Scale Issues in Remote Sensing: A Review on Analysis, Processing and Modeling <s> The scaling methods for retrieval models <s> Abstract Models used in remote sensing are based mostly on a relationship between a parameter and the measured reflectance at a given wavelength. They are generally built on the basis of physical considerations at a scale enough small enough so that the media can be considered to be homogeneous. The general practice in remote sensing is to apply these models to reflectances measured from satellites on heterogeneous surfaces, without considering making any correction in order to compensate for the change of scale. It is shown that this usual practice generates errors, the lower and upper bounds of which are given in the form of two functions associated with the physical model. Finally, a general method is proposed to build a new model connected to the physical one, reducing the error introduced by the heterogeneity of the analysed surface. An application to the role of resolution change for the quantification of covered area is also considered. <s> BIB001 </s> Scale Issues in Remote Sensing: A Review on Analysis, Processing and Modeling <s> The scaling methods for retrieval models <s> Models used in remote sensing are based on the assumed relationship between measured radiances and the physical parameters of the soil surface. A number of them are semi-empirical, in the sense that they contain constants adjusted to take account of in situ experiments, generally carried out on a given type of medium. Once validated in situ, these models are applied to large scale areas with global radiances. In this paper, we demonstrate that constants taken to be appropriate for in situ experiments are no longer appropriate for global radiances. For remote sensing applications, we demonstrate that these constants must be determined by a new least square cost function. Finally, we evaluate, for various simulated cases using the LAI and the APAR models, the magnitude of gain of the new method for space applications, when compared to classical examples. <s> BIB002 </s> Scale Issues in Remote Sensing: A Review on Analysis, Processing and Modeling <s> The scaling methods for retrieval models <s> The problem of how the scale, or spatial resolution, of reflectance data impacts retrievals of vegetation leaf area index (LAI) is addressed in this article. We define the goal of scaling as the process by which it is established that LAI values derived from coarse resolution sensor data equal the arithmetic average of values derived independently from fine resolution sensor data. The increasing probability of land cover mixtures with decreasing resolution is defined as heterogeneity, which is a key concept in scaling studies. The effect of pixel heterogeneity on spectral reflectances and LAI retrievals is investigated with 1-km Advanced Very High Resolution Radiometer (AVHRR) data aggregated to different coarse spatial resolutions. It is shown that LAI retrieval errors at coarse resolution are inversely related to the proportion of the dominant land cover in such pixel. Further, large errors in LAI retrievals are incurred when forests are minority biomes in non-forest pixels compared to when forest biomes are mixed with one another, and vice versa. A physically based scaling with explicit spatial resolution-dependent radiative transfer formulation is developed. The successful application of this theory to scaling LAI retrievals from AVHRR data of different resolutions is demonstrated. These principles underlie our approach to the production and validation of LAI product from the Moderate Resolution Imaging Spectroradiometer (MODIS) and the Multi-angle Imaging Spectroradiometer (MISR) aboard the TERRA platform. <s> BIB003 </s> Scale Issues in Remote Sensing: A Review on Analysis, Processing and Modeling <s> The scaling methods for retrieval models <s> The monitoring of earth surface dynamic processes requires global observations of the structure and the functioning of vegetation. Moderate resolution sensors (with pixel size ranging from 250 m to 7 km) provide frequent estimates of biophysical variables to characterize vegetation such as the leaf area index (LAI). However, the computation of LAI from moderate resolution remote sensing data induces a scaling bias on the LAI estimate if the moderate resolution pixel is heterogeneous and if the transfer function that relates remote sensing data to LAI is non-linear. This study provides a model to evaluate and correct the scaling bias. The model is built first for a univariate semi-empirical transfer function relating LAI directly to NDVI. The scaling bias is a function of (i) the degree of non-linearity of the transfer function quantified by its second derivative and (ii) the spatial heterogeneity of the moderate resolution pixel quantified by the variogram of the high spatial resolution (20 m) NDVI image. Then, the model is extended to a bivariate transfer function where LAI is related to red and near infrared reflectances. The scaling bias depends on (i) the Hessian matrix of the transfer function and (ii) the variograms and cross variogram of the red and near infrared reflectances. The scaling bias is investigated on several distinct landscapes from the VALERI database. Adjusting for scaling bias is critical on crop sites which are the most heterogeneous sites at the landscape level. Regarding the univariate transfer function, the magnitude of the scaling bias increases rapidly with pixel size until this size is larger than the typical spatial scale of the data. For the bivariate transfer function, it results from the addition of several components that may add up or cancel each other out. It is thus more difficult to analyze. The accuracy of the model to estimate the scaling bias is discussed. It depends mainly on the ability of the variograms and cross variogram to represent the local dispersion variances and covariance within the moderate resolution pixel. The model is generally highly accurate at 1000 m spatial resolution for the univariate transfer function and less accurate for the bivariate transfer function. <s> BIB004 </s> Scale Issues in Remote Sensing: A Review on Analysis, Processing and Modeling <s> The scaling methods for retrieval models <s> The paper revises and clarifies definitions of the term scale and scaling conversions for imaging spectroscopy of soil and vegetation. We demonstrate a new four-dimensional scale concept that includes not only spatial but also the spectral, directional and temporal components. Three scaling remote sensing techniques are reviewed: (1) radiative transfer, (2) spectral (un)mixing, and (3) data fusion. Relevant case studies are given in the context of their up- and/or down-scaling abilities over the soil/vegetation surfaces and a multi-source approach is proposed for their integration. ::: Radiative transfer (RT) models are described to show their capacity for spatial, spectral up-scaling, and directional down-scaling within a heterogeneous environment. Spectral information and spectral derivatives, like vegetation indices (e.g. TCARI/OSAVI), can be scaled and even tested by their means. Radiative transfer of an experimental Norway spruce (Picea abies (L.) Karst.) research plot in the Czech Republic was simulated by the Discrete Anisotropic Radiative Transfer (DART) model to prove relevance of the correct object optical properties scaled up to image data at two different spatial resolutions. Interconnection of the successive modelling levels in vegetation is shown. A future development in measurement and simulation of the leaf directional spectral properties is discussed. ::: We describe linear and/or non-linear spectral mixing techniques and unmixing methods that demonstrate spatial down-scaling. Relevance of proper selection or acquisition of the spectral endmembers using spectral libraries, field measurements, and pure pixels of the hyperspectral image is highlighted. An extensive list of advanced unmixing techniques, a particular example of unmixing a reflective optics system imaging spectrometer (ROSIS) image from Spain, and examples of other mixture applications give insight into the present status of scaling capabilities. ::: Simultaneous spatial and temporal down-scaling by means of a data fusion technique is described. A demonstrative example is given for the moderate resolution imaging spectroradiometer (MODIS) and LANDSAT Thematic Mapper (TM) data from Brazil. Corresponding spectral bands of both sensors were fused via a pyramidal wavelet transform in Fourier space. New spectral and temporal information of the resultant image can be used for thematic classification or qualitative mapping. <s> BIB005
Generally speaking, we may only know the retrieval model at a specific scale. That is to say, we need to scale the retrieval models to the other scales through the appropriate assumptions and simplifications in order to not only consider the scaling effects but also provide scale invariant algorithms. Raffy BIB001 proposed a general method, the Computational Geometry Method (CGM), to reduce the scaling effects introduced by the heterogeneity of measurements and the nonlinearity of a retrieval model at a local scale. The method takes advantage of the convex hull of computational geometry to determine the interval where the distributed result may exist. Considering all possible distributions of measurements, the distributed result always falls into the interval of the lower and upper bounds of the retrieval model which measures the maximum error due to the scaling. Here, the bounding functions can be interpreted as a measure for the non-linearity of the model  . Subsequently, Raffy and Gregoire BIB002 used such a concept combined with a least square method to determine the coefficients of the semi-empirical model that is validated at a small scale and upscale the model to the large scale with global radiances. If there is not more information available other than the domains of measurements and the retrieval model at the local scale, one can expect to reduce the scaling effects by employing the model spatialization method, which is the function of the convex and concave function of the retrieval model at a local scale. Garrigues et al. BIB004 argued that the assumption that the retrieval model follows a uniform distribution in the interval between the convex and concave function is inappropriate within a moderate resolution pixel. This may be one of the reasons limiting the application of the spatialization model. For the specific retrieval model, a more specific scaling method would be used, the Physical Scaling Method (PSM) based on the radiative transfer theory. Tian et al. BIB003 developed this method with an explicit spatial resolution dependent in order to upscale LAI retrieved from AVHRR data to the coarser resolutions. Malenovsky et al. BIB005 discussed PSM and gave examples of the radiative transfer scaling in a Norway spruce forest stand.