Landscape level indices of spatial structure are used to examine forest and agricultural land cover transitions in time series data under conditions of varying spatial scale. Working with four coterminous classified TM images (August 1988, '89, '90 and '91) of an altered landscape north of the city of Manaus in northern Brazil, conditional probability matrices of land cover transition are compared to measures of landscape structure. A commercial GIS database and algorithms for modeling structural composition are used to define landscape elements and provide an estimate of the probabilistic behavior of pixels. While no strong relationship is found in the data presented, the conceptual framework and the GIS models developed in this investigation link probability estimates of pixel change with spatial measures of patch structure. This approach suggests a means of linking the probabilistic behavior of fine scale dynamics to the pattern observed at larger spatial scales.

The deforestation and fragmentation of tropical forests in the Amazon Basin of South America have become the focus of a global debate on international environmental policy (Porter and Brown 1991). The NASA Mission to Planet Earth - Earth Observing System (MTPE-EOS) research program provides the opportunity to investigate the hydrology, biogeochemistry and forest dynamics of the Amazon with multi-scaled remote sensing data (Richey et al. 1990). As an element of that program, this investigation explores a probabilisitic spatial model of landscape transition from measures of landscape structure where remotely sensed data and geographic information systems are integrated. This investigation explores the link between landscape patch structure and the individual pixel transition of intrapatch heterogeneity. Specifically, this investigation develops tools and tests the hypothesis that a probabilistic statement for the transition of a pixel, once the initial state of the pixel in a land cover time series dataset is known, can suggest the spatial structure of the landscape patch to which that pixel belongs.

It may help to illustrate this hypothesis with the following example. To an observer, a small cluster of contiguous cells classified as open canopy forest surrounded by a much larger and relatively homogeneous patch of primary forest may be recognized as a single landscape element of primary forest having some degree of intrapatch variety and a complex spatial structure. This investigation explores the hypothesis that the pixels of open canopy constitute pixels of fine scale intrapatch heterogeneity and that their probabilistic behavior for land cover change will be influenced by the compositional variety and spatial configuration of the larger scale pattern of the forest patch. Standard GIS functions are used to construct a four year time series spatial database of patches, probabilities of land cover transitions, and measures of spatial structure. This approach is discussed as a tool for modeling large scale spatial patterns from knowledge of finer scale dynamics.

Multi-scale research has likewise received considerable attention of late. The non-cartographic meaning of the term "spatial scale" refers to geographic extent (window size) and resolution (the degree to which spatial objects are distinguishable). Scale then becomes a complex variable which captures the dynamics of change in both space and time. Theories and conceptual models of ecological scale (Allen and Star 1982, Meentemeyer and Box 1987, Lord and Norton 1990) have led to investigations of the simulation of multi-scale research (Delcourt and Delcourt 1988, Rastetter et al. 1992, Wessman 1992, Perestrello de Vasconcelos et al. 1993, King et al. 1991). Others have gone on to look at the application of various methods for modeling multi-scale data (Gaines & Denny 1993, Smith & Urban 1988, Carlile et al. 1989, Turner and Gardner 1991, O'Neill et al. 1991).

The essential goal of modeling and monitoring environmental change from remotely sensed data is to compare images at a spatial and temporal resolution appropriate to the ecological scale of the processes of interest. Satellite remote sensing instruments provide measurements at a variety of pixel resolutions, spatial extents and temporial scales. However, due to variability in illumination, atmospheric effects, and instrument calibration, conventional supervised or unsupervised classification techniques have difficulty providing pixel to pixel comparisons between images from different times. Classification of any given pixel into a discrete land cover class for the purpose of determining change requires that these variables be considered (Adams et al. in press).

After arriving at comparable classes, current work in change detection has begun to go beyond simple descriptive summations of change. LaGro and DeGloria (1992) applied multiple regression to modeling land use dynamics where the proportions of change in each of five classes of land use/land cover were used as the dependent variable. The error term for each linear regression was attributed to sampling errors, variables not used in the analysis, or stochastic effects. While land cover change has received less attention in spatially explicit models than other biophysical processes, this and other works have suggested that current models are insufficient for the complexity of anthropogenic impacts. Where anthropogenic forces are at work land cover change should be thought of in probabilistic terms.

This general concept can be expressed clearly in two statements. The first recognizes the influence of patch stucture on the future behavior of individual member pixels along with the influence of all other non-defined processes (E), while the second recognizes that the transition is governed by random variables and may be described only in probabilistic terms.

m m LC = f(C[patch ] , E) (2.1) t+1 i,k t and also, m m m m P(LC = j | LC = i, LC = i , ..., LC = i ) t+1 t t-1 1 0 0 m m = P{LC =j|LC = i} = P (2.2) t+1 t i,j

m = 1 ..., n cells (pixels) i = 1 ..., 8 patch type (classes) j = 1 ..., 8 patch type (classes) k = 1 ...,lpatches t = t, 1, 2, ... time

m LC is the land cover class of pixel m at time t (2.3) t and: m LC = 1 ..., 8 for all m t

m C[patch ] is the structure of the kth patch of class i (2.4) i,k t where pixel m is a member at time t Where: C = f(D, VAR) Such that: 2 ln(.25 perimeter) D = ------------------- ln area and: VAR is the unique number of i's in the patchEquation 2.2 expresses the Markov assumption that the future behavior of a pixel is determined once the state of the process at the present time is known. Pi,j stand for the conditional probability for change of a given pixel in state i at time t to state j at time t+1. Pi,j is derived from the initial change matrix by simply totalling the rows of the joint probabilities and dividing each element in the row by its own row total. When the probabilities have been normalized, the conditional probabilities form the transition, or stochastic, matrix. The transition mechanism of individual pixels is captured in the matrix of conditional probabilities.

The two earlier conceptual statements (2.1 and 2.2) express patch structure (2.4) as a strong determinant in the future state of a given pixel. This investigation tests the general hypothesis that patch structure can be used to model landcover transitions. More specifically it tests the hypothesis that the probability of transition will increase as patch structure becomes more complex. Or,

m as C[patch ] increases, then P -> 1 (2.6) i,k t i,j

The investigation begins with four spatially coterminous land cover classified Landsat Thematic Mapper (TM) images (August 1988, '89, '90 and '91) of the Basin of Greater Manaus in northern Brazil (3° South, 60° West). The classified land cover dataset (Figure 1) is provided by the NASA supported EOS-Amazon project team and derived outside the scope of this investigation (see Adams et al. in press). Each pixel was assigned to a land cover class through application of a spectral mixing model. This procedure provides the fractional composition (percent of each pixel) for one of four end-members (non-photosynthetic vegetation, photosynthetic vegetation, soil and shade). The classification step concludes by assigning one of eight recognizable land cover "labels" to each pixel.

Land Cover Classification

The GIS dataset is constructed by clipping these images to a common spatial extent and converting them to raster GIS data model (grids) with a cell size of 30m X 30m. The study area is in a region of the Amazon basin where forest clearing for agriculture began in the late 1970's. This area, referred to as Fazenda Dimona, began as a cattle ranch and now includes the research areas of the Biological Dynamics of Forest Fragments Project (BDFFP) of the National Institute for Amazon Research (INPA) and the Smithsonian Institution (Lovejoy and Bierregaard 1990). The management history of the site (Bierregaard et. al. 1992) has resulted in a variety of land covers from well-maintained pasture through regrowth to primary forest (Lucas et al. 1993).

The value of each cell in the land cover grids represents that cell's land cover value at the first time step (eq. 2.3). The spatial connectivity of these values and a variable for the allowable size of a landscape patch are used to construct landscapes of largely homogeneous patches with attributes of intrapatch heterogeneity. In this way the resolution, a component of spatial scale, is defined by the analyst. In this investigation the minimum patch size is one hectare.

Landscape patches are constructed through a step model which uses a number of map algebra steps to evaluate the size of discrete regiongroups (patches) and, based upon a user defined value for the allowable size of a minimum sized patch, creates patches which can contain intrapatch heterogeneity. In this way patches are recognized as landscape elements containing heterogenous land cover types. For example, a pasture may represent as a single landscape element but recognized as a patch containing small groupings of non-pasture land cover. The land cover value for individual pixels (eq. 2.3) of patches that fall below the allowable minimum patch size are replaced with the land cover value of the nearest landscape patch. New regiongroups are defined and a spatial union is performed to link the newly defined patches to their original patch identities. In this way only those cells which were in patches that fell below the minimum value will have different values of patch identity in this final step. Small patches and isolated cells are therefore assigned to the nearest landscape patch (Figure 2).

Landscape Patches (minimum allowable size of 1 hectare)

Various measures of spatial structure, including the value for D in equation 2.4, can be obtained simply from software applications such as Fragstats ver. 2.0 (McGarigal and Marks 1994). However, in this investigation values of both terms in equation 2.4 are generated by a macro using grid operators and the functions of zonalarea, zonalperimeter, and zonalvariety, where the zone was defined as a regiongroup and forms unique patches. A spatial union joins the grids on a cell by cell basis, while preserving in its attribute table the patch-ID and values for the terms D and VAR of each patch.

The initial change matrix is constructed from the attribute table of a grid resulting from a spatial union between each time step. A C program then creates the matrix of conditional probabilities (eq. 2.2) by totalling the rows of the joint probabilities and dividing each element in the row by its own row total.

The final operation is to join the database files for the grids containing the values of patch structure with the grid containing unique transitions within each patch, on the common item of patch-ID.

Land Cover Transitions (number of land cover values within the dataset)

The measure of patch structure (2.4) includes a term for patch configuration (D) described here as a simple 2-dimensional fractal measure in an area to perimeter ratio. Each patch is modeled as a homogeneous patch composed of one of eight land cover types. A value of D is unique to each patch. A value of 1.0 corresponds to simple shapes (circles or squares) and increases towards 2.0 as the shape becomes more highly convoluted (Figure 4).

Characteristics of D by Land Cover Types

The matrix of conditional probabilities is derived from the land cover change matrix. The matrices in Table 1 display the likelihood of transition between states based upon the relative frequency in one time step, when that time step was one year (1988 to 1989). It is characteristic of land cover transition matrices that the probabilities on the matrix diagonal (no change in state) are usually greater than o.5. This reflects that for many land cover types there is a relatively high likelihood that no change will occur. Table 1(a) reports the estimates of conditional probability based upon the behavior of all cells in the landscape, while Table 1(b) reflects that probability from only those cells which belonged to patches that were below the allowable patch size and were therefore modeled as cells of intrapatch heterogeneity.

Conditional Probabilities Matrices (Pi,j)

(one year time step)

1989 1 2 3 4 5 6 7 8 ------------------------------------------------ 1| 0.913 0.056 0.013 0.002 0.008 0.004 0.001 0.003 2| 0.066 0.803 0.048 0.019 0.023 0.032 0 0.009 3| 0.084 0.479 0.225 0.041 0.085 0.064 0.002 0.020 1988 4| 0.027 0.318 0.221 0.209 0.138 0.081 0.005 0 5| 0.022 0.045 0.092 0.115 0.577 0.118 0.030 0.002 6| 0.028 0.149 0.041 0.291 0.300 0.184 0.006 0.001 7| 0.024 0.002 0.022 0.022 0.467 0.009 0.453 0 8| 0.250 0.250 0 0.250 0 0 0 0.250 (a) Conditional probabilities derived from all landscape cells 1989 1 2 3 4 5 6 7 8 ------------------------------------------------ 1| 0.559 0.126 0.045 0.054 0.072 0.117 0.018 0.009 2| 0.157 0.525 0.104 0.071 0.062 0.058 0 0.023 3| 0.134 0.431 0.249 0.047 0.089 0.036 0.004 0.009 1988 4| 0.034 0.301 0.234 0.159 0.161 0.105 0.004 0.001 5| 0.118 0.165 0.220 0.090 0.317 0.047 0.040 0.003 6| 0.044 0.189 0.048 0.305 0.215 0.189 0.008 0.002 7| 0.028 0.003 0.031 0.031 0.515 0.009 0.383 0 8| 0.250 0.250 0 0.250 0 0 0 0.250 (b) Conditional probabilities derived from cells assigned as cells of intrapatch heterogeneity

The goal of this investigation was to explore the relationship between patch structure and the probability of land cover transition of individual cells (Pi,j). These probabilities are spatially linked to activity between cells in the GIS database and therefore to patch membership. Figures 5(a) and 5(b) graph the values of landscape patch composition (VAR) and configuration (D) against the conditional probability of making a transition during a one year time step, calculated from cells of intrapatch heterogeneity. In this way, the probability estimate reflects the finer spatial scale dynamics, while models of patch composition and configuration represent the large scale structure.

Comparison of Patch Structure and Probability of Cell Transition

(one year time step)

One important issue not addressed in this investigation is the nonstationary (non-temporally homogeneous) nature of many ecological Markov models. The ecological processes that are reflected by transitions may not be constant in space or time (Bell 1974). Therefore, we would expect the estimate of conditional probabilities calculated from observations of a three year time step (Table 2) to differ from those of a one year time step (Table 1) raised to the third power. In this investigation the nonstationary nature of the processes is ignored and the matrices are used as estimates of probability for discrete time steps. Table 2 and Figure 6 are the results of calculating conditional probabilities from a three year time step and comparisons to patch structure at time t.

Conditional Probability Matrix (Pi,j)

(Three year time step)

1991 1 2 3 4 5 6 7 8 ------------------------------------------------ 1| 0.840 0.082 0.012 0.011 0.011 0.039 0 0.004 2| 0.082 0.806 0.070 0.010 0.020 0.011 0 0 3| 0.147 0.639 0.155 0.017 0.037 0.006 0 0 1988 4| 0.036 0.609 0.205 0.053 0.088 0.005 0.004 0 5| 0.040 0.301 0.240 0.085 0.248 0.073 0.012 0 6| 0.030 0.568 0.194 0.054 0.100 0.050 0.005 0 7| 0.041 0.119 0.205 0.039 0.421 0.026 0.149 0 8| 1.000 0 0 0 0 0 0 0 (a) Conditional probabilities derived from all landscape cells 1991 1 2 3 4 5 6 7 8 ------------------------------------------------ 1| 0.622 0.171 0.081 0.018 0.063 0.045 0 0 2| 0.141 0.635 0.125 0.029 0.047 0.023 0 0 3| 0.230 0.551 0.152 0.020 0.038 0.009 0 0 1988 4| 0.049 0.584 0.228 0.044 0.083 0.006 0.006 0 5| 0.193 0.503 0.189 0.012 0.065 0 0.037 0 6| 0.034 0.504 0.167 0.086 0.155 0.046 0.008 0 7| 0.037 0.154 0.231 0.025 0.420 0.028 0.105 0 8| 1.000 0 0 0 0 0 0 0 (b) Conditional probabilities derived from cells assigned as cells of intrapatch heterogeneity

Comparison of Patch Structure and Probability of Cell Transition

(Three year time step)

The conceptual framework presented and the GIS models developed in this investigation have linked probability estimates of pixel change with spatial measures of patch structure. This approach provides a means to link the probabilistic behavior of fine scale dynamics to the spatial pattern observed at larger spatial scales. Future work should explore the temporal homogeneous nature of the conditional probability matrix if the Markovian property is to be used as a predictive tool. Finally, different minimum allowable patch sizes and indices of patch structure may result in a stronger relationship between pixel transition and landscape structure.

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Miles G. Logsdon

EOS-Amazon Project

University of Washington

School of Oceanography

P.O. Box 357940

Seattle, WA 98195-7940

Telephone:(206)543-5334

e-mail: mlog@u.washington.edu