# Damage Diagnosis for Offshore Wind Turbine Foundations Based on the Fractal Dimension
## 3 Damage Diagnose Strategy
### Fractal Dimension  
Fractal geometry was proposed by [PERSON] [29] and it is a relatively new mathematics discipline which has found a lot of applications in bio-science [30; 31; 32], engineering [33] and many other fields [34].  
Euclidean geometry describes common geometric forms like lines, planes, spheres or rectangular volumes. Each of the geometric objects considered so far has an integer dimension (\(D\)), either \(1,2,\) or \(3\). However, many natural shapes do not harmonize with the integer-based idea of dimension.  
In order to give meaning to noninteger dimensions, a more mathematical description of dimension proposed by [PERSON] [35] is based on \"how the _size_ of an object behaves as the linear dimension increase\". More precisely, consider, for instance, three objects with dimensions \(D=1\) (a line segment); \(D=2\) (a square); and \(D=3\) (a cube). If the line segment, the square and the cube are linearly scaled by a factor of 2, then the results are 2 copies, 4 copies, and 8 copies of the initial objects, respectively. In other words, the length (characteristic size) of the line segment is doubled (Figure 1(a)), the area (characteristic size) of the square increases by a factor of 4 (Figure 1(b)), and the volume (characteristic size) of the cube increases by a factor of 8 (Figure 1(c)).  
The relation between the scaling factor \(S\), the dimension \(D\) and the number of generated copies \(N\) (increasing size) can be generalized and expressed as:  
\[N=S^{D}, \tag{4}\]  
which is equivalent to  
\[D=\frac{\log(N)}{\log(S)}. \tag{5}\]  
Since \(D\) is defined in terms of \(N\) and \(S\) in Equation (5), it is possible to find the dimension, for instance, of the famous Koch curve [36]. In the case of the Koch curve, at each step, we divide the line segment into \(S=3\) segments of equal length and we draw an equilateral triangle that has the middle as its base and points outward. Therefore, we have created \(N=4\) copies (the two external sides of the original line segment and the two sides of the triangle). Consequently, the fractal dimension \(D_{\text{Koch}}\) of the Koch curve is:  
\[D_{\text{Koch}}=\frac{\log(4)}{\log(3)}\approx 1.2619.\]  
As it is very well known, fractals are self-similar subsets of the Euclidean space where the fractal dimension defined in Equation (5) surpasses their topological dimension. Fractals have the same appearance at different scales. In this sense, many time series of different processes can be considered as fractals, since many parts taken from these time series, scaled by proper factors, are similar to the whole series. Considering that the fractal dimension is, somehow, a measure of the complexity that is repeating on each scale, it seems very interesting to compute the fractal dimension of a time series. In this regard, there are several algorithms that can be applied to estimate the fractal dimension of a time series. The approach used in this paper to estimate the fractal dimension is [PERSON]'s algorithm, that is summarized in Section 3.2.1.  
Figure 2: When objects of different dimension (e.g., (**a**) line, (**b**) square, and (**c**) cube) are linearly scaled by a factor of 2, their characteristic size will have different results associated to their dimension.