For two-dimensional objects with arbitrary cross-section, polygons can be used to approximate the shape of their cross-section. As long as the coordinates (ξ, ζ) of each corner of a polygon are given, its gravity anomaly can be calculated by analytical formula. Obviously, its accuracy depends on how close a polygon is to any shape.
Figure 2-6- 10 Polygon Approximation Method Schematic Diagram
Fig. 2-6- 10 is a cross-sectional view of a two-dimensional body represented by polygon ABCDEFGA. First, we calculate the anomaly of the second-degree horizontal cylinder at the O point represented by the △OAB section surrounded by the AB edge and the O point (which coincides with the calculated point). According to the basic formula:
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Reference polar coordinates ξ = rcos θ, ζ = rsin θ, d ξ d ζ = rd θ dr
The above formula is
Considering the DRS in θ = dζ zeta, there are:
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According to formula (2-6-35), the anomaly of △ OAB at point O should be
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In the above formula, zeta is a function of θ, which can be obtained by using the linear equation of point A and point B, zeta = zeta cotθ:
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Where I and I+ 1 represent the numbers of two endpoints in the clockwise direction of the ith side. Substitute the zeta expression into the formula (2-6-36).
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The gravity anomalies of △OBC and △ OCD are calculated and added in the same way. Due to the circulation integration along the same direction, the anomalies outside the polygon AB…A cancel each other out, and the final result is only the gravity anomalies generated by the two-dimensional body represented by the polygon at the origin. If the polygon * * * has n sides, the expression for calculating gravity anomaly is:
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When applying Equation (2-6-38), it should be noted that when the number of edges is n, ξ n+ 1 = ξ 1, ζ n+1= ζ1; When ξ i+ 1 > ξ i, the arc tangent function value is between 0 and π, otherwise it is between-π and 0. Since the above formula is derived when the origin coincides with the calculation point, it is only necessary to change ξi and ζi into ξ i-x and ζ i-z when calculating the anomaly of any point P(x, 0, z).
In this way, vertical steps, inclined steps, various plates and so on. The above can be calculated by the coordinates of corner points, which is very simple. When infinite or infinite extension is involved, it can be expressed by a large ξi and ξ I+ 1 or ζi and ζ I+ 1. Moreover, when Δ σ is a constant, the anomalies (such as fault structures) produced by multiple two-dimensional bodies at the same time can be calculated. The specific method is to connect the corner points of adjacent two-dimensional bodies in the same direction (such as clockwise) with straight lines to form a closed single two-dimensional body. See figure 2-6- 1 1.
(2) Forward modeling of 3D entities with arbitrary shapes.
1. cuboid element method
Three-dimensional objects with arbitrary shapes are divided into three groups of planes parallel to rectangular coordinate planes, so that the objects are divided into a series of rectangular units. According to the coordinates of eight corner points (Figure 2-6- 12) and referring to the basic formula, we can get that the gravity anomaly caused by a cuboid element at the coordinate origin is
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Fig. 2-6- 1 1 Polygon Approximation Method for Calculating 2D Anomaly of Faults and Complex Bodies.
or
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Where r = (ξ 2+η 2+ζ 2) 1/2.
Add up the anomalies of all cuboid elements to get the approximate anomaly of the object at the origin. Note: the value range of the arctangent function in the formula should be 0 ~ π. If the calculation point is any point P(x, y, z), then (ξ, η, ζ) in the formula should be replaced by (ξ-x, η-y, ζ-z) respectively.
2. Panel method
Cut an object with a set of parallel vertical planes (or horizontal planes) and divide it into several vertical (or horizontal) slices, each of which is approximately a polygon. Analytic method is used to calculate the "action value" of the polygon on the calculation point, and finally the "action value" of all the slices is added up by numerical integration.
Fig. 2-6- 12 schematic diagram of rectangular element method
Fig. 2-6- 13 schematic diagram of vertical panel method
The calculation formula of vertical panel method is as follows, as shown in figure 2-6- 13. Let the coordinate origin be the calculation point, and the X axis is orthogonal to the tangent plane. According to the basic formula, the exception generated by the whole object can be expressed as
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formula
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It represents the "action value" of a slice. For ξ = ξ=ξj piece, N-sided polygons are used to approximate its cross-sectional shape.
Let the coordinates of each corner of the polygon be (ξj, ξ k, ζk), k = 1, 2, …, n, then S(ξj) can be written as:
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The linear equation on the k-th side is
ζ=Ckη+Dk
Where:,.
Substitute zeta into formula (2-6-42) for integration. Because η n+ 1 = η 1, η n+1= ζ1,the second result on the right side of the formula is zero, so there are:
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Where:.
After S(ξj) is obtained, the anomaly of the whole object at the calculation point can be obtained by applying numerical integration according to formula (2-6-4 1).
Fig. 2-6- 14 schematic diagram of vertical line element method
3. Line element method
Cut a three-dimensional body with a set of vertical planes parallel to the X axis and a set of vertical planes parallel to the Y axis, and divide it into vertical rectangular columns. When the two sides of the cross section of a cylinder are longer than the length of the cylinder, the gravity anomaly can be regarded as a material line segment whose mass is uniformly compressed on its vertical central axis. Figure 2-6- 14 is the coordinate of the k-th vertical rectangular cylinder with o as the calculation point. (ξk, ηk, ζk, 1) and (ξk, ζk, ζ k, 2) respectively represent the upper and lower endpoint coordinates of the shaft, and λ represents the residual linear density. According to the basic formula, the abnormal δg of the material line segment can be obtained by integrating zeta, that is
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Where:,.
Therefore, the gravity anomaly caused by the whole three-dimensional body at point O is the sum of the anomalies caused by all segmented line elements.
It can be seen from the formula (2-6-44) that when Rk, 1 = 0, δg? ∞, this happens when the column is just below the calculation point and close to the ground. In order to reduce the error, an upright cylinder with equal cross-sectional area is used instead. Right above the central axis of an upright cylinder, the expression of gravity anomaly is
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Where r shall satisfy:
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Obviously, the above calculation methods are approximate, and the finer the segmentation, the better the natural approximation of the true value, which depends on the needs of working accuracy.