Landscape evolution is driven by abiotic, biotic, and anthropic factors. The interactions among these factors and their influence at different scales create a complex dynamic. Landscapes have been shown to exhibit numerous scaling laws, from Horton’s laws to more sophisticated scaling of heights in topography and river network topology. This scaling and multiscaling analysis has the potential to characterise the landscape in terms of the statistical signature of the measure selected. The study zone is a matrix obtained from a digital elevation model (DEM) (map 10 × 10 m, and height 1 m) that corresponds to homogeneous region with respect to soil characteristics and climatology known as “
Each landscape unit is defined by primary physiographic characteristics. In the landscape, several abiotic and biotic factors, as well as anthropic factors, interact to generate a characteristic dynamic over time. The focus of this study is an alluvial surface of arkose resulting from the erosion of the granite of the Sierra del Guadarrama produced by the factors cited above. These factors, along with their interactions at different scales, produce a strong modelling effect through erosion. The universal equation of hydraulic erosion presented by Wischmeyer and Smith (1978) [
A digital elevation model (DEM) provides the information basis used for many geographic applications, for example, topographic studies, geomorphologic studies, and landscape analysis with geographic information systems (GIS). The ability of a DEM to represent the earth’s surface depends on the surface roughness and the resolution used [
In the general mathematical framework of fractal geometry, many analytical methods have been developed. For example, textural homogeneity has been characterised using the fractal dimension [
Motivated by the fractal geometry of sets [
The acquisition of remotely sensed multiple spectral images is thus a unique source of data for determining the scaleinvariant characteristics of the radiant fields related to many factors such as the chemical composition of soil and bedrock, their moisture content, and their surface temperature [
There are scientific debates over what is fractal. However, a surface does not need to be multifractal to admit a multifractal analysis (MFA). The most important issues are whether MFA is a reliable method for determining fractal parameters and how the results of the MFA are to be interpreted in a given context [
The study area is represented by a 1024 × 1024 data matrix obtained from a DEM with a resolution of 10 × 10 m at each point and a height resolution of 1 m, which correspond with a region known as “
Visualization of DEM (1024 × 1024 data points) at the area studied (a) and the localization of the reservoir in the map at different filling levels (b) from emptiness (576 m) to full capacity (630 m).
The criteria of the selection of the study area were to delimit a homogeneous area with respect to soil characteristics and climatology, and then the topographic factor acquires a main role. Regarding vegetation cover, Mediterranean forest is present with some areas influenced by pasture characteristics as a consequence of historical use for hunting and a minimum soil management. With regard to anthropic factors, these have been much less than in the surrounding areas which have been cultivated, producing a high reduction in the original trees and shrubs of the area. However, in 1973, the construction of the reservoir on Manzanares River modified the water level equilibrium of some local streams at the same time than the main river in this area. A direct consequence was an alteration of the dynamic processes that shape this landscape.
A multifractal analysis is basically the measurement of a statistic distribution and therefore gives useful information on a selfsimilar behaviour [
A monofractal object can be measured by counting the number
There are several methods for implementing multifractal analysis; in this section, the selected moment method is explained [
Applying a nonoverlapping covering by boxes in an “upscaling” partitioning process, we obtain the partition function
The singularity index (
The number of cells of size
Multifractal spectrum (MFS), that is, a graph of
Schertzer and Lovejoy [
The characteristics of both functions have been discussed in detail by Schertzer and Lovejoy [
In the case that
In addition, the relationships between their model and the multifractal formalism based on
According to numerous analyses of remote sensing images, the value of
Then, the original measure was replaced by
First, a preliminary fractal analysis was performed to study how a change in the altitude threshold would affect the fractal dimension (
Boxcounting dimension (
As the threshold increased, the value of
The altitude frequencies for different water levels of the reservoir are shown in Figure
Frequencies distribution, with the water reservoir at different filling levels of (a) altitudes and (b) absolute gradient (frequency in logarithmic scale).
The original measure (altitude) was analysed by first calculating the mass exponent function (
(a) Mass exponent function (
The MF spectra for the five water levels analysed show that the differences among the five spectra with respect to altitude and frequency amplitude are almost null (see Table
Parameters extracted from the multifractal spectrum based on the original measure (altitude) with the water reservoir at different filling levels. Holder exponent at










El Pardo_576  1.995  2.001  1.999  2.005  0.010  1.986  1.989  0.003 
El Pardo_600  1.995  2.001  1.999  2.005  0.010  1.986  1.989  0.003 
El Pardo_610  1.995  2.000  1.999  2.005  0.010  1.986  1.990  0.004 
El Pardo_620  1.994  2.001  1.999  2.006  0.012  1.983  1.989  0.006 
El Pardo_630  1.995  2.001  2.000  2.004  0.009  1.987  1.991  0.004 
The same type of analysis was applied after the data were transformed (see (
Absolute gradient,
These differences are even more pronounced if the frequencies of the absolute gradient are plotted for each case study (see Figure
The nonlinearity observed in
Parameters extracted from the multifractal spectrum based on the average absolute differences of altitudes with the water reservoir at different filling levels. Holder exponent at










El Pardo_576  1,539  2,094  1,930  3,213  1,674  1,033  0,357  −0,676 
El Pardo_600  1,539  2,098  1,932  3,226  1,687  1,033  0,409  −0,624 
El Pardo_610  1,509  2,092  1,950  3,580  2,071  0,098  0,938  −0,840 
El Pardo_620  1,527  2,045  1,960  2,745  1,218  0,873  0,528  −0,345 
El Pardo_630  1,294  2,124  1,980  3,645  2,351  0,744  0,186  −0,558 
(a) Mass exponent function (
The value of the Hölder exponent at the box dimension
The general increase in
If we transform the multifractal spectrum into a moment scaling function (
Moment scaling function (
The goal of this study was to examine the multiscale statistical properties of the altitude and the absolute gradient in an area of homogeneous soil. In this area, the topography and the reservoir constructed on the river played a main role. Such characterisation is related to the spatial organisation of the landscape and could shed light on its evolution.
Several clear results have emerged from this analysis. First, topographic altitude exhibits a weak multiscale statistical structure and a negligible deviation from scale invariance or monoscaling when a multifractal spectrum is obtained. Second, if the original measure (altitude) is replaced by the mean absolute gradient (or mean absolute difference), the multiscale analysis reveals a higher degree of multifractality, allowing a more informative analysis of the influence of the water level of the reservoir.
By addressing the issues of structure and scale, the multifractal formalism, unlike classical geomorphometrical tools, provides scaleinvariant attributes for characterising topography and landscapes. The results of this study show that the use of the multifractal approach with mean absolute gradient data is a useful tool for analysing the topography represented by the digital elevation model.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The data provided by Guadarrama Monitoring Network Initiative (GuMNet) through the Project of CEI Campus Moncloa is greatly appreciated.