The CA-based model was carried out in the Dinamica EGO software (an acronym for Environment for Geoprocessing Objects), developed by the Remote Sensing Center of the Federal University of Minas Gerais (CSR/UFMG). Data spatialization was performed in QGIS software.
4.2. Model Calibration and Validation
Model calibration consists of four steps: calculating transition matrices; calculating weights of evidence coefficients; map correlation analysis and adjustment and execution of the simulation model. The transition matrix provides the percentage of land that will change from one class to another. This information is obtained by cross-tabulation between the initial (2009) and the final (2013) class maps [12
As the input maps (2009 and 2013 maps) are categorical, that is, they have states defined by numbers, the Dinamica EGO cellular automata system understands these numbers as representing a state of a given cell this case is the pixel. Then, when calculating the transition matrix, the automata can identify how many pixels in state 1 in 2009 changed to state 2 in 2013.
Based on the initial and final maps, we calculated the transition rates in a single step (total period of 4 years) and multiple steps (annualized). From this, it was possible to generate, respectively, a transition matrix with the changes that occurred in the total of 4 years between the years 2009 and 2013 and four transition matrices with annual changes: (i) from 2009 to 2010, (ii) 2010 to 2011, (iii) 2011 to 2012 and (iv) 2012 to 2013. All these transition matrices are derived from an ergodic matrix, that is, a matrix that has real eigenvalues and eigenvectors [26
In the stage of determining the weights of evidence, the static variable and dynamic variables are inserted in the model. The difference map (static variable) was obtained by a matrix subtraction in QGIS, specifically, between the maps of the year 2009 and 2013. The distances for the consumption classes that have undergone transitions (dynamic variables) were calculated within the simulation model of the EGO Dynamics software, using the “Calc to Distance Map” (“Distance”) function. Thus, both the static variable and the dynamic enter the simulation model from the stage of determining the weights of evidence (Figure 3
The “Calc to Distance Map” (“Distance”) function receives the categorical map corresponding to the initial year (2009), and also receives the values corresponding to the consumption classes for which it must calculate distances (Euclidean distances), based on class position information (pixels) from the 2009 map. In our simulation, there were consumption transitions for classes 1, 2, 3, 4, and 6, from 2009 to 2013, these were the numbers entered in the “Calc to Distance Map” function for calculating distances. With that, it generated five maps (in the case of our simulation) of border distance (closest distance) of the cells corresponding to consumption classes 1, 2, 3, 4, and 6, based on the 2009 map. The function identifies consumer class 1 pixels on the initial map and calculates the distances between them and the surrounding areas. Thus, the function takes into account the proximity of consumption areas 1 in relation to the probability of a new consumption class 1.
The resulting maps have their distances represented in meters. The map format has the same dimensions as the categorical map (2009 map). Thus, the Euclidean distances calculated in a two-dimensional plane are , for two-dimensional points P = (px, py) e Q = (qx, qy).
The final map (2013) is not used in the “Calc Distance Map” function; it is only used in the “Weights of Evidence Ranges” and “Determine Weights of Evidence Coefficients” functions, as data to assist in the determination of evidence weights, that is, from the 2013 map, it is verified whether these ranges of distances calculated from the 2009 map influenced a class change registered between 2009 and 2013, according to the proximity between the consumption classes (pixels). If it is proven that these distances (dynamic variables) influence the change of class by proximity, they receive an influence value (weights of evidence); the closer an area is to a particular class that influences, the greater the value that weight.
In this sense, the weights of evidence method are applied in Dinamica EGO to produce a transition probability map, which represents the most favorable areas for a change [27
]. This method consists of a Bayesian method, in which the effect of a spatial variable on a transition is calculated regardless of a combined solution. Weights of evidence represent each influence on a variable in the spatial probability of a transition
, and are calculated as follows:
is the weight of evidence for the occurrence of event D, given a spatial pattern B. The a posteriori probability of a transition
, given a set of spatial data (B, C, D, …, N), is expressed as follows:
where B, C, D and N are the values of the spatial variables that are measured at the location x
and represented by their weights
For the application of the weights of evidence method in the simulation model, it was necessary to categorize the variable maps applied in the simulation model, since the weights of evidence are applied only to categorical data. Using both the QGIS and the difference map (between the years 2009 and 2013), it was possible to categorize the static variable. Meanwhile, the simulation model in Dynamics EGO categorized the dynamic variable, where a line generalization algorithm defined the categorical ranges [26
], that is, the best fit curve delimits the categorical ranges of change from a series of straight lines segments.
The only necessary assumption for the weights of evidence method is that the input maps must be spatially independent. Then, correlation analysis was applied to analyze the spatial interdependence of the maps. We performed this analysis using two tests, the Cramer’s V (V) test and the joint information uncertainty test (JIU).
According to Bonham-Carter [28
], the closer V and JIU are to 1, the higher the spatial dependence between the considered pair of variables. Variables with a correlation above 0.5 (50%) should be neglected or combined, to replace the correlated pair in the model.
Model parameterization also includes the adjustment of the simulation model. Dinamica EGO uses as a local cellular automaton rule, a transition mechanism composed of two complementary transition functions: Patcher and Expander (“Updated Landscape” box) (Figure 4
). The Patcher function generates new patches with a seeding mechanism, while the Expander function accounts for the expansion or contraction of previously existing patches of a given class [27
In addition to defining the Patcher/Expander ratio, both mean and variance of the patch size and the patch isometry must be set. We applied R Studio software to calculate these variables. These parameters are used to standardize patch sizes that either appear or expand from existing ones as transformations occur. The patch isometry ranges from 0 to 2, increasing as the patches take on a more isometric form. The degree of the patch size fragmentation is inversely proportional to the isometry [29
Thus, the Patcher function creates new patches for a given consumption class according to the trend of change, and the Expander function only expands the existing patches of a specific consumption class around it, according to an increasing trend for that particular class.
After defining the parameters, annual simulations were generated from the initial and final maps. Model performance was validated through the fuzzy similarity index, developed by the Remote Sensing Center of the Federal University of Minas Gerais (CSR/UFMG). This index is based on the fuzzy similarity index created by Hagen [30
]. Two maps of difference were compared: the first obtained from the initial and observed final maps, and the second, obtained from the initial map and the simulated final map (Figure 5
). An exponential decay function with a window size of 11 × 11 and a constant decay function, calculated with the following window sizes: 1 × 1, 3 × 3, 5 × 5, 7 × 7, 9 × 9 and 11 × 11, were adopted. The window sizes are grids of pixels, where each pixel corresponds to an area of 900 m2
(30 m × 30 m).
After validating the model, two future water demand scenarios were simulated, using the same parameters defined in calibration and validation steps, changing only the number of iterations. In this step, 2021 and 2025 trend scenarios were generated.
Thus, short and medium-term intervals were adopted to define future scenarios. The 2017 scenario was simulated, in order to compare the consumption under average conditions (simulated map) with the water use after drought (observed map), since Fortaleza was affected by drought between 2013 and 2017.