UDC 550.837

GEOPHYSICS

B. S. ENENSHTEIN

INTERPRETATION OF THREE-LAYER CURVES OF FREQUENCY ELECTROMAGNETIC SOUNDINGS OF TYPES $A$ AND $H$

(Presented by Academician A. V. Peive, September 23, 1965)

In (¹) a method is described for interpreting three-layer curves of electromagnetic soundings of types $K$ and $Q$, obtained with large spacings $r$, based on the use of a curve-transformation procedure. This procedure will also be used for interpreting three-layer curves of types $A$ and $H$, obtained with large $r$.

However, as is shown below, in order to ensure a satisfactory interpretation of three-layer curves of types $A$ and $H$, it is not sufficient to use only the transformation procedure, as for curves of types $K$ and $Q$; it is necessary, in addition, to make use of the symmetry property of frequency soundings, which consists in the fact that curves of type $H$ with parameters $\rho_1; h_1; h_2/h_1=\nu_2^H; \rho_2/\rho_1=\mu_2^H; \mu_{13}^H=\rho_3/\rho_1$ correspond to curves of type $K$ reflected in mirror fashion with respect to the horizontal axis $\rho=1$, with parameters $\rho_1; h_1; \nu_2^K=\nu_2^H/\mu_2^H; \mu_2^K=1/\mu_2^H; \mu_3^K=1/\mu_3^H$. The same holds between three-layer curves of types $A$ and $Q$ (², ³).

The method of transforming three-layer curves of electromagnetic soundings of types $K$ and $Q$, described in (¹), consists in the following: the interpreted curve, representing a section in which the lower bed has a finite resistivity, is replaced by a curve representing the same section but with the lower bed having a resistivity equal to zero. After such a transformation of the curves, it becomes possible to carry out a quantitative interpretation and determine the values of the parameters of the section.

A different situation arises in the interpretation of three-layer curves of type $A$ (Fig. 1, curve $I$) and type $H$. In this case it proves impossible to carry out the interpretation by using only the one procedure developed for three-layer curves of types $K$ and $Q$.

Indeed, if one uses the transformation procedure described in (¹) for curves of types $K$ and $Q$, then with the aid of a three-layer chart of type $A$, constructed according to the same principle as the $K$ chart described in (¹), it is possible to transform curve $I$ into curve $II$, characterizing a geoelectric section with the same parameters of the two upper layers, but with the value $\rho_3 \approx \infty$.

Interpretation of the left branch of curve $I$ or $II$ by the method described in (⁴) makes it possible, from the value $\rho_1 = 12$ ohm·m, taken directly from the left asymptotes of the curves, to determine the value $h_1 = 194$ m.

For the interpretation of the entire curve $II$, with the aim of determining the quantities $h_2$ and $\rho_2$, we choose an arbitrary value $\rho_l$ (for example, 30 ohm·m). We superpose the horizontal axis $(\rho=1)$ of the two-layer chart of type $\lambda_1/h_{1\infty}$ with the value $\rho_l=30$ ohm·m on the ChZ form in such a way that the right branches of curve $II$ and of the chart curve $\rho_2/\rho_1\approx\infty$ (dashed line $III$) coincide. As a result of superposing the right branches of the curves, the abscissa of the chart $\lambda_1/h_1=8$ (dashed line $IV$) coincides with the value $f_1\approx0.55$ Hz of the horizontal axis of curve $II$. Substituting the values $\rho_l=30$ ohm·m and $f_1=0.55$ Hz into the expression $\lambda/h=8=\sqrt{10\rho_l}/h\sqrt{f_1}=\sqrt{300}/8\sqrt{0.55}$, we find $h\approx2920$ m.

If, for interpretation, instead of \(\rho_1 = 30\ \Omega\cdot\mathrm{m}\) one chooses another value, for example \(B\rho_1 = 40\ \Omega\cdot\mathrm{m}\), then we find that, as a result of matching the right-hand branches of the curves, the abscissa of the master curve \(\lambda_1/h_1 = 8\) coincides with another abscissa of the curve \(f_2 = B^{-2/\tan\alpha} f_1\) \((f_2 = 0.41\ \mathrm{Hz})\), where \(\alpha\) is the angle between the right-hand branch of curve \(II\) and the abscissa axis, equal to \(63^\circ 30'\) for \(\rho_3 \approx \infty\). The new value \(h^*\) will be

\[ h^*=\frac{\sqrt{10B\rho_1}}{8\sqrt{B^{-2/\tan\alpha}f_1}} =\frac{\sqrt{B}}{\sqrt{B^{-2/\tan\alpha}}}\, \frac{\sqrt{10\rho_1}}{\sqrt{f_1}} =B^{(\tan\alpha+2)/2\tan\alpha}h =B^k h \]

for \(\alpha = 63^\circ 30'\); \(\tan\alpha = 2\), \(k = 1\), and \(h^* = Bh\).

Thus, when the value \(\rho_1\) chosen for interpretation is changed by a factor of \(B\), the obtained value of \(h\) also changes by a factor of \(B\). Consequently, direct interpretation of a curve of type \(A\) does not make it possible to determine the values \(h_2\) and \(\rho_2\) \((h\) and \(\rho_1)\), but it does make it possible to determine the value \(S\), equal to \(h/\rho_1\). The data presented apply equally to the interpretation of curves of type \(H\).

It seems possible, however, to interpret three-layer FES curves of types \(A\) and \(H\) by the transformation method; but for this purpose it is necessary, as indicated above, to use the symmetry property and transform curves of types \(A\) and \(H\), respectively, into curves of types \(Q\) and \(K\), and then transform them into curves of the analogous type but with the resistivity of the lower layer equal to zero \((\rho_3=0)\). A curve of type \(A\), transformed in this way into a curve of type \(Q\), is interpreted in accordance with the procedure described in (1).

Fig. 1

Let us explain the proposed procedure for interpreting three-layer curves of types \(A\) and \(H\) using a concrete example (Fig. 1, curve \(I\)). We read the value \(\rho_1\) directly from the left asymptote of the curve \((12\ \Omega\cdot\mathrm{m})\). Using the value \(\rho_1 = 12\ \Omega\cdot\mathrm{m}\) and the procedure described in (1), we determine \(h_1 = 194\ \mathrm{m}\). On the FES sheet of Fig. 1 we plot curve \(V\), symmetric to curve \(I\) with respect to the horizontal axis \(\rho_1 = 12\ \Omega\cdot\mathrm{m}\). By this we have transformed the curve of type \(A\) into a curve of type \(Q\). We match the entire curve \(V\) with the most suitable curve, one of the three-layer master curves of type \(Q\) (Fig. 2), and, in accordance with the transformation principle, transfer to the FES sheet the master-curve curve with the same values of \(\rho_1\), \(h_1\), \(h_2\), and \(\rho_2\), but for \(\rho_3 = 0\) (Fig. 1, curve \(VI\))*.

* The master sheet presented in Fig. 2 is constructed according to the same principle as that used to construct the master curves of type \(K\) described in (1). The left group of curves \((1–5)\) corresponds to the values \(\rho_2 = \frac14\rho_1\), \(h_2=\frac12 h_1\), and \(\rho_3 = 0,\ \frac{1}{32},\ \frac{1}{16},\ \frac18,\ \frac14\). The following

We choose for the interpretation an arbitrary value of \(\rho\) (for example, \(6\ \Omega\cdot\text{m}\)) and align the right branch of curve \(VI\) with the curve \(\rho_2/\rho_1=0\) (dashed line \(VII\)) of the two-layer palette in such a way that the ordinate of the palette \(\rho=1\) coincides with the value \(\rho=6\ \Omega\cdot\text{m}\). Using the procedure described in (1), we find that the abscissa of the palette \(\lambda_1/h_1=8\) (dashed line \(VIII\)) coincides with the abscissa of the curve \(f=0.7\ \text{Hz}\). Substituting the values \(\rho=6\ \Omega\cdot\text{m}\) and \(f=0.7\ \text{Hz}\), we obtain

\[ h=\sqrt{10\rho}/8\sqrt{f}=\sqrt{60}/8\sqrt{0.7}=1150\ \text{m}. \]

Since \(h_1=194\ \text{m}\), it follows that \(h_2=956\ \text{m}\) and \(\nu_{12}^{Q}=h_2/h_1=4.8\).

Fig. 2

In accordance with the conditions of the symmetric transformation of the curves, the true value \(\nu_{12}^{A}\) for curve \(II\) will be equal to \(\nu_{12}^{Q}/\mu_{12}^{Q}\), obtained from curve \(VI\). Since the value \(\nu_{12}^{Q}\) is known to us (4.8), it remains to find \(\mu_{12}^{Q}=\rho_2/\rho_1\).

To determine \(\mu_{12}^{Q}\), we use the procedure described in (1). From the known values \(\nu_{12}^{Q}=4.8\) and \(\mu_{13}=0\), we find, among the group of three-layer palettes of type \(Q\), the curve that best coincides with curve \(VI\). Since the palettes differ in the parameter \(\rho_2\), the value of \(\rho_2\) is thereby determined. In our case \(\rho_2\) turns out to be equal to \(\sim 1/4\). Hence we find

\[ \nu_{12}^{A}=\nu_{12}^{Q}/\mu_{12}^{Q}=4.8/0.25=19.2;\qquad \mu_{12}^{A}=1/\mu_{12}^{Q}\approx 4. \]

\[ h_2=h_1\nu_{12}^{A}=194\cdot 19.2\approx 3720\ \text{m};\qquad \rho_2=\rho_1\mu_{12}^{A}=12\cdot 4=48\ \Omega\cdot\text{m}. \]

The described procedure for interpreting curves of type \(A\), in which palettes of type \(Q\) are used, is equally suitable for interpreting curves of type \(H\) using palettes of type \(K\).

Owing to the different resolving power of the electromagnetic sounding method with respect to layer thicknesses and their resistivities, the accuracy of determining the quantities \(h_2\) and \(\rho_2\) by the method described in (1) and in the present paper is not the same. Analysis of three-layer theoretical sounding curves of various types shows that the accuracy of determining the thickness of the second layer \(h_2\) from curves \(K\) and \(Q\) depends on the accuracy with which the transformation of the original curve is performed and can be achieved within 5%. The accuracy of determining \(\rho_2\), however, can on average be achieved to the order of 10%, decreasing for small \(\nu_2\).

The groups of curves of the palette correspond to the same \(\rho_2\) and \(\rho_3\), but to different \(h_2\), namely: the 2nd group \((h_2=h_1)\), the 3rd group \((h_2=2h_1)\), the 4th group \((h_2=3h_1)\), the 5th group \((h_2=5h_1)\), the 6th group \((h_2=9h_1)\), and the 7th group \((h_2=24h_1)\), to which curves 6–10 correspond. The palettes differ from one another in the value of \(\mu_{12}\).

Since, in the method described, to determine the value of $h_2$ from the $A$ and $H$ curves it is necessary first to find the value of $\varphi_2$, the accuracy of determining $h_2$ from these curves cannot be greater than 10%.

In conclusion we note that the methods of interpretation developed in the present article and in (1) are also valid for curves of magnetotelluric soundings and for the left-hand parts of frequency-sounding and field-build-up curves obtained with small spacings $(r = 3h - 4h)$.

Geological Institute
Academy of Sciences of the USSR

Received
22 IX 1965

REFERENCES

1. B. S. Epshtein, Dokl. Akad. Nauk SSSR, 168, No. 4 (1966).
2. L. L. Vanyan, Zhurn. prikl. geofiz., vol. 21 (1957).
3. M. V. Kalmykov, N. P. Vladimirov, Izv. Akad. Nauk SSSR, ser. geofiz., No. 4 (1961).
4. B. S. Epshtein, Izv. Akad. Nauk SSSR, ser. geofiz., No. 9 (1962).