Coefficient Matrix Properties of Downward Continuation for Potential Fields and Barzilai-Borwein Downward Continuation Method
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摘要: 位场的向下延拓不仅仅能够提高地球物理数据解释的可靠性，在导航方面也有着重要的作用. 为了进一步提高计算精度和速度，提出了位场向下延拓的Barzilai-Borwein （BB）法. 首先证明了位场向下延拓的系数矩阵为对称的双重Toeplitz系统矩阵（block-Toeplitz-Toeplitz-block，BTTB）；其次，假定该系数矩阵为正定的条件下，采用BB法迭代求解下延方程组，并约束其迭代步长确保算法收敛；最后，分别通过理论模型无噪声数据和实际资料对BB法进行检验，并与积分迭代法进行对比. 结果表明：理论模型验证时，同一收敛精度条件下，BB法的计算速度是积分迭代法的2倍以上；实际资料检验时，在相同计算次数下，BB法与积分迭代法的平均相对误差分别为6.1%与7.7%.
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关键词:
- 向下延拓 /
- 系数矩阵 /
- Barzilai-Borwein法 /
- 收敛性
Abstract: Downward continuation of potential fields can not only improve the reliability of geophysical data interpretation, but also plays an important role in navigation. In order to further enhance the calculation accuracy and speed of downward continuation, the Barzilai-Borwein (BB) downward continuation method is proposed. First, the coefficient matrix of downward continuation for potential fields is proved to be a symmetric block-Toeplitz-Toeplitz-block matrix (BTTB). Next, assuming that the coefficient matrix is positive definite, the BB method is used to solve the equations of downward continuation, and the iterative step is limited to ensure the convergence of the algorithm. Finally, the BB method is validated by noise-free data of the theoretical model and practical data, and compared with the integral iterative method. The results show that under the same convergence accuracy, in the case of the theoretical model, the calculation speed of BB method is more than 2 times that of integral iteration method. In the case of practical data, the average relative errors of BB method and integral iteration method are 6.1% and 7.7%, respectively.-
Key words:
- downward continuation /
- coefficient matrix /
- Barzilai-Borwein method /
- convergence
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图 1 理论重力异常与不同方法向下延拓结果对比
Figure 1. Theoretical gravity anomaly compared with downward continuation results of different methods
图 3 不同高度面上的磁异常等值线（单位：nT）
Figure 3. Magnetic anomaly isolines at different planes (unit： nT)
图 4 理论磁异常向下延拓后的计算结果（单位：nT）
Figure 4. Calculation results of theoretical gravity anomaly after downward continuation (unit： nT)
表 1 三维组合圆球型磁异常体参数
Table 1. Magnetic anomaly parameters of 3D combination sphere
模型编号 ${x_0}/$km ${{{y}}_0}/$km ${ { {h} }_0}/$km ${ { { {\textit{r} } } }_0}/$km ${M_0}/$（A•m−1） ${I_0}/$（°） ${A_0}/$（°） $I'/$（°） $A'/$（°） 1 13.5 6.5 2.5 0.5 1.0 50 60 30 60 2 6.5 6.5 2.5 0.5 1.2 50 60 30 60 3 10.0 10.0 2.5 0.5 1.0 50 60 30 60 4 13.5 13.5 2.5 0.5 1.0 50 60 30 60 
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