Modeling Solar Radiation at the Earth’s Surface

# Measuring the Fractal Dimension of Signals

4.1 A Survey of Existing Methods

Many natural processes described by time series (e. g., noises, economical and de­mographic data, electric signals... etc.) are also fractals in the sense that their graph is a fractal set (Maragos and Sun 1993). Thus, modeling fractal signals is of great interest in signal processing.

Considering the importance of this index and the impact of its use in practice, the precision of its estimate is necessary. Methods of Box-counting and of Minkowski- Bouligand prove then ineffective due to the fact that they suffer from inaccuracy as we already mentioned. Inspired by the Minkowski-Bouligand method, a class of approaches to compute the fractal dimension of signal curves or one-dimensional profiles called “covering methods” is then proposed by several researchers.

These methods consist in creating multiscale covers around the signal’s graph. Indeed, each covering is formed by the union of specified structuring elements. In the method of Box-counting, the structuring element used is the square or limp, that of Minkowski-Bouligand uses the disk.

Dubuc et al. (1989 and Tricot et al. (1988) proposed a new method called “Varia­tion method”. This one criticizes the standard methods of fractal dimension estima­tion namely: Box-counting and Minkowski-Bouligand. Indeed, “Variation method” applied to various fractal curves showed a high degree of accuracy and robustness.

Maragos and Sun (1993) generalized the method of Minkowski-Bouligand by proposing the “Morphological covering method” which uses multiscale morpholog­ical operations with varying structuring elements. Thus, this method unifies and im­proves other covering methods. Experimentally, “Morphological covering method” demonstrated a good performance, since it has experimentally been found to yield average estimation errors of about 2%-4% or less for discrete fractal signals whose fractal dimension is theoretically known (Maragos and Sun 1993). For determinis­tic fractal signals (these signals will be detailed further in this chapter) Maragos and Sun developed an optimization method which showed an excellent performance, since the estimation error was found between 0 % and 0.07 %.

## Modeling Solar Radiation at the Earth’s Surface

### Quality Assessment Based Upon Comparison with Models

Many models based on the physics of radiation transfer through the clear atmo­sphere have been developed (Lacis and Hansen 1974; Atwater and Ball 1978; Hoyt 1978; Bird and Hulstrom 1981a, …

### Solar Horizontal Diffuse and Beam Irradiation on Clear Days

There exist a number of models to determine the solar horizontal diffuse irradia­tion on a clear day (Kondratyev 1969) but they are complex and have very stringent conditions. Similarly, there …

### Foreword

Reading the twenty chapters of this book caused me mixed reactions, though all were positive. My responses were shaped by several factors. Although I have main­tained a “watching brief’ on …

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