# Quantitative morphometric analysis of lakes using GIS: rectangularity R, ellipticity E, orientation O, and the rectangularity vs. ellipticity index, REi.

Introduction

It has being long recognized that lake shape and orientation can tell us much about the processes behind lake formation (Reeves 1968). For example, in the Arctic region, where many lakes are geometric and share a similar orientation, it has been suggested that their origin is due to constant winds (Carson and Hussey 1962; Cote and Bum 2002). According to Allenby (1989), the presence of rectangular lakes in the Old Crow region in Canada has been associated with underlying tectonic control. Likewise, the rectangular shape and the straightness of the shorelines of many lakes in the Bolivian Amazon (Plafker 1964) and in the northeast of Brazil (Bezerra et al. 2001) are interpreted as the result of bedrock fractures. The analysis of shape and orientation patterns can help understand the geomorphological agents behind lake formation and monitor lake evolution. To do this, reliable and easy-to-use methods to quantitatively assess shape and orientation of the lakes are needed.

Although shape and orientation are very basic and intuitive concepts, they are difficult to quantify. Several methods of shape and orientation recognition have been developed in the field of computer vision (Gartner and Schonherr 1997; Rosin 2003; Stojmenovic and Zunic 2008; Stojmenovic and Nayak 2007; Rosin 1999; Zunic, Martinez-Ortiz, and Zunic 2012; Martin and Rosin 2004); however, their implementation within a geographic information system (GIS) software requires complex programming, is time consuming, and therefore beyond the reach of many earth scientists. On the other hand, as GIS tools for morphometric analysis are scant, morphometric analyses must rely on a few, often insufficient, parameters.

Up to now, the analysis of lake geometry has relied almost exclusively on the analysis of basic parameters such as perimeter, area, shore development (the ratio between lake perimeter and the circumference of a circle of the same area of the lake), and the geometric characteristics of the lake's best fitting ellipse (BFE, the best ellipsoidal approximation of the lake's shape) (Frohn, Hinkel, and Eisner 2005; Hinkel et al. 2005; Morgenstem et al. 2011; Sjoberg, Hugelius, and Kuhry 2012; Barba 2003). The most common metrics based on the BFE are the ratio between the minor and the major axes of the ellipse (on which asymmetry and elongation are calculated) and the orientation of the major axis of the ellipse, which estimates the orientation of the lake (see, e.g., Hinkel et al. 2005; Gonzales and Aydin 2008). In these studies, the parameters describing the BFE are determined by GIS software using canonical analysis, where the first eigenvalue gives the major axis of the ellipse and the minor axis is given by the second eigen-value. However, while the use of the BFE is satisfactory in many cases, it is subject to incorrect orientation estimates in the case of compact or symmetric shapes (Rosin 1999, 2003; Zunic, Rosin, and Kopanja 2006). For example, when a lake resembles a square, the ellipse approximates a circle. Consequently, small protrusions in the lake's shape can cause important orientation errors (see example in Figure la). Another problem of estimating the orientation of lake basins based on the direction of the major axis of the BFE is that this method is not very sensitive to the presence of straight shores, which are important lake characteristics as they can be the expression of faults or other geomorphic agents. Moreover, the ratio between the major and minor axes of the BFE does not allow differentiating between lakes of different shapes. To overcome these limitations, Cote and Bum (2002) have determined lake shape by comparing the area of each lake with the area of the ellipse, rectangle, triangle, and circle inscribed in the lake's minimum bounding rectangle (MBR). The MBR is the rectangle of smallest area enclosing a polygon. A clear advantage of using the MBR instead of the BFE is that the MBR is far more sensitive to the presence of straight edges and therefore provides a better approximation in the case of rectangular or square lakes. However, the MBR yields big errors if the lake has important protrusions, as in the case of Figure lb.

[FIGURE 1 OMITTED]

In this article, an alternative easy-to-use method to measure rectangularity, ellipticity, and orientation is proposed. This is a variant of the rectangular discrepancy measure [R'.sub.D] of Rosin (1999, 2003) and can be fully implemented using the built-in functions of standard GIS software. In addition, a rectangularity vs. ellipticity index (REi) is derived from the measures of rectangularity and ellipticity.

Here, this method is applied to the case study of oriented lakes in the Bolivian Amazon using ArcGis[R] 10. Many of these lakes have been noted for their rectangular shape and markedly uniform SW-NE orientation.

Although they vary considerably in size, they are characterized by being very shallow, usually less than 2 m deep, and having a flat bottom. Many different mechanisms have been proposed for their formation, including scouring caused by large-scale flooding (Campbell, Frailey, and Arellano 1985); paleo deflation (Clapperton 1993); human agency (Belmonte and Barba 2011); and combined paleo deflation and wind/wave action (Dumont and Fournier 1994; Langstroth 1996). However, the most accepted hypothesis to date is that of tectonic control, as proposed by Plafker almost 50 years ago (Allenby 1988; Gonzales and Aydin 2008; Plafker 1964, 1974; Price 1968). According to Plafker, the lakes' rectangular shape results from the propagation of bedrock fractures through unconsolidated sediments. This tectonic model has been recently challenged by Lombardo and Veit (2014) who found that at least the three rectangular and oriented lakes studied were not formed by tectonic displacements. Despite the fact that the rectangularity of these lakes has been greatly stressed, many of them are, in fact, elliptical. According to Plafker, lakes form as rectangles but become more elliptical because of wave action against the shores. The impact of wave action should be more evident in the case of larger lakes where wave action is stronger, assuming that there is no correlation between size and time of lake formation. To test this hypothesis, we need to assess whether there is a relation between lake size and its rectangularity/ellipticity.

New methods to measure rectangularity (R), ellipticity (E), orientation (O), and REi are here developed and used to unveil underlying patterns and to explore the relation between lake size and its rectangularity/ellipticity to test the Plafker's hypothesis of wave action. Results are compared with those obtained using previously proposed methods (Rosin 1999, 2003; Cote and Burn 2002).

Rectangularity (R), ellipticity (E) orientation (O), and rectangularity vs. ellipticity index REi

Rectangularity R is here defined as the highest value between D and D':

R = max[D,D]: D = 1 - [2(A - [A.sub.cBFE])/A]; D' = 1 - [2(A - [A.sub.cMBR])/A] (1)

where A is the area of the lake, [A.sub.cBFE] is the clip between the lake and the best fitting rectangle (BFR) oriented as the major axis of the BFE, and [A.sub.cMBR] is the clip between the lake and the BFR oriented as the MBR. R is a variant of Rosin's discrepancy method ([R'.sub.D]) (Rosin 1999, 2003). Rosin's [R.sub.D] calculates the normalized positive and negative differences between the lake and its BFR:

[R.sub.D] = 1 - [[[A.sub.1] + [A.sub.3] - 2[A.sub.2]]/[A.sub.3]] (2)

where [A.sub.1] is the lake area, A2 is the clipped area, and [A.sub.3] is the area of the BFR. The BFR is the best rectangular approximation of the lake's shape. As the BFR is estimated using the BFE, which is prone to error in the case of compact shapes (Figure la), [R.sub.D] is not always reliable. Rosin (1999) suggests to measure the value of [R.sub.D] for both the estimated orientation given by the major axis of the BFE and its 45[degrees] offset ([R.sub.D45]). The highest value between [R.sub.D] and [R.sub.D45] is then retained as [R'.sub.D].

R in Equation (1) differs from Rosin's [R'.sub.D] in the following: (1) the area of the BFR is equal to the area of the lake; (2) the length/width ratio of the BFR equals to the ratio of the BFE's axes; and (3) instead of rotating the BFR 45[degrees], the BFR is here given the same orientation as the lake's MBR.

[FIGURE 2 OMITTED]

Ellipticity (E) can be calculated in a similar way as D.

E = 1 - [2(A - [A.sub.E])/A] (3)

where [A.sub.E] is the clip between the lake and the BFE.

E and R can vary between 1, for, respectively, a perfect ellipse or rectangle and tend to be -1 when the clip tends to 0. However, the latter is a very unlike case. When considering only rectangular (or square) and elliptical (or circular) forms, the orientation (O) of the lake is defined by the orientation of the BFE if (D or E) > D' or by the orientation of the lake's MBR if (D or E) < D'. Figure 2 shows two examples in which the BFR oriented as the MBR provides a better approximation of the lake's shape than the BFR oriented as the BFE.

The REi is defined as

REi = [[R - E]/[R + E]] (4)

REi will be positive for rectangular lakes and negative for elliptical ones. As R and E can vary between 1 and -1, REi can, from a purely mathematical point of view, vary between -[infinity] and +[infinity]. However, as R and E are correlated, REi will always be close to 0.

In the following section, a step-by-step workflow for measuring R, E, O, and REi is described using built-in tools of ArcGis[R] 10.0. Spatial Analyst extension is needed.

Measuring E, R, O, and REi in ArcGis[R]

The first step consists in creating the BFE, which is used to calculate E. Then, the BFE is used to create the BFR, which is used to calculate R. Once E and R have been calculated, 0 and REi are easily derived from them.

Measuring E

The BFE can be created as a table (containing the length of the major and minor axes and the orientation of the major axis) with the Zonal Geometry tool. The Zonal Geometry tool builds the ellipses in such a way that the centroid of each ellipse coincides with the centroid of the corresponding original feature (the lake in this case) and its area is equal to the area of the lake it represents (A). The major and minor axes coincide with the first two canonical components; the orientation of the ellipses is that of the first component (Ebdon 1985). The polygonal features class of the ellipses can now be created with the Table to Ellipse tool, taking care to first convert the angles calculated by the Zonal Geometry tool (zero in the east, values increasing counterclockwise) to Azimuth (zero in the north, values increasing clockwise). The Table to Ellipse tool's output is a polyline features class that can be converted to polygons using the Feature to Polygon tool. The polygon features class of the lakes can now be clipped using the polygon features class of the BFE just created and [A.sub.E], the clip between the lake and the BFE, is obtained. Once A and [A.sub.E] are known, calculating E is straightforward (see Equation (3)).

Measuring R, O, and REi

To measure the rectangularity R, D, and D' must be calculated (see Equation (1)). To calculate D, we need to derive the BFR from the BFE. The rectangle must have the same area as the lake and the centroids of both must coincide. One way to build the BFR is to first create a second ellipse (small_e) for which the major and minor axes are, respectively, [square root of [pi][a.sup.2]] and fnb2, where a and b are the major and the minor axes of the BFE. The BFR is then created as the MBR of small_e. This can be obtained with the Minimum Bounding Geometry tool, setting Geometry type as "RECTANGLE BY AREA" (default) and using the option "Add geometry characteristics as attributes to output." To calculate Dsmall_e is rotated with the Table to Ellipse tool using the orientation of the MBR; then the BFR is created as the MBR of the rotated small_e. The rectangularity R is then determined by the highest value between D and D' (Equation (1)). Once both ellipticity (E) and rectangularity (R) are known, the REi can also be calculated using Equation (4).

[FIGURE 3 OMITTED]

Once E, D, and D' are known, the lake is assigned the orientation of the MBR if (D or E) < D 'or the orientation of the BFE if (D or E) > D'. This ensures that rectangular lakes with a BFR, which is oriented as the MBR (see example in Figure 2), are given the orientation of the MBR and not that of the BFE.

Creating the database of the Bolivian oriented lakes

An ArcGis feature class has been created to store the polygons of the geometric and oriented lakes of the Llanos de Moxos in the Bolivian Amazon. These polygons have been generated based on a supervised classification of Landsat images (Figure 3). Eleven Landsat 7 ETM + level-1 geoTIFF scenes have been downloaded from the USGS Earthexplorer (http://earthexplorer.usgs. gov/): LE70010692000146EDC00, LE70010701999287 COA00, LE70010711999223COAO1, LE723106920012 23AGS00, LE72310702001223AGSOO LE723107120012 23AGS00, LE72320692001262EDC00, LE7232070200 1262EDC00, LE72320712001262EDC00, LE72330692 001173EDC00, LE7233O7O2001237EDCO0, and LE7233 0711999232EDC00. Images have been classified using Maximum Likelihood Supervised classification (Richards and Jia 1999) in three classes (water, forest, and savannah) using bands 7, 5, 4, and 3. Color RGB (5,4,3) composites have been displayed as background and used as reference to identify training areas. The water class has been vectorialized into 30,014 polygons representing the totality of water bodies. Overlapping polygons have been manually corrected and their topology validated. Based on the visual analysis of the images and after a "trial and error" assessment aimed at retaining the majority of rectangular and oriented lakes and minimizing the amount of noise and irregular lakes, the lakes smaller than 36 hectares or with a Shape Index greater than 1.6 have been excluded (Figure 3). The Shape Index is defined as [shape length]/ (4 x sqr([shape area]) (Hinkel et al. 2005). To filter out all the small polygons, mostly representing fragments of rivers and water ponds, all the polygons smaller than 36 hectares have been deleted. Further filtering has been performed by deleting all the polygons that intersected rivers (line feature with a 300 m buffer). Polygons representing oxbows have been identified as those with a Shape Index greater than 1.6. After all the water bodies that do not fulfill the criteria for this study were deleted, 504 polygons were left for the morphometric analysis (Figure 3). To remove the effect of the 30 x 30 m pixels on the lake borders, the polygons have been slightly smoothed. The existence of a correlation between the shape of the lake and its size hypothesized by Plafker is tested using the Spearman's rank correlation coefficient rho. Rho measures the tendency for Y to either increase or decrease when X increases; it varies between -1, a perfect negative correlation, and 1, a perfect positive correlation; zero indicates no correlation (Ebdon 1985).

Results and discussion

For the 504 selected lakes, rectangularity R, ellipticity E, and orientation O have been calculated.

The methods proposed in this article are applied for the morphometric analysis of the "rectangular and oriented" lakes of the Llanos de Moxos, Bolivia. These new tools are successfully used to (1) test the Plafker's hypothesis of the existence of a correlation between lake size and shape and (2) unveil underlying patterns that would have been very difficult to identify with other existing methods.

To compare the accuracy of R in relation to Rosin's discrepancy method ([R'.sub.D]) (Rosin 1999, 2003), the clipped area [A.sub.c] of Equation (1) has been created for three different orientations of the BFR: oriented as the BFE (D), as the MBR (D j and as the BFE plus 45[degrees] as in Rosin's [R'.sub.D]. Out of the 504 lakes, D provided the highest values in 189 cases, D' in 303 and the BFE plus 45[degrees] only in 12. This result shows that R represents an improvement over ([R'.sub.D]).

To assess whether a lake is rectangular or elliptical, ellipticity (E) has been calculated and compared with rectangularity (R). Results show that 397 of the lakes are elliptical and 107 are rectangular. Of the latter, 75 are oriented as the MBR and 32 are oriented as the BFE. When applying the Cote and Bum (2002) method of lake classification, only 2 lakes are classified as rectangular, 474 lakes are classified as elliptical, and 28 lakes are classified as triangular. Hence, failing to capture the most striking aspect of these Bolivian lakes: the fact that an important part of them are rectangular. A simple visual analysis of the lakes shows that the method here proposed can discriminate between rectangular and elliptical lakes more accurately than the method of inscribed geometries used by Cote and Burn (2002).

Out of the total number of 504 lakes, the fact that D' > D for 75 of the 107 rectangular lakes means that in 15% of the cases lake orientation is better estimated by the MBR than by the BFE. The difference between the orientation of the MBR and that of the BFE is greater than 5[degrees] in 43 out of the 75 rectangular lakes and greater than 10[degrees] in 23 of them. The orientations of the BFEs (Figure 4a) are slightly less clustered (higher standard deviations) than the orientations based on R (Figure 4b). The difference between the two is most visible when looking at the statistics calculated for the 75 rectangular lakes with D' > D (Figure 4c and d), which show a far smaller standard deviation for the MBR orientations than for the BFEs, particularly in the case of the lakes clustered around 45[degrees]. Hence, the method developed in this article, which combines both the MBR and the BFE, provides a better estimation of lake orientation than what is achieved if the BFE is used on its own, as currently done in most studies of geometric lakes (Barba 2003; Gonzales and Ay din 2008; Hinkel et al. 2005; Morgenstem et al. 2011).

To test the Plafker's hypothesis that a correlation exists between the shape of the lake and its size, Spearman's rank correlation coefficient rho is calculated between REi and the size of the lakes. The rho value for the 504 selected lakes is almost zero (0.03403, p-value = 0.4629), indicating no correlation between the lakes' shape (elliptical vs. rectangular) and their size. The data suggest that the Plafker's hypothesis that there was a correlation between shape and size of the lakes caused by a stronger wave action on the shores of the larger lakes is incorrect. However, in the present study, only lakes larger than 36 hectares have been included in the analysis. Hence, the inclusion of smaller lakes into the analysis could alter this result.

[FIGURE 4 OMITTED]

Lakes with high values of E also show high values of R, and vice versa (Figure 5), because perfect rectangles and ellipses have the same high value (about 0.91) of, respectively, E and R. In the case of very irregular shapes that do not resemble a rectangle or an ellipse at all, when E and R are small, the REi can become unstable and should be used with caution. For high values of both indexes, more "regular" lakes can be expected. These "regular" lakes share similar orientations, suggesting a correlation between the regularity of the shape and the orientation. This correlation has been explored separately for ellipticity and rectangularity. Figure 6 shows that "irregular" lakes, with low ellipticity or rectangularity, are almost randomly oriented (Figure 6a and b), while "regular" lakes are more likely to share a similar orientation. The clustering of lakes is greater among those lakes with high rectangularity than among lakes with high ellipticity (Figure 6c and d). The interpretation of this fact is not straightforward. It could be that the factors that control the shape of the lakes impact their orientation differently. This could provide an argument in favor of the Plafker's tectonic hypothesis, with the orientation of the faults being more consistent than the orientation of the winds that would have later reshaped the lakes. It could also indicate that the lakes with higher rectangularity have a tectonic origin while the rest formed in a different way, but, by coincidence, all of them were "reoriented" in the direction of the faults by wind action. Further statistical analysis is needed to better interpret the data; however, this is beyond the scope of this article. Future studies could combine the methods proposed here with spatial statistics and field data to help uncover other underlying patterns that can shed new light on the possible origin and evolution of the Bolivian rectangular lakes.

[FIGURE 5 OMITTED]

Conclusions

New morphometric parameters that can be easily calculated using common GIS packages are proposed here: rectangularity R, ellipticity E, orientation O and the REi. They are compared with previously used methods and applied to analyze the oriented lakes of the Bolivian Amazon. Results show that R provides an improved estimate of rectangularity compared to Rosin's discrepancy method (R'D). The use of R and E also provide an improved classification of lakes, based on their shape, than the classification method used by Cote and Bum. The new, combined use of the MBR and BFE to measure lake orientation improves the conventional measures of orientation, which are based only on the BFE. The use of the proposed shape index REi for the morphometric analysis of the Bolivian oriented lakes has allowed, for the first time, to test previously formulated hypothesis about the mechanisms behind lake formation. Moreover, it has been shown that the use of R, E, O, and REi within GIS packages has great potential to help uncover patterns that are very difficult to identify otherwise.

[FIGURE 6 OMITTED]

http://dx.doi.org/10.1080/15230406.2014.919540

Acknowledgments

This work was supported by the Swiss National Science Foundation [grant number 200020-141277/1], I would like to thank H. Veitu for his support and guidance, and D. Garcia-Castellanos for the discussions that helped structure the ideas for this paper. Further thanks are also due to E. Canal-Beeby and two anonymous reviewers for their valuable suggestions.

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Umberto Lombardo *

Institute of Geography, University of Bern, Bern, Switzerland

(Received 29 September 2013; accepted 26 April 2014)

* Email: lombardo@giub.unibe.ch

It has being long recognized that lake shape and orientation can tell us much about the processes behind lake formation (Reeves 1968). For example, in the Arctic region, where many lakes are geometric and share a similar orientation, it has been suggested that their origin is due to constant winds (Carson and Hussey 1962; Cote and Bum 2002). According to Allenby (1989), the presence of rectangular lakes in the Old Crow region in Canada has been associated with underlying tectonic control. Likewise, the rectangular shape and the straightness of the shorelines of many lakes in the Bolivian Amazon (Plafker 1964) and in the northeast of Brazil (Bezerra et al. 2001) are interpreted as the result of bedrock fractures. The analysis of shape and orientation patterns can help understand the geomorphological agents behind lake formation and monitor lake evolution. To do this, reliable and easy-to-use methods to quantitatively assess shape and orientation of the lakes are needed.

Although shape and orientation are very basic and intuitive concepts, they are difficult to quantify. Several methods of shape and orientation recognition have been developed in the field of computer vision (Gartner and Schonherr 1997; Rosin 2003; Stojmenovic and Zunic 2008; Stojmenovic and Nayak 2007; Rosin 1999; Zunic, Martinez-Ortiz, and Zunic 2012; Martin and Rosin 2004); however, their implementation within a geographic information system (GIS) software requires complex programming, is time consuming, and therefore beyond the reach of many earth scientists. On the other hand, as GIS tools for morphometric analysis are scant, morphometric analyses must rely on a few, often insufficient, parameters.

Up to now, the analysis of lake geometry has relied almost exclusively on the analysis of basic parameters such as perimeter, area, shore development (the ratio between lake perimeter and the circumference of a circle of the same area of the lake), and the geometric characteristics of the lake's best fitting ellipse (BFE, the best ellipsoidal approximation of the lake's shape) (Frohn, Hinkel, and Eisner 2005; Hinkel et al. 2005; Morgenstem et al. 2011; Sjoberg, Hugelius, and Kuhry 2012; Barba 2003). The most common metrics based on the BFE are the ratio between the minor and the major axes of the ellipse (on which asymmetry and elongation are calculated) and the orientation of the major axis of the ellipse, which estimates the orientation of the lake (see, e.g., Hinkel et al. 2005; Gonzales and Aydin 2008). In these studies, the parameters describing the BFE are determined by GIS software using canonical analysis, where the first eigenvalue gives the major axis of the ellipse and the minor axis is given by the second eigen-value. However, while the use of the BFE is satisfactory in many cases, it is subject to incorrect orientation estimates in the case of compact or symmetric shapes (Rosin 1999, 2003; Zunic, Rosin, and Kopanja 2006). For example, when a lake resembles a square, the ellipse approximates a circle. Consequently, small protrusions in the lake's shape can cause important orientation errors (see example in Figure la). Another problem of estimating the orientation of lake basins based on the direction of the major axis of the BFE is that this method is not very sensitive to the presence of straight shores, which are important lake characteristics as they can be the expression of faults or other geomorphic agents. Moreover, the ratio between the major and minor axes of the BFE does not allow differentiating between lakes of different shapes. To overcome these limitations, Cote and Bum (2002) have determined lake shape by comparing the area of each lake with the area of the ellipse, rectangle, triangle, and circle inscribed in the lake's minimum bounding rectangle (MBR). The MBR is the rectangle of smallest area enclosing a polygon. A clear advantage of using the MBR instead of the BFE is that the MBR is far more sensitive to the presence of straight edges and therefore provides a better approximation in the case of rectangular or square lakes. However, the MBR yields big errors if the lake has important protrusions, as in the case of Figure lb.

[FIGURE 1 OMITTED]

In this article, an alternative easy-to-use method to measure rectangularity, ellipticity, and orientation is proposed. This is a variant of the rectangular discrepancy measure [R'.sub.D] of Rosin (1999, 2003) and can be fully implemented using the built-in functions of standard GIS software. In addition, a rectangularity vs. ellipticity index (REi) is derived from the measures of rectangularity and ellipticity.

Here, this method is applied to the case study of oriented lakes in the Bolivian Amazon using ArcGis[R] 10. Many of these lakes have been noted for their rectangular shape and markedly uniform SW-NE orientation.

Although they vary considerably in size, they are characterized by being very shallow, usually less than 2 m deep, and having a flat bottom. Many different mechanisms have been proposed for their formation, including scouring caused by large-scale flooding (Campbell, Frailey, and Arellano 1985); paleo deflation (Clapperton 1993); human agency (Belmonte and Barba 2011); and combined paleo deflation and wind/wave action (Dumont and Fournier 1994; Langstroth 1996). However, the most accepted hypothesis to date is that of tectonic control, as proposed by Plafker almost 50 years ago (Allenby 1988; Gonzales and Aydin 2008; Plafker 1964, 1974; Price 1968). According to Plafker, the lakes' rectangular shape results from the propagation of bedrock fractures through unconsolidated sediments. This tectonic model has been recently challenged by Lombardo and Veit (2014) who found that at least the three rectangular and oriented lakes studied were not formed by tectonic displacements. Despite the fact that the rectangularity of these lakes has been greatly stressed, many of them are, in fact, elliptical. According to Plafker, lakes form as rectangles but become more elliptical because of wave action against the shores. The impact of wave action should be more evident in the case of larger lakes where wave action is stronger, assuming that there is no correlation between size and time of lake formation. To test this hypothesis, we need to assess whether there is a relation between lake size and its rectangularity/ellipticity.

New methods to measure rectangularity (R), ellipticity (E), orientation (O), and REi are here developed and used to unveil underlying patterns and to explore the relation between lake size and its rectangularity/ellipticity to test the Plafker's hypothesis of wave action. Results are compared with those obtained using previously proposed methods (Rosin 1999, 2003; Cote and Burn 2002).

Rectangularity (R), ellipticity (E) orientation (O), and rectangularity vs. ellipticity index REi

Rectangularity R is here defined as the highest value between D and D':

R = max[D,D]: D = 1 - [2(A - [A.sub.cBFE])/A]; D' = 1 - [2(A - [A.sub.cMBR])/A] (1)

where A is the area of the lake, [A.sub.cBFE] is the clip between the lake and the best fitting rectangle (BFR) oriented as the major axis of the BFE, and [A.sub.cMBR] is the clip between the lake and the BFR oriented as the MBR. R is a variant of Rosin's discrepancy method ([R'.sub.D]) (Rosin 1999, 2003). Rosin's [R.sub.D] calculates the normalized positive and negative differences between the lake and its BFR:

[R.sub.D] = 1 - [[[A.sub.1] + [A.sub.3] - 2[A.sub.2]]/[A.sub.3]] (2)

where [A.sub.1] is the lake area, A2 is the clipped area, and [A.sub.3] is the area of the BFR. The BFR is the best rectangular approximation of the lake's shape. As the BFR is estimated using the BFE, which is prone to error in the case of compact shapes (Figure la), [R.sub.D] is not always reliable. Rosin (1999) suggests to measure the value of [R.sub.D] for both the estimated orientation given by the major axis of the BFE and its 45[degrees] offset ([R.sub.D45]). The highest value between [R.sub.D] and [R.sub.D45] is then retained as [R'.sub.D].

R in Equation (1) differs from Rosin's [R'.sub.D] in the following: (1) the area of the BFR is equal to the area of the lake; (2) the length/width ratio of the BFR equals to the ratio of the BFE's axes; and (3) instead of rotating the BFR 45[degrees], the BFR is here given the same orientation as the lake's MBR.

[FIGURE 2 OMITTED]

Ellipticity (E) can be calculated in a similar way as D.

E = 1 - [2(A - [A.sub.E])/A] (3)

where [A.sub.E] is the clip between the lake and the BFE.

E and R can vary between 1, for, respectively, a perfect ellipse or rectangle and tend to be -1 when the clip tends to 0. However, the latter is a very unlike case. When considering only rectangular (or square) and elliptical (or circular) forms, the orientation (O) of the lake is defined by the orientation of the BFE if (D or E) > D' or by the orientation of the lake's MBR if (D or E) < D'. Figure 2 shows two examples in which the BFR oriented as the MBR provides a better approximation of the lake's shape than the BFR oriented as the BFE.

The REi is defined as

REi = [[R - E]/[R + E]] (4)

REi will be positive for rectangular lakes and negative for elliptical ones. As R and E can vary between 1 and -1, REi can, from a purely mathematical point of view, vary between -[infinity] and +[infinity]. However, as R and E are correlated, REi will always be close to 0.

In the following section, a step-by-step workflow for measuring R, E, O, and REi is described using built-in tools of ArcGis[R] 10.0. Spatial Analyst extension is needed.

Measuring E, R, O, and REi in ArcGis[R]

The first step consists in creating the BFE, which is used to calculate E. Then, the BFE is used to create the BFR, which is used to calculate R. Once E and R have been calculated, 0 and REi are easily derived from them.

Measuring E

The BFE can be created as a table (containing the length of the major and minor axes and the orientation of the major axis) with the Zonal Geometry tool. The Zonal Geometry tool builds the ellipses in such a way that the centroid of each ellipse coincides with the centroid of the corresponding original feature (the lake in this case) and its area is equal to the area of the lake it represents (A). The major and minor axes coincide with the first two canonical components; the orientation of the ellipses is that of the first component (Ebdon 1985). The polygonal features class of the ellipses can now be created with the Table to Ellipse tool, taking care to first convert the angles calculated by the Zonal Geometry tool (zero in the east, values increasing counterclockwise) to Azimuth (zero in the north, values increasing clockwise). The Table to Ellipse tool's output is a polyline features class that can be converted to polygons using the Feature to Polygon tool. The polygon features class of the lakes can now be clipped using the polygon features class of the BFE just created and [A.sub.E], the clip between the lake and the BFE, is obtained. Once A and [A.sub.E] are known, calculating E is straightforward (see Equation (3)).

Measuring R, O, and REi

To measure the rectangularity R, D, and D' must be calculated (see Equation (1)). To calculate D, we need to derive the BFR from the BFE. The rectangle must have the same area as the lake and the centroids of both must coincide. One way to build the BFR is to first create a second ellipse (small_e) for which the major and minor axes are, respectively, [square root of [pi][a.sup.2]] and fnb2, where a and b are the major and the minor axes of the BFE. The BFR is then created as the MBR of small_e. This can be obtained with the Minimum Bounding Geometry tool, setting Geometry type as "RECTANGLE BY AREA" (default) and using the option "Add geometry characteristics as attributes to output." To calculate Dsmall_e is rotated with the Table to Ellipse tool using the orientation of the MBR; then the BFR is created as the MBR of the rotated small_e. The rectangularity R is then determined by the highest value between D and D' (Equation (1)). Once both ellipticity (E) and rectangularity (R) are known, the REi can also be calculated using Equation (4).

[FIGURE 3 OMITTED]

Once E, D, and D' are known, the lake is assigned the orientation of the MBR if (D or E) < D 'or the orientation of the BFE if (D or E) > D'. This ensures that rectangular lakes with a BFR, which is oriented as the MBR (see example in Figure 2), are given the orientation of the MBR and not that of the BFE.

Creating the database of the Bolivian oriented lakes

An ArcGis feature class has been created to store the polygons of the geometric and oriented lakes of the Llanos de Moxos in the Bolivian Amazon. These polygons have been generated based on a supervised classification of Landsat images (Figure 3). Eleven Landsat 7 ETM + level-1 geoTIFF scenes have been downloaded from the USGS Earthexplorer (http://earthexplorer.usgs. gov/): LE70010692000146EDC00, LE70010701999287 COA00, LE70010711999223COAO1, LE723106920012 23AGS00, LE72310702001223AGSOO LE723107120012 23AGS00, LE72320692001262EDC00, LE7232070200 1262EDC00, LE72320712001262EDC00, LE72330692 001173EDC00, LE7233O7O2001237EDCO0, and LE7233 0711999232EDC00. Images have been classified using Maximum Likelihood Supervised classification (Richards and Jia 1999) in three classes (water, forest, and savannah) using bands 7, 5, 4, and 3. Color RGB (5,4,3) composites have been displayed as background and used as reference to identify training areas. The water class has been vectorialized into 30,014 polygons representing the totality of water bodies. Overlapping polygons have been manually corrected and their topology validated. Based on the visual analysis of the images and after a "trial and error" assessment aimed at retaining the majority of rectangular and oriented lakes and minimizing the amount of noise and irregular lakes, the lakes smaller than 36 hectares or with a Shape Index greater than 1.6 have been excluded (Figure 3). The Shape Index is defined as [shape length]/ (4 x sqr([shape area]) (Hinkel et al. 2005). To filter out all the small polygons, mostly representing fragments of rivers and water ponds, all the polygons smaller than 36 hectares have been deleted. Further filtering has been performed by deleting all the polygons that intersected rivers (line feature with a 300 m buffer). Polygons representing oxbows have been identified as those with a Shape Index greater than 1.6. After all the water bodies that do not fulfill the criteria for this study were deleted, 504 polygons were left for the morphometric analysis (Figure 3). To remove the effect of the 30 x 30 m pixels on the lake borders, the polygons have been slightly smoothed. The existence of a correlation between the shape of the lake and its size hypothesized by Plafker is tested using the Spearman's rank correlation coefficient rho. Rho measures the tendency for Y to either increase or decrease when X increases; it varies between -1, a perfect negative correlation, and 1, a perfect positive correlation; zero indicates no correlation (Ebdon 1985).

Results and discussion

For the 504 selected lakes, rectangularity R, ellipticity E, and orientation O have been calculated.

The methods proposed in this article are applied for the morphometric analysis of the "rectangular and oriented" lakes of the Llanos de Moxos, Bolivia. These new tools are successfully used to (1) test the Plafker's hypothesis of the existence of a correlation between lake size and shape and (2) unveil underlying patterns that would have been very difficult to identify with other existing methods.

To compare the accuracy of R in relation to Rosin's discrepancy method ([R'.sub.D]) (Rosin 1999, 2003), the clipped area [A.sub.c] of Equation (1) has been created for three different orientations of the BFR: oriented as the BFE (D), as the MBR (D j and as the BFE plus 45[degrees] as in Rosin's [R'.sub.D]. Out of the 504 lakes, D provided the highest values in 189 cases, D' in 303 and the BFE plus 45[degrees] only in 12. This result shows that R represents an improvement over ([R'.sub.D]).

To assess whether a lake is rectangular or elliptical, ellipticity (E) has been calculated and compared with rectangularity (R). Results show that 397 of the lakes are elliptical and 107 are rectangular. Of the latter, 75 are oriented as the MBR and 32 are oriented as the BFE. When applying the Cote and Bum (2002) method of lake classification, only 2 lakes are classified as rectangular, 474 lakes are classified as elliptical, and 28 lakes are classified as triangular. Hence, failing to capture the most striking aspect of these Bolivian lakes: the fact that an important part of them are rectangular. A simple visual analysis of the lakes shows that the method here proposed can discriminate between rectangular and elliptical lakes more accurately than the method of inscribed geometries used by Cote and Burn (2002).

Out of the total number of 504 lakes, the fact that D' > D for 75 of the 107 rectangular lakes means that in 15% of the cases lake orientation is better estimated by the MBR than by the BFE. The difference between the orientation of the MBR and that of the BFE is greater than 5[degrees] in 43 out of the 75 rectangular lakes and greater than 10[degrees] in 23 of them. The orientations of the BFEs (Figure 4a) are slightly less clustered (higher standard deviations) than the orientations based on R (Figure 4b). The difference between the two is most visible when looking at the statistics calculated for the 75 rectangular lakes with D' > D (Figure 4c and d), which show a far smaller standard deviation for the MBR orientations than for the BFEs, particularly in the case of the lakes clustered around 45[degrees]. Hence, the method developed in this article, which combines both the MBR and the BFE, provides a better estimation of lake orientation than what is achieved if the BFE is used on its own, as currently done in most studies of geometric lakes (Barba 2003; Gonzales and Ay din 2008; Hinkel et al. 2005; Morgenstem et al. 2011).

To test the Plafker's hypothesis that a correlation exists between the shape of the lake and its size, Spearman's rank correlation coefficient rho is calculated between REi and the size of the lakes. The rho value for the 504 selected lakes is almost zero (0.03403, p-value = 0.4629), indicating no correlation between the lakes' shape (elliptical vs. rectangular) and their size. The data suggest that the Plafker's hypothesis that there was a correlation between shape and size of the lakes caused by a stronger wave action on the shores of the larger lakes is incorrect. However, in the present study, only lakes larger than 36 hectares have been included in the analysis. Hence, the inclusion of smaller lakes into the analysis could alter this result.

[FIGURE 4 OMITTED]

Lakes with high values of E also show high values of R, and vice versa (Figure 5), because perfect rectangles and ellipses have the same high value (about 0.91) of, respectively, E and R. In the case of very irregular shapes that do not resemble a rectangle or an ellipse at all, when E and R are small, the REi can become unstable and should be used with caution. For high values of both indexes, more "regular" lakes can be expected. These "regular" lakes share similar orientations, suggesting a correlation between the regularity of the shape and the orientation. This correlation has been explored separately for ellipticity and rectangularity. Figure 6 shows that "irregular" lakes, with low ellipticity or rectangularity, are almost randomly oriented (Figure 6a and b), while "regular" lakes are more likely to share a similar orientation. The clustering of lakes is greater among those lakes with high rectangularity than among lakes with high ellipticity (Figure 6c and d). The interpretation of this fact is not straightforward. It could be that the factors that control the shape of the lakes impact their orientation differently. This could provide an argument in favor of the Plafker's tectonic hypothesis, with the orientation of the faults being more consistent than the orientation of the winds that would have later reshaped the lakes. It could also indicate that the lakes with higher rectangularity have a tectonic origin while the rest formed in a different way, but, by coincidence, all of them were "reoriented" in the direction of the faults by wind action. Further statistical analysis is needed to better interpret the data; however, this is beyond the scope of this article. Future studies could combine the methods proposed here with spatial statistics and field data to help uncover other underlying patterns that can shed new light on the possible origin and evolution of the Bolivian rectangular lakes.

[FIGURE 5 OMITTED]

Conclusions

New morphometric parameters that can be easily calculated using common GIS packages are proposed here: rectangularity R, ellipticity E, orientation O and the REi. They are compared with previously used methods and applied to analyze the oriented lakes of the Bolivian Amazon. Results show that R provides an improved estimate of rectangularity compared to Rosin's discrepancy method (R'D). The use of R and E also provide an improved classification of lakes, based on their shape, than the classification method used by Cote and Bum. The new, combined use of the MBR and BFE to measure lake orientation improves the conventional measures of orientation, which are based only on the BFE. The use of the proposed shape index REi for the morphometric analysis of the Bolivian oriented lakes has allowed, for the first time, to test previously formulated hypothesis about the mechanisms behind lake formation. Moreover, it has been shown that the use of R, E, O, and REi within GIS packages has great potential to help uncover patterns that are very difficult to identify otherwise.

[FIGURE 6 OMITTED]

http://dx.doi.org/10.1080/15230406.2014.919540

Acknowledgments

This work was supported by the Swiss National Science Foundation [grant number 200020-141277/1], I would like to thank H. Veitu for his support and guidance, and D. Garcia-Castellanos for the discussions that helped structure the ideas for this paper. Further thanks are also due to E. Canal-Beeby and two anonymous reviewers for their valuable suggestions.

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Umberto Lombardo *

Institute of Geography, University of Bern, Bern, Switzerland

(Received 29 September 2013; accepted 26 April 2014)

* Email: lombardo@giub.unibe.ch

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Author: | Lombardo, Umberto |
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Publication: | Cartography and Geographic Information Science |

Article Type: | Report |

Geographic Code: | 3BOLI |

Date: | Sep 1, 2014 |

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