考虑地表起伏的不规则区域重力梯度场模型构建方法——以云南地区为例

doi:10.3969/j.issn.0253-4967.2023.05.006

地震地质 ›› 2023, Vol. 45 ›› Issue (5): 1129-1146.DOI: 10.3969/j.issn.0253-4967.2023.05.006

考虑地表起伏的不规则区域重力梯度场模型构建方法——以云南地区为例

刘金钊¹^,²⁾(), 梁星辉²⁾^,^*(), 叶周润³⁾, 陈兆辉¹⁾, 胡敏章⁴⁾, 韩宇飞⁵⁾, 王青华⁶⁾, 刘东⁶⁾, 郝洪涛⁴⁾, 张双喜¹⁾, 陈铭¹⁾

¹⁾ 中国地震局第一监测中心, 天津 300180
²⁾ 中国科学院精密测量科学与技术创新研究院, 大地测量与地球动力学国家重点实验室, 武汉 430077
³⁾ 合肥工业大学, 合肥 230009
⁴⁾ 中国地震局地震研究所, 地震大地测量重点实验室, 武汉 430071
⁵⁾ 中国地震台网中心, 北京 100045
⁶⁾ 云南省地震局, 昆明 650224

收稿日期:2022-12-26 修回日期:2023-05-31 出版日期:2023-10-20 发布日期:2023-11-23
通讯作者: *梁星辉, 男, 1983年生, 博士, 副研究员, 主要从事动态重力测量的理论、方法与应用研究, E-mail: lxh_whigg@asch.whigg.ac.cn。
作者简介:
刘金钊, 男, 1986年生, 2015年于中国科学院测量与物理研究所获大地测量学与测量工程专业博士学位, 副研究员, 主要从事重力测量、建模及应用方面的研究, E-mail: whiggliujinzhao@126.com。
基金资助:
国家自然科学基金(41704084); 国家自然科学基金(41904010); 安徽省基金(2008085MD115)

STUDY ON THE REGIONAL GRAVITY GRADIENT FIELD MODELING IN IRREGULAR AREA WITH CONSIDERATION OF SURFACE FLUCTUATION: A CASE STUDY OF YUNNAN PROVINCE

LIU Jin-zhao¹^,²⁾(), LIANG Xing-hui²⁾^,^*(), YE Zhou-run³⁾, CHEN Zhao-hui¹⁾, HU Min-zhang⁴⁾, HAN Yu-fei⁵⁾, WANG Qing-hua⁶⁾, LIU Dong⁶⁾, HAO Hong-tao⁴⁾, ZHANG Shuang-xi¹⁾, CHEN Ming¹⁾

¹⁾ The First Monitoring and Application Center, China Earthquake Administration, Tianjin 300180, China
²⁾ State Key Laboratory of Geodesy and Earth's Dynamics, Innovation Academy for Precision Measurement Science and Technology. Chinese Academy of Science, Wuhan 430077, China
³⁾ Hefei University of Technology, Hefei 230009, China
⁴⁾ Key Laboratory of Earthquake Geodesy, Institute of Seismology, China Earthquake Administration, Wuhan 430071, China
⁵⁾ China Earthquake Networks Center, Beijing 100045, China
⁶⁾ Yunnan Earthquake Administration, Kunming 650224, China

Received:2022-12-26 Revised:2023-05-31 Online:2023-10-20 Published:2023-11-23

摘要/Abstract

摘要：

文中推导并给出了基于非格网分布的起伏面扰动重力或重力异常解算区域扰动重力梯度场模型的数值计算公式。基于澳大利亚West Arnhem Land地区的格网重力数据, 以频谱域(二维快速傅里叶变换)解算的扰动重力梯度全张量作为“基准值”, 然后利用基于推导公式的最小二乘配置方法(LSC)对相同区域非规则范围的重力数据进行扰动重力梯度模型解算, 将结果作为“评估值”。对比“基准值”与“评估值”之差, 研究发现: 1)基于推导公式的最小二乘配置方法解算得到的扰动重力梯度值与频谱域方法得到扰动梯度“基准值”各分量在空间形变变化上是一致的; 2)统计扰动重力梯度各分量的差值 $Δ δ Γ x x f f t - l s c 、 Δ δ Γ x y f f t - l s c 、 Δ δ Γ x z f f t - l s c 、 Δ δ Γ y y f f t - l s c 、 Δ δ Γ y z f f t - l s c 和 Δ Γ z z f f t - l s c,$ “基准值”与“评估值”差值的标准差分别为5.54E、 5.30E、 1.85E、 6.55E、 2.09E和9.67E(1E=1×10^-9s^-2), 远低于国际上实测重力梯度与解算模型差值的研究结果。最后, 基于云南地区实测地表差分重力值, 文中首次给出了该区域半波长约20km的重力梯度场年际变化模型。文中的思路和方法提高了广泛分布的重力数据(主要为重力异常和扰动重力)的使用效率, 可为地球物理学、地质学研究更好地理解和解释重力数据、重力梯度数据及其与场源的关系提供数据基础。

关键词: 协方差函数, 起伏地形, 云南地区, 非格网重力值, 区域重力梯度场构建

Abstract:

The gravity gradient full tensor can sense the slight changes of the gravity vector in different directions, and because of the large number of components(6 components), it can reflect more information on different sides of the same field source than the gravity vector. On the other hand, the ground gravity survey has accumulated a lot of basic gravity vector data, and using the mathematical relationship between the gravity vector and the full tensor of gravity gradient to build a full tensor gravity gradient field model is helpful to the depth mining of existing gravity data information.

Based on the mathematical relation and covariance function relation of disturbance potential, gravity anomaly, gravity disturbance and gravity gradient disturbance, along with the Least Square Configuration(LSC)algorithm, in this paper, the numerical formulas have been derived in detail for modeling the full tensor of regional gravity gradient disturbance field from gravity disturbance or gravity anomaly which distributed on undulating surface with non-grid pattern. The full tensor of gravity gradient derived from the grid distributed gravity anomaly data in West Arnhem Land, Australia by using the spectral domain(two-dimensional fast Fourier transform)method were taken as the “reference value”, Then, the full tensor of gravity gradient, in the same area with irregular regional gravity data distribution, derived form the LSC algorithm based on derived formulas were taken as the “evaluated value”.

By comparing the differences between the “reference value” and the “evaluated value” of the gravity gradient, we found that: 1)The “evaluated value” obtained by the LSC method is consistent in spatial pattern of variation with the “reference value” derived from the spectral domain method. The vertical component of the gravity gradient disturbance describes the boundary characteristics of the field source more precisely than the vertical component of the gravity anomaly, and other gravity gradient disturbance components provide the information representation of the same field source, which provides inspiration for the in-depth interpretation of the field source. 2)Each difference component ΔδΓxxfft-lsc,ΔδΓxyfft-lsc,ΔδΓxzfft-lsc,ΔδΓyyfft-lsc,ΔδΓyzfft-lsc and ΔΓzzfft-lsc between the “evaluated value” and the “reference value” of the gravity gradient has systematic deviation, which is 5.54E, 5.30E, 1.85E, 6.55E, 2.09E and 9.67E respectively. It is much lower than the difference between measured gravity gradient and that from constructed model in previous studies. The results show that the accuracy of the two methods is within the allowable range, but the accuracy of the modeling method based on the Least Square Configuration needs to be further verified by the measured data.

Finally, based on the measured surface differential gravity anomaly values in regional area of Yunnan province, the annul gravity gradient field variation model for this region, about 20km in half wavelength, is firstly constructed and presented by using the LSC with the derived formulas in this paper. The different gravity gradient component models show more abundant interannual scale signal variation characteristics, and demonstrate more local signal characteristics from different sensitive directions in Yunnan region during this time period. In addition, in the vicinity of Zhaotong city in northeast Yunnan province, because there is no gravity points available, both the annual difference results of gravity value and the annual change model of gravity gradient show uniform signal characteristics, indicating that the modeling method in this paper does not introduce additional false “anomalies”. In southern Yunnan province, where gravity points are relatively sparse, both the annual difference results of gravity values and the annual change models of gravity gradients are dominated by long-band signals, and no additional signals will be added due to the increase of expansion order of the LSC algorithm, which is also consistent with our intuitive cognition. These provide support for further research on the relationship between crustal material migration, hydrological changes and earthquakes in Yunnan region.

The procedure and method proposed in this paper can improve the efficiency of using measured gravity data(mainly gravity anomaly and gravity disturbance). Moreover, it can provide basic data for better understanding and interpretation of gravity data, gravity gradient data and their relationship with different field sources in geophysics and geology.

Key words: Co-variance function, Rugged topography, Non-grid gravity, Construction of regional gravity gradient field, Yunnan province

刘金钊, 梁星辉, 叶周润, 陈兆辉, 胡敏章, 韩宇飞, 王青华, 刘东, 郝洪涛, 张双喜, 陈铭. 考虑地表起伏的不规则区域重力梯度场模型构建方法——以云南地区为例[J]. 地震地质, 2023, 45(5): 1129-1146.

LIU Jin-zhao, LIANG Xing-hui, YE Zhou-run, CHEN Zhao-hui, HU Min-zhang, HAN Yu-fei, WANG Qing-hua, LIU Dong, HAO Hong-tao, ZHANG Shuang-xi, CHEN Ming. STUDY ON THE REGIONAL GRAVITY GRADIENT FIELD MODELING IN IRREGULAR AREA WITH CONSIDERATION OF SURFACE FLUCTUATION: A CASE STUDY OF YUNNAN PROVINCE[J]. SEISMOLOGY AND GEOLOGY, 2023, 45(5): 1129-1146.

图/表 12

表1 自由空气重力异常统计

Table 1 Statistics of free air gravity anomaly

起始经度	起始纬度	终止经度	终止纬度	经向点数	纬向点数	最小值	最大值	分辨率
132.96443484°	-12.7732721555°	133.78443484°	-12.2532721555°	42	27	27.59mGal	75.85mGal	0.02°

图1 西澳地区自由空气重力异常及相应频谱方法解算的重力梯度全张量 a 扰动重力梯度δΓxx; b 扰动重力梯度δΓxy; c 扰动重力梯度δΓxz; d 扰动重力梯度δΓyy; e 扰动重力梯度δΓyz; f 扰动重力梯度δΓzz; g 重力异常Δg

Fig. 1 Free air gravity anomalies and the FFT-derived full tensor of gravity gradient of western Australia.

图2 西澳地区裁剪得到的自由空气重力异常及重力梯度全张量 a 扰动重力梯度δΓxx; b 扰动重力梯度δΓxy; c 扰动重力梯度δΓxz; d 扰动重力梯度δΓyy; e 扰动重力梯度δΓyz; f 扰动重力梯度δΓzz; g 重力异常Δg

Fig. 2 Tailored free air gravity anomalies and the full tensor of gravity gradients tailored of western Australia.

图3 基于推导公式的最小二乘配置方法与频谱方法解算的重力梯度扰动结果的比较(δΓxx, δΓxy, δΓxz分量) a 解算的扰动重力梯度δ Γ x x l s c; b 扰动重力梯度差值δ Γ x x f f t-δ Γ x x l s c; c 扰动重力梯度差值δ Γ x x f f t-δ Γ x x l s c的直方图; d 解算的扰动重力梯度δ Γ x y l s c; e 扰动重力梯度差值δ Γ x y f f t-δ Γ x y l s c; f 扰动重力梯度差值δ Γ x y f f t-δ Γ x y l s c的直方图; g 解算的扰动重力梯度δ Γ x z l s c; h 扰动重力梯度差值δ Γ x z f f t-δ Γ x z l s c; i 扰动重力梯度差值δ Γ x z f f t-δ Γ x z l s c的直方图

Fig. 3 Comparison of LSC-derived gravity gradient and FFT-derived gravity gradient(component δΓxx, δΓxy, δΓxz).

图4 协方差函数-最小二乘配置与频谱方法解算的重力梯度结果的比较(δΓyy、 δΓyz、 δΓzz分量) a 解算的扰动重力梯度δ Γ y y l s c; b 扰动重力梯度差值δ Γ y y f f t-δ Γ y y l s c; c 扰动重力梯度差值δ Γ y y f f t-δ Γ y y l s c的直方图; d 解算的扰动重力梯度δ Γ y z l s c; e 扰动重力梯度差值δ Γ y z f f t-δ Γ y z l s c; f 扰动重力梯度差值δ Γ y z f f t-δ Γ y z l s c的直方图; g 解算的扰动重力梯度δ Γ z z l s c; h 扰动重力梯度差值δ Γ z z f f t-δ Γ z z l s c; i 扰动重力梯度差值δ Γ z z f f t-δ Γ z z l s c的直方图

Fig. 4 Comparison of lsc-derived gravity gradient and FFT-derived gravity gradient(component δΓyy, δΓyz, δΓzz).

表2 基于推导公式的最小二乘配置方法与频谱域方法解算重力梯度结果的差值统计(1E=1×10-9s-2)

Table 2 The statistical difference of calculated gravity gradient between LSC method and FFT method(1E=1×10-9s-2)

差值	最大值/E	最小值/E	平均值/E	标准差/E
δ $Γ x x f f t$ -δ $Γ x x l s c$	31.53	-7.73	4.42	5.54
δ $Γ x y f f t$ -δ $Γ x y l s c$	16.76	-17.12	1.66	5.30
δ $Γ x z f f t$ -δ $Γ x z l s c$	9.23	-8.09	-0.004	1.85
δ $Γ y y f f t$ -δ $Γ y y l s c$	36.76	-10.09	-0.73	6.55
δ $Γ y z f f t$ -δ $Γ y z l s c$	9.77	-5.33	-1.16	2.09
δ $Γ z z f f t$ -δ $Γ z z l s c$	11.32	-68.22	-3.54	9.67

表2 基于推导公式的最小二乘配置方法与频谱域方法解算重力梯度结果的差值统计(1E=1×10-9s-2)

Table 2 The statistical difference of calculated gravity gradient between LSC method and FFT method(1E=1×10-9s-2)

差值	最大值/E	最小值/E	平均值/E	标准差/E
δ $Γ x x f f t$ -δ $Γ x x l s c$	31.53	-7.73	4.42	5.54
δ $Γ x y f f t$ -δ $Γ x y l s c$	16.76	-17.12	1.66	5.30
δ $Γ x z f f t$ -δ $Γ x z l s c$	9.23	-8.09	-0.004	1.85
δ $Γ y y f f t$ -δ $Γ y y l s c$	36.76	-10.09	-0.73	6.55
δ $Γ y z f f t$ -δ $Γ y z l s c$	9.77	-5.33	-1.16	2.09
δ $Γ z z f f t$ -δ $Γ z z l s c$	11.32	-68.22	-3.54	9.67

图5 云南地区重力网实测点位分布及联测路线

Fig. 5 The distribution of measured point and survey route of gravity network in Yunnan province.

表3 研究区域测网的重力数据、仪器参数及平差后的结果统计(单位: μGal)

Table 3 Statistics of the survey points, instrument parameters and gravity data after adjustment of regional survey network(Unit:μGal)

绝对重力测点数	相对重力点数	相对重力测段数	相对重力仪编号及格值参数	点值精度平均值	观测年月
9	249	274	CG5-C169, 1.000089	7.80	2020-03
9	249	274	CG5-C170, 0.999928	7.80	2020-03
9	249	274	CG5-C169, 1.000089	10.30	2021-03
9	249	274	CG5-C170, 0.999928	10.30	2021-03

表4 研究区域测点的高程分布及重力值变化统计

Table 4 The statistics of elevation distribution of survey points and gravity variation in the study area

数据类型	最大值	最小值	平均值	标准差
测点高程/m	3330.00	0.00	1574.00	584.08
重力变化/μGal	30.69	-48.70	-10.27	14.78

图6 云南地区实测地表重力值的年度差分结果(2020年3月—2021年3月)

Fig. 6 Regional gravity changes from Mar. 2020 to Mar. 2021 in Yunnan province.

图7 基于推导公式的最小二乘配置方法解算的云南地区地表重力梯度年度变化模型(2020年3月—2021年3月)

Fig. 7 Annual variation model of surface gravity gradient in Yunnan province by using the least squares configuration method based on the derived data from Mar. 2020 to Mar. 2021.

表5 研究区域地表重力梯度年度变化模型量值统计(1E=1×10-9s-2)

Table 5 Statistics of the annual variation of surface gravity gradient model in the study area(1E=1×10-9s-2)

重力梯度年度变化	最大值/E	最小值/E	平均值/E	标准差/E
ΔΓ_xx	52.36	-76.26	2.91	15.57
ΔΓ_xy	78.75	-42.00	0.47	11.79
ΔΓ_xz	74.59	-89.26	-2.62	22.09
ΔΓ_yy	49.36	-67.62	2.64	13.61
ΔΓ_yz	82.68	-2.27	-4.92	31.99
ΔΓ_zz	64.65	-49.35	-5.55	17.23

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考虑地表起伏的不规则区域重力梯度场模型构建方法——以云南地区为例

STUDY ON THE REGIONAL GRAVITY GRADIENT FIELD MODELING IN IRREGULAR AREA WITH CONSIDERATION OF SURFACE FLUCTUATION: A CASE STUDY OF YUNNAN PROVINCE

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