In this letter we work with truly multitemporal change detection in multilooked, multifrequency polarimetric synthetic aperture radar (polSAR) data in the covariance matrix formulation. We apply recent general results on better approximations than the usual chi-squared distribution for the probability distributions associated with maximum likelihood ratio test statistics for equality of several block-diagonal covariance matrices with complex Wishart distributed blocks. We demonstrate the superiority of the new approximations by means of generated data and airborne EMISAR data from four time points covering an agricultural region in Denmark. Results from the generated data show the importance of applying the new approximations in the no change situation. This use is more important for low equivalent number of looks (ENL) and for long time series (i.e., high number of degrees of freedom). Results from the generated data example are confirmed by results from the case with EMISAR data.

We report on a complex Wishart distribution based test statisticQfor block-diagonality in Hermitian matrices such as the ones analysed in polarimetric synthetic aperture radar (polSAR) image data in the covariance matrix formulation. We also give an improved probability measurePassociated with the test statistic. This is used in a case with simulated data to demonstrate the superiority of the new expression forPand to illustrate the dependence of results on the choice of covariance matrix, its dimensionality, the equivalent number of looks, and two parameters in the improvedPmeasure. We also give two cases with acquired data. One case is with airborne F-SAR polarimetric data, where we test for reflection symmetry, another case is with (spaceborne) dual pol Sentinel-1 data, where we test if the data are diagonal-only. The absence of block-diagonal structure occurs mostly for man-made objects. In the example with Sentinel-1 data some objects (e.g., buildings, cars, aircraft and ships) are detected, others (e.g., bridges) are not.

In multilook polarimetric synthetic aperture radar (polSAR) data dihedral targets unaligned with the radar line of sight cause cross-polarized reflections which can be confused with vegetation. To avoid this, rotation can be performed. For the covariance representation of polSAR data this is described in two different ways, namely 1. by minimization of the covariance matrix cross-polarization term $\left\langle {{S_{hv}}S_{hv}^{\ast}} \right\rangle $ , and 2. by means of the argument of the circular covariance matrix term $\left\langle {{S_{rr}}S_{ll}^{\ast}} \right\rangle $ .

Unlike the original Cloude–van Zyl decomposition of reflection symmetric polarimetric synthetic aperture radar (polSAR) data, a recently suggested version of the decomposition for full/quad pol data relies on the Cloude–Pottier mean alpha angle (${\bar {\alpha }}$) to characterize the scattering mechanism.${\bar {\alpha }}$can be calculated from the eigenvectors of the coherency matrix. By means of the eigenvector-eigenvalue identity (EEI), we can avoid the calculation of the eigenvectors. The EEI finds${\bar {\alpha }}$by means of eigenvalues of the$3\,\,{\times }$3 coherency matrix and its$2\,\,{\times }$2 minor(s) only and is well suited for fast array-based computer implementation. In this letter with focus on computational aspects, we demonstrate fast EEI-based determination of${\bar {\alpha }}$on X-band Flugzeug synthetic aperture radar (F-SAR) image data over Vejers, Denmark, including a detailed example of calculations and computer code.

We describe calculation of eigenvalues of 2×2 and 3×3 Hermitian matrices as used in the analysis of multilook polarimetric SAR data. The eigenvalues are calculated as the roots of quadratic or cubic equations. The methods are well suited for fast matrix oriented computer implementation and we obtain speed-ups over calculations based on a built-in eigenproblem solver in for-loops over rows and columns in an image by a factor of 350 (for dual pol) and 175 (for full/quad pol).

We use the combined Wishart-Loewner method to successfully detect change and direction of change in truly multitemporal, multilooked quad/full polarization synthetic aperture radar image data in the covariance matrix representation. Based on in situ data interpretations of the obtained results are given for three selected fields. Histograms of the Wishart test statistics in a wooded no-change area shows good agreement with the theoretical distributions.

With a growing number of different satellite sensors, data fusion offers great potential in many applications. In this work, a convolutional neural network (CNN) architecture is presented for fusing Sentinel-1 synthetic aperture radar (SAR) imagery and the Advanced Microwave Scanning Radiometer 2 (AMSR2) data. The CNN is applied to the prediction of Arctic sea ice for marine navigation and as input to sea ice forecast models. This generic model is specifically well suited for fusing data sources where the ground resolutions of the sensors differ with orders of magnitude, here 35 km × 62 km (for AMSR2, 6.9 GHz) compared with the 93 m × 87 m (for sentinel-1 IW mode). In this work, two optimization approaches are compared using the categorical cross-entropy error function in the specific application of CNN training on sea ice charts. In the first approach, concentrations are thresholded to be encoded in a standard binary fashion, and in the second approach, concentrations are used as the target probability directly. The second method leads to a significant improvement in R2measured on the prediction of ice concentrations evaluated over the test set. The performance improves both in terms of robustness to noise and alignment with mean concentrations from ice analysts in the validation data, and an R2 value of 0.89 is achieved over the independent test set. It can be concluded that CNNs are suitable for multisensor fusion even with sensors that differ in resolutions by large factors, such as in the case of Sentinel-1 SAR and AMSR2.

We describe the calculation of eigenvalues of 2 × 2 or 3 × 3 Hermitian matrices as used in the analysis of multilook polarimetric synthetic aperture radar (SAR) data. The eigenvalues are calculated as the roots of quadratic or cubic equations. We also describe the pivot-based calculation of the Loewner order for the partial ordering of differences between such matrices. The methods are well suited for fast matrix-oriented computer implementation, and the speed-up over simpler calculations based on built-in eigenproblem solvers is enormous.

When the covariance matrix formulation is used for multilook polarimetric synthetic aperture radar (SAR) data, the complex Wishart distribution can be used for change detection between acquisitions at two or more time points. Here, we are concerned with the analysis of change between two time points and the “direction” of change: Does the radar response increase, decrease, or does it change its structure/nature between the two time points? This is done by postprocessing/coprocessing the detected change with the Loewner order which calculates the definiteness of the difference of the covariance matrices at the two time points. We briefly describe the theory. Two case studies illustrate the technique on Sentinel-1 data covering the international Frankfurt Airport, Germany, and on Radarsat-2 data covering Bonn, Germany, and surroundings. We successfully demonstrate our “direction” of change approach to detected change areas.

The Infrared Atmospheric Sounding Interferometer (IASI) on board the MetOp satellite series provides important measurements for Numerical Weather Prediction (NWP). Retrieving accurate atmospheric parameters from the raw data provided by IASI is a large challenge, but necessary in order to use the data in NWP models. Statistical models performance is compromised because of the extremely high spectral dimensionality and the high number of variables to be predicted simultaneously across the atmospheric column. All this poses a challenge for selecting and studying optimal models and processing schemes. Earlier work has shown non-linear models such as kernel methods and neural networks perform well on this task, but both schemes are computationally heavy on large quantities of data. Kernel methods do not scale well with the number of training data, and neural networks require setting critical hyperparameters. In this work we follow an alternative pathway: we study transfer learning in convolutional neural nets (CNN s) to alleviate the retraining cost by departing from proxy solutions (either features or networks) obtained from previously trained models for related variables. We show how features extracted from the IASI data by a CNN trained to predict a physical variable can be used as inputs to another statistical method designed to predict a different physical variable at low altitude. In addition, the learned parameters can be transferred to another CNN model and obtain results equivalent to those obtained when using a CNN trained from scratch requiring only fine tuning.

In this paper we present a combined strategy for the retrieval of atmospheric profiles from infrared sounders. The approach considers the spatial information and a noise-dependent dimensionality reduction approach. The extracted features are fed into a canonical linear regression. We compare Principal Component Analysis (PCA) and Minimum Noise Fraction (MNF) for dimensionality reduction, and study the compactness and information content of the extracted features. Assessment of the results is done on a big dataset covering many spatial and temporal situations. PCA is widely used for these purposes but our analysis shows that one can gain significant improvements of the error rates when using MNF instead. In our analysis we also investigate the relationship between error rate improvements when including more spectral and spatial components in the regression model, aiming to uncover the trade-off between model complexity and error rates.

Based on an omnibus likelihood ratio test statistic for the equality of several variance-covariance matrices following the complex Wishart distribution with an associated p-value and a factorization of this test statistic, change analysis in a time series of 19 multilook, dual polarization Sentinel-1 SAR data in the covariance matrix representation (with diagonal elements only) is carried out. The omnibus test statistic and its factorization detect if and when change occurs.

Data-driven classification algorithms have proved to do well for automatic target recognition (ATR) in synthetic aperture radar (SAR) data. Collecting data sets suitable for these algorithms is a challenge in itself as it is difficult and expensive. Due to the lack of labeled data sets with real SAR images of sufficient size, simulated data play a big role in SAR ATR development, but the transferability of knowledge learned on simulated data to real data remains to be studied further. In this letter, we show the first study of Transfer Learning between a simulated data set and a set of real SAR images. The simulated data set is obtained by adding a simulated object radar reflectivity to a terrain model of individual point scatters, prior to focusing. Our results show that a Convolutional Neural Network (Convnet) pretrained on simulated data has a great advantage over a Convnet trained only on real data, especially when real data are sparse. The advantages of pretraining the models on simulated data show both in terms of faster convergence during the training phase and on the end accuracy when benchmarked on the Moving and Stationary Target Acquisition and Recognition data set. These results encourage SAR ATR development to continue the improvement of simulated data sets of greater size and complex scenarios in order to build robust algorithms for real life SAR ATR applications.

Based on an omnibus likelihood ratio test statistic for the equality of several variance-covariance matrices following the complex Wishart distribution with an associated p-value and a factorization of this test statistic, change analysis in a (short) time series of multilook, polarimetric SAR data in the covariance matrix representation is carried out. The omnibus test statistic and its factorization detect if and when change(s) occur. The technique is demonstrated on airborne EMISAR C-band data but may be applied to ALOS, COSMO-SkyMed, RadarSat-2, Sentinel-1, TerraSAR-X, and Yoagan or other dual- and quad/full-pol data also.

We present the likelihood ratio test statistic for the homogeneity of several complex variance-covariance matrices that may be used in order to assess whether at least one change has taken place in a time series of SAR data. Furthermore, we give a factorization of this test statistic into a product of test statistics that each tests simpler hypotheses of homogeneity up to a certain point and that are independent if the hypothesis of total homogeneity is true. This factorization is used in determining the (pixelwise) time points of change in a series of six L-band EMISAR polarimetric SAR data. The pixelwise analyses are applied on homogeneous subareas covered with different vegetation types using the distribution of the observed p-values.

Canonical correlation analysis (CCA) is an established multi-variate statistical method for finding similarities between linear combinations of (normally two) sets of multivariate observations. In this contribution we replace (linear) correlation as the measure of association between the linear combinations with the information theoretical measure mutual information (MI). We term this type of analysis canonical information analysis (CIA). MI allows for the actual joint distribution of the variables involved and not just second order statistics. While CCA is ideal for Gaussian data, CIA facilitates analysis of variables with different genesis and therefore different statistical distributions and different modalities. As a proof of concept we give a toy example. We also give an example with one (weather radar based) variable in the one set and eight spectral bands of optical satellite data in the other set.

A test statistic for the equality of several variance-covariance matrices following the complex Wishart distribution is introduced. The test statistic is applied successfully to detect change in C-band EMISAR polarimetric SAR data over four time points.

The goal of this paper is to develop an efficient method for forest change detection using multitemporal stereo panchromatic imagery. Due to the lack of spectral information, it is difficult to extract reliable features for forest change monitoring. Moreover, the forest changes often occur together with other unrelated phenomena, e.g., seasonal changes of land covers such as grass and crops. Therefore, we propose an approach that exploits kernel Minimum Noise Fraction (kMNF) to transform simple change features into high-dimensional feature space. Digital surface models (DSMs) generated from stereo imagery are used to provide information on height difference, which is additionally used to separate forest changes from other land-cover changes. With very few training samples, a change mask is generated with iterated canonical discriminant analysis (ICDA). Two examples are presented to illustrate the approach and demonstrate its efficiency. It is shown that with the same amount of training samples, the proposed method can obtain more accurate change masks compared with algorithms based on k-means, one-class support vector machine, and random forests.

Based on the established methods kernel canonical correlation analysis and multivariate alteration detection we introduce a kernel version of multivariate alteration detection. A case study with SPOT HRV data shows that the kMAD variates focus on extreme change observations.

Based on the original, linear minimum noise fraction (MNF) transformation and kernel principal component analysis, a kernel version of the MNF transformation was recently introduced. Inspired by we here give a simple method for finding optimal parameters in a regularized version of kernel MNF analysis. We consider the model signal-to-noise ratio (SNR) as a function of the kernel parameters and the regularization parameter. In 2-4 steps of increasingly refined grid searches we find the parameters that maximize the model SNR. An example based on data from the DLR 3K camera system is given.

This paper introduces a nonlinear feature extraction method based on kernels for remote sensing data analysis. The proposed approach is based on the minimum noise fraction (MNF) transform, which maximizes the signal variance while also minimizing the estimated noise variance. We here propose an alternative kernel MNF (KMNF) in which the noise is explicitly estimated in the reproducing kernel Hilbert space. This enables KMNF dealing with non-linear relations between the noise and the signal features jointly. Results show that the proposed KMNF provides the most noise-free features when confronted with PCA, MNF, KPCA, and the previous version of KMNF. Extracted features with the explicit KMNF also improve hyperspectral image classification.

This paper introduces kernel versions of maximum autocorrelation factor (MAF) analysis and minimum noise fraction (MNF) analysis. The kernel versions are based upon a dual formulation also termed Q-mode analysis in which the data enter into the analysis via inner products in the Gram matrix only. In the kernel version, the inner products of the original data are replaced by inner products between nonlinear mappings into higher dimensional feature space. Via kernel substitution also known as the kernel trick these inner products between the mappings are in turn replaced by a kernel function and all quantities needed in the analysis are expressed in terms of this kernel function. This means that we need not know the nonlinear mappings explicitly. Kernel principal component analysis (PCA), kernel MAF, and kernel MNF analyses handle nonlinearities by implicitly transforming data into high (even infinite) dimensional feature space via the kernel function and then performing a linear analysis in that space. Three examples show the very successful application of kernel MAF/MNF analysis to: 1) change detection in DLR 3K camera data recorded 0.7 s apart over a busy motorway, 2) change detection in hyperspectral HyMap scanner data covering a small agricultural area, and 3) maize kernel inspection. In the cases shown, the kernel MAF/MNF transformation performs better than its linear counterpart as well as linear and kernel PCA. The leading kernel MAF/MNF variates seem to possess the ability to adapt to even abruptly varying multi and hypervariate backgrounds and focus on extreme observations.

Principal component analysis (PCA) is often used for general feature generation and linear orthogonalization or compression by dimensionality reduction of correlated multivariate data, see Jolliffe for a comprehensive description of PCA and related techniques. Schölkopf et al. introduce kernel PCA. Shawe-Taylor and Cristianini is an excellent reference for kernel methods in general. Bishop and Press et al. describe kernel methods among many other subjects. The kernel version of PCA handles nonlinearities by implicitly transforming data into high (even infinite) dimensional feature space via the kernel function and then performing a linear analysis in that space. In this paper we shall apply a kernel version of maximum autocorrelation factor (MAF) analysis to irregularly sampled stream sediment geochemistry data from South Greenland and illustrate the dependence of the kernel width. The 2,097 samples each covering on average 5 km2are analyzed chemically for the content of 41 elements.

The IR-MAD components show changes in the agricultural areas as well as in the mine, and the kMAF components focus on extreme changes in the mine. Due to lack of change in the spectral signal (the change occurs in the height of the surface only) excavation of material (here brown coal) leaving the same material behind is not detected.

A kernel version of maximum autocorrelation factor (MAF) analysis is described very briefly, and applied to change detection in remotely sensed hyperspectral image (HyMap) data. The kernel version is based on a dual formulation also termed Q-mode analysis in which the data enter into the analysis via inner products in the Gram matrix only. In the kernel version the inner products are replaced by inner products between nonlinear mappings into higher dimensional feature space of the original data. Via kernel substitution also known as the kernel trick these inner products between the mappings are in turn replaced by a kernel function and all quantities needed in the analysis are expressed in terms of this kernel function. This means that we need not know the nonlinear mappings explicitly. Kernel PCA and MAF analyses handle nonlinearities by implicitly transforming data into high (even infinite) dimensional feature space via the kernel function and then performing a linear analysis in that space. An example shows the successful application of kernel MAF analysis to change detection in HyMap data covering a small agricultural area near Lake Waging-Taching, Bavaria, Germany.

Many change detection algorithms work by calculating the probability of change on a pixel-wise basis. This is a disadvantage since one is usually looking for regions of change, and such information is not used in pixel-wise classification - per definition. This issue becomes apparent in the face of noise, implying that the pixel-wise classifier is also noisy. There is thus a need for incorporating local homogeneity constraints into such a change detection framework. For this modelling task Markov Random Fields are suitable. Markov Random Fields have, however, previously been plagued by lack of efficient optimization methods or numerical solvers. We here address the issue of efficient incorporation of local homogeneity constraints into change detection algorithms. We do this by exploiting recent advances in graph based algorithms for Markov Random Fields. This is combined with an IR-MAD change detector, and demonstrated on real data with good results.

A method is proposed for pixel-level satellite image fusion derived directly from a model of the imaging sensor. By design, the proposed method is spectrally consistent. It is argued that the proposed method needs regularization, as is the case for any method for this problem. A framework for pixel neighborhood regularization is presented. This framework enables the formulation of the regularization in a way that corresponds well with our prior assumptions of the image data. The proposed method is validated and compared with other approaches on several data sets. Lastly, the intensity-hue-saturation method is revisited in order to gain additional insight of what implications the spectral consistency has for an image fusion method.

This paper describes new extensions to the previously published multivariate alteration detection (MAD) method for change detection in bi-temporal, multi- and hypervariate data such as remote sensing imagery. Much like boosting methods often applied in data mining work, the iteratively reweighted (IR) MAD method in a series of iterations places increasing focus on "difficult" observations, here observations whose change status over time is uncertain. The MAD method is based on the established technique of canonical correlation analysis: for the multivariate data acquired at two points in time and covering the same geographical region, we calculate the canonical variates and subtract them from each other. These orthogonal differences contain maximum information on joint change in all variables (spectral bands). The change detected in this fashion is invariant to separate linear (affine) transformations in the originally measured variables at the two points in time, such as 1) changes in gain and offset in the measuring device used to acquire the data, 2) data normalization or calibration schemes that are linear (affine) in the gray values of the original variables, or 3) orthogonal or other affine transformations, such as principal component (PC) or maximum autocorrelation factor (MAF) transformations. The IR-MAD method first calculates ordinary canonical and original MAD variates. In the following iterations we apply different weights to the observations, large weights being assigned to observations that show little change, i.e., for which the sum of squared, standardized MAD variates is small, and small weights being assigned to observations for which the sum is large. Like the original MAD method, the iterative extension is invariant to linear (affine) transformations of the original variables. To stabilize solutions to the (IR-)MAD problem, some form of regularization may be needed. This is especially useful for work on hyperspectral data. This paper describes ordinary two-set canonical correlation analysis, the MAD transformation, the iterative extension, and three regularization schemes. A simple case with real Landsat Thematic Mapper (TM) data at one point in time and (partly) constructed data at the other point in time that demonstrates the superiority of the iterative scheme over the original MAD method is shown. Also, examples with SPOT High Resolution Visible data from an agricultural region in Kenya, and hyperspectral airborne HyMap data from a small rural area in southeastern Germany are given. The latter case demonstrates the need for regularization.

A method for detecting clutter in weather radar images by information fusion is presented. Radar data, satellite images, and output from a numerical weather prediction model are combined and the radar echoes are classified using supervised classification. The presented method uses indirect information on precipitation in the atmosphere from Meteosat-8 multispectral images and near-surface temperature estimates from the DMIHIRLAM-S05 numerical weather prediction model. Alternatively, an operational now casting product called 'Precipitating Clouds' based on Meteosat-8 input is used. A scale-space ensemble method is used for classification and the clutter detection method is illustrated on a case of severe sea clutter contaminated radar data. Detection accuracies above 90 % are achieved and using an ensemble classification method the error rate is reduced by 40 %.

In order to facilitate the update of the building theme in photogrammetrically derived GIS databases, we investigate spectra, textural, and shape features of areas of aerial photos previously registered as buildings. In following steps, these features are used in a classification tree characterisation of the entire photo, and a simple classification routine (minimum Mahalanobis distance). The classification results are to he used for (1) verification of the GIS database and (2) change detection for subsequent update work. The classification trees show to be hard to prune (generalize) without large loss of precision. This indicates that the problem at hand is far from trivial: the building and background classes are hard to separate. Hence, the method described here cannot stand alone: additional algorithms focusing on different aspects of the problem (and finally combined in a fusion step) could be one way to go; another, technically more promising, but economically less viable, would be to enter high resolution height data, derived from photogrammetry or laser scanning, into the input data set.

A test statistic for equality of two covariance matrices following the complex Wishart distribution has previously been used in new algorithms for change detection, edge detection and segmentation in polarimetric SAR images. Previously, the results for change detection and edge detection have been quantitatively evaluated. This paper deals with the evaluation of segmentation. A segmentation performance measure originally developed for single-channel SAR images has been extended to polarimetric SAR images, and used to evaluate segmentation for a merge-using-moment algorithm for polarimetric SAR data.

When working with multilook fully polarimetric synthetic aperture radar (SAR) data, an appropriate way of representing the backscattered signal consists of the so-called covariance matrix. For each pixel, this is a 3/spl times/3 Hermitian positive definite matrix that follows a complex Wishart distribution. Based on this distribution, a test statistic for equality of two such matrices and an associated asymptotic probability for obtaining a smaller value of the test statistic are derived and applied successfully to change detection in polarimetric SAR data. In a case study, EMISAR L-band data from April 17, 1998 and May 20, 1998 covering agricultural fields near Foulum, Denmark are used. Multilook full covariance matrix data, azimuthal symmetric data, covariance matrix diagonal-only data, and horizontal-horizontal (HH), vertical-vertical (VV), or horizontal-vertical (HV) data alone can be used. If applied to HH, VV, or HV data alone, the derived test statistic reduces to the well-known gamma likelihood-ratio test statistic. The derived test statistic and the associated significance value can be applied as a line or edge detector in fully polarimetric SAR data also.

Finding the edges between different regions in an image is one of the fundamental steps of image analysis, and several edge detectors suitable for the special statistics of synthetic aperture radar (SAR) intensity images have previously been developed. In this paper, a new edge detector for polarimetric SAR images is presented using a newly developed test statistic in the complex Wishart distribution to test for equality of covariance matrices. The new edge detector can be applied to a wide range of SAR data from single-channel intensity data to multifrequency and/or multitemporal polarimetric SAR data. By simply changing the parameters characterizing the test statistic according to the applied SAR data, constant false-alarm rate detection is always obtained. An adaptive filtering scheme is presented, and the distributions of the detector are verified using simulated polarimetric SAR images. Using SAR data from the Danish airborne polarimetric SAR, EMISAR, it is demonstrated that superior edge detection results are obtained using polarimetric and/or multifrequency data compared to using only intensity data.

A newly developed test statistic for equality of two complex covariance matrices following the complex Wishart distribution and an associated asymptotic probability for the test statistic has been used in a segmentation algorithm. The segmentation algorithm is based on the MUM (merge using moments) approach, which is a merging algorithm for single channel SAR images. The polarimetric version described in this paper uses the above-mentioned test statistic for merging. The segmentation algorithm has been applied to polarimetric SAR data from the Danish dual-frequency, airborne polarimetric SAR, EMISAR. The results show clearly an improved segmentation performance for the full polarimetric algorithm compared to single channel approaches.

This paper describes two- and multiset canonical correlations analysis (CCA) for data fusion, multisource, multiset, or multitemporal exploratory data analysis. These techniques transform multivariate multiset data into new orthogonal variables called canonical variates (CVs) which, when applied in remote sensing, exhibit ever-decreasing similarity (as expressed by correlation measures) over sets consisting of 1) spectral variables at fixed points in time (R-mode analysis), or 2) temporal variables with fixed wavelengths (T-mode analysis). The CVs are invariant to linear and affine transformations of the original variables within sets which means, for example, that the R-mode CVs are insensitive to changes over time in offset and gain in a measuring device. In a case study, CVs are calculated from Landsat Thematic Mapper (TM) data with six spectral bands over six consecutive years. Both Rand T-mode CVs clearly exhibit the desired characteristic: they show maximum similarity for the low-order canonical variates and minimum similarity for the high-order canonical variates. These characteristics are seen both visually and in objective measures. The results from the multiset CCA R- and T-mode analyses are very different. This difference is ascribed to the noise structure in the data. The CCA methods are related to partial least squares (PLS) methods. This paper very briefly describes multiset CCA-based multiset PLS. Also, the CCA methods can be applied as multivariate extensions to empirical orthogonal functions (EOF) techniques. Multiset CCA is well-suited for inclusion in geographical information systems (GIS).