**2. Method**

*Current Practice in Fluvial Geomorphology - Dynamics and Diversity*

other variables, such as sediment size.

stress. A channel is characterized by its width, depth, and slope. The regime theory relates these characteristics to the water and sediment discharge transported by the channel empirically. Empirical measurements are taken on channels and attempts are made to fit empirical equations to the observed data. The channel characteristics are related primarily to the discharge, but allowance is also made for variations in

For practical purposes, rivers are preserved to be in equilibrium (in regime) or in quasi-equilibrium with these characteristics, which have not changed over a long period of time. Canals usually maintain constant discharge, and regime relations may, therefore, be established using field data. However, field measurements for

If you use a ruler of k = 1000 m, you will need K rulers to run the entire river meander curvature. If you use a ruler of l = 500 m, you need L rulers and succes-

The number of rulers necessary to measure a meander curvature line M is proportional to the length of ruler m with an exponent D, where D is a constant that defines the dependence between the number of rules and the length of ruler and is

It is intended to calculate fractal dimension slightly undulating line. It is found one code from net on boxcounting method (by [2]) and used for slightly undulating surface that is not given correct answer. Having x and z value of corresponding line.

A characteristic feature of fractals is their fine structure. An object is known to have fine structure if it has irregularities at arbitrarily small scale. 'Fractal dimension' attempts to quantify the fine structure by measuring the rate at which the increased detail becomes apparent as we examine a fractal ever more closely. Fractal dimension indicates the complexity of the fractal and of the amount of space it occupies when viewed at high resolution. All definitions of dimension depend on

A fractal, strictly speaking, has no "physical meaning." It is like asking about some curve we see on some Cartesian 2D coordinate frame-"what is its physical meaning? Or the curve, the function which we may have available to help us understand it and the frame of reference are all constructs of what we can now say… for purposes of brevity…is our intuition and our urge to express ourselves in ways that somehow help us deal with or cope with actions we have to take either now or

Thus, the lines we see on the graph paper have no physical meaning, per se. But that does not mean they have no "use." In fact, "use" is perhaps the best notion of "meaning." Their use if those who may be able to co further with those constructs and incorporate them into models they might work with in regard to various inquiries in science. Unfortunately, there has been little inquiry into just how and in which way and why fractals may be of use. We only tend to "look at the computer screens" and think that we "are seeing" something beyond some interesting calcula-

In the end, complex numbers and their spaces are of far more use than real numbers and Euclidean-style geometries. Hopefully, we will be able to hone our intuitions to make use of them and of fractals in a wide range of pursuits, and, among them would be those understandings of ourselves and matters of human engagement that cannot begin to be approached with real number spaces and

"Reality" itself is an entirely flawed concept, which is rooted in our intuitions and imaginations being locked into a limited "real number/Euclidean/Cartesian" model for thinking and expressing ourselves. When we then speak of "reality," we

rivers are not usually suitable for establishing laws for rivers in regime.

known as a fractal dimension for measuring a river meander curvature.

Is it possible to calculate from these values by any software/code [2]?

sively. What is the physical meaning of fractal dimension?

measuring fractals in some way at increasingly fine scales.

**84**

in the future.

tions in complex number space.

Euclidean assumptions about "reality."

The body is where most of the objects found in nature possess irregular shapes that cannot be quantified with the help of standard Euclidian geometry. In many cases, these objects have a peculiar character of self-similarity where a part of the object looks like the whole [4]. Such objects are known as fractals and the associated degree of complexity of shape, structure, and texture is quantified in terms of fractal dimension (**Figure 1**). Natural fractals do exhibit self-similarity and scale invariance; however, this is present to a limited extent [5]. For example, a part of a cauliflower may look like the whole, but if further division is made, the resulting part may not resemble much the original cauliflower after several steps. The concept of fractal was first introduced by Mandelbrot in the year 1980; he showed that the concept of fractal can be used to quantify the complexity of shape associated with irregular geometry [2].

**Figure 1.** *Examples of naturally occurring fractal patterns in nature [5].*

#### **3. River meander curvature fractals**

Fractal dimension of the curve is found from the slope of the best fitting straight line to the data (fractal dimension = 1 – m), where m is the slope of the straight lime.

Richardson's plot technique using rulers or segments of different sizes [6].

It is seen from **Figure 2** that for a given line with irregular shape, the number of segments or rulers of a given size increases as the size of the ruler is decreased. This results in different measures of the length of the curved line and the complexity of

the shape is related to this difference. For a straight line, the measurements made using different sizes of rulers or line segments result in the same length, whereas for complex curves, the measured distance is larger and larger as smaller and smaller ruler sizes are used. The fractal dimension is related to the complexity of shape associated with the curve and a higher fractal dimension stands for a higher degree of complexity of the pattern analyzed.

If the object can be represented by a two-dimensional binary images in a computer screen or a matrix, which can be input from a digital camera or an image scanner, the fractal dimension estimation can be described as follows:

For an object in two-dimensional Euclidean space, the mass-radius (MR) relation is expressed as the mass included is proportional to the square of the circle of radius r or.

$$\mathbf{M}(\mathbf{r}) = \mathbf{r}^2 \tag{1}$$

As an example, the area of a square measured by using circular discs of increasing sizes is directly proportional to the square of the radius of the disc used for the measurement. The power law exponent "2" is therefore the Euclidean dimension (a square is two-dimensional); however, the mass of a fractal object changes with a fractional exponent such that (1 < D < 2):

$$\mathbf{D}\,\mathbf{M}(\mathbf{r}) = \mathbf{r} \tag{2}$$

From this power law, the fractal dimension "D" of the object can be found as log(r):

$$\log\left(\mathbf{M}(\mathbf{r})\right) = \mathbf{D}'' \tag{3}$$

**87**

*River Plume in Sediment-Laden Rivers*

**Figure 3.**

*DOI: http://dx.doi.org/10.5772/intechopen.90089*

law of a measuring distance (ruler size) r:

*Irregular shape of a line is analyzed using the ruler method.*

where D is the fractal dimension.

emphasizes the local behavior of the curve.

**4. Self-similarity (concept)**

thus are strictly self-similar.

which they are strictly self-similar [3].

Here, D is the slope of the straight line describing the log(M(r)) versus log(r). The two-point correlation function (C(r)) is related to the MR relation that can be used to determine the fractal dimension. For a fractal, C(r) decays as per the power

Since the ruler has a finite length, the details of the curve that are smaller than the ruler get skipped over and therefore the length we measure is normally less than the actual length of the curve. This can be seen in **Figure 3** where three rulers of different lengths are used to determine the length of the curve. The fractal dimension is estimated by measuring the length L of the curve at various scales. Also, it is true that as has been discussed in the use of ruler method the starting point or origin position affects the count or number of boxes required; here too, the estimated value of L may vary depending on the starting position. It is recommended that the same procedure be repeated at different starting positions [6]. This method of determining the fractal dimension of a boundary or a curve is also referred as "structured walk." Longley and Batty [8] discuss number of variants of this basic procedure. Normant and Tricot [3] have described an alternative estimation algorithm, termed the "constant deviation variable step (CDVS) method" that

The term self-similarity came into existence about 40 years ago that too in a relation to fractals and fractal geometry [6]. Fractal structures are said to be self-similar, when part of the object looks like the whole object under fractal dimension and self-similarity appropriate scaling, that is, the structure looks like a reduced copy of the full set on a different scale of magnification. The beauty of these clusters is that each of these smaller clusters again is composed of still smaller ones, and those again of even smaller ones. The second, third, and all the following generations are essentially scaled down versions of the previous ones. However, this scaling cannot be indefinitely extended; after certain stage, the smaller pieces may not perfectly represent the original shape; and this is the characteristic of natural fractals. In general, this is termed as self-similarity or statistical self-similarity. Thus, natural fractals exhibit self-similarity over a limited range and naturally occurring fractals usually exhibit statistical self-similarity [8], whereas mathematical fractals exhibit self-similarity at all length scales and

Fractals are also strictly self-similar if they can be expressed as a union of sets. Geometric fractals may be composed of exact replicas of the whole object with

**Figure 4** exhibits the fractal properties of self-similarity, where MATLAB was

used to create a binarized version of the image (**Figure 5**) [2].

D C(r) = R (4)

**Figure 2.** *River meander curvature fractals [1].*

*River Plume in Sediment-Laden Rivers DOI: http://dx.doi.org/10.5772/intechopen.90089*

*Current Practice in Fluvial Geomorphology - Dynamics and Diversity*

of complexity of the pattern analyzed.

fractional exponent such that (1 < D < 2):

radius r or.

log(r):

the shape is related to this difference. For a straight line, the measurements made using different sizes of rulers or line segments result in the same length, whereas for complex curves, the measured distance is larger and larger as smaller and smaller ruler sizes are used. The fractal dimension is related to the complexity of shape associated with the curve and a higher fractal dimension stands for a higher degree

If the object can be represented by a two-dimensional binary images in a computer screen or a matrix, which can be input from a digital camera or an image

For an object in two-dimensional Euclidean space, the mass-radius (MR) relation is expressed as the mass included is proportional to the square of the circle of

M(r) = r

From this power law, the fractal dimension "D" of the object can be found as

As an example, the area of a square measured by using circular discs of increasing sizes is directly proportional to the square of the radius of the disc used for the measurement. The power law exponent "2" is therefore the Euclidean dimension (a square is two-dimensional); however, the mass of a fractal object changes with a

<sup>2</sup> (1)

D M(r) = r (2)

log (M(r)) = D″ (3)

scanner, the fractal dimension estimation can be described as follows:

**86**

**Figure 2.**

*River meander curvature fractals [1].*

**Figure 3.** *Irregular shape of a line is analyzed using the ruler method.*

Here, D is the slope of the straight line describing the log(M(r)) versus log(r). The two-point correlation function (C(r)) is related to the MR relation that can be used to determine the fractal dimension. For a fractal, C(r) decays as per the power law of a measuring distance (ruler size) r:

$$\mathbf{D}\,\mathbf{C}(\mathbf{r}) = \mathbf{R} \tag{4}$$

where D is the fractal dimension.

Since the ruler has a finite length, the details of the curve that are smaller than the ruler get skipped over and therefore the length we measure is normally less than the actual length of the curve. This can be seen in **Figure 3** where three rulers of different lengths are used to determine the length of the curve. The fractal dimension is estimated by measuring the length L of the curve at various scales. Also, it is true that as has been discussed in the use of ruler method the starting point or origin position affects the count or number of boxes required; here too, the estimated value of L may vary depending on the starting position. It is recommended that the same procedure be repeated at different starting positions [6]. This method of determining the fractal dimension of a boundary or a curve is also referred as "structured walk." Longley and Batty [8] discuss number of variants of this basic procedure. Normant and Tricot [3] have described an alternative estimation algorithm, termed the "constant deviation variable step (CDVS) method" that emphasizes the local behavior of the curve.

### **4. Self-similarity (concept)**

The term self-similarity came into existence about 40 years ago that too in a relation to fractals and fractal geometry [6]. Fractal structures are said to be self-similar, when part of the object looks like the whole object under fractal dimension and self-similarity appropriate scaling, that is, the structure looks like a reduced copy of the full set on a different scale of magnification. The beauty of these clusters is that each of these smaller clusters again is composed of still smaller ones, and those again of even smaller ones. The second, third, and all the following generations are essentially scaled down versions of the previous ones. However, this scaling cannot be indefinitely extended; after certain stage, the smaller pieces may not perfectly represent the original shape; and this is the characteristic of natural fractals. In general, this is termed as self-similarity or statistical self-similarity. Thus, natural fractals exhibit self-similarity over a limited range and naturally occurring fractals usually exhibit statistical self-similarity [8], whereas mathematical fractals exhibit self-similarity at all length scales and thus are strictly self-similar.

Fractals are also strictly self-similar if they can be expressed as a union of sets. Geometric fractals may be composed of exact replicas of the whole object with which they are strictly self-similar [3].

**Figure 4** exhibits the fractal properties of self-similarity, where MATLAB was used to create a binarized version of the image (**Figure 5**) [2].

**Figure 4.**

*Both the ht-index and fractal dimensions, characterizing fractals from different perspectives [4].*

**Figure 5.** *Sierpinski triangles and Koch curve [7].*
