Figure 1

Layer $M$ contains the source point ${\mathbf{r}}^{\prime }=(x^{\prime },y^{\prime },z^{\prime })$ and layer $L$ contains the observation point $\mathbf{r}=(x,y,z)$. The dipole source $\mathcal{L}$ can be either electric or magnetic.

Figure 2

The incident modes ($i$,I and $i$,II subscripts), Type I/II reflected modes due to the incident Type I ($s$I,I and $s$I,II subscripts) and Type II modes ($s$II,I and $s$II,II subscripts), and Type I/II transmitted modes due to the incident Type I ($t$I,I and $t$I,II subscripts) and Type II modes ($t$II,I and $t$II,II subscripts) are shown.

Figure 3

Schematic depicting the canonical three-layer medium for which the corresponding GRM and 3TM, associated with down-going incident fields in region 1^′, are calculated.

Figure 4

Typical $k_x$ plane features present when evaluating Eq. (2.13). “Radiation BC Map” and “Program BC Map” refer to the branch cuts associated with the radiation condition at infinity and the computer program's square root convention (respectively). The encircled “X” symbols represent the branch points and the red “X” symbols represent slab- and interface-guided mode poles. For $K$ extrapolation intervals used, the red contour represents the integration path extending to $k_x=\pm {\xi }_{K+1}$.

Figure 5

Phase-apparent resistivity log comparison with Fig. 2 of Ref. [3] (homogeneous medium): $R_h=10\Omega \mathrm{m},\beta =0^{\circ },f=2\mathrm{MHz},L_1=25\mathrm{in},L_2=31\mathrm{in}$. In Figs. 5–5 the respective tool dip angles are as follows: $0^{\circ },{30}^{\circ },{45}^{\circ },{60}^{\circ },{75}^{\circ },{90}^{\circ }$.

Figure 6

Magnitude-apparent resistivity log comparison with Fig. 3 of Ref. [3] (homogeneous medium): $R_h=10\Omega \mathrm{m},\beta =0^{\circ },f=2\mathrm{MHz},L_1=25\mathrm{in},L_2=31\mathrm{in}$. In Figs. 6–6 the respective tool dip angles are as follows: $0^{\circ },{30}^{\circ },{45}^{\circ },{60}^{\circ },{75}^{\circ },{90}^{\circ }$.

Figure 7

Apparent conductivity log comparison with Fig. 2 of Ref. [1] (homogeneous medium). $\kappa =\sqrt{5},R_h=1\Omega \mathrm{m},\beta =0^{\circ },f=25\mathrm{kHz},L=1\mathrm{m}$.

Figure 8

Apparent conductivity log comparison with Fig. 3 of Ref. [1]. $\left\{{\kappa }_n\right\}=\sqrt{5}$ and $\left\{{\alpha }_n\right\}=\left\{{\beta }_n\right\}=0^{\circ }$ in all beds; $f=25$ kHz, $L=0.4$ m, ${\sigma }_h=$$\{$1.0, 0.1, 1.0, 0.1, 1.0, 0.1, 1.0, 0.1, 1.0, 0.1, 1.0, 0.1, 1.0$\}$S/m, $z_B=$$\{$0.0, 0.2, 4.2, 4.7, 8.7, 9.7, 13.7, 15.7, 19.7, 22.7, 26.7, 31.7$\}$m.

Figure 9

Magnitude-apparent resistivity log comparison with Fig. 6 of Ref. [2]: ${\kappa }_1=1,{\kappa }_2=\sqrt{20},R_{h1}=2\Omega \mathrm{m},R_{h2}=0.5\Omega \mathrm{m},{\beta }_2=0^{\circ },f=2\mathrm{MHz},L=40\mathrm{in},z_B=0\mathrm{ft}$. In Figs. 9–9 the respective dip angles of the bottom formation are $0^{\circ }$ and ${60}^{\circ }$.

Figure 10

Magnitude-apparent resistivity log comparison with Fig. 7 of Ref. [2]: ${\kappa }_1=1,{\kappa }_2=\sqrt{20},R_{h1}=2\Omega \mathrm{m},R_{h2}=25\Omega \mathrm{m},{\beta }_2=0^{\circ },f=2\mathrm{MHz},L=40\mathrm{in},z_B=0\mathrm{ft}$. In Figs. 10–10 the respective dip angles of the bottom formation are $0^{\circ }$ and ${60}^{\circ }$.

Figure 11

Magnitude-apparent resistivity log comparison with Fig. 8 of Ref. [2]: ${\kappa }_1=1,{\kappa }_2=\sqrt{20},R_{h1}=100\Omega \mathrm{m},R_{h2}=0.5\Omega \mathrm{m},{\beta }_2=0^{\circ },f=2\mathrm{MHz},L=40\mathrm{in},z_B=0\mathrm{ft}$. In Figs. 11–11 the respective dip angles of the bottom formation are $0^{\circ }$ and ${60}^{\circ }$.

Figure 12

Magnitude-apparent resistivity log comparison with Fig. 9 of Ref. [2]: ${\kappa }_1=1,{\kappa }_2=\sqrt{20},R_{h1}=100\Omega \mathrm{m},R_{h2}=25\Omega \mathrm{m},{\beta }_2=0^{\circ },f=2\mathrm{MHz},L=40\mathrm{in},z_B=0\mathrm{ft}$. In Figs. 12–12 the respective dip angles of the bottom formation are $0^{\circ }$ and ${60}^{\circ }$.

Figure 13

Magnitude-apparent resistivity log comparison with Fig. 11 of Ref. [2]: ${\kappa }_1={\kappa }_3=1,{\kappa }_2=5,R_{h1}=R_{h3}=40\Omega \mathrm{m},R_{h2}=2\Omega \mathrm{m},{\beta }_2=0^{\circ },f=2\mathrm{MHz},L=40\mathrm{in},z_B=\{2.5,-2.5\}\mathrm{ft}$. In Figs. 13–13 the respective dip angles of the center formation are as follows: $0^{\circ },{45}^{\circ },{60}^{\circ },{70}^{\circ },{80}^{\circ },{90}^{\circ }$.

Figure 14

Magnitude-apparent resistivity log comparison with Fig. 13 of Ref. [2]: ${\kappa }_1={\kappa }_3=1,{\kappa }_2=5,R_{h1}=R_{h3}=40\Omega \mathrm{m},R_{h2}=2\Omega \mathrm{m},{\beta }_2=0^{\circ },f=2\mathrm{MHz},L=40\mathrm{in},z_B=\{10,-10\}\mathrm{ft}$. In Figs. 14–14 the respective dip angles of the center formation are as follows: $0^{\circ },{45}^{\circ },{60}^{\circ },{70}^{\circ },{80}^{\circ },{90}^{\circ }$.

Figure 15

Field component intensities from a $y$-directed horizontal electric dipole (HED), which is radiating at $f=8$ MHz (${\lambda }_o=37.5$m), centered at the origin, and supported on a grounded dielectric substrate 4${\lambda }_o$ thick with free space above. The substrate's dielectric constant is ${\epsilon }_r=3.3(1+0.01i)$, while $y-y^{\prime }=0$m and $z-z^{\prime }=3$ m. Only $|H_z|$ reference data were published in Ref. [39].

Figure 16

Convergence towards the solution comprising the field contribution from “Region 1”. The reference field values are computed using $p=31$ for both figures, as well as

$-{log}_2\left(h\right)=9$ for Fig. 16 and $-{log}_2\left(h\right)=11$ for Fig. 16. The reference field values computed for Figs. 16 and 16 use different $h$ because in the latter scenario, $H_z$ converges more slowly and thus necessitates smaller $h$ values in the nonreference field results to show a meaningful decay in error. As a result, one also requires an even smaller $h$ for the reference field result from which the relative error is computed.

Figure 17

Convergence towards the solution comprising the field contribution from “Region 2”. The reference field values are computed using $\text{LGQ}=30$ for both figures, as well as $B=150$ for Fig. 17 and $B=1000$ for Fig. 17. The reference field values computed for Figs. 17 and 17 use different $B$. This is because in the latter scenario, as can be observed, $H_z$ converges more slowly; indeed, while $H_z$ in case two levels off more rapidly than in case one, it fails to reach accuracy near to machine precision within the same range of $B$ exhibited for both cases. Thus similar reasoning applies as that behind using smaller $h$ for the reference and nonreference field results in Fig. 16 [versus Fig. 16].