Modeling vegetation diversity types in mexico based upon topographic features

Azucena Pérez, Jean François Mas, Alejandro Velázquez and Lorenzo Vázquez

Azucena Pérez Vega. Ph.D. student in Geography, Universidad Nacional Autónoma de México (UNAM). Address: Centro de Investigaciones en Geografía Ambiental (CIGA), UNAM. Antigua Carretera a Pátzcuaro, Col. Ex-Hacienda de San José de La Huerta Nº 8701 C.P. 58190 Morelia, Michoacán, México. e-mail: azuperezvega@yahoo.com

Jean-François Mas. Ph.D. in Remote Sensing, Université Paul Sabatier, France. Researcher, CIGA, UNAM, Mexico.

Alejandro Velázquez Montes. Ph.D. in Landscape Ecology, University of Amsterdam, The Netherlands. Researcher, CIGA, UNAM, Mexico.

Lorenzo Vázquez Selem. Ph.D. in Geography, Arizona State University, EEUU. Researcher, Instituto de Geografía, UNAM, Mexico.

SUMMARY

The role played by topography as a surrogate variable to explain vegetation diversity types (VDT) was evaluated. Using GIS, a Simpson vegetation diversity index for the entire Mexican territory was computed based on land use and vegetation maps. Then, the correlation between VDT and topographical attributes (elevation, range of elevation, slope, roughness and diversity of aspect) was calculated using different sizes of analysis windows. A significant correlation between topographical variables and vegetation diversity was found (Spearman coefficient of correlation >0.4, p=0.01 with three variables: elevation average and range, and slope), with an optimal window of 80×80km². Subsequently, modeling VDT from the topographical attributes using an artificial neural network approach was attempted. The comparison between the modeled and the observed VDT maps showed that the model produced a reasonable estimate of vegetation diversity. From error analysis it may be deduced that VDT cannot be totally explained by topographical attributes alone, although these play a primary role in VDT at regional to continental scale. As results have shown that land cover and vegetation accurately portrays biodiversity distribution patterns, VDT modeling is a promising approach to assess biodiversity. It is concluded that topographical variables, globally available at a 3-arc-second resolution (~90m) through the Shuttle Radar Topography Mission (SRTM) project, may be used to portray regional biodiversity patterns. This is crucial in tropical countries that harbor high biodiversity but often lack accurate and updated land cover databases.

Se evaluó el papel de la topografía como variable sustituta para explicar patrones de diversidad de tipos de vegetación (DTV). Se calculó el índice de diversidad de vegetación de Simpson para todo el territorio mexicano utilizando mapas de uso del suelo y vegetación por medio de SIG. Se determinó la correlación entre diversidad de vegetación y algunos atributos topográficos (altitud, rango de elevación, pendiente, rugosidad y exposición), encontrándose correlación significativa entre las variables topográficas y la diversidad de vegetación (Coeficiente de Spearman >0,4, p=0,01 con tres variables: rango y promedio de elevación, y pendiente) con una ventana óptima de 80×80km². Luego se modeló los DTV con base en los atributos topográficos usando un enfoque de redes neurales artificiales. La comparación entre los mapas de DTV modelados y reales mostró que el modelo es una buena estimación de la diversidad de vegetación. El análisis de errores sugiere que los DTV no pueden explicarse totalmente por atributos topográficos, aunque éstos juegan un papel fundamental en los DTV a escala regional y continental. Las coberturas del suelo y la vegetación reflejan los patrones de distribución de la biodiversidad, por lo que la modelación de los DTV es un enfoque prometedor para evaluar la biodiversidad. La conclusión principal es que las variables topográficas, disponibles en diferentes escalas de resolución para la mayor parte del mundo, pueden ser utilizadas para representar patrones regionales de biodiversidad. Este aspecto es crucial en países tropicales, los cuales presentan alta biodiversidad pero frecuentemente carecen de bases de datos sobre coberturas del suelo confiables y actualizadas.

Modelado da diversidade de tipos de vegetação no méxico baseado em atributos topográficos

RESUMO

Avaliou-se o papel da topografia como variável substituta para explicar padrões de diversidade de tipos de vegetação (DTV). Calculou-se o índice de diversidade de vegetação de Simpson para todo o território mexicano utilizando mapas de uso do solo e vegetação por meio de SIG. Determinou-se a correlação entre diversidade de vegetação e alguns atributos topográficos (altitude, faixa de elevação, inclinação, rugosidade e exposição), encontrando-se correlação significativa entre as variáveis topográficas e a diversidade de vegetação (Coeficiente de Spearman >0,4, p=0,01 com três variáveis: faixa e média de elevação, e inclinação) com uma ventana otima de 80×80km². Logo se modelou os DTV com base nos atributos topográficos usando um enfoque de redes neurais artificiais. A comparação entre os mapas de DTV modelados e reais mostrou que o modelo é uma boa estimação da diversidade de vegetação. A análise de erros sugere que os DTV não podem explicar-se totalmente por atributos topográficos, ainda que estes joguem um papel fundamental nos DTV a escala regional e continental. As coberturas do solo e a vegetação refletem os padrões de distribuição da biodiversidade, pelo que a modelação dos DTV é um enfoque prometedor para avaliar a biodiversidade. A conclusão principal é que as variáveis topográficas, disponíveis em diferentes escalas de resolução para a maior parte do mundo, podem ser utilizadas para representar padrões regionais de biodiversidade. Este aspecto é crucial em países tropicais, os quais apresentam alta biodiversidade mas, freqüentemente carecem de bases de dados sobre coberturas do solo confiáveis e atualizadas.

Thorough documentation of habitat patterns is crucial to the understanding of long term biogeographical and ecological processes, to reconcile alternatives between productive systems and maintenance of environmental services, to establish protected areas for conservation, to optimize allocation of financial support to natural resource management and, ultimately, to reach ecologically sound rural development without compromising socio-cultural justice and rights (Brechin et al., 2003). Mayr (2000) clearly documented the unprecedented rates of loss of biodiversity. Habitat deterioration, by most accounts, tops the list of threats to biodiversity (Wilshusen et al., 2002). Habitat mapping, therefore, has become a critical issue (e.g., in journals such as Biodiversity Conservation, Conservation Biology, Nature and Science). Biodiversity databases usually over-represent a few species records and largely misrepresent the environmental heterogeneity prevailing in tropical countries (Terborgh et al., 2002). Models of species distribution are based on overlapping climate, geology, elevation and land cover patterns (Ceballos et al., 2005) and are confronted with the problem of input data scale as well as the unevenness of biodiversity measurements. Currently, abundant literature describes species distribution patterns and its relation with environmental variables (see conservation biology and biodiversity and conservation journals www.jstor.org/journals and www.springer.com/west/home). Second level biodiversity patterns, especially cartographically portrayed are scanty. Research has shown that vegetation diversity types accurately portrays biodiversity distribution patterns (Burnett et al., 1998; Priego-Santander, 2004). Geomorphology, in turn, has been pinpointed as a key attribute to understand land cover distribution patterns (Harner and Harper, 1976), as tested in the USA (Nichols et al., 1998; Coblentz and Riitters, 2004) and in Germany (Müller et al., 2004). The latter authors state that biodiversity patterns are predictable, based upon existing topographic data. Rather than simply acknowledging that biodiversity may be explained by geomorphology alone, it is crucial to evaluate how much of the vegetation diversity types (VDT) is controlled by measurable aspects of the topographic "character" of the landscape. These types of studies are yet to be conducted and are urgently needed in Neotropical countries (Mas et al., 2004b). In addition, the degree of certainty in predicting VDT ought to be measured in order to assess model performance.

The artificial neural networks (ANN) are non-linear mapping structures based on the function of the human brain. Advantages of the ANN approach include the ability to handle non-linear functions which relate input (exploratory) variable(s) and output (dependent) variables, to perform model-free function estimation, to learn from data relationships that are not known otherwise and to generalize to new situations. ANNs have been shown to be universal and highly flexible function approximators for any data. Therefore, ANNs make powerful tools for models, especially when the underlying data relationships are unknown (Lek et al., 1996; Lek and Guégan, 1999). The application of ANNs in environmental sciences is increasingly common. Some recent utilizations are models predicting species distribution, abundance or diversity as a function of environmental variables (Lek and Guégan, 1999; Manel et al., 1999, Foody and Cutler, 2006), land use/cover distribution modeling (Mas, 2004), rice crop damage by flamingos (Tourenq et al., 1999), streamflow and flash floods (Kim and Barros, 2001), and ecosystem characteristics from remotely sensed data (Paruelo and Tomasel, 1997; Jensen et al., 1999). Nonetheless, few applications have been reported in the literature aimed at predicting land cover and vegetation patterns (Hilbert and Ostendorf, 2001; Aitkenhead et al., 2004; Corne et al., 2004). Due to their characteristics, ANNs are a promising tool to predict complex and spatially explicit patterns such as vegetation diversity.

Mexico encompasses a continental territory of nearly 2×10⁶km² and is one of the five biologically richest countries, therefore considered as megadiverse (Groombridge and Jenkins, 2000). A combination of topographic, geologic and climatic factors have contributed to its exceptional biodiversity. It comprises complex geomorphologic patterns and heterogeneous topography (Lugo, 1990) as can be observed in Figure 1. This complexity, along with the variety of climates and cultural history of Mexico, have been considered as key attributes to explain the formation of a mosaic of environmental conditions that enabled the evolution of a large variety of habitats and lifeforms. A transitional biogeographic zone runs through Mexico forming a bridge between the Nearctic (North American) and Neotropical (Central and South American) realms. This transitional zone was formed when the land masses of North and South America made contact through the emergence of the Isthmus of Panama, ~5×10⁶ years ago (Haug and Tiedemann, 1998). As a result, Mexico represents a compound biogeographical area, where rich mixtures of fauna and flora with different biogeographical filiations meet. In addition, Mexico has experienced severe climatic changes in the past. During the late Quaternary the entire territory was episodically under the influence of colder climates, drier in the central and southern parts, and moister in the northern part (Metcalfe et al., 2000); glaciers formed on the high mountains (Vázquez-Selem and Heine, 2004) and vegetation communities experienced marked changes and geographic shifts (Lozano-García et al., 2005). This enabled the movement of many species and at the same time caused the extinction of many tropical species in a large part of their original areas of distribution, forcing them to restricted zones where conditions remained favorable. This isolation resulted in new species, which in many cases extended the areas of distribution after the glaciers receded. This process led to a considerable increase in the number of relatively new and endemic species.

In order to assess vegetation diversity, a digital land use and vegetation map at scale 1:250000 from the Instituto Nacional de Geografía, Estadística e Informática (INEGI, 2005) was used. The map is based on photo-interpretation of aerial photographs dated from 1968 to 1986, plus a considerable amount of fieldwork. The classification scheme was derived from the vegetation classification by Miranda and Hernandez (1963). It includes 49 categories of natural vegetation types based upon physiognomic, floristic and phenologic attributes (Table I). Figure 2 shows this map at subformation level. The digital version of this map was obtained by digitizing the 122 sheets covering the Mexican territory. A more recent version of this map, based on remotely-sensed images, is available. However, the older version was used as it was considered more accurate and representing a higher proportion of natural land cover, due to the subsequent high rates of deforestation in Mexico (Mas et al. 2004b). This map, which includes more than 52500 polygons of natural vegetation (minimum mapping unit of 2.5×10⁵m²), was rasterized at 200m cell size. The pixel size was determined according to O´Neill et al. (1996), who recommended that the grain of the data should be 2-5 times smaller than the feature of interest in order to avoid biasing the estimation of small patch size.

A digital elevation model (DEM) with a spatial resolution of 200m was obtained by interpolation of contour lines from topographic maps at scale 1:250000 by INEGI (1980). The DEM was used to create slope and aspect maps. The software Arc/info was used to manage the GIS database and to calculate diversity and topographical indices.

Maps of vegetation diversity types (VDT) were created by computing a diversity index from the observed proportions of different vegetation types within a window (support region or kernel), which was translated across the study area in steps of one pixel. Water and man-made covers (urban, agriculture and pasture) pixels were ignored in order to focus on the diversity of natural vegetation types. The windows that presented <70% natural cover were not taken into account in subsequent analyses, because significant land cover conversion can lead to the elimination, or to a dramatic reduction, of some specific vegetation types and consequently may bias the diversity estimate.

Windows of different sizes ranging from 1 to 10000km² were used. The window was centered on each pixel of the original land use and vegetation map, and the diversity index was calculated within the window and stored in a new VDT map at the location of the subject pixel. Therefore, each pixel of the diversity map represents the vegetation diversity within the surrounding blocks of the original data set.

The vegetation diversity was based upon the Simpson’s index (Simpson, 1949). Vegetation type proportions replaced the species proportions used in the original equation. This vegetation index (Eq. 1) was calculated from the observed proportions (P_i) of the i different vegetation types in a window as

(1)

The index ranges from 0 to 1, a larger value indicating greater diversity; it is easy to visualize and its properties are well known by ecologists. It takes into account the number of types of vegetation present, as well as the relative abundance of each vegetation type, and represents the probability that two randomly selected polygons in the area belong to different vegetation types. Compared to most other diversity indices, the Simpson’s index is relatively more sensitive to changes in abundant vegetation types and less sensitive to changes in rare types. This characteristic may be misleading as a measure of biodiversity. However, this index is adequate to create a first-order proxy of biodiversity and is less sensitive to classification errors in the maps.

The topographic variables used in this study have been chosen because they correlated with vegetation diversity in a North American mountain range (Coblentz and Riiters, 2004). The DEM was used to produce slope and aspect maps. The circular variable (0-360º) of the slope aspect was reclassified into four categories (north, south, east and west aspects). Then, these maps were used to elaborate several topographic variables using the same window method used to calculate the VDT. Windows of different sizes were centered on each pixel of the DEM in order to calculate: 1) the mean elevation, 2) the range of elevation (maximum elevation minus minimum elevation within the window), 3) the average slope, 4) the roughness (variance of slope), and 5) the diversity of aspect (Simpson´s index based on the proportions of the four aspect categories).

The analysis of the correlation between vegetation diversity and topographical attributes was performed using GIS and summarized on tables that indicate, for each pixel, the value of the neighbouring VDT and topographical variables for different window sizes. Then, the correlations were analyzed statistically using the Spearman´s correlation coefficient. Sperman’s rank correlation tests the direction and strength of the relationship between two variables. This analysis permitted the evaluation of the strength of the relationship between VDT and topographical variables with the different window sizes and, therefore, the estimation of the window size that presents the strongest relation and the best possibility to model VDT from topographical variables. A confidence interval for the values of the coefficient was constructed using Fisher’s z transformation, which converts the correlation coefficients into a normally distributed variable (Hollander and Wolfe, 1999). Finally, the significance for selecting the best window size was evaluated with a Student t-test which allowed to determine if the correlations derived from two window sizes are significantly different.

The multi-layer perceptron (MLP) is one of the most popular ANN architecture in use today (Bishop, 1995; Lek and Guégan, 1999) due to its simplicity and because it is based on a supervised procedure, i.e. the network constructs a model based on examples of data with known outputs. The training is done solely from the examples presented, which are assumed to implicitly contain the information necessary to establish the relation.

The architecture of the MLP is a layered neural network in which the non-linear elements (neurons) are arranged in successive layers, and the information flows unidirectionally, from the input layer to the output layer through the hidden layer(s): when the network is executed, the input variable values are placed in the input units, and then the hidden and output layer units are progressively executed. Each calculates its activation value by taking the weighted sum of the outputs of the units in the preceding layer. The activation value is passed through the activation function (typically linear functions in the input and output layers, and logistic functions in the hidden layers) to calculate the output value of the neuron. When the entire network has been executed, the outputs of the output layer act as the output of the entire network (Figure 3).

The learning procedure is based on a simple concept: if the network gives the wrong answer, then the weights are corrected and the error is lessened so that future responses of the network are more likely to be correct. The best-known example of MLP-training algorithm is back-propagation. In this algorithm, a training pattern is presented to the network and the signals are fed forwards as described above. Then, the network output is compared with the desired output by computing the root mean square (RMS) error. The error is then back-propagated through the network and the weights of the connections are altered according to what is known as the generalized delta rule (Rumelhart et al., 1986; Bishop, 1995):

δW'_ij (t+1) = ηe_jo_i + δαW_ij (t)

where w_ij(t): connection weight from input i to neuron j at time t, η: learning rate, e_j: error at processing unit j, and α: momentum parameter. The learning rate controls the size of weight changes made by the algorithm. The momentum allows the back-propagation algorithm to ‘‘pick up speed’’ if a number of consecutive steps change the weights in the same direction. During the training, this process of feeding forward signals and back-propagating the error is repeated iteratively until the error of the network as a whole is minimized or reaches an acceptable magnitude. A detailed description of ANNs architecture and training can be found in Bishop (1995), Hewitson and Crane (1994), and Levi and Varela (2003).

Training was done using the back-propagation algorithm. As the network is trained to minimize the error on the training set, a major issue is over-fitting to the training data. In order to avoid over-fitting, cross-verification was used: some of the training cases (verification set) were not actually used for training but kept as an independent check on the progress of training. As training progresses, the training error progressively drops. If the verification error stops dropping, or starts to rise, this indicates that the network is starting to over-fit the data, and training should stop. A third subset, the test set, was not used at all during training but allowed to track the network’s error performance and to identify the more efficient networks.

In order to evaluate the importance of the input variables, a sensitivity analysis was carried out. This analysis rates variables according to the deterioration in performance that occurs if that variable is no longer available to the model. It indicates which input variables are considered most important by a particular network (Saltelli et al., 2004).

A map of predicted VDT was elaborated using the MLP outputs. In order to identify and analyze the errors of the model, this map was compared with the original VDT map.

The effect of analysis window size on the strength of the relationship between topographic attributes and VDT was assessed. The Figure 4 shows the Spearman´s correlation coefficients found between VDT and the topographical attributes as a function of window sizes, along with the 95% confidence interval of the estimations. The relationship between VDT and the topographical variables resembled typical species-area curves with an asymptote for a window size of 6000km². Wickham et al. (1995) reported a similar relationship between land cover richness (number of vegetation types in the window) and elevation range. It can be observed that, for small window sizes (<400km²) the relationship between VDT and topographical attributes is less robust (correlation coefficient of 0.05-0.17 for 1×1km² windows) although it remains significant (p=0.01). The mean area of polygons of natural cover (at scale 1:250000, as used in this research) is 21km², which means that, generally, small windows are located entirely inside a polygon and diversity is null. When the window is larger or is located on the boundary between polygons with different land cover categories, the diversity index increases. Therefore, the diversity index calculated within small windows is very sensitive to edge-effect (Figure 5). With larger window sizes, this "edge effect" disappears and correlation increases. With a window size of 80×80km², correlation between elevation range and VDT reaches 0.53 (significant at p=0.01). For windows larger than 80×80km², the correlation presents a small increase or even a small decrease, depending on the topographical variable, because these windows lead to an excessive generalization. A T-test showed that the improvement of the correlation from window size of 80×80km² to 100×100km² is significant only for one variable (elevation range) at p=0.01. There is no significant increase of correlation for three variables (elevation, roughness and slope) and a significant decrease for aspect diversity (p=0.05). Therefore, the window of 80×80km² was considered as the optimal one in order to model VDT using topographical variables, because it is a tradeoff between the edge effect related with too small windows and excessive generalization due to too large windows.

Physical attributes depicting a complex topography, such as elevation range, roughness, slope type and diversity of aspect, correlated positively with VDT. Heterogeneous topography creates a patch-work-like pattern suitable for multiple habitats within a relatively small geographical space, which leads to large vegetation diversity (Hoersch et al., 2002).

The results based upon the Spearman correlation coefficient between topographical variables and VDT at the appropriate window size (80×80km²) suggest that VDT can be predicted from topographical attributes. The P coefficient of correlation between topographical variables was calculated to assess their statistical independence. Table II shows that mean elevation and aspect diversity have an intermediate to high level of correlation (0.3 to 0.5) with the other topographical variables, whereas the range of elevation, the roughness and the slope are very highly correlated (Pearson >0.8). Therefore, it is clear that the mean elevation, the aspect diversity and one of the highly correlated attributes incorporate relevant information about topography and that not all the variables are needed to derive a predicted VDT.

The MLP was trained by back-propagation using data divided in three sections: the training set, the verification set and the test set, following the proportion 1/2, 1/4 and 1/4, respectively (8964, 4482 and 4482 cases). The "best" network found used four topographical variables as inputs, with a hidden layer comprising eight neurons. It was trained in 48 complete iterations of the training set and presented a RMS error of 0.1666 (based on the test data).

As shown by the sensitivity analysis (Table III) the more important explanatory variables of the model are the range of elevation, the roughness and the mean elevation. The hierarchy is not exactly the same as the one given by the correlation coefficient due to the correlation between explanatory variables. The model used two out of the three variables best correlated to the VDT index (elevation range and roughness), and the two variables less correlated with the other topographical variables (mean elevation and aspect diversity).

The values of VDT predicted by the MLP were used to construct a map of diversity (Figure 6). In order to visualize the spatial distribution of the error in estimating VDT from topographical attributes, the difference between the VDT values derived from the land use and vegetation map and those obtained by the ANN model was calculated for each window. Figure 7 illustrates the spatial distribution of the error and shows that it is not randomly distributed (over and underestimation of diversity are spatially aggregated) and, therefore, that there are underlying factors which influence the pattern distribution error.

In order to pinpoint the limitations of the model, the error map (bias of diversity estimation) was visually compared with maps of variables such as the floristic zoning (Rzedowsky and Reyna-Trujillo, 1990), the biogeographical regions (CONABIO, 1997), the natural regions (Cervantes-Zamora et al., 1990) and the climatic zoning (García, 1998). The comparison shows that

– Some of the areas where the model underestimates VDT are flat coastal areas where the presence of large areas of swamps and mangroves is associated with the marine influence and does not depend upon the topography (areas indicated as 1 in Figure 7). This situation does not occur in the Pacific coast where the mountains are close to the coast.

– Other areas where diversity was underestimated by the model correspond to natural regions such as the Desert of San Sebastian Vizcaino in the Baja California peninsula (Cervantes-Zamora et al., 1990). This region presents high diversity, as it is located between two climatic systems that originate two types of desert (2a in Figure 7). Other underestimated areas in the north-eastern part of the territory are located at the limit between the coastal plain of the Gulf of Mexico and the high plateau, which corresponds to a change of climatic zone from temperate to tropical conditions (2b in Figure 7).

– Many areas where the model overestimates diversity correspond to the core area of biogeographical regions, such as the central high plateau (3a), the Yucatán Peninsula (3b) and the highest portion of the Sierra Madre Occidental (3c in Figure 7). As the data used to train the model were derived from windows which, due to their size, often comprise several natural regions, the model accounts with diversity due to the inclusion of different regions. Thus, the model tends to overestimate diversity in regions that belong to one natural region only.

– In some cases, overestimated areas correspond to areas dominated by only one type of vegetation: chaparral (4a), succulent-dominated scrubland (4b), tropical deciduous forest (4c) and, evergreen tropical forest (4d). Some of these areas are natural regions (4d) as Sierra Lacandona (Cervantes-Zamora et al., 1990) or biogeographical regions and floristic regions (4a) (CONABIO, 1997; Rzedowsky and Reyna-Trujillo, 1990).

– In at least one case, a region where the model apparently underestimates diversity, the model actually underestimates vegetation diversity. This is due to error labelling of polygons at the boundary between two sheets, which produces a fictitious increase of vegetation types (5).

Many studies suggest that land cover and vegetation accurately portray biodiversity distribution patterns (see Burnett et al., 1998; Priego-Santander, 2004). Therefore, VDT maps can be used as a first approximation of biodiversity distribution. However, some vegetation types, such as evergreen tropical forests, cover extensive areas and are very diverse, whereas other vegetation types might form small patches within the landscape and be species-poor. Moreover, the classification scheme can be more detailed for certain kinds of vegetation, leading to an increase of the number of vegetation types and, therefore, increasing the types of vegetation diversity artificially. The present approach depends, therefore, upon the scale and the detail of the classification scheme. As a consequence, vegetation diversity must be extrapolated to biodiversity with caution. However, the VDT map shows some of the areas known as biodiversity hotspots.

It has been extensively documented that biodiversity sampling has been biased towards accessible areas (for Mexico, see Bojórquez-Tapia et al., 1994), so that surrogate attributes are urgently needed to accurately portray biodiversity patterns (Velázquez et al., 2003). The major advantage of the present approach is that spatially referenced topographic data are available for large areas at 3-arc-second resolution like the Shuttle Radar Topographic Mission (STRM), thus offering a more reliable and uniform predictor database compared with direct climatic, geomorphological or edaphic data. Thus, modeling diversity from topographical variables can be a practical approach that does not depend on the availability of accurate spatial data on climatic, edaphic or geomorphological data. This is particularly important for tropical countries with high biodiversity where accurate geographical databases, at the appropriate scale, are often lacking.

Azucena Pérez-Vega received a doctoral fellowship from CONACyT, Mexico. This study was carried out with financial support from the PAPIIT program (project IN112803), Universidad Nacional Autónoma de México.

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