5. 1. 1 Newton iteration and fractal
The most basic method of nonlinear iteration is Newton iteration method. That is to say, the function is expanded into Taylor series, the higher-order terms are omitted, and the modified increment and Jacobian matrix are proposed from the first term to form a linear equation group. Newton iterative method converges quickly, but the convergence depends on the initial guess.
In 1988, Petigen and Saupe published an interesting experimental result. He considered the following simple nonlinear equation.
z3- 1=0 (5. 1. 1)
One real root of this equation is z= 1, and two complex roots are
z=exp( 2πi/3) (5. 1.2)
Newton iterative scheme
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To approach, is it a real root or a complex root?
Of course, the initial value z0 can be any point in the complex plane z=x+iy. It can be guessed that z0 can be divided into several regions on the complex plane. After iteration with formula (5. 1.3), z0 converges to one region and to the complex roots of another region. People who are used to linear thinking will think that there are clear boundaries between these areas. For example, in the neighborhood where z0 is equal to 1, Newton's iteration will converge to the real root z= 1, and its area accounts for about 1/3 of the z plane, and other areas will converge to the complex root. That was not the case. The convergence domain of the initial value z0 is fractal, as shown in Figure 5. 1. As can be seen from Figure 5. 1, the area of the black area is indeed 1/3 of the initial value area (-2≤x≤2, -2≤y≤2), but its boundary is fractal, that is, it contains all scales and is self-similar. Why does a simple iterative format like formula (5. 1. 1) produce such a complex fractal image? Why does a small change in the initial value on this boundary make the iteration converge to a completely different root?
Fig. 5. The black region of the complex plane Z with1real imaginary axis in the range of (-2,2) converges to the initial value region of the real root z= 1 after Newton iteration, and the white region converges to the complex root.
The problem comes down to the nonlinearity of equation (5. 1. 1), which is a necessary condition for the system to move towards chaos. For nonlinear systems, the small change of initial value will make the system state bounce between several "attractors", and its geometry is fractal.
5. 1.2 Fractal Earth Model
This book regards the earth parameters as a set of real functions, that is, elements of Hilbert space, which is a deterministic model. The deterministic model implies the assumption of orderly distribution of the earth's materials, while the stochastic model implies the assumption of random distribution of the earth's materials. We further assume that the material distribution of the earth is self-similar or self-affine and has a multi-scale hierarchical structure, which leads to the fractal model of the earth.
The basis of describing the earth from the fractal point of view is that the earth is a scale-free complex object, and its scale can range from a few millimeters of microcracks to tens of thousands of kilometers of earth diameter, and the phenomena between different scales are similar.
People have a characteristic scale, that is, height, which is about 1.6 meters or 5 feet. Therefore, man-made things also have characteristic scales, such as the height of trains is about 2 meters, and the average height of ships and tall buildings is tens of meters. This characteristic scale is called scale.
Natural phenomena generally have multi-scale characteristics, and there is no characteristic scale. Fractal geometry relates phenomena of different scales to scaling laws.
p(λt)=λαp(t)，0 < α < 1 (5. 1.4)
Where p(t) is the scale of a certain level, p(λt) is the scale after it is enlarged by λ times, and α is the scale index. but
D0=2-α (5. 1.5)
Equal to Mandelbrot fractal dimension.
Dimension refers to the number of independent coordinates placed by a point in a geometric object. For example, a point on the earth's surface is expressed by latitude and longitude, and its dimension is 2. In fractal geometry, the dimension can be a fraction, and the dimension of a fraction is called fractal dimension.
For the two-dimensional case, if each side of a square is magnified 3 times (scaled up), it will become 9 original squares with
2=l n9/l n3
For geometric objects with integer dimension d, each direction is enlarged by L times, resulting in N original objects, including
d=lnN/lnL
Magnifying L times in each direction is equivalent to reducing the measurement scale (or measurement unit) in this direction to the original ε= 1/L times. Therefore, generally speaking, when studying the scale change of an object with a small unit of measurement ε, we can define
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This is Mandelbrot fractal dimension.
1992 Korvin compiled a book called "fractal models in earth science", which listed many fractal models related to earth science. It is mentioned that in 1984, the US Geological Survey dispatched dozens of fire engines to flush the rock outcrop area in Nevada, and then made a detailed map of its faults. The average fractal dimension of the fault system in this area was 1.744. Using the large-scale regional fault structure map, the fractal dimension of the fault system in this area is 1.773, which proves that there is self-similarity among different levels of the earth fault system. Chen Qing and Turcotte's monographs also have wonderful descriptions of this.
Discussion on fractal geometry and other fractal dimensions (such as correlation dimension D2, information dimension D 1, etc.). ), please refer to the relevant monographs. The following only introduces the calculation method of fractal dimension D0 of time series. The traditional method of introducing D0 fractal dimension mostly uses the power spectrum calculation of time series. Because the power spectrum of geophysical data contains a lot of noise in high frequency band, this calculation method can hardly be used. We only study the following algorithm, and achieved good results in the processing of reflected seismic data.
For plane curves, the total length is
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Where: ε is the unit of measurement (feet); N is the number of feet measured; F is the remaining length after the ruler is measured (f 0, the information loss of each iteration is greater than zero, and the system entropy increases continuously, leading to chaos. Fig. 5.2(b) shows the change of λ with R in the second iteration, and compares it with bifurcation and chaos of the system. As can be seen from the figure, when λ < 0, the corresponding system is stable, and the system bifurcates at λ=0, while λ > 0 corresponds to chaos. Therefore, Lyapunov is an important scalar parameter indicating the state.
(2) the regularity of the overall behavior. Although the specific state of the system in the future is uncertain and unpredictable, "seemingly crazy and messy, in fact it has its own rules" (Shakespeare). In the figure of the phase space formed by the trajectory of all system evolution, there are several small spaces (called attractors) that attract the trajectory, making the trajectory shrink into it or jump to another attractor. This phenomenon shows that the overall behavior is still holistic.
The regularity of the whole behavior is also manifested in the similarity (fractal) of different levels of movement. Feigenbaum proved that any chaotic motion is xn+ 1=f(xn), and the scaling characteristics of its transformation to chaos are controlled by two universal constants, which further shows that chaos theory has overall regularity.
The form is periodic, and the occurrence of chaotic state sometimes appears repeatedly, but this repetition is uncertain. For example, the occurrence of major earthquakes is frequent, including high-frequency recurrence, and there is no accurate cycle.
The all-round development of nonlinear scientific research was still in the 1990s. The theoretical framework of linear science was established in19th century and developed into a complete system in 20th century. However, the theoretical framework of nonlinear science will be established in 2 1 century. The research on positive problems is still the same, while the research on nonlinear problems is sporadic. Next, some examples of nonlinear inversion based on chaos theory are introduced.