ) 2 +(s - s ′ ) 2 , and 2σ indicates the two-way atten- uation. If we now let s = b cos θ i , r = b sin θ i , and Ψ(s ′ ) = χ(s ′ ) exp(-2σ(s ′ 1 - s ′ )), then, under the Fresnel zone ap- proximation [8], eq. (1) can be simplified, after normalization to non-essential factors, as 2660 0-7803-9050-4/05/$20.00 ©2005 IEEE. 2660