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Fractal analysis of the complexity of Panama City coastlines, Central America.

Resumen

El proposito de esta investigacion estuvo centrado en analizar la complejidad de cuatro lineas costeras geologicamente diferentes de la ciudad de Panama, discriminando estadisticamente cada una de ellas a traves de los conceptos de la teoria fractal. El analisis de estas formas naturales fue llevado a cabo utilizando el metodo del divisor el cual consiste en determinar la longitud de una linea costera para un rango de longitudes de segmentos establecidos. El analisis fractal realizado sobre las imagenes digitalizadas y corregidas (1:20,000) del Instituto Geografico Nacional Tommy Guardia, permitio conocer los diferentes niveles de complejidad de estas costas: para la linea costera I resulto elevada debido a la presencia de elementos geologicos de tipo volcanico (D = 1.196 [+ o -] 0.019); intermedia para la linea costera II con elementos geologicos de tipo volcanico y areniscas (D = 1.140 [+ o -] 0.018); baja para la linea costera III (D = 1.017 [+ o -] 0.001) caracterizada por la presencia de aluviones y la accion de factores humanos (crecimiento economico actual de la ciudad), y relativamente baja para la linea costera IV (D = 1.031 [+ o -] 0.004) la cual se encuentra influenciada por la presencia de aluviones, rellenos y pantanos.

Palabras clave: lineas costeras, analisis fractal, ciudad de Panama.

Abstract

The purpose of this investigation was centered in analyzing the complexity of four coastlines geologically different of Panama City, discriminating statistically each one of them through the concepts of the fractal theory. The analysis of these natural ways was carried out using the divider method which consists on determining the longitude of a coastline for a range of longitudes of established segments. The analysis carried out on the digitized and corrected images (1:20,000) of the Instituto Geografico Nacional Tommy Guardia, allowed to know the different levels of complexity of these coasts: for the coastline I it was high due to the presence of geologic elements of volcanic type (D = 1.196 [+ o -] 0.019); it was intermediate for the coastline II with volcanic and sandstone geologic elements (D = 1.140 [+ o -] 0.018); it was low for the coastline III (D = 1.017 [+ o -] 0.001) characterized by the presence of alluviums and the action of human factors (current economic growth of the city), and it was relatively low for the coastline IV (D = 1.031 [+ o -] 0.004) which is influenced by the presence of alluviums, fillers and swamps.

Key words: Coastlines, Fractal analysis, Panama City.

Introduction

The study of the zones of dynamic contact among the air, the earth and the sea or coastlines has been documented in a considerable way (Hanson, 1988; Burke et al., 2001; Woodroffe, 2003; Davis and Fitzgerald, 2004). The geometry in these natural ways depends on the dynamic that governs them, which is also associated with certain kind of factors such as: (1) terrestrial factor that is characterized by geologic, geomorphologic and climatic or sub-aerial processes; (2) the marine factor where it becomes present the tides, waves in deep and superficial waters and the induced currents; (3) the biological factor where it is characterized the vegetation and the marine fauna; and the (4) human factor where the economic growth and population (Hanson, 1988) show up.

When the geometry of the coastlines turns to be affected, a level of complexity or of ruggedness that can be quantified through the concepts of the Fractal Geometry shows up thanks to the calculation of its fractal dimension (D). These ideas were postulated by B.B. Mandelbrot when he studied the self-similar curves that the fractional dimensions have between 1 and 2 (Mandelbrot, 1967; 1983; 1995)subsequently there have been realized a series of investigations which associate the fractal dimensions of the coastlines with their geologic, tectonic, and geomorphologic context (Goodchild, 1980; Phillips, 1986; Leonardi et al., 1994; Ringrose, 1994, Andrle, 1996; Xiaohua, 2004; Tanner et al., 2006; Schwimmer, 2006) and with their ecological characteristics too (Bradbury and Reicheit, 1983; Lorimer et al., 1994; Beck, 1998; Zhang et al., 2002; Burrough, 2002). On the other hand there is not enough information related to the negative effects of the human activities upon the fractal dimension of a coastline (Lorimer et al., 1994). The objective of this research was focused on analyzing the complexity of fours coastlines with different geologic characteristics of Panama City using the concepts of fractal theory.

Coastlines of Panama City

The city of Panama is located near the Bay of the same name, in the narrowest area of the isthmus just as it is shown in Figure 1. This geographical position is historically framed with the arrival of the Spaniards to the American continent at the end of the XV century. The election of the place during this period obeyed certain factors of regional position such as the topography, the climate and the vegetation; even with the existence of marsh and swamps that made difficult the arrival of big ships to the coast (Rubio, 1999).

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Besides this the existence of rocky formations for the construction of big buildings in some areas was another factor that impelled the election of the place. Then, at the end of the XIX century there were carried out the first geologic studies of the central area of the Isthmus of Panama, as a consequence of the construction of the canal (Stewart et al., 1980) they propose a detailed map of the geology of the canal including the city of Panama. Figure 2 shows a geological map prepared by (Cedeno, 2007) which is based on geological work mentioned above, and the coastal area of Panama City divided in 4 areas.

The deep rocks of Panama City are influenced by the Panama Formation where there become present two phases: the Volcanic and Marine. The coastlines studied were selected according to the geologic elements that compose them; the coastlines I and II are affected by the Panama Formation, while the coastlines III and IV are affected for the Lajas Formation. Although the first two coastlines are framed under the same geologic formation, the coastline I is affected by the Volcanic phase whose geologic components are the agglomerates, the andesite in tuffs of tine grain and conglomerates. The coastline II is influenced by the Volcanic and Marine phases, being characterized this last one for the presence of tuffaceous sandstone, tuffaceous silts and limestone with fossils. On the other hand, in the Lajas Formation there become present the alluviums, consolidated silts and swamps. It fits to highlight that throughout the coastal areas that contain the lines I, II and III there have been developed construction activities which are the product of the economic growth; the area that contains the line IV still preserves its natural environment.

In relation to the climatic conditions of Panama, it experiences periods of rain and drought annually. The rainy station (May-December) generates variable winds, soft currents and strong periodic rains. The average temperature of the air is in the order of the 28[degrees]C, the humidity average is on 87%, the temperature of the ocean is on (average = 28[degrees]C, range = 26[degrees]C-32[degrees]C) and salinity of 33.35 ppt; as a characteristic type of the tropical areas. The strong and intense rains cease during the dry time (January-April) and it is characterized by strong winds in North direction (30.40 km/h), which can produce remarkable surf, currents and turbulent waters. According to (D'Croz and Robertson, 1997) in the pacific side of the Isthmus of Panama, a high range of tides corresponds to one of the most important characteristics in the coastal environment. The tides are semi-diurnal and they can reach a range of magnitude higher than 6 m. During more than one year the layer of superficial water in the tropical east pacific corresponds to the mass of superficial tropical water. Two regular events affect the coastal oceanography of the Bay of Panama: a directed wind, of seasonal type and an episodic event (interval of 4-9 years) of the ocean due to the south oscillations of El Nino phenomenon. In this same aspect the studies carried out by (Fiedler and Talley, 2006) demonstrate that the transportation of water steam through the Isthmus of Panama toward the West also contributes to the net high precipitation in this area. On the other hand, the new urban landscape that nowadays presents Panama City is due to the current economic development of the country which executes some pressure on its own coastal characteristics and in the same way a change in its geometry. All these factors have contributed to the formation of the coasts of Panama City generating in them, certain rank of complexity that is possible to study through the fractal theory.

The Hausdorff-Besicovitch dimension

The definition of some natural ways as those that we have been analyzing in this work can be made through the concepts of Fractal Geometry, mainly starting from the definition of groups. The fractal sets can be considered as objects that are obtained from an inductive process in microscopic successive ways (Massopust, 1994) and a way to measure them is through a new class of well-known measurement called Hausdorff measurement. Starting from this concept it is obtained the dimension of the group called Hausdorff-Besicovitch dimension which represents one of the most applied definitions nowadays since it is based on the concept of measurement and its applicability to any group (Falconer, 1990). Let's define a sub-set [LAMBDA] of the Euclidian space [R.sup.n], a positive constant [eta] and {[[psi].sub.i]} as a collection of the subsets that belong [R.sup.n] of lower diameter than T capable of cover the sub-set [LAMBDA]. The Hausdorff-Besicovitch dimension of [LAMBDA] is defined as follows:

[dim.sub.H] ([LAMBDA]) = inf{[eta]: [N.sup.n]([LAMBDA]) = 0}

Where [N.sup.n] ([LAMBDA]) corresponds to the [eta]-dimensional Hausdorff measure of [LAMBDA] and this one is given by:

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII]

In this equation [N.sup.n.sub.[gamma]] ([LAMBDA]) represents the Hausdorff measure whose mathematical formalism can be expressed as:

[N.sup.n.sub.[gamma]] ([LAMBDA]) = inf [[summation].sup.[infinity].sub.i=0] [[absolute value of [[PSI].sub.i]].sup.[eta]]

The infimum (inf) is taken over all countable [gamma]-coverings of A. For a fractal set the calculus of the Hausdorff-Besicovitch dimension is not evident due to the requirement of balls with minor diameters to [gamma] capable of cover this set; to do this; let's think over that [LAMBDA] is overlaid with similar diameters to [gamma]. Under this assumption, it is necessary to define the inferior and superior box dimension of [LAMBDA] as:

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII]

In this set of equations [N.sub.[gamma]] ([LAMBDA]) corresponds a small number of sets with diameter equal [gamma] that could cover [LAMBDA]. If both limits are the same, then the common value it is known with the name of box dimension of [LAMBDA]. Finally, you have that:

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII]

These concepts were used to carry out the calculation of the four coastlines chosen in this research study.

Method, data and results

In this work the divider method was used to quantify the fractal dimension of the coastlines of Panama City already described previously. This method has been applied for some authors to calculate the fractal dimension of irregular curves (Mandelbrot, 1967; Carr and Benzer, 1991; Andrle, 1996; Kuchta, 2002; Uthayakumar and Paramanathan, 2007). This method consists on defining a group of segments of length [gamma], that cover the irregular line under study; this process will give as result a total number N of segments anda remainder of length [lambda]. The total length [tau] of the coastline will be given as [tau] = [gamma]N + [lambda], which can also be expressed as [tau] = [gamma](N + [phi]), where [phi] = [lambda]/[gamma]). When increasing the length of [gamma] the term

(N + [phi]) will go decreasing. (Carr and Benzer, 1991) suggesting a potential relationship between both variables, so that:

r = [[gamma].sup.D] (N + [phi])

[FIGURE 3 OMITTED]

This equation can be written like this:

(N + [phi]) = [tau][[gamma].sup.-D]

In this equation, the numerical value of D is associated with the fractal dimension of the set under study. From this last equation, it is obtained:

D = log(N + [phi])/-log[gamma]

This means that the slope of the log (N + [phi]) and log ([gamma]) will give information referring to the dimension of the coastlines studied.

The aerial images of scale 1:20000 given by the Instituto Geografico Nacional Tommy Guardia were geo-referenced and corrected. With Geographical Information System tools was possible to digitalize the coastlines and to establish the length range of the segments used for each coastline, which was determined between 40 and 1,000m. The Figure 3 presents the result of least squares regression of data obtained on the four coastlines under research using Grapher 8.0 of Golden Software.

With the aim of verifying the level of reliability of the four regressions accomplished in this study, it was carried out a residual analysis of the obtained results. The plots of the residual analysis shown in Figure 4 don't reveal systematic curves; this fact shows a good fit in the linear pattern of the bi-logarithmic representations and therefore, the characteristic of statistical self-similarity is evidenced in these coastlines.

[FIGURE 4 OMITTED]

Table 1 summarizes the experimental results of the four coastlines together with some geologic characteristics.

Discussion

The values of the correlation coefficient ([R.sup.2]) obtained for each regression they are higher than 0.999 and this fact demonstrates that the coastlines of the city of Panama correspond to statistical fractals and in the same way, they have characteristic of statistical self-similarity. The plot of Figure 5 allows visualizing the differences among the ranges of values of the fractal dimension obtained for each coastline under study.

[FIGURE 5 OMITTED]

The coastline I presents the biggest range of irregularity and therefore, the highest value in the fractal dimension in comparison to the other coasts studied as well. (D = 1,196 [+ o -] 0.019); this fact is associated to the presence of fragments of big, rough and angular rocks which keep relationship with the flows of lava (agglomerated) characteristic of the Volcanic Phase of the Panama Formation. On the other hand, the coastline II has a relatively high value in its fractal dimension in comparison to the coastlines III and IV (D = 1.140 [+ o -] 0.018), this is due to the fact that coastline 11 has similar geologic characteristic to coastline I but also, it is influenced by the Marine phase of the Panama Formation where there is showed up tuffaceous sandstone and tuffaceous siltstone, generating a coastal area with less ruggedness and as a result, a smaller level of complexity in comparison to coastline I.

In the same way, the coastline III presents the smallest value in its fractal dimension (D = 1.017 [+ o -] 0.001) which shows a low level of rugosity. This is associated to the geologic elements that characterize it: alluviums and consolidated siltstone; also, this coastal area has limited or scarce energy where the affectation of the waves is almost null. Besides that, the human activity has also affected its form because of the important urban projects developed around it. These factors have generated a very low level of complexity if it is compared to the other coastlines under analysis. Finally, coastline IV presents a higher level than coastline III (D = 1.031 [+ o -] 0.004) even taking into account that both of them become part of the same geologic formation, in coastline IV prevails the swamps environment whose ecosystem offers protection to the coast so that this area keeps regulated and protected of storms and erosions generated by winds and waves. This fact situates this kind of coastlines with a higher level of complexity if compared to the other coasts that present a flatter or less regular morphology, as the case of coastline III.

If we analyze these results (Figure 5) inside a context of geologic formations, it can be observed a notable difference among the group of coastal areas that constitute the Panama Formation (coastlines I and II) and those that constitute the Lajas Formation (coastlines III and IV).

Conclusions

The results of this investigation show that it is possible to carry out the characterization of a coastline through the Fractal Geometry. The four coastlines of Panama City could be differentiated in a statistical way using the divider method, which is based on fractal concepts. With this methodology it could be proven that each of the coastlines studied has a dimension comprehended between 1 and 2 that is why they have a property of statistical self-similarity. The geologic composition of each one of these coastlines is related to the fractal character of themselves. The coastlines of volcanic character (coastlines I and II) are characterized to have higher values in their fractal dimension due to the high level of compression of the present agglomerates, which were formed for eruptions of a variety of volcanic structures boiler and cones type, located to few kilometers to the North of the city; due to these characteristics, the geometry and the levels of complexity of these coastal areas have not been very affected by the meteorological characteristics proper of the rainy season present in the tropic. In this same aspect, the contribution of the tertiary silts of the Marine phase of the coastline II has generated a level of inferior complexity to the coastline I. Contrary to this, another type of less compact materials, characteristic of the coastlines III and IV mainly compound of Holocene sediments, and not differentiated but mainly marine, swamp, alluvium or fillers have contributed to the low level of complexity that these coastal areas possess and as a consequence the values of their inferior fractal dimensions as well. Contrary to coastlines I and II, these last ones not only can turn affected by the action of the winds, tides and rains (which reign during great part of the year) but also for the pressure created by the development; the Pacific coast of Panama is characterized to have a bigger development and population's growth in relation to the Atlantic coast and for this reason it is important the structuring of integrated programs to manage the coasts and their preservation.

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Alexis Mojica*

Carlos A. Ho*

Maria Gonzalez**

Leomar Acosta***

* Laboratorio de Ingenieria Aplicada, Centro Experimental de Ingenieria, Universidad Tecnologica de Panama, Ave. Tocumen, Extension de Tocumen, 0819-07289 Panama, Rep. de Panama, correo electronico: alexis.mojica@utp.ac.pa

** Escuela de Fisica, Universidad de Panama, Ave. Simon Bolivar, correo electronico: meg010101@hotmail.com

*** Facultad de Ciencias Computacionales y Telecomunicaciones, Universidad Latina de Panama, Ave. Hector Alejandro Santacoloma, correo electronico: lacosta@ns.ve.ulat.ac.pa
Table 1
Fractal dimension, correlation coefficient and geological formation
of each coastline studied in this work

                  Fractal dimension
Coastline           and [R.sup.2]                    Geology

I           1.196 [+ or -] 0.019 (0.9996)   Panama Form. Volcanic
                                              phase
II          1.140 [+ or -] 0.018 (0.9996)   Panama Form. Marine phase
III         1.017 [+ or -] 0.001 (0.9999)   Lajas Form. (filled area)
IV          1.031 [+ or -] 0.004 (0.9999)   Lajas Form. (mangroves)
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Author:Mojica, Alexis; Ho, Carlos A.; Gonzalez, Maria; Acosta, Leomar
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Date:Jan 1, 2011
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