Proceedings of the Symposium on Global and Environmental Monitoring: Proceedings of the Symposium on Global and Environmental Monitoring

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Persistent identifier:: 856665355

Title:: Proceedings of the Symposium on Global and Environmental Monitoring

Sub title:: techniques and impacts ; September 17 - 21, 1990, Victoria Conference Centre, Victoria, British Columbia, Canada

Year of publication:: 1990

Place of publication:: Victoria, BC

Publisher of the original:: [Verlag nicht ermittelbar]

Identifier (digital):: 856665355

Language:: English

Document type:: Multivolume work

Volume

Persistent identifier:: 856669164

Title:: Proceedings of the Symposium on Global and Environmental Monitoring

Sub title:: techniques and impacts; September 17 - 21, 1990, Victoria Conference Centre, Victoria, British Columbia, Canada

Scope:: XIV, 912 Seiten

Year of publication:: 1990

Place of publication:: Victoria, BC

Publisher of the original:: [Verlag nicht ermittelbar]

Identifier (digital):: 856669164

Illustration:: Illustrationen, Diagramme, Karten

Signature of the source:: ZS 312(28,7,1)

Language:: English

Usage licence:: Attribution 4.0 International (CC BY 4.0)

Editor:: International Society for Photogrammetry and Remote Sensing, Commission of Photographic and Remote Sensing Data

Publisher of the digital copy:: Technische Informationsbibliothek Hannover

Place of publication of the digital copy:: Hannover

Year of publication of the original:: 2016

Document type:: Volume

Collection:: Earth sciences

Chapter

Title:: [THP-3 ENVIRONMENTAL IMPACT ASSESSMENT]

Document type:: Multivolume work

Structure type:: Chapter

Chapter

Title:: MAPPING VEGETATION CHANGES IN DUTCH HEATHLAND, USING CALIBRATED LANDSAT TM IMAGERY. E. J. Van Kootwijk and H. van der Voet

Document type:: Multivolume work

Structure type:: Chapter

Van Kootwijk c.a. (1990) these referenees arc discussed and six methods for predicting ground cover composition from pixel values are compared. Only the corrected inverse regression method (IRe) is presented in this paper. For a better understanding, first the generalised least squares (GLS) method is discussed. GLS-prediclion is directly based on the theoretically expected relation (1). It is assum ed that the deviations r,, = y^-E^) are indepen dent drawings from a normal distribution with mean 0 and variance o^. In order to guarantee that jpredictions respect the unit-sum constraint ^k"=i x ik = l the no-intercept model (1) is rewrit ten as a model with intercept for p = K-l cover fractions: E(yij) = 1 X ikM-kj = m<j 1 X ik(m<j - I- L Kj) or: E(yij) = a j + X£ =1 Xj k (3 K j (2) The excluded class can be chosen arbitrarily (here the last class, K, has been chosen), and the predicted value for this class is simply the complement x ik . It is useful to write the model in matrix notation as Y = al’ + XB + R (3) where Y is the n*q matrix with pixel values for n pixels, X is the n*p matrix with pixel compositions, a is vector of length q, B is a p*q matrix, 1 is a vector (here of length n) with all elements equal to 1, and R is the n*q matrix of residuals, the rows of which have an expected covariance matrix £. The use of this model in multivariate calibration is standard (e.g. Brown 1982): first ly, ordinary least squares analysis applied to the training data provides estimates for a, B and the residuals R. From the latter an esti mate for £ is easily obtained. Then predictions for a set of m pixels with pixel values collec ted in a m*q matrix Y are made as X’gls = (Br 1 B’)- 1 Br 1 (Y’-al’) (4) This predictor is known as the generalised least squares (GLS) estimator or the classical estimator. It can be derived by considering a and B as known and X as a matrix of un known parameters (hence the usual designation "estimator" instead of "predictor"). The estimat ed covariance matrix of the predictions can be shown to be (BSr'B’)' 1 . It can further be noted that the predictor qnly exists provided that p<q (otherwise BL'B’is not invertible), that is the number of cover classes should not exceed the number of spectral bands by more than 1. Prediction method IR (for Inverse Regres sion) is computationally simple, as it consists of calculating the ordinary least squares regres sion equations for predicting the columns of X from the columns of Y: X’ 1R = pF + QY’ (5) where p is a vector of length K with regressi on constants, and Q is a K*q matrix with each row containing the regression coefficients for one cover class. In many calibration situations X is fixed, and only the variation in the pixel values y, is considered stochastic. Inverse re gression may then seem unnatural because it would involve a model specifying stochasticity for x, at fixed values yj. However, equation (5) can also be derived without this consideration. The inverse regression predictor is the optimal linear predictor if we consider the mean squared error of prediction (MSEP), if only means and covariances of x, in the training set are equal to those in the population of all pixels to be predicted (Lwin and Maritz 1982). Moreover, the inverse predictor can be shown to be MSEP-superior to the GLS predictor if only the means in the training set are repre sentative for the population and the spreaa of the cover percentages is not smaller than that in the population (Sundberg 1985). Of course according to standard regression theory it would also be the optimal linear predictor if round cover data were collected at pixels aving prespecified pixel value combinations. Both linear predictors, GLS and IR, give predicted cover fractions that sum to 1 over classes, but individual values may be outside the interval |0,1 ]. Three approaches have been considered to remedy this: (1) a posterior correction procedure, (2) linear modelling on another, unrestricted scale, and (3) generalised linear modelling. All three approaches make the predictors in effect nonlinear functions of the pixel values. A simple posterior correction procedure consists of setting negative predictions to 0 followed by rescaling to sum 1. This has been applied to the results of the inverse regression method, and we shall refer to the correspon ding predictor as IRc. The other procedures are dealt with in Van Kootwijk e.a. (1990). TRAINING SET FORMATION A major problem when compiling a training set of individual mixels with continuous varying amounts of heather and grass is to find the position in the image corresponding to the position in the field where data were collected. In the following, the area on the ground cor responding with a pixel will be called a ground element. Exact delineation in the field of the ground element corresponding to a single pixel is impossible, mainly because of the limited accuracy of the geometric correcti on and due to the point spread function of the sensor. In hcathland vegetation, where spatial varia tion is high, selecting pixels that correspond to measured ground elements is critical: shifts of one pixel yield significantly different training sets, regression coefficients and cover estima tes. To overcome this problem, training pixels and ground elements were arranged in linear arrays of about 40 elements. In the field, for 686

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