基于非结构化有限元的三维井地电阻率法约束反演

王智, 王程, 方思南

Constraint inversion of three-dimensional borehole-to-surface resistivity based on unstructured finite element

WANG Zhi, WANG Cheng, FANG Si-Nan

GN-CG反演算法

1) 给定初始模型m₀与参考模型m_ref,设置收敛系数

ε

,正则化因子

λ

和惩罚因子

μ

,给定光滑度矩阵W_m;
2) 开始GN-CG 迭代, 设k=1,2,…,N_maxGN;
3) 计算

Δ d_{k} = d_{o b s} - F (m_{k})

,并计算

r_{0} = - g_{k}

;

g_{k} = - J_{k}^{T} W_{d}^{T} W_{d} Δ d_{k} + λ W_{m}^{T} W_{m} (m - m_{r e f}) + μ [m i n (m - m_{m i n}, 0) - m i n (m_{m a x} - m, 0)]

;
4) 开始CG 迭代, 令

Δ m = 0, p_{0} = r_{0}

,设i=1,2,…,N_maxcg;
4-1)计算H_k=

J_{k}^{T} W_{d}^{T}

W_dJ_k+λ

W_{m}^{T}

W_m+μW_c, 得到H_kp_i;
4-2)计算

t_{i} = \frac{r_{i}^{T} r_{i}}{p_{i}^{T} H p_{i}}

;
4-3)计算

Δ m_{i + 1} = Δ m_{i} + t_{i} p_{i}

;
4-4)计算

r_{i + 1} = r_{i} - t_{i} H_{k} p_{i}

;
4-5)计算

β_{i}^{F R} = \frac{r_{i + 1}^{T} r_{i + 1}}{r_{i}^{T} r_{i}}

;
4-6)计算

p_{i + 1} = r_{i + 1} + β_{i} p_{i}

;
5) CG迭代结束, 求得

Δ m

,根据Wofe-Powell条件(式12)进行一维搜索得到最优步长α_k,更新模型:

m_{k + 1} = m_{k} + α_{k} Δ m_{k}

;
6) 如果RMS<

ε

则停止GN-CG迭代,输出m_k,否则令k=k+1 转到步骤3);