﻿ 确定排列片横纵比的方法——85%法则再分析
 石油地球物理勘探  2019, Vol. 54 Issue (5): 947-953  DOI: 10.13810/j.cnki.issn.1000-7210.2019.05.001 0
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### 引用本文

ZHANG Hua, WANG Meisheng, LI Kanghu, NING Hongxiao, HAN Zhi-xiong, GU Yong. A method for determining the aspect ratio of patch: reanalysis of 85% rule. Oil Geophysical Prospecting, 2019, 54(5): 947-953. DOI: 10.13810/j.cnki.issn.1000-7210.2019.05.001.

### 文章历史

A method for determining the aspect ratio of patch: reanalysis of 85% rule
ZHANG Hua , WANG Meisheng , LI Kanghu , NING Hongxiao , HAN Zhi-xiong , GU Yong
Acquisition Technology Center, BGP Inc., CNPC, Zhuozhou, Hebei 072751, China
Abstract: On the premise of satisfying the needs of geological tasks, the paper proposes a method to determine the size and aspect ratio of patches for a more efficient and convenient 3D geometry design.With the center of the patch as the origin, and the spatial coordinates (x, y) as the independent variables, a rectangular coordinate system is constructed.With the center of the patch to the designed maximum offset as the radius, a circle is drawn.Receivers in the circle are considered effective, and those outside the circle are considered invalid.Taking the effective information and invalid information and their inter-relation information as dependent variables, the functional relations of the effective information and invalid information with the spatial coordinate position are calculated.The spacial coordinates determine the length and width of a 3D patch, and their proportion (the aspect ratio).At the same time, a template and a table for the horizontal and vertical parameter selection of the patch are generated.This method is convenient and flexible, and can be used for a reference in the 3D survey design.
Keywords: aspect ratio    85% rule    mathematical model    mathematical analysis    area integral    directional derivative    broadband    wide-azimuth    high-density (BWH)
0 引言

Cordsen等[16]给出了一种“折中”方法，这就是一直指导大家进行三维地震勘探观测系统设计的“85%法则”——排列片纵向长度选择为最大炮检距长度的85%，横向长度又选择为纵向长度的85%。李庆忠[17]随后指出“Andreas Cordsen主张排列片的最佳纵、横比为85%，而没有深入讨论其缺点”，显然此处的“缺点”有更宽泛的地球物理内涵。然而，Cordsen等二十多年前提出的85%法则，至今尚未见到有人深入分析其合理性并详细论证何种排列片才是更经济高效的？

1 数学模型

 图 1 通用数学模型

2 数学分析 2.1 坐标点M(x′，y′)在圆内或圆周上

 $S_{\mathrm{rect}}=2 R \sin \theta \times 2 R \cos \theta=2 R^{2} \sin 2 \theta$ (1)
 图 2 点在圆周上的矩形排列片数学模型

θ=45°时，排列片为正方形，M(x′，y′)点坐标为$\left(\frac{\sqrt{2}}{2} R, \frac{\sqrt{2}}{2} R\right)$，排列片面积Srect=2R2，达到最大值。该值约为外接圆面积的64% $\left(\frac{2 R^{2}}{\pi R^{2}}=\frac{2}{\pi}\right)$

2.2 坐标点M(x′，y′)在圆外

 图 3 点在圆外的矩形排列片数学模型

 \begin{aligned} S_{\text {inv }} &=\int_{\sqrt{1-y^{2}}}^{x^{\prime}}\left(y^{\prime}-\sqrt{1-x^{2}}\right) \mathrm{d} x \\ &=x^{\prime} y^{\prime}-\frac{x^{\prime} \sqrt{1-x^{2}}}{2}-\frac{y^{\prime} \sqrt{1-y^{2}}}{2}-\\ & \frac{\arcsin x^{\prime}}{2}+\frac{\arcsin \sqrt{1-y^{2}}}{2} \end{aligned} (2)

 $\begin{array}{l} {S_{\rm{v}}} = {x^\prime }{y^\prime } - {S_{{\rm{inv}}}} = \frac{{{x^\prime }\sqrt {1 - {x^2}} }}{2} + \frac{{{y^\prime }\sqrt {1 - {y^2}} }}{2} + \\ \frac{{\arcsin {x^\prime }}}{2} - \frac{{\arcsin \sqrt {1 - {y^2}} }}{2} \end{array}$ (3)

 \begin{aligned} S_{\text {v-inv}}=& x^{\prime} y^{\prime}-2 S_{\text {inv }}=x^{\prime} \sqrt{1-x^{2}}+y^{\prime} \sqrt{1-y^{2}}+\\ & \arcsin x^{\prime}-\arcsin \sqrt{1-y^{\prime}}-x^{\prime} y^{\prime} \end{aligned} (4)

2.3 数据分析

 图 4 无效信息面积变化函数图

 图 5 有效信息面积变化函数图

 图 6 无效信息与有效信息面积变化叠合图

 图 7 有效信息面积与无效信息面积差函数(Sv-inv)图

 图 8 Sv-inv函数等高线平面图
3 确定排列片横、纵参数

Cordsen等提出的85%法则在图 8中如红线交点所示，排列参数对应的Sv =0.5999，Sinv=0.0142，Sv-inv=0.5857。排列片有效面积相对于R圆面积为4Sv/π=76.4%，该比值并不等于Cordsen等提出的78%[16]，而85%法则点和原点构成的矩形面积与四分之一圆面积比为0.85×0.852/(π/4)=78%，该比值显然包含了无效信息的面积，可见Cordsen在提出85%法则时，并没有经过严谨的微积分计算；其所有数值都是基于几何学关系近似求得，“85%法则”虽然较为理想，但数学论证并不充分、系统。

M点移动至C点时，排列片相当于最大炮检距半径圆的外接正方形(图 2)，这时Sv=0.785，Sinv=0.215，则Sv-inv=0.570，排列片无效信息面积相对于R圆面积为4Sinv/π=27%。

 图 9 Sinv/v函数图

Sv-inv函数等高线与Sinv/v函数等高线叠合起来(图 10)，取名为排列片横纵参数量板，图中蓝线为Sinv/v函数等值线，色标值为Sv-inv值(为显示效果，图件在前文等高线图基础上抽稀一半)。为最大限度增加有效信息面积和减少无效信息面积，M点最佳选择区域应为Sv-inv等值线密集且穿越Sinv/v(蓝)函数等值线值尽可能小、线数尽可能少的区域，即理想区域是等高线与OC交点附近(T点前)，最佳位置点在等值线与OC交点上。

 图 10 排列片横纵参数量板

 图 11 Sinv/v(蓝)与Sv-inv函数等高线叠合图

4 结束语

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