Segmented Sampling Least Squares Algorithm for Green's Function of Arbitrary Layered Soil

Influence analysis of calculated horizontally layered soil parameters on grounding parameters

“…The size of current measurement electrodes is much smaller than the entire soil space, C 1 and C 2 can be approximated as the point sources. The position of C 1 is taken as the origin and a cylindrical coordinate system is established, the apparent soil resistivity is $$${\rho }_{\mathrm{m}}=\frac{4\pi D}{1+\frac{2D}{\sqrt{{D}^{2}+4{h}_{0}^{2}}}-\frac{D}{\sqrt{{D}^{2}+{h}_{0}^{2}}}}\left(\varphi \left(D,{h}_{0}\right)-\varphi \left(2D,{h}_{0}\right)\right)$$$ where ρ m is the measured apparent soil resistivity, $$\varphi \left(D,{h}_{0}\right)$$ and $$\varphi \left(2D,{h}_{0}\right)$$ are the measured potential of electrodes P 1 and P 2 [20]. Besides, the h 0 is usually much less than D , so the Equation () can be further simplified as: $$${\rho }_{\mathrm{m}}=2\pi D(\varphi (D,0)-\varphi (2D,0))$$$ …”

“…The solution method of $$\varphi (D,0)$$ is $$$\varphi (D,0)=\frac{{\rho }_{1}}{2\pi }\int \nolimits_{0}^{\infty }{\alpha }_{1}{J}_{0}(\lambda D)\mathrm{d}\lambda $$$ where, $${\alpha }_{1}$$ is the integral coefficient that is related to the soil parameters $${\rho }_{1},{{\cdots}},{\rho }_{n},{h}_{1},{{\cdots}},{h}_{n-1}$$, the detailed solution process of $${\alpha }_{1}$$ is shown in Ref. [20, 21]. The complex image method is used to process Equation (), and the processed results are substituted into Equation () [22]: $$${\rho }_{\mathrm{c}}{\,=\,\rho }_{1}D\sum\limits _{j=1}^{2n}{a}_{j}^{2}{\left({b}_{j}^{2}+{D}^{2}\right)}^{-\tfrac{1}{2}}$$$ where $${\rho }_{\mathrm{c}}$$ is the calculated apparent soil resistivity, a j and b j are coefficients of the complex image method, and n is the number of soil layers.…”

“…where ρ m is the measured apparent soil resistivity, φðD; h 0 Þ and φð2D; h 0 Þ are the measured potential of electrodes P 1 and P 2 [20]. Besides, the h 0 is usually much less than D, so the Equation ( 1) can be further simplified as:…”

“…where, α 1 is the integral coefficient that is related to the soil parameters ρ 1 ; ⋯; ρ n ; h 1 ; ⋯; h n−1 , the detailed solution process of α 1 is shown in Ref. [20,21]. The complex image method is used to process Equation (3), and the processed results are substituted into Equation (2) [22]:…”