Fractal statistical model

Set the fractal statistical model:

Fractal Chaos and Mineral Prediction

Where r represents the characteristic scale, c > 0 is called the proportional constant, d > 0 is called the fractal dimension, and N(r) represents the number of scales greater than or equal to r (when the sign before fractal dimension d is negative, it is recorded as N(≥r)) or less than or equal to r (when the sign before fractal dimension d is positive, it is recorded as N(≤r)).

For the convenience of research, equation (3.3. 1) can be decomposed into the following two equations:

Fractal Chaos and Mineral Prediction

Many geological phenomena are characterized by scale invariance, such as rock fragments, faults, earthquakes, volcanic eruptions, mineral deposits and oil wells. The distribution between the frequency and magnitude of these phenomena is scale-invariant. The characteristics of fractal distribution require that the number is greater than or equal to a certain scale, and there is a power function relationship with the size of the object, that is, the relationship of (3.3. 1). For example, r can represent the gold grade, n (. R can also represent the radius of a circle, and N(≤r) represents the number of ore bodies falling into a circle with radius R. 。

The characteristics of fractal distribution require that the number of objects is larger than a certain scale, and there is a power function relationship with the size of objects. In the statistical distribution of geological phenomena, the fractal distribution of power function (namely, power function distribution, Pareto distribution and Chepov distribution) is not unique, and there are other types such as logarithmic distribution. But the fractal distribution of power function is the only distribution without characteristic scale. In this way, these distributions can be applied to geological phenomena with scale invariance.

The establishment of model is actually the establishment of fractal (similar) model. Based on the similarity principle, the model unit is established, and the prediction unit is fractal processed and predicted.

In order to find the fractal dimension d, the observed data (N(r 1), N(r2), …, N(rn)) and (r 1, r2, …, rn) are drawn on double logarithmic coordinate paper. If the scattered points are roughly distributed on a straight line, the fractal dimension d can be obtained by using the slope of the straight line. N(rn)) and (r 1, r2, …, rn) are substituted into the formula (3.3. 1), and then the logarithm of both sides is taken and (3.3. 1) is transformed into a linear regression model:

Fractal Chaos and Mineral Prediction

The estimation of slope d by least square method is fractal dimension. At present, this method (traditional method) is almost always used to find fractal dimension D. Although this method is simple, the result may be incorrect (Bethea et al., 1985), and the parameters C and D should be estimated by nonlinear regression model.

In fact, equation (3.3. 1) is a nonlinear regression model, in which c and d are unknown parameters, and the estimator of parameter d in equation (3.3. 1) directly obtained by the least square method of nonlinear regression model is also a fractal dimension. The fractal dimension D obtained by this new method is better than the traditional method (3.3.655).

The new method has the following advantages:

(1) to find the fractal dimension d by the traditional method, the original data (N(r 1), N(r2), …, N(rn)) and (r 1, r2, …, rn) should be transformed at the same time, but in most cases, the original data is special.

(2) Using the new method, the approximate deviation and variance of fractal dimension estimator can be obtained, and the approximate prediction deviation and variance can also be obtained. The traditional method can't get the above results (the deviation and variance of d in the formula (3.3.2) obtained by the new method are fundamentally different from those of d in the formula (3.3. 1)).

(3) When fitting the fractal model, the parameter estimator obtained by the new method is better than the traditional method, that is, the sum of squares of residuals is smaller (sum of squares of residuals is a quantitative index to measure the goodness of fitting), and the parameter estimator is more stable.

3.3.2 Simulation Research on Fractal Statistical Model

We generated 1000 random numbers of uniform distribution, standard normal distribution and lognormal distribution with the interval of [0, 1] on the computer, and divided each random number into 10 groups (that is, each group has 1000 random numbers, and * * has 30 groups) for fractal.

Arrange each group of 1000 random numbers from small to large, divide the total interval of random number distribution into k subintervals, and count the frequency NFi(i= 1, 2, …, k) at which r is a positive integer.

In this way, the data (N(r 1), N(r2), …, N(rn)) and (r 1, r2, …, rn) are obtained. Substituting these data into the fractal statistical model (3.3. 1 "), the fractal dimension can be obtained by least square method.

See table 3- 1, table 3-2 and table 3-3 for the specific calculation results.

Table 3- 1 fractal dimension estimator of uniform distribution d

Table 3-2 Fractal Dimension Estimation of Normal Distribution D

Table 3-3 Fractal Dimension Estimation of Lognormal Distribution

In Tables 3- 1, 3-2 and 3-3: ① For uniformly distributed random numbers, k= 150, n=26, ri=2i(i= 1, 2, ..., 26); ② For random numbers with normal distribution, k=80, n=2 1, ri=2i+ 10(i= 1, 2, …, 21); ③ Lognormal random number, k= 100, n=2 1, ri=2i(i= 1, 2, …, 21); ④ For random data with different distributions, the range of values of k and r is different, mainly according to the data (N(r 1), N(r2), …, N(rn)) and (r 1, r2, …, rn). In this range, there are no scale areas and statistical requirements. ⑤ Randomly select 1000 samples to meet the requirements of statistical inference.

From the data in Table 3- 1, Table 3-2 and Table 3-3, the following results can be deduced:

(1) The fractal dimension estimation obtained by the new method tends to be more stable than that obtained by the traditional method, because the standard deviation is a quantitative description of the degree of data dispersion. The smaller the standard deviation, the more concentrated the data is near the average value.

(b) The value of fractal dimension d can represent the structure between random numbers or samples. According to the fractal statistical model (3.3. 1 "), it can be seen that the smaller the value of d, the smaller the difference between random numbers or samples, that is, the better the uniformity. Conversely, the greater the value of d, the greater the difference between random numbers or samples. That is, the uniformity is poor. The fractal dimension of random numbers with uniform distribution (good uniformity) (average 0. 1287) is smaller than that of normal distribution (moderate uniformity) (average 0.6853) and lognormal distribution (poor uniformity) (average 0.9762). The above conclusions are consistent with the actual situation.

3.3.3 Application example

Metallogenic prediction of Luobusha chromite deposit in Tibet.

The Luobusa chromite deposit in Tibet is the largest chromite deposit known in China at present, with nearly 5 million tons of proven chromite reserves, accounting for more than one third of the proven reserves in China. Therefore, it is of great significance to predict the mineralization of Luobusha chromite deposit in Tibet.

Luobusa ophiolite is located in the east of the famous Yarlung Zangbo River ophiolite belt and the south of Gangdise volcano-magma arc. The rock mass protrudes northward in an arc shape and is distributed between flysch formation, which contains a small amount of crystalline limestone and spilite porphyry in the Late Triassic, marine volcanic rocks, radiolarian cherts and Neogene intermontane molasses formations in the Late Cretaceous. The plane shape of rock mass is like a lens, and some of it is staggered by faults. The main body extends from east to west, about 30km long.

Through the study of the deposit, it is considered that the spatial distribution of surface ore bodies, ore groups and deposit reserves has good fractal structural characteristics, that is, self-similarity, and the fractal statistical model (3. 3. 1') can be used as a prediction model of ore bodies, ore groups and their resources in the exploration area under Quaternary coverage.

1. Geological conditions

The study shows that although the exposed elevation and position of ore bodies in Luobusha and Xianggashan ore blocks are slightly different, and the chemical composition and physical properties of rocks and ores are also different, they are all in the same mantle peridotite and belong to the same product of diagenesis and mineralization. The original structural ore-bearing complex belt is unified, and the structural analysis shows that the ore bodies in the whole area are in the same structural ore-bearing complex belt. Therefore, it is geologically feasible to extend the model area to the whole two ore sections.

2. Mathematical conditions

The parameters of the self-similar system of Luobusa chromite deposit show self-similarity. The observed data (N(r 1), N(r2), …, N(rn)) and (r 1, r2, …, rn) are plotted in a log-log coordinate system (i.e. lgn (r)-lgr). That is, there is a scale-free area. Self-similarity is the property that things do not change with the observation scale in a certain scale range (scale-free area), and the conclusion obtained in the 55cm part of scale-free area can be extrapolated to the whole scale-free area. The spatial distribution of surface ore bodies, ore groups and their reserves in the model area has a good fractal structure, that is, self-similarity. Therefore, in the self-similar system of sediments, the upper limit of scale-free region can be extrapolated.

3. Similarity and analogy between model area and prediction area.

The predicted ore bodies and ore groups are those on the bedrock surface under the Quaternary coverage area, and the predicted resource amount corresponds to the C+D reserves in the model area. Quaternary research shows that the Quaternary in the prediction area is residual slope accumulation and a small amount of moraine, so it is considered that the surface weathering and erosion conditions and ore body preservation conditions in the model area and the prediction area are similar, and the model can be extrapolated.

4. Data source and parameter estimation

(1) surface ore body

There are *** 152 surface ore bodies in the model area, which are distributed in the structural ore-bearing complex belt marked on 1: 1000 topographic and geological map (see figure 3- 1 and figure 3-2), and each ore body is regarded as a point represented by its center (figure 3-3 and figure 3). Put a point in the LGR-LGN (r) coordinate, and use the least square method to fit the straight line, and get the equation of the straight line as follows:

Fractal Chaos and Mineral Prediction

Taking D(0)=0.784 1 and C(0)= 100.9348=8.6060 as the initial iteration values, the least squares estimator (fractal dimension) and the least squares estimator (fractal dimension) in the fractal statistical model (3. 3. 1') are obtained by a new method. 9.4039)= 1 14.6947 < Q0(0.784 1，8.6060) = 125.3347。 Maximum inherent curvature γ n = 0.034 18 < 0.220 = 60. 0.05)) (see Appendix A). At this time, the intrinsic nonlinear strength of the fractal statistical model (3. 3. 1') is very weak and can be ignored. Therefore, the fractal statistical model (3. 3. 1') can be used as a mathematical model for predicting surface ore bodies. Namely:

Fractal Chaos and Mineral Prediction

Table 3-4 Surface Ore Body Data

(2) Surface mineral groups

On the topographic and geological map of 1∶ 10000, there are 14 ore groups in the ore-bearing complex belt in the model area. Each ore group can be regarded as a point represented by its center. The definition of center and radius is similar to that of surface ore body, and the R value is still the same as that of surface ore body. If the center of the circle is fixed, draw a circle with different radii r and calculate the ore that falls into the circle every time.

Fractal Chaos and Mineral Prediction

Taking D(0)=0.8982 and c (0) =10-0.3130 = 0.4864 as the initial iteration values, the least squares estimator (fractal dimension) in the fractal statistical model (3. 3. 1') is obtained by a new method. 0.4864)= 1.38, and the maximum inherent curvature γ n = 0.04715 < 0.2205 =1/(2f1/2 (2,6,0.05)) (see appendix a). At this time, the fractal statistical model (.

Fractal Chaos and Mineral Prediction

Table 3-5 Data of Surface Mineral Group

Figure 3- 1 Structural Schematic Diagram of Luobusa Area

Figure 3-2 Luobusa-Zhang Ga Structural Profile

(3) Deposit reserve

On the topographic map of 1∶ 10000, the distribution data of C+D grade chromite reserves of 42118t in the structural ore-bearing complex zone in the model area are studied. The definition of center and radius is similar to that of surface ore bodies, and the center is fixed. Therefore, it is calculated that it falls into the sphere at different R radii (actually because the distribution of reserves is projected on the topographic and geological map of 1: 10000), and the reserves of C+D grade ore are recorded as N(r) (Table 3-6), and the reserves of LGR-LGN are calculated by least square method.

Fractal Chaos and Mineral Prediction

Taking D(0)=0.7048 and c (0) =105.51= 323668.176 as the initial iteration values, the fractal statistical model (3.3./kloc-0) is obtained by a new method. 367 166.9)= 0.3350 16 1× 10 12 < Q0(0.7048， 323668.176) = 0.3551732×1012 Maximum inherent curvature γn = 0.00 At this time, the inherent nonlinear strength of the fractal statistical model (3. 3. 1') is very strong.

Namely:

Fractal Chaos and Mineral Prediction

Table 3-6 Surface Ore Reserve Data

Figure 3-3 is a schematic diagram of the projection point of the mine on the horizontal projection plane.

5. Interpretation of prediction results and parameter significance

With the largest ore body in the ore group as the center, r = 5, 10, 15, …, 50,55 cm (on the topographic and geological map of1:10000), substitute r into the above formula (3.3.3). The total resources minus the known resources at r=55cm is the predicted resources under Quaternary in Xiangshan ore block. The results are: 43 "surface" ore bodies, 4 "surface" ore groups (rounded), and the resources (chromite):1071815.342t. The average grade of total platinum group elements is 0.497g/t, and the total resource is 532.692kg. The predicted results are in good agreement with those calculated by conventional methods (see Figure 3-5, Figure 3-6 and Figure 3-7).

The density is defined as ρ=N(r)/(πr2)=(C/π)rD-2.

When D=2.0, the density ρ=C/π, indicating that the density is uniform;

When d > 2.0, the density ρ increases with the increase of r;

When d < 2.0, the density ρ decreases with the increase of r;

When r= 1.0, C =ρρ= N( 1).

0.6654 (fractal dimension of ore deposits) < 0.7568 (fractal dimension of surface ore bodies) < 0.9298 (fractal dimension of surface ore groups) < 2 shows that with the increase of R, the density of ore deposits, surface ore bodies and surface ore groups gradually decreases.

Therefore, fractal dimension D quantitatively expresses the density change trend of ore body distribution, and C represents the initial value of ore body distribution, which has important guiding significance for mineral resources exploration, prediction and evaluation.

Figure 3-4 Schematic Diagram of Projection Point of Ore Body Center on East-West Vertical Projection Plane

Figure 3-5 Fitting Diagram of Original Data Curve of Ore Body

Figure 3-6 Curve Fitting of Original Data of Surface Mineral Groups

Figure 3-7 Curve Fitting of Original Reserves Data