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On the size-distribution of solution dolines in carbonate karst: Lognormal or power model?
Journal Pre-proof On the size-distribution of solution dolines in carbonate karst: Lognormal or power model?
Eulogio Pardo-Igúzquiza, Peter A. Dowd, Tamás Telbisz PII:
S0169-555X(19)30463-5
DOI:
https://doi.org/10.1016/j.geomorph.2019.106972
Reference:
GEOMOR 106972
To appear in:
Geomorphology
Received date:
14 October 2019
Revised date:
21 November 2019
Accepted date:
21 November 2019
Please cite this article as: E. Pardo-Igúzquiza, P.A. Dowd and T. Telbisz, On the sizedistribution of solution dolines in carbonate karst: Lognormal or power model?, Geomorphology(2018), https://doi.org/10.1016/j.geomorph.2019.106972
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© 2018 Published by Elsevier.
Journal Pre-proof
On the size-distribution of solution dolines in carbonate karst: lognormal or power model? Eulogio Pardo-Igúzquizaa,*, Peter A. Dowdb and Tamás Telbiszc a
Instituto Geológico y Minero de España (IGME). Ríos Rosas, 23, 28003
Madrid Spain. Faculty of Engineering, Computer and Mathematical Sciences, The University
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b
Eötvös University Department of Physical Geography, 1117 Budapest,
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Pázmány sétány 1/C., Hungary.
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c
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of Adelaide, Australia.
28003, Madrid, Spain.
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*Corresponding author at: Instituto Geológico y Minero de España, Ríos Rosas, 23,
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E-mail address: [email protected] (Eulogio Pardo).
Abstract
Dolines, or closed karst depressions, are the most characteristic feature of karst landscapes. The two main types of doline are those formed by the removal of material by solution and those formed by the geological processes of collapse or suffosion. Solution dolines are most frequently found in carbonate karst in high relief karst massifs. Various algorithms can be used to identify and delineate dolines in highresolution digital elevation models (DEM) where the lower size limit is determined by the resolution of the DEM, i.e., the size of the cell or the DEM pixel in terrain units. The work presented here addresses the size distribution of dolines, which is a significant 1
Journal Pre-proof component of the geomorphometric analysis of dolines. On the basis of experimental results, various authors have advocated the lognormal or the power distribution as the best fit to experimental data. A power distribution is consistent with a fractal structure implying the fractal nature of the sizes of dolines which, in turn, may reflect a relationship between size and the structural controls of fracture networks and the fractal fragmentation of the terrain. This paper presents case studies of four karst landscapes that are representative of karst massifs in Spain. Two of these are located in the
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Pyrenees in the north of Spain and two are located in the Betic mountains in the south of
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Spain. In all four cases, a lognormal distribution provides a bad fit while a power law
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distribution provides a good fit in three cases and in the fourth case provides a good fit
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for dolines that have a diameter greater than 50 m.
Dolines
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Power distribution
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Karst depressions
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Keywords:
Lognormal distribution
1. Introduction
Karst depressions are scale-invariant landforms with sizes that cover several orders of magnitude. The study of these features has significantly improved with the use of modern digital elevation models (Telbisz, 2010), which have overcome the limitations of manual delineation on topographic maps (Angel et al., 2004; de Carvalho et al., 2014; Bauer, 2015; Wall et al., 2017; Zumpano et al., 2019). Karst depressions include dolines (or sinkholes), which are the most typical and representative landform of karst
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Journal Pre-proof landscapes (Sauro, 2003; Veress, 2017). Dolines can be identified and mapped by using high resolution digital elevation models obtained from LiDAR data (Obu and Podobnikar, 2013; Launspach, 2013; Pardo-Igúzquiza et al., 2013, 2014, 2016b; Zhu et al., 2014; Kobal et al., 2015; Wu et al., 2016; Hofierka et al., 2018). The geomorphometric study of dolines has resulted in the definition of many morphometric parameters (Williams, 1966, 1971; Jennings, 1975; Kemmerly, 1982; Day, 1983; Bondesan et al., 1992; Šušteršič, 2006; Ford and Williams, 2007; Telbisz et al., 2009;
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Jeanpert et al., 2016; Öztürk et al., 2018). The work reported here focusses on one of
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these morphometric parameters: the size-distribution (or area-distribution) of dolines,
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which has variously been reported as log-normal (Telbisz et al., 2009) or power-law
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(Galve et al., 2011; Wall and Bohnenstiehl, 2014; Pardo-Igúzquiza et al., 2016a; Yizhaq et al., 2017). The log-normal distribution (Aitchinson and Brown, 1957) has often been
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assumed in geoscience applications (Ahrens, 1954) although its validity has been
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questioned (Reimann and Filzmoser, 2000). Power law and lognormal distributions are related (Mitzenmacher, 2003). Shen (2011) showed that, under certain conditions, the
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lognormal distribution shows fractal properties. Reams (1992) showed the fractal character of sinkhole perimeters. This work investigates the size-distribution of dolines by using exhaustive doline maps from four karst massifs in Spain.
2. Methodology Two aspects are considered in this section: the identification and delineation of dolines and the statistical methods used to study the size-distribution of dolines. Many papers have compared the identification of depressions by eye from aerial photographs or topographic maps with automatic procedures using numerical processes 3
Journal Pre-proof and digital elevation models (DEM). A comparison of automatic methods is outside the scope of this paper. The method used in this paper is described in detail in PardoIgúzquiza et al. (2013) and has been verified in the field using the GPS locations of dolines in test areas. The number of dolines observed in the field was equal to the number of dolines mapped by the automatic procedure and no doline was unidentified except for captured dolines, which are not closed depressions. The methodology uses the difference between the depression-free DEM, and the original DEM and provides
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maps of dolines together with a map of the depth of the dolines with respect to their rim.
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This procedure is based on the algorithm of Jenson and Domingue (1988) for pit
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removal in DEMs. The basic Jenson and Domingue (1988) algorithm for removing
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depressions can be summarised as using computed flow directions and iterative spatial techniques (region-growing procedures) that extend the algorithm beyond a
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neighbourhood operation. The good performance of the procedure is illustrated in
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Pardo-Igúzquiza et al. (2013, 2016b).
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For the size distribution of dolines, a statistical method is required to test the validity of adopting a given theoretical model for a specific size-distribution. In our particular problem there are two competing size-distributions: lognormal and power distributions. These distributions, although related (Clauset et al., 2009; Mitzenmacher, 2003), are quite different. In particular, a power law size-distribution implies that the size of karst depressions is fractal as has been shown, for example, for the size of lakes (Downing et al., 2006; Seekel et al., 2013; Cael and Seekel, 2016). To test whether the size-distribution of dolines follows a lognormal distribution is equivalent to testing whether the logarithm of the size (area) of dolines follows a normal distribution. Many statistical tests have been designed for this test including the Shapiro-Wilk, Kolmogorov-Smirnov and Anderson-Darling, (Razali and Wah, 2011). 4
Journal Pre-proof Testing whether a given data set follows a normal distribution is a routine practice in applied statistics. Graphical methods, together with a standard hypothesis test, can be used to test whether the size-distribution of karst depressions follows a power law (fractal distribution). A fractal distribution requires that the number of objects, 𝑁, with a size greater than 𝑧, follows a power law: 𝑁 = 𝐴𝑧 −𝐷 ,
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where 𝐴 is a constant and 𝐷 is the fractal dimension.
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(1)
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Equation (1) is known as the Korcak law (Mandelbrot, 1975; Turcotte, 1977) or Zipf’s
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law (Laverty, 1987).
Equation (1) can be written formally in probability terms as the complementary
𝑧
𝑧min
−(𝛼−1)
)
,
(2)
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P(𝑍 > 𝑧) = (
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cumulative power law distribution as (Newman, 2005):
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where 𝑧min is the minimum value of the experimental data or the smallest value for which the power-law behaviour holds (Newman, 2005). The relationship between the parameter 𝛼 in Eq. (4), in the complementary cumulative distribution, and the parameter D in Eq. (1), in the probability density function, is 𝐷 = 𝛼 − 1.
(3)
Taking logarithms of both sides of equation (2) gives: ln 𝑁 = ln 𝐴 − 𝐷 ln 𝑧,
(4)
which is the equation of a straight line with a slope equal to the fractal dimension. This slope is the least squares estimate, 𝐷𝐿𝑆 , of the parameter 𝐷. 5
Journal Pre-proof In addition to this graphical method of checking whether a log-log plot of the experimental data follows a straight line, a standard statistical test can be applied. For example, the Kolmogorov-Smirnov test can be applied to the power law distribution where the null hypothesis (𝐻0 ) is that the experimental data follow a power law distribution while the alternative hypothesis (𝐻1 ) is that the experimental data do not follow a power law distribution. A test statistic (𝑇) is defined and its distribution is calculated on the basis that the null hypothesis is true. The experimental value of the
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test statistic (𝑇 ∗ ) is calculated from the data. The 𝑝 − value is calculated as the
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probability of 𝑇 ∗ if the null hypothesis is true. This value is easily calculated from the
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distribution of 𝑇. A small 𝑝 − value (e.g., less than 0.01 with a confidence level of
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99%), is evidence that the null hypothesis is not true. A large 𝑝 − value indicates that
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there is no evidence for rejecting the null hypothesis and the null hypothesis is accepted. Given the sorted experimental data values of the areas of karst depressions 𝑧1 ≤ 𝑧2 ≤
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𝑧3 ≤ ⋯ ≤ 𝑧𝑛 , the Kolmogorov-Smirnov statistic is defined as (Conover, 1999): (5)
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𝑇 ∗ = max𝑧 |𝐹 ∗ (𝑧) − 𝐹𝑛 (𝑧)|
where 𝐹 ∗ (𝑧) is the hypothesized distribution function (power law distribution) and 𝐹𝑛 (𝑧) is the empirical cumulative distribution function. An alternative to the previous methods is to estimate the parameter 𝛼 of the power law distribution by maximum likelihood (Clauset et al., 2009). The log-likelihood function (ln L) of the power-law distribution is: 𝑛
𝑧𝑖 ln 𝐿(𝛼) = 𝑛 ln(𝛼 − 1) − 𝑛 ln(𝑧min ) − 𝛼 ∑ ln ( ) 𝑧min 𝑖=1
and the maximum likelihood estimate of the parameter 𝛼 is given by
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(6)
Journal Pre-proof 𝜕ln𝐿(𝛼) = 0, 𝜕𝛼
(7)
which gives the maximum likelihood (ML) estimate (Newman 2005): 𝑛
−1
𝑧𝑖
𝛼𝑀𝐿 = 1 + 𝑛 [∑ ln ( )] 𝑧min
.
(8)
𝑖=1
The maximum likelihood estimate of the fractal dimension, 𝐷𝑀𝐿 , is given by equation
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(3) as 𝐷𝑀𝐿 = 𝛼𝑀𝐿 − 1.
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The standard error of the maximum likelihood estimates of 𝛼 and D, is (Clauset et al.,
√𝑛
.
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𝛼−1
(9)
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𝜎𝛼 = 𝜎𝐷 =
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2009):
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used in equation (9).
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As 𝛼, the true value of the parameter, is unknown, its maximum likelihood estimate is
3. Case studies
We use four different karst massifs in Spain (Fig. 1) as case study areas that cover a range of conditions under which dolines form. Two of the karst massifs are located in southern Spain in the Betic mountain range and are typical of Mediterranean karst massifs. One of these, the Sierra de las Nieves (SN in Fig. 1) karst massif, is a highrelief Mediterranean karst massif. The other is the Sierra Gorda karst massif (SG in Fig. 1), which is a medium-relief Mediterranean karst. The other two karst massifs are located in the Pyrenees mountain range in the north of Spain (Fig. 1). The Sierra de Cotiella (CO in Fig. 1) and the Sierra de Tendeñera (TEN in Fig. 1) are similar to 7
Journal Pre-proof Alpine karsts. Both Cotiella and Tendeñera are high-relief karsts. The digital elevation models of the four karst massifs are shown in Fig. 2 in which the range of altitudes and their spatial distributions are evident. The Sierra Cotiella and Sierra Tendeñera, in the Pyrenees, have maximum altitudes of around 3000 m. In the South of Spain, the Sierra de las Nieves has maximum altitudes of around 2000 m and the Sierra Gorda maximum altitudes are less than 1700 m. The Sierra de las Nieves karst massif consists of a succession of Triassic marbles and dolostones, Jurassic limestones and a Tertiary
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carbonic breccia (Liñan-Baena, 2005). The sequence is folded by a NE-SW trending
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overturned syncline (Matín-Algarra, 1987). The Sierra Gorda karst massif is a NW-SE
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oriented elliptic massif and is typical of Mediterranean karst systems developed on
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Jurassic limestones and dolostones (Pezzi, 1977; López-Chicano, 1992). The Sierra de Cotiella karst massif is oriented WNW-ESE and is formed by a succession of Mesozoic
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limestones physically deformed by significant glacial and periglacial activity
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(Belmonte-Ribas, 2014). The Sierra de Tendeñera karst massif is oriented E-W and comprises Upper Cretaceous limestones and sandstones and Tertiary limestones and
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dolostones in an almost vertical sequence (Teixel, 1992). The glacial and periglacial deformation is also evident in the Sierra de Cotiella. The DEMs for each of the four karst massifs were provided by the Instituto Geográfico Nacional of Spain (www.ign.es). The data are in the public domain and all were obtained by LiDAR and have a spatial resolution of 5 m. Thus, all four DEMs have the same quality and resolution. The dolines in each of the four karst massifs were identified and delineated using the automatic procedure described in the methodology section. The doline maps, as well as the detail of some dolines, are shown in Figs. 3 to 6 for Sierra de las Nieves, Sierra Gorda, Sierra de Cotiella and Sierra de Tendeñera, respectively. Doline is used here in 8
Journal Pre-proof the broad sense of solution dolines, which in some cases may be uvalas (i.e. coalescences of dolines) or small poljes. The area of each identified depression was calculated, and the histogram and basic statistics of the areas are given in Figs. 7a to 7d for the four karst massifs; each histogram is highly positively skewed. Although sophisticated criteria can be used to determine the histogram bin width (e.g., minimization of the mean integrated square error; Wand, 1979), an ad hoc selection has
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been used here to show detail in the left tail of the histogram.
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The histograms and basic statistics of the logarithms of the areas are given in Figs. 8a to 8d. Even after the logarithm transform, the histograms of ln(area) remain positively
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skewed with high kurtosis. These histograms clearly show that the distributions of the
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original variables are not lognormal. The results of applying the Kolmogorov-Smirnov
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test to the ln(area) for the four karst massifs are given in Table 1 from which it is clear that the null hypothesis of normality of the logarithm of the area of karst depressions
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cannot be accepted and thus the lognormal character of the size-distribution of karst
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depressions must be rejected as well.
The first method for testing the fit of a power law to the size-distributions of dolines is to assess how well Eq. 4 fits the plotted numbers of depressions of given sizes. For each given area, the number of dolines with a surface larger than that area is counted and the results are plotted on log-log scale. The results are given in Fig. 9a to 9d. Some of these graphs show deviations from a straight line at the extremes. At the left extreme, artefacts due to truncation will contribute to the deviation while at the right extreme artefacts due to censoring will contribute to the deviation. Fig. 9 was calculated by starting at 100 m2 and using a step of 100 m2 to calculate Eq. 4. For the Pyrenees karst massifs, Sierra de Cotiella (Fig. 9c) and Sierra de Tendeñera (Fig. 9d), a straight line is a good fit to all the points, but that is not the case for the karst massifs in Southern 9
Journal Pre-proof Spain. For the Sierra de las Nieves (Fig. 9a), a few points in the right tail were disregarded in fitting the line while in Sierra Gorda (Fig. 9b) there is a significant departure from the fitted line in the left tail. In Fig. 9, a straight line has been fitted to the blue points and the red points have been disregarded. The slopes of the straight lines provide the least squares estimates of the fractal dimension as given in Table 2. The fractal dimension estimated by maximum
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likelihood is also reported in Table 2. The fractal dimension results are similar for the
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three high-relief karst systems (Sierra de las Nieves, Sierra de Cotiella and Sierra de Tendeñera) whereas the results for the Sierra Gorda medium-relief Mediterranean karst
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are different and are discussed further in the discussion section. To fit a power-law
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distribution to the sizes of the dolines in Sierra Gorda, the value of 𝑧min was increased
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from 100 m2 to 700 m2, which is shown graphically in Fig. 9b. The first few points in the graph were disregarded in order to capture the linear behaviour of most of the
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points. The Kolmogorov-Smirnov test was applied to test the power law fit and the
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results are shown in Table 2. The null hypothesis of a power law for the sizedistributions of karst depressions cannot be rejected for the Sierra de las Nieves, Sierra de Cotiella and Sierra de Tendeñera karst massifs and it cannot be accepted for Sierra Gorda (taking into account all the doline sizes). This confirms the previous result from the graphical method. Fig. 10 shows a good fit of the empirical cumulative distribution function and the hypothesized cumulative distribution function for the Sierra de las Nieves karst massif while Fig. 11 shows a poor fit for the Sierra Gorda karst massif.
4. Discussion
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Journal Pre-proof There are different types of closed depressions in karst terrains such as poljes, uvalas, dolines, water pools, grikes, potholes and kamenitzas (solution pans). The main genetic classes of dolines (the most common type of karst depression at the metric scale) are solution dolines, subsidence dolines and collapse dolines (Ford and Williams, 2007). Different karst depressions have different shapes and morphometries. This study deals only with solution dolines in a broad sense and may include uvalas and small poljes. They occur in four carbonate high-relief karst massifs in Spain that developed in hard
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rocks (limestones, dolostones, marbles and carbonate breccias) and were formed by a
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long process (several hundreds of thousands of years) that, although it is still active,
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does not change the number of dolines from year to year. This case is thus different to
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collapse sinkholes, the number of which varies from year to year (Taheri et al., 2015;
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Yizhaq et al., 2017).
The focus of the work presented here is not on shapes or geometries and
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morphometries, but on an aspect that involves only the area or surface of the solution
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dolines: their size-distribution. The size-distribution of depression-related bodies is a research problem in many other areas, for example, lakes (Downing et al., 2006; Seekell et al., 2013; Cael and Seekell, 2016) for which the references have shown that the distribution of the sizes of lakes follows a power-law distribution. It is interesting, and perhaps significant, to note that Downing et al (2006) found that the exponent of the power law for the sizes of different sets of lakes ranged from -0.66 to -1.334, with a mean of -0.88 and a median of -0.79. The lakes are in terrain depressions that are impervious to water. In the study presented here the power law exponents for the four doline data sets range from -0.78 to -1.14, with a mean of 0.91 and a median of 0.87. The power-law is a scale-free distribution that represents the spatial distribution of phenomena that do not have a characteristic size but one that varies across several 11
Journal Pre-proof orders of magnitude. On the other hand, the lognormal distribution has a characteristic scale and the dispersion of the values depends on the variance of the distribution. The work presented here demonstrates that, for four different karst massifs in Spain, the lognormal distribution is not a suitable model for the size-distribution of solution dolines. It has also shown that the power law is a good model for the size-distribution of solution dolines in three of the four karst massifs presented in this study. The three karst massifs with power laws for the sizes of their solution dolines are high-relief karst. The
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Sierra Gorda karst massif has the lowest altitudes and neither a lognormal law nor a
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power law is a suitable model. Although the Sierra Gorda karst massif has the highest
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number of mapped solution dolines, the log-log plot fit to both models is the worst,
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particularly for solution dolines with relatively small areas. This could be explained by the fact that many dolines in this karst massif are coalescing and may have been counted
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as a single depression (uvala) whereas there are clearly two or more dolines (Fig. 12). In
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addition, the dolines in Sierra Gorda may have a characteristic size at which they form
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polygonal karst such as the one in the Google Earth image in Fig. 13.
5. Conclusions
The availability of high resolution DEMs has expanded the possibilities for the morphometric analysis of dolines. If a suitable method of identification and delineation is used, the number of dolines in a karst massif can be obtained and morphometric analysis can be applied. An interesting problem is to resolve the question of whether a power law or a lognormal distribution is the better fit to the size distribution of dolines. The lognormal distribution was popular in previous morphometric analyses where the identification and delineation of dolines was made by hand from topographic maps and
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Journal Pre-proof only relatively large dolines were identified. The study presented in this paper shows that a power-law distribution is a better fit for karst massifs in Spain, especially for Alpine karst massifs with many solution dolines that have developed on glaciokarst (Veress, 2017). The power law distribution implies a fractal character of the sizedistribution of dolines. This is logical given that the main structural control is the fracture and faults network, which has been shown to be fractal (Ross, 1986; Walsh and Watterson, 1993). It is also logical given that the enlargement of solution dolines is a
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positive feedback process (Ford and Williams, 2007) that encourages further
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development, which is a mechanism known to produce a power-law distribution
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(Newman, 2005).
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This analysis can be included in the toolkit of the geomorphologist and can be used to
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compare the characteristics and evolution of different karst massifs. More research is required on the size-distribution of dolines to gain a better understanding of the
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implications of these results.
Acknowledgements
This work was supported by research project CGL2015-71510-R of the Ministerio de Economía, Industria y Competitividad of Spain. We thank the reviewers for providing positive criticism that has helped to improve the final version of the paper.
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Journal Pre-proof Martín-Algarra, A., 1987. Evolución geológica alpine del contacto entre las Zonas Internas y las Zonas Externas de la Cordillera Bética. PhD thesis, Universidad de Granada, 1171 pp. Mitzenmacher, M., 2003. A Brief History of Generative Models for Power Law and Lognormal Distributions. Internet Mathematics 1, 226-251.
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density on Taurus Mountains, Turkey. Environmental Earth Sciences 77, 536. Pardo-Igúzquiza, E., Dowd, P.A., 1998. Maximum likelihood inference of spatial covariance parameters of soil properties. Soil Science 163, 272-219. Pardo-Igúzquiza, E., Dowd, P.A., 2003. Assessment of the uncertainty of spatial covariance parameters of soil properties and its use in applications. Soil Science 168, 769-782. Pardo-Igúzquiza, E., Dowd, P.A., Grimes, D.I.F., 2005. An automatic moving window approach for mapping meteorological data. International Journal of Climatology 25, 665-678.
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Journal Pre-proof Pardo-Igúzquiza, E., Durán, J.J., Dowd, P.A., 2013. Automatic detection and delineation of karst terrain depressions and its application in geomorphological mapping and morphometric analysis. Acta Carsologica 42, 17-24. Pardo-Igúzquiza, E., Durán, J.J., Luque-Espinar, J.A., Martos-Rosillo, S., 2014. Análisis del relieve kárstico mediante el modelo digital de elevaciones. Aplicaciones a la Sierra de las Nieves (provincia de Málaga). Boletín Geológico
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Pardo-Igúzquiza, E., Durán, J.J., Robledo-Ardila, P.A., 2016a. Modelado fractal de la
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Journal Pre-proof Ross, B., 1986. Dispersion in fractal fracture networks, Water Resources Research 22, 823–827. Sauro, U., 2003. Dolines and sinkholes: aspects of evolution and problems of classification. Acta Carsologica 32, 41–52. Seekell, D.A., Pace, M.L., Tranvik, L.J. and Verpoorter, C., 2013. A fractal-based approach to lake size-distributions. Geophysical Research Letters 40, 517–521.
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Šušteršič, F., 2006. A power function model for the basic geometry of solution dolines:
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considerations from the classical karst of south‐central Slovenia. Earth Surface
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Processes and Landforms 31, 293-302.
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Taheri, K., Gutiérrez, F., Mohseni, H., Raeisi, E., Taheri, M., 2015. Sinkhole susceptibility mapping using the analytical hierarchy process (AHP) and
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magnitude-frequency relationships: A case study in Hamadan province, Iran. Geomorphology 234, 64–79. Teixell, R., 1992. Estructura alpina en la transversal de la terminación occidental de la Zona Axial Pirenaica. PhD thesis, Universidad de Barcelona, 252 p Telbisz, T., Dragušica, H., Nagy, B., 2009. Doline Morphometric Analysis and Karst Morphology of Biokovo Mt (Croatia) based on Field Observations and Digital Terrain Analysis.- Croatian Geographical Bulletin, 71, 2−22. Telbisz, T., 2010. Morphology and GIS-analysis of closed depressions in Sinjajevian Mts (Montenegro). Karst Development 1, 41–47.
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Journal Pre-proof Turcotte, D.L., 1997. Fractals and Chaos in Geology and Geophysics, Cambridge University Press, Cambridge, Second edition. Veress, M. 2017. Solution DOLINE development on GLACIOKARST in alpine and Dinaric areas. Earth-Science Reviews 173, 31-48. Wall, J., Bohnenstiehl, D., 2014. Power-law relationship of sinkholes and depressions within karst geology. Geological Society America Abstract Programs, 46, 10.
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Natural Hazards 85, 729–749.
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Walsh, J.J. and Watterson, J., 1993. Fractal analysis of fracture patterns using the standard box-counting technique: valid and invalid methodologies. Journal of
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Structural Geology 15, 1509-1512.
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Wand, M.P., 1997. Data-based choice of histogram bin width. The American Statistician 51, 59-64.
Wu, Q., Deng, C., Chen, Z., 2016. Automated delineation of karst sinkholes from LiDAR-derived digital elevation models. Geomorphology 266, 1-10. Yizhaq, H., Ish-Shalom, C., Raz, E., Ashkenazy, Y., 2017. Scale-free distribution of Dead Sea sinkholes: Observations and modeling. Geophysical Research Letters 44, 4944–4952.
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Journal Pre-proof Zhu, J., Taylor, T.P., Currens, J.C., Crawford, M.M., 2014. Improved karst sinkhole mapping in Kentucky using LiDAR techniques: a pilot study in Floyds Fork Watershed. Journal of Cave and Karst Studies 76, 207. Zumpano, V., Pisano, L., Parise, M., 2019. An integrated framework to identify and
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analyze karst sinkholes. Geomorphology 332, 213–225.
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Journal Pre-proof Number of data D statistics p-value Sierra de las Nieves
2250
0.2477
0.0
Sierra Gorda
3100
0.0544
0.0
Sierra de Cotiella
2457
0.2436
0.0
792
0.2834
0.0
Sierra de Tendeñera
𝑧min
𝑛
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𝐷𝐿𝑆
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Table 1. Results of the Kolmogorov-Smirnov test for the null hypothesis that the natural logarithm of the area of karst depressions follows a Gaussian distribution.
𝛼𝑀𝐿
𝐷𝑀𝐿
𝜎𝑀𝐿
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(m2) 0.874
100 2250 1.874 0.825 0.032
Sierra Gorda
1.144
700 1210 2.144 0.885 0.025
0.857
100 2457 1.857 0.927 0.034
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Sierra de Cotiella
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Sierra de las Nieves
Sierra de Tendeñera
0.777
100
792 1.777 0.905 0.065
Table 2. Results of the least-squares and maximum likelihood estimation for the fractal dimension of the doline size-distributions of the four karst massifs. 𝐷𝐿𝑆 is the least squares estimate of the fractal dimension as in Fig. 9. 𝑧min is the minimum value used for fitting a power-law by maximum likelihood using equation (8). 𝑛 is the number of experimental data used in the maximum likelihood estimation. 𝛼𝑀𝐿 is the maximum likelihood estimation of the parameter 𝛼 of the power-law distribution. 𝐷𝑀𝐿 is the maximum likelihood estimate of the fractal dimension calculated by equation (3). 𝜎𝑀𝐿 is the uncertainty (standard error) of the maximum likelihood estimates of the parameters 𝛼 and 𝐷 according to equation (9).
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Journal Pre-proof Number of data D statistics p-value Sierra de las Nieves
2250
0.0206
0.309
Sierra Gorda
3100
0.1486
0.0
Sierra de Cotiella
2457
0.0111
0.915
792
0.0223
0.774
Sierra de Tendeñera
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Table 3. Results of the Kolmogorov-Smirnov test for the null hypothesis that the area of karst depressions follows a power-law distribution.
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Fig. 1. Locations of the four karst massifs: Sierra de las Nieves (SN), Sierra Gorda (SG), Sierra de Cotiella (CO) and Sierra de Tendeñera (TEN).
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Fig. 2. Digital elevation models (DEM) of (a) Sierra de las Nieves (SN); (b) Sierra Gorda (SG); (c) Cotiella karst massif (CO); and (d) Sierra de Tendeñera (TEN).
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Fig. 3. (a) Map with 2250 dolines in Sierra de las Nieves; (b) Detail of the dolines in the yellow rectangle in (a). The scale bar shows the depth (in metres) of the closed karst depressions with respect to the rim of the depression. 27
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Fig. 4. (a) Map with 3100 karst dolines in Sierra Gorda; (b) Detail of the dolines in the yellow rectangle in (a). The scale bar shows the depth (in metres) of the closed karst depressions with respect to the rim of the depression.
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Fig. 5. (a) Map with 2457 karst dolines in Sierra de Cotiella; (b) Detail of the dolines in the yellow rectangle in (a). The scale bar shows the depth (in metres) of the closed karst depressions with respect to the rim of the depression.
Fig. 6. (a) Map with 792 karst dolines in Sierra de Tendeñera; (b) Detail of the dolines in the yellow rectangle in (a). The scale bar shows the depth (in metres) of the closed karst depressions with respect to the rim of the depression.
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Fig. 7. Histogram and basic statistics of the sizes (areas) of dolines in (a) Sierra de las Nieves; (b) Sierra Gorda; (c) Sierra de Cotiella; and (d) Sierra de Tendeñera.
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Fig. 8. Histogram and basic statistics of the natural logarithm of the sizes (areas) of dolines in (a) Sierra de las Nieves; (b) Sierra Gorda; (c) Sierra de Cotiella; and (d) Sierra de Tendeñera.
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Fig. 9. Calculation of the fractal dimension fitting a power law to the size-distribution of the dolines in (a) Sierra de las Nieves; (b) Sierra Gorda; (c) Sierra de Cotiella; and (d) Sierra de Tendeñera.
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Fig. 10. Comparison of the empirical cumulative distribution function (cdf) and the model cdf with the parameter 𝛼 = 1.82 estimated by maximum likelihood. Example for Sierra de las Nieves.
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Fig. 11. Comparison of the empirical cumulative distribution function (cdf) and the model cdf with 𝛼 = 1.35 estimated by maximum likelihood using all the dolines for Sierra Gorda, and with 𝑧min = 25 m2 .
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Fig. 12. Detail of the dolines of the Sierra Gorda karst massif where small dolines have coalesced to give apparently larger depressions (uvalas), which reduces the number of small depressions.
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Fig. 13. Oblique Google Earth image of a polygonal karst in Sierra Gorda where the characteristic doline size is a diameter between 30 and 50 m.
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Journal Pre-proof Declaration of interests
☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
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Journal Pre-proof Highlights
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The paper presents an analysis of the size-distribution of dolines in four high relief karsts. In the four cases studied, the lognormal distribution does not fit the experimental data. In three of the four cases, the power law provides a good fit to the data. In the remaining case the power law is a good model for dolines larger than 50 m. The power law implies the fractal character of the sizes of dolines.
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42
Eulogio Pardo-Igúzquiza, Peter A. Dowd, Tamás Telbisz PII:
S0169-555X(19)30463-5
DOI:
https://doi.org/10.1016/j.geomorph.2019.106972
Reference:
GEOMOR 106972
To appear in:
Geomorphology
Received date:
14 October 2019
Revised date:
21 November 2019
Accepted date:
21 November 2019
Please cite this article as: E. Pardo-Igúzquiza, P.A. Dowd and T. Telbisz, On the sizedistribution of solution dolines in carbonate karst: Lognormal or power model?, Geomorphology(2018), https://doi.org/10.1016/j.geomorph.2019.106972
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2018 Published by Elsevier.
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On the size-distribution of solution dolines in carbonate karst: lognormal or power model? Eulogio Pardo-Igúzquizaa,*, Peter A. Dowdb and Tamás Telbiszc a
Instituto Geológico y Minero de España (IGME). Ríos Rosas, 23, 28003
Madrid Spain. Faculty of Engineering, Computer and Mathematical Sciences, The University
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Eötvös University Department of Physical Geography, 1117 Budapest,
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Pázmány sétány 1/C., Hungary.
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c
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of Adelaide, Australia.
28003, Madrid, Spain.
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*Corresponding author at: Instituto Geológico y Minero de España, Ríos Rosas, 23,
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E-mail address: [email protected] (Eulogio Pardo).
Abstract
Dolines, or closed karst depressions, are the most characteristic feature of karst landscapes. The two main types of doline are those formed by the removal of material by solution and those formed by the geological processes of collapse or suffosion. Solution dolines are most frequently found in carbonate karst in high relief karst massifs. Various algorithms can be used to identify and delineate dolines in highresolution digital elevation models (DEM) where the lower size limit is determined by the resolution of the DEM, i.e., the size of the cell or the DEM pixel in terrain units. The work presented here addresses the size distribution of dolines, which is a significant 1
Journal Pre-proof component of the geomorphometric analysis of dolines. On the basis of experimental results, various authors have advocated the lognormal or the power distribution as the best fit to experimental data. A power distribution is consistent with a fractal structure implying the fractal nature of the sizes of dolines which, in turn, may reflect a relationship between size and the structural controls of fracture networks and the fractal fragmentation of the terrain. This paper presents case studies of four karst landscapes that are representative of karst massifs in Spain. Two of these are located in the
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Pyrenees in the north of Spain and two are located in the Betic mountains in the south of
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Spain. In all four cases, a lognormal distribution provides a bad fit while a power law
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distribution provides a good fit in three cases and in the fourth case provides a good fit
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for dolines that have a diameter greater than 50 m.
Dolines
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Power distribution
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Karst depressions
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Keywords:
Lognormal distribution
1. Introduction
Karst depressions are scale-invariant landforms with sizes that cover several orders of magnitude. The study of these features has significantly improved with the use of modern digital elevation models (Telbisz, 2010), which have overcome the limitations of manual delineation on topographic maps (Angel et al., 2004; de Carvalho et al., 2014; Bauer, 2015; Wall et al., 2017; Zumpano et al., 2019). Karst depressions include dolines (or sinkholes), which are the most typical and representative landform of karst
2
Journal Pre-proof landscapes (Sauro, 2003; Veress, 2017). Dolines can be identified and mapped by using high resolution digital elevation models obtained from LiDAR data (Obu and Podobnikar, 2013; Launspach, 2013; Pardo-Igúzquiza et al., 2013, 2014, 2016b; Zhu et al., 2014; Kobal et al., 2015; Wu et al., 2016; Hofierka et al., 2018). The geomorphometric study of dolines has resulted in the definition of many morphometric parameters (Williams, 1966, 1971; Jennings, 1975; Kemmerly, 1982; Day, 1983; Bondesan et al., 1992; Šušteršič, 2006; Ford and Williams, 2007; Telbisz et al., 2009;
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Jeanpert et al., 2016; Öztürk et al., 2018). The work reported here focusses on one of
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these morphometric parameters: the size-distribution (or area-distribution) of dolines,
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which has variously been reported as log-normal (Telbisz et al., 2009) or power-law
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(Galve et al., 2011; Wall and Bohnenstiehl, 2014; Pardo-Igúzquiza et al., 2016a; Yizhaq et al., 2017). The log-normal distribution (Aitchinson and Brown, 1957) has often been
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assumed in geoscience applications (Ahrens, 1954) although its validity has been
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questioned (Reimann and Filzmoser, 2000). Power law and lognormal distributions are related (Mitzenmacher, 2003). Shen (2011) showed that, under certain conditions, the
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lognormal distribution shows fractal properties. Reams (1992) showed the fractal character of sinkhole perimeters. This work investigates the size-distribution of dolines by using exhaustive doline maps from four karst massifs in Spain.
2. Methodology Two aspects are considered in this section: the identification and delineation of dolines and the statistical methods used to study the size-distribution of dolines. Many papers have compared the identification of depressions by eye from aerial photographs or topographic maps with automatic procedures using numerical processes 3
Journal Pre-proof and digital elevation models (DEM). A comparison of automatic methods is outside the scope of this paper. The method used in this paper is described in detail in PardoIgúzquiza et al. (2013) and has been verified in the field using the GPS locations of dolines in test areas. The number of dolines observed in the field was equal to the number of dolines mapped by the automatic procedure and no doline was unidentified except for captured dolines, which are not closed depressions. The methodology uses the difference between the depression-free DEM, and the original DEM and provides
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maps of dolines together with a map of the depth of the dolines with respect to their rim.
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This procedure is based on the algorithm of Jenson and Domingue (1988) for pit
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removal in DEMs. The basic Jenson and Domingue (1988) algorithm for removing
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depressions can be summarised as using computed flow directions and iterative spatial techniques (region-growing procedures) that extend the algorithm beyond a
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neighbourhood operation. The good performance of the procedure is illustrated in
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Pardo-Igúzquiza et al. (2013, 2016b).
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For the size distribution of dolines, a statistical method is required to test the validity of adopting a given theoretical model for a specific size-distribution. In our particular problem there are two competing size-distributions: lognormal and power distributions. These distributions, although related (Clauset et al., 2009; Mitzenmacher, 2003), are quite different. In particular, a power law size-distribution implies that the size of karst depressions is fractal as has been shown, for example, for the size of lakes (Downing et al., 2006; Seekel et al., 2013; Cael and Seekel, 2016). To test whether the size-distribution of dolines follows a lognormal distribution is equivalent to testing whether the logarithm of the size (area) of dolines follows a normal distribution. Many statistical tests have been designed for this test including the Shapiro-Wilk, Kolmogorov-Smirnov and Anderson-Darling, (Razali and Wah, 2011). 4
Journal Pre-proof Testing whether a given data set follows a normal distribution is a routine practice in applied statistics. Graphical methods, together with a standard hypothesis test, can be used to test whether the size-distribution of karst depressions follows a power law (fractal distribution). A fractal distribution requires that the number of objects, 𝑁, with a size greater than 𝑧, follows a power law: 𝑁 = 𝐴𝑧 −𝐷 ,
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where 𝐴 is a constant and 𝐷 is the fractal dimension.
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(1)
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Equation (1) is known as the Korcak law (Mandelbrot, 1975; Turcotte, 1977) or Zipf’s
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law (Laverty, 1987).
Equation (1) can be written formally in probability terms as the complementary
𝑧
𝑧min
−(𝛼−1)
)
,
(2)
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P(𝑍 > 𝑧) = (
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cumulative power law distribution as (Newman, 2005):
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where 𝑧min is the minimum value of the experimental data or the smallest value for which the power-law behaviour holds (Newman, 2005). The relationship between the parameter 𝛼 in Eq. (4), in the complementary cumulative distribution, and the parameter D in Eq. (1), in the probability density function, is 𝐷 = 𝛼 − 1.
(3)
Taking logarithms of both sides of equation (2) gives: ln 𝑁 = ln 𝐴 − 𝐷 ln 𝑧,
(4)
which is the equation of a straight line with a slope equal to the fractal dimension. This slope is the least squares estimate, 𝐷𝐿𝑆 , of the parameter 𝐷. 5
Journal Pre-proof In addition to this graphical method of checking whether a log-log plot of the experimental data follows a straight line, a standard statistical test can be applied. For example, the Kolmogorov-Smirnov test can be applied to the power law distribution where the null hypothesis (𝐻0 ) is that the experimental data follow a power law distribution while the alternative hypothesis (𝐻1 ) is that the experimental data do not follow a power law distribution. A test statistic (𝑇) is defined and its distribution is calculated on the basis that the null hypothesis is true. The experimental value of the
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test statistic (𝑇 ∗ ) is calculated from the data. The 𝑝 − value is calculated as the
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probability of 𝑇 ∗ if the null hypothesis is true. This value is easily calculated from the
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distribution of 𝑇. A small 𝑝 − value (e.g., less than 0.01 with a confidence level of
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99%), is evidence that the null hypothesis is not true. A large 𝑝 − value indicates that
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there is no evidence for rejecting the null hypothesis and the null hypothesis is accepted. Given the sorted experimental data values of the areas of karst depressions 𝑧1 ≤ 𝑧2 ≤
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𝑧3 ≤ ⋯ ≤ 𝑧𝑛 , the Kolmogorov-Smirnov statistic is defined as (Conover, 1999): (5)
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𝑇 ∗ = max𝑧 |𝐹 ∗ (𝑧) − 𝐹𝑛 (𝑧)|
where 𝐹 ∗ (𝑧) is the hypothesized distribution function (power law distribution) and 𝐹𝑛 (𝑧) is the empirical cumulative distribution function. An alternative to the previous methods is to estimate the parameter 𝛼 of the power law distribution by maximum likelihood (Clauset et al., 2009). The log-likelihood function (ln L) of the power-law distribution is: 𝑛
𝑧𝑖 ln 𝐿(𝛼) = 𝑛 ln(𝛼 − 1) − 𝑛 ln(𝑧min ) − 𝛼 ∑ ln ( ) 𝑧min 𝑖=1
and the maximum likelihood estimate of the parameter 𝛼 is given by
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(6)
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(7)
which gives the maximum likelihood (ML) estimate (Newman 2005): 𝑛
−1
𝑧𝑖
𝛼𝑀𝐿 = 1 + 𝑛 [∑ ln ( )] 𝑧min
.
(8)
𝑖=1
The maximum likelihood estimate of the fractal dimension, 𝐷𝑀𝐿 , is given by equation
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(3) as 𝐷𝑀𝐿 = 𝛼𝑀𝐿 − 1.
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The standard error of the maximum likelihood estimates of 𝛼 and D, is (Clauset et al.,
√𝑛
.
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𝛼−1
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𝜎𝛼 = 𝜎𝐷 =
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2009):
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used in equation (9).
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As 𝛼, the true value of the parameter, is unknown, its maximum likelihood estimate is
3. Case studies
We use four different karst massifs in Spain (Fig. 1) as case study areas that cover a range of conditions under which dolines form. Two of the karst massifs are located in southern Spain in the Betic mountain range and are typical of Mediterranean karst massifs. One of these, the Sierra de las Nieves (SN in Fig. 1) karst massif, is a highrelief Mediterranean karst massif. The other is the Sierra Gorda karst massif (SG in Fig. 1), which is a medium-relief Mediterranean karst. The other two karst massifs are located in the Pyrenees mountain range in the north of Spain (Fig. 1). The Sierra de Cotiella (CO in Fig. 1) and the Sierra de Tendeñera (TEN in Fig. 1) are similar to 7
Journal Pre-proof Alpine karsts. Both Cotiella and Tendeñera are high-relief karsts. The digital elevation models of the four karst massifs are shown in Fig. 2 in which the range of altitudes and their spatial distributions are evident. The Sierra Cotiella and Sierra Tendeñera, in the Pyrenees, have maximum altitudes of around 3000 m. In the South of Spain, the Sierra de las Nieves has maximum altitudes of around 2000 m and the Sierra Gorda maximum altitudes are less than 1700 m. The Sierra de las Nieves karst massif consists of a succession of Triassic marbles and dolostones, Jurassic limestones and a Tertiary
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carbonic breccia (Liñan-Baena, 2005). The sequence is folded by a NE-SW trending
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overturned syncline (Matín-Algarra, 1987). The Sierra Gorda karst massif is a NW-SE
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oriented elliptic massif and is typical of Mediterranean karst systems developed on
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Jurassic limestones and dolostones (Pezzi, 1977; López-Chicano, 1992). The Sierra de Cotiella karst massif is oriented WNW-ESE and is formed by a succession of Mesozoic
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limestones physically deformed by significant glacial and periglacial activity
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(Belmonte-Ribas, 2014). The Sierra de Tendeñera karst massif is oriented E-W and comprises Upper Cretaceous limestones and sandstones and Tertiary limestones and
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dolostones in an almost vertical sequence (Teixel, 1992). The glacial and periglacial deformation is also evident in the Sierra de Cotiella. The DEMs for each of the four karst massifs were provided by the Instituto Geográfico Nacional of Spain (www.ign.es). The data are in the public domain and all were obtained by LiDAR and have a spatial resolution of 5 m. Thus, all four DEMs have the same quality and resolution. The dolines in each of the four karst massifs were identified and delineated using the automatic procedure described in the methodology section. The doline maps, as well as the detail of some dolines, are shown in Figs. 3 to 6 for Sierra de las Nieves, Sierra Gorda, Sierra de Cotiella and Sierra de Tendeñera, respectively. Doline is used here in 8
Journal Pre-proof the broad sense of solution dolines, which in some cases may be uvalas (i.e. coalescences of dolines) or small poljes. The area of each identified depression was calculated, and the histogram and basic statistics of the areas are given in Figs. 7a to 7d for the four karst massifs; each histogram is highly positively skewed. Although sophisticated criteria can be used to determine the histogram bin width (e.g., minimization of the mean integrated square error; Wand, 1979), an ad hoc selection has
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The histograms and basic statistics of the logarithms of the areas are given in Figs. 8a to 8d. Even after the logarithm transform, the histograms of ln(area) remain positively
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skewed with high kurtosis. These histograms clearly show that the distributions of the
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original variables are not lognormal. The results of applying the Kolmogorov-Smirnov
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test to the ln(area) for the four karst massifs are given in Table 1 from which it is clear that the null hypothesis of normality of the logarithm of the area of karst depressions
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cannot be accepted and thus the lognormal character of the size-distribution of karst
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depressions must be rejected as well.
The first method for testing the fit of a power law to the size-distributions of dolines is to assess how well Eq. 4 fits the plotted numbers of depressions of given sizes. For each given area, the number of dolines with a surface larger than that area is counted and the results are plotted on log-log scale. The results are given in Fig. 9a to 9d. Some of these graphs show deviations from a straight line at the extremes. At the left extreme, artefacts due to truncation will contribute to the deviation while at the right extreme artefacts due to censoring will contribute to the deviation. Fig. 9 was calculated by starting at 100 m2 and using a step of 100 m2 to calculate Eq. 4. For the Pyrenees karst massifs, Sierra de Cotiella (Fig. 9c) and Sierra de Tendeñera (Fig. 9d), a straight line is a good fit to all the points, but that is not the case for the karst massifs in Southern 9
Journal Pre-proof Spain. For the Sierra de las Nieves (Fig. 9a), a few points in the right tail were disregarded in fitting the line while in Sierra Gorda (Fig. 9b) there is a significant departure from the fitted line in the left tail. In Fig. 9, a straight line has been fitted to the blue points and the red points have been disregarded. The slopes of the straight lines provide the least squares estimates of the fractal dimension as given in Table 2. The fractal dimension estimated by maximum
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likelihood is also reported in Table 2. The fractal dimension results are similar for the
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three high-relief karst systems (Sierra de las Nieves, Sierra de Cotiella and Sierra de Tendeñera) whereas the results for the Sierra Gorda medium-relief Mediterranean karst
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are different and are discussed further in the discussion section. To fit a power-law
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distribution to the sizes of the dolines in Sierra Gorda, the value of 𝑧min was increased
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from 100 m2 to 700 m2, which is shown graphically in Fig. 9b. The first few points in the graph were disregarded in order to capture the linear behaviour of most of the
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points. The Kolmogorov-Smirnov test was applied to test the power law fit and the
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results are shown in Table 2. The null hypothesis of a power law for the sizedistributions of karst depressions cannot be rejected for the Sierra de las Nieves, Sierra de Cotiella and Sierra de Tendeñera karst massifs and it cannot be accepted for Sierra Gorda (taking into account all the doline sizes). This confirms the previous result from the graphical method. Fig. 10 shows a good fit of the empirical cumulative distribution function and the hypothesized cumulative distribution function for the Sierra de las Nieves karst massif while Fig. 11 shows a poor fit for the Sierra Gorda karst massif.
4. Discussion
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Journal Pre-proof There are different types of closed depressions in karst terrains such as poljes, uvalas, dolines, water pools, grikes, potholes and kamenitzas (solution pans). The main genetic classes of dolines (the most common type of karst depression at the metric scale) are solution dolines, subsidence dolines and collapse dolines (Ford and Williams, 2007). Different karst depressions have different shapes and morphometries. This study deals only with solution dolines in a broad sense and may include uvalas and small poljes. They occur in four carbonate high-relief karst massifs in Spain that developed in hard
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rocks (limestones, dolostones, marbles and carbonate breccias) and were formed by a
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long process (several hundreds of thousands of years) that, although it is still active,
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does not change the number of dolines from year to year. This case is thus different to
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collapse sinkholes, the number of which varies from year to year (Taheri et al., 2015;
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Yizhaq et al., 2017).
The focus of the work presented here is not on shapes or geometries and
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morphometries, but on an aspect that involves only the area or surface of the solution
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dolines: their size-distribution. The size-distribution of depression-related bodies is a research problem in many other areas, for example, lakes (Downing et al., 2006; Seekell et al., 2013; Cael and Seekell, 2016) for which the references have shown that the distribution of the sizes of lakes follows a power-law distribution. It is interesting, and perhaps significant, to note that Downing et al (2006) found that the exponent of the power law for the sizes of different sets of lakes ranged from -0.66 to -1.334, with a mean of -0.88 and a median of -0.79. The lakes are in terrain depressions that are impervious to water. In the study presented here the power law exponents for the four doline data sets range from -0.78 to -1.14, with a mean of 0.91 and a median of 0.87. The power-law is a scale-free distribution that represents the spatial distribution of phenomena that do not have a characteristic size but one that varies across several 11
Journal Pre-proof orders of magnitude. On the other hand, the lognormal distribution has a characteristic scale and the dispersion of the values depends on the variance of the distribution. The work presented here demonstrates that, for four different karst massifs in Spain, the lognormal distribution is not a suitable model for the size-distribution of solution dolines. It has also shown that the power law is a good model for the size-distribution of solution dolines in three of the four karst massifs presented in this study. The three karst massifs with power laws for the sizes of their solution dolines are high-relief karst. The
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Sierra Gorda karst massif has the lowest altitudes and neither a lognormal law nor a
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power law is a suitable model. Although the Sierra Gorda karst massif has the highest
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number of mapped solution dolines, the log-log plot fit to both models is the worst,
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particularly for solution dolines with relatively small areas. This could be explained by the fact that many dolines in this karst massif are coalescing and may have been counted
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as a single depression (uvala) whereas there are clearly two or more dolines (Fig. 12). In
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addition, the dolines in Sierra Gorda may have a characteristic size at which they form
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polygonal karst such as the one in the Google Earth image in Fig. 13.
5. Conclusions
The availability of high resolution DEMs has expanded the possibilities for the morphometric analysis of dolines. If a suitable method of identification and delineation is used, the number of dolines in a karst massif can be obtained and morphometric analysis can be applied. An interesting problem is to resolve the question of whether a power law or a lognormal distribution is the better fit to the size distribution of dolines. The lognormal distribution was popular in previous morphometric analyses where the identification and delineation of dolines was made by hand from topographic maps and
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Journal Pre-proof only relatively large dolines were identified. The study presented in this paper shows that a power-law distribution is a better fit for karst massifs in Spain, especially for Alpine karst massifs with many solution dolines that have developed on glaciokarst (Veress, 2017). The power law distribution implies a fractal character of the sizedistribution of dolines. This is logical given that the main structural control is the fracture and faults network, which has been shown to be fractal (Ross, 1986; Walsh and Watterson, 1993). It is also logical given that the enlargement of solution dolines is a
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positive feedback process (Ford and Williams, 2007) that encourages further
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development, which is a mechanism known to produce a power-law distribution
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(Newman, 2005).
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This analysis can be included in the toolkit of the geomorphologist and can be used to
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compare the characteristics and evolution of different karst massifs. More research is required on the size-distribution of dolines to gain a better understanding of the
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implications of these results.
Acknowledgements
This work was supported by research project CGL2015-71510-R of the Ministerio de Economía, Industria y Competitividad of Spain. We thank the reviewers for providing positive criticism that has helped to improve the final version of the paper.
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Journal Pre-proof Number of data D statistics p-value Sierra de las Nieves
2250
0.2477
0.0
Sierra Gorda
3100
0.0544
0.0
Sierra de Cotiella
2457
0.2436
0.0
792
0.2834
0.0
Sierra de Tendeñera
𝑧min
𝑛
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Table 1. Results of the Kolmogorov-Smirnov test for the null hypothesis that the natural logarithm of the area of karst depressions follows a Gaussian distribution.
𝛼𝑀𝐿
𝐷𝑀𝐿
𝜎𝑀𝐿
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(m2) 0.874
100 2250 1.874 0.825 0.032
Sierra Gorda
1.144
700 1210 2.144 0.885 0.025
0.857
100 2457 1.857 0.927 0.034
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Sierra de Cotiella
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Sierra de las Nieves
Sierra de Tendeñera
0.777
100
792 1.777 0.905 0.065
Table 2. Results of the least-squares and maximum likelihood estimation for the fractal dimension of the doline size-distributions of the four karst massifs. 𝐷𝐿𝑆 is the least squares estimate of the fractal dimension as in Fig. 9. 𝑧min is the minimum value used for fitting a power-law by maximum likelihood using equation (8). 𝑛 is the number of experimental data used in the maximum likelihood estimation. 𝛼𝑀𝐿 is the maximum likelihood estimation of the parameter 𝛼 of the power-law distribution. 𝐷𝑀𝐿 is the maximum likelihood estimate of the fractal dimension calculated by equation (3). 𝜎𝑀𝐿 is the uncertainty (standard error) of the maximum likelihood estimates of the parameters 𝛼 and 𝐷 according to equation (9).
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Journal Pre-proof Number of data D statistics p-value Sierra de las Nieves
2250
0.0206
0.309
Sierra Gorda
3100
0.1486
0.0
Sierra de Cotiella
2457
0.0111
0.915
792
0.0223
0.774
Sierra de Tendeñera
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Table 3. Results of the Kolmogorov-Smirnov test for the null hypothesis that the area of karst depressions follows a power-law distribution.
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Fig. 1. Locations of the four karst massifs: Sierra de las Nieves (SN), Sierra Gorda (SG), Sierra de Cotiella (CO) and Sierra de Tendeñera (TEN).
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Fig. 2. Digital elevation models (DEM) of (a) Sierra de las Nieves (SN); (b) Sierra Gorda (SG); (c) Cotiella karst massif (CO); and (d) Sierra de Tendeñera (TEN).
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Fig. 3. (a) Map with 2250 dolines in Sierra de las Nieves; (b) Detail of the dolines in the yellow rectangle in (a). The scale bar shows the depth (in metres) of the closed karst depressions with respect to the rim of the depression. 27
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Fig. 4. (a) Map with 3100 karst dolines in Sierra Gorda; (b) Detail of the dolines in the yellow rectangle in (a). The scale bar shows the depth (in metres) of the closed karst depressions with respect to the rim of the depression.
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Fig. 5. (a) Map with 2457 karst dolines in Sierra de Cotiella; (b) Detail of the dolines in the yellow rectangle in (a). The scale bar shows the depth (in metres) of the closed karst depressions with respect to the rim of the depression.
Fig. 6. (a) Map with 792 karst dolines in Sierra de Tendeñera; (b) Detail of the dolines in the yellow rectangle in (a). The scale bar shows the depth (in metres) of the closed karst depressions with respect to the rim of the depression.
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Fig. 7. Histogram and basic statistics of the sizes (areas) of dolines in (a) Sierra de las Nieves; (b) Sierra Gorda; (c) Sierra de Cotiella; and (d) Sierra de Tendeñera.
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Fig. 8. Histogram and basic statistics of the natural logarithm of the sizes (areas) of dolines in (a) Sierra de las Nieves; (b) Sierra Gorda; (c) Sierra de Cotiella; and (d) Sierra de Tendeñera.
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Fig. 9. Calculation of the fractal dimension fitting a power law to the size-distribution of the dolines in (a) Sierra de las Nieves; (b) Sierra Gorda; (c) Sierra de Cotiella; and (d) Sierra de Tendeñera.
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Fig. 10. Comparison of the empirical cumulative distribution function (cdf) and the model cdf with the parameter 𝛼 = 1.82 estimated by maximum likelihood. Example for Sierra de las Nieves.
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Fig. 11. Comparison of the empirical cumulative distribution function (cdf) and the model cdf with 𝛼 = 1.35 estimated by maximum likelihood using all the dolines for Sierra Gorda, and with 𝑧min = 25 m2 .
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Fig. 12. Detail of the dolines of the Sierra Gorda karst massif where small dolines have coalesced to give apparently larger depressions (uvalas), which reduces the number of small depressions.
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Fig. 13. Oblique Google Earth image of a polygonal karst in Sierra Gorda where the characteristic doline size is a diameter between 30 and 50 m.
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Journal Pre-proof Declaration of interests
☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
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The paper presents an analysis of the size-distribution of dolines in four high relief karsts. In the four cases studied, the lognormal distribution does not fit the experimental data. In three of the four cases, the power law provides a good fit to the data. In the remaining case the power law is a good model for dolines larger than 50 m. The power law implies the fractal character of the sizes of dolines.
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