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On the electromagnetic detection of the thickness of a coal or lignite seam with slate backing
On the Electromagnetic Detection of the Thickness of a Coal or Lignite Seam with Slate Backing bye. N.SAHALOS Department
and G.KYRIACOU
of Electrical
Engineering,
University of Thrace,
Xanthi,
Greece
ABSTRACT : The use of two small current loops for probing the thickness of a two-layer earth structure is discussed. The thickness and the conductivity contrast can be found by zeroing the secondary magnetic3eld on the axis of the receiving loop. The inversionfrom the measured data to the layer thickness proves that our procedure is simple. The whole method can be incorporated into an automated mining machine for sensing a coal or lignite seam with a slate backing.
I. Zntroduction In the development of better mining machines, it is important to have sensors which are capable of detecting the thickness of a mineral. In our case, we are talking about sensing a thin slab of coal or lignite, say a maximum of 1 m in thickness, backed by a reflecting material such as draw slate. Most of the methods previously reported are based upon radiation in front of the coal or lignite surface. The radiation problem of an electric or magnetic dipole in the presence of the earth is not new. Sommerfeld (1) in his classic 1909 paper presented the problem of a vertical electric dipole (VED) located at the interface separating two regions. Various investigators wrote papers on VED, horizontal electric dipole (HED), vertical magnetic dipole (VMD) or horizontal magnetic dipole (HMD) located in a multiple-layered earth, Bafios (2) presented a complete and systematic investigation for VED, HED, VMD and HMD by using the saddle-point method of integration and by deriving many expressions for a variety of ranges and parameters. An excellent extension on multiple-layered earth was produced by Wait (3, 4). Wait’s work can be applied to the detection of fragmented rock over a sedimentary rock base, underground water and mineral strips. Electric and magnetic dipoles are used as models of small current elements and loops. The input impedance of one dipole or the mutual coupling between pairs of dipoles can give the electric characteristics and the thickness of the ground. The induction methods, characterized by the use of dipoles in geophysical prospecting, have been given by Wait (4) in his overview of the most actively used techniques. These techniques are separated in two categories : the surface and the subsurface. In subsurface techniques the electrodes are placed below the surface of the ground and resistivity in the vicinity of the hole is measured.
OThe Franklin Institute 00164032/85$3.00+0.00
83
J. N. Sahalos and G. Kyriacou
Surface methods include magnetotelluric resistivity, induced polarization, wave tilt and radio wave methods. This study deals with the surface methods and discusses the measurement of the two small current loops in a two-layer earth. Various authors have dealt with this problem : see Maley (5), Chang (6) and Chang and Wait (7). The mutual coupling of two small current loops is well detailed by Ralston and Wait (8) and Wilson (9). However, these works could not lend themselves to a simple automated mining machine. Wilson showed a more desirable approach to finding the layer thickness by using a mini-computer incorporated into a mining device. In his inversion scheme, Wilson tried to find the layer thickness. However, we believe that the high cost of the method made it unfeasible. Our procedure generates theoretical curves, and matches the zero coupling data with the thickness of the layer. In order to make the method more compatible with an automated mining machine, we use a least square method which gives the thickness and the conductivity contrast of the layers. II. Formulation
A small horizontal current loop is located at a height h above an air-earth interface. The current distribution is essentially uniform and thus can be represented as a vertical magnetic dipole (VMD) with moment, i.e. m = ilds.
(1)
The configuration to be analyzed is shown in Fig. 1. The wavenumbers of a plane wave in a region are kj = wJ(P&J
(2)
d 1
I
(~zlQ*Jo)
FIG. 1. A VMD above a two-layer half-space model. Journal
84
of the Franklin Institute Pergamon Press Ltd.
Thickness of Coal or Lignite Seam where (3) E, is the relative dielectric constant and .sOis the free space permittivity. The electric and magnetic fields are formulated in terms of the magnetic Hertzian vector potential II*. The total Hertzian vector fjr* is the sum of the primary field If*P and the secondary field i=f*’ composed of reflections and transmissions due to the presence of loose earth. i’i*j’ has the i direction and satisfies the following equation : (V” + k;)ll*P = -Ids@ The solution
of Eq. (4) has the following n*P=
_._
Ids
m
J47L s 0
-X0). :
expression
e-juolz-hl J,(ap)c
Using Eq. (5) as a guide, we seek the general solution Ids -jG
s
.Ids
HI* = I
-J471
(4)
z> 0
> z > -d
s0
ada T(a) @2zJ0(ap) -,-d>z. uo uj are the transverse
wavenumbers
uj =
(6)
defined by
(ky -a2)1’2 -j(a’
(5)
of the form (9) :
m o Ce- bolz-hl +R(a) e- jub +“1 JO(ap) $, co [a(a)~“+B(a)e-‘““1J,(ap)~,O
da_
R,(k$ < a2
- kT)l12 R,(kT) < a2
From the continuity of the tangential interfaces, we can find that
electric
j = o 1 2 ’ ’ ’
and magnetic
(7) fields at the two
(8) A(a) =
B(a)
(l+~)+(l-~)R(a)e~juo”
= (l-~)+(l+f+(a)e-‘oh
and qa)
=
e
-juoh
. e_iwd
{[(l+~)+(l-$)R(a)]eplY1d
Vol. 320, No. 2, pp. 83-101, August 1985 Printed in Great Britain
85
J. N. Sahalos and G. Kyriacou
The magnetic field can be found from II* via the following equation : If = V(V - II*) + k%* J *
(12)
We know that the primary field II*p is the free space field due to a magnetic dipole, and can be expressed by n*p = E
eeikoRO
4n
(13)
R,
where R, = [p2+(z-h)2]1/2. From (12) and (13) the total magnetic field can be expressed by
+p p OS
omR(a)e- iuo(zfhfJ,(ap)a2 da
H,=g(z-h)y
3jk, e-ikoRo -!- +--R; R$
+z p OS
om R(a)e-
kg R; )
>
0
R(a) e-iuo(z+h)Jo(ap)$ s 0
da
(14)
1 _!_+!!k+ 1
iua(z+h)J,(ap)a2 da
co -.i
1
1
0
.
I
Our particular scheme is based on the definition of the mutual coupling between two small current loops. Suppose that we have one horizontal current loop and one loop in an arbitrary direction (Fig. 2). Both loops are placed at a height h above the air-earth interface and the separation between the loops is po. To have a zero secondary coupling between the two loops we must have a zero secondary magnetic field on the axis of the current loops. If we assume that the horizontal is the transmitting loop then the components of the total transmitting to the other loop are H”z .
H; Hz=:
(15) a3'l
J,(ap,) u. da R(a) e- jZuoh
and
f&=H”,=~
s cc
R(a) e- j2”OhJ,(apo)a2 da.
0
From H, we can subtract Hfl because this shows the primary field. The subtraction in practice will be by zeroing the ammeter ; an indication that the two loops are parallel 86
Journal
of the Franklin Institute Pergamon Press Ltd.
Thickness of Coal or Lignite Seam
-
-f h
FIG. 2. The nonparallel
loop configuration.
on the same height and very far from the earth. When we put the loops near the earth, the ammeter will give an indication other than zero. This creates the H”, field. By swivelling the receiving loop until we find a null position there results l? = Hi cos 8 + Hi sin 8 = 0
(17)
or R = -tan
8 = H”,/H;.
(18)
It is easy to measure the angle 0 and as we know we can find the ratio of H”,/H”, as a function of the thickness d and the contrast conductivity. Equations (15Hl8) agree with the results given by Wilson (9). For the quasi-static case we suppose that k, -+ 0, and as we can see in Refs. (8) and (9), R is given in the following form : (Gl)(A2+1)-1’2+(B*+l)-i/2 R = (l-K)A(A2+1)-1/2-B(B2+1)-1/2+K where K is the conductivity
contrast
A = 2(h+d)/p
(1%
~,/a,, and
B = 2h/p.
(20)
Equation (19) is used in (9) as the first approximation of the value of R. We must say that for low frequencies R converges nicely to its quasi-static limit. In this problem we use the maximum “low frequency” because the magnitude of the field components are proportional to the frequency. Hence, the lower frequency needs more sensitive equipment. We also found that R is “independent” of frequency for loop height h = O-O.4 m and frequency f = &500 kHz. This is shown in Figs. 3 and 4 where we have the amplitude of R as a function of frequency for various loop heights and two different Vol. 320, No. 2, pp. 83-101, August 1985 Printed in Chat Britain
87
J. N. Sahalos and G. Kyriacou
layer thicknesses. From Figs. 3 and 4 we can see that a frequency of 500 kHz would be a. practical choice. To find the thickness d we would start from the quasi-static value of R. By using the inverse method given in (9) we can find d. The only problem was the high cost of computer time which made the whole method impractical. An easy method to use can be made if we get analytic results of R as a function of d,
D=0.6
m
-17.
3a
25
15
10
5
1
,
3
I
4
I
I
I
I
7
8
9
10
FREQ
(HZ)
FIG. 3. Ratio R as a function of frequency for h = 0.1,0.4,0.6 and 1.1 m for D = 0.6 m.
88
Journal of the Franklin
Institute Pergamon Press Ltd.
Thickness of Coal or Lignite Seam
35
KHz
30
25
c 5 II Dz 20
F-
L
h=o.4
15
10
h=o.lm
5
1
I
-r 2
3
4
5
II
I
,
I
I
I
6
7
6
9
hii (f)
FREQ
(HZ)
FIG. 4. Ratio R as a function of frequency for h = 0.1,0.4,0.6 and 1.1 m for D = 1.1 m. h and K. These results, with the help of the least square simple expression of R.
method,
can produce
a
III. Numerical Results One of the main decisions to be made before starting the computation is. the definition of the separation distance (po) between the two loops. We calculate the Vol. 320, No. 2, PP. 83-101, August 1985 Printed in Great Britain
89
J. N. Sahalos and G. Kyriacou ratio R and the angle 6 of the receiving loop for the quasistatic case vs the layer thickness (d) and the height (h). From the results, we found that for h < 0.4 m ad/ah and M/ad increase as the separation distance decreases. From a practical point of view we must have the maximum possible difference in the angle f3for two different heights. To achieve our purpose we choose a separation distance equal to 0.2 m. This distance gives the best combination of discrimination and construction ability. For coal and lignite seams we would expect conductivity ai in the neighborhood of 1O-3 mhos/m. We use at first a conductivity equal to 10e3 mhos/m and we make all the calculations for the ratio R. In Fig. 5 we can see the ratio R as a function of K for h = 0.1-1.1. Figure 5 is given for a layer thickness equal to d = 0.1 m. Similar figures can be taken for any value of d. From a series of results we can see that there is a linear relation between R and K for h = 0.1-0.4 m. We suppose that R(K) = A+B.K (21) where A, B must be functions of d and h. To find the form of those functions we make plots of A(h) and B(h) vs h for given values of d. Figure 6 shows the above plots. As we can see A(h) and B(h) are of the following form : A(h) = a, + a,h
(22)
B(h) = bl + b,h + b,h.
The coefficients ai and bi are functions of d and are given in Figs. 3 and 8. Both figures show that ai and hi(d) must be of the form : f(d) = cl +c,d+c,d’.
(23)
We note that a, and b, are related with d in a third degree polynomial expression. For the approximation (23) and the third degree polynomial we find that the accuracy is approximately the same. From the above discussion the expression R(d, K, h) is of the form R(d, K, h) = a,(d) + a,(d) * h + [b,(d) + b,(d) * h + b,(d). h’]K. By the least square method
we obtain
R(d, K, h) = 1.398 -4.079d
(24)
values of a, bi and we have
+ 4.837d’ + (17.848 + 23.278d
- 22.982d2)h2 + [O.11+ 0.0054d - 0.063d2 + (- 0.654 + 4.845d - 2.867d’)h + (0.753 - 8.959d + 6.685d2)h2]K.
(25)
Equation (25) gives the ratio R as a function of d, h and K for a given conductivity ol. Since the conductivity is in the neighborhood of 10m3 mhos/m we can make calculations for the ratio R in the two boundary values of the conductivity. Thus in Fig. 9 we have R vs K for a given h and different d. From a series of results similar to those of Fig. 9, by using the least square method, we can find that the real ratio R, R,, is given by R,(d, K,h,o,) 90
= [l +62.64(10-3-~a,)]R(d,
K, h).
(26)
Journal of the Franklin Institute Pergamon Press Ltd.
Thickness
of Coal or Lignite Seam
h=l .I
hz.9 h= .a h=
.7
-h=.6
hz.5 h=
.4
h=,s
hz.2
5---
hz.1
0,
4
5
6
7
6
9
10
KZ=
02
FIG. 5. Ratio R as a function of contrast conductivity K for h = 0.1-1.1 m and D = 0.1 m.
IV. A Scheme to Extract the Layer Thickness The definition of d and K is given if we know two different different heights h and the exact value of ol. Vol. 320, No. 2, pp. 83-101, August Printed in Great Britain
1985
values of R for two
91
J. N. Sahalos and G. Kyriacou
, dzl.1
t
1.
d= d-.9
BP
A(h:
dz.8 dx.3 d= .2
.:
d= .7
dz1.l d= .I d=
8
.:
.6
d=s
d=
6
.4
dz.3
.l
\ dz.2
\ 1
I
3
8
.l
.2
.3
.4
.a h(m)
.l
d=.l .2
.3
.4
h(m)
FIG. 6. A(h) and B(h) coefficients vs h for different d.
As we said before we want a simple procedure which can be applied in a microcomputer. If we try to solve for d and K a system of two equations as in (26), there results a fourth degree equation which can only be solved numerically. To overcome this difficulty we use Figs. 10-12 which represent R(K) vs K for h = 0.1,0.2 and 0.3 m with d from 0.1 to 1.1 m. These figures give the ratio R for (rr = 10e3 mhos/m. We measured the values of R, for h = 0.1,0.2 and 0.3 m. In dividing those values by [l +62.64(10e3 -a,)], we find the value of R which must correspond to c1 = 10m3 mhos/m. With the help of Figs. l&12 we found an initial pair of d, and K,. We define as R, and R, the divided values for hl = 0.1 m and h, = 0.2 m. By using the initial values da, K, from Eq. (25) we can find two theoretical ratios R which for 92
Journal of the Franklin Institute Pergamon Press L.td.
Thickness
of Coal or Lignite Seam
2.5
1.5
al,bl,bz
k t 1
.5
.9 d (ml
FIG. 7. The coefficients a,, b,, b, as functions of the layer thickness d. h, and h, will be R; and R;. The ratios will be given from (25) in the following form :
, R; = “l+Bldo+y1K,+61do~io+e,d~+r,K,d~ R’, = a2 +&do +y2K, + S,d,K,, + E2d? + c2K0d$
I
(27)
The coefficients cli,fib yi, 6, q, ci are functions of the height hi and are given by Eli= 1.3998 + 17.848 hi /Ii = - 4.0794 + 23.278 hi yi = 0.11-0.6544
hi + 0.754 hf
6i = 0.0542 + 4.8454 hi- 8.9592 hi2 ei = 4.8368 -22.982
I
(28)
hi
li = - 0.0626 - 2.867 hi + 6.685 hf. Vol. 320, No. 2, pp. 83-101, August 1985 Printed in Great Britain
93
J. N. Sahalos and G. Kyriacou We write that Xi = pi + 6iK, + 2&id, + 21;id,K, yi = yj + 6id, + [id; (29)
ARi = R,-R; i = 1,2. For a first degree approximation
I the exact values of d and K are
d = d,+Ad
K = K,+AK.
and
(30)
(a)
24
2:
2:
a2 2’
I
I
I
I
I
I
.2
,3
9
,5
.6
,7
I
,S
1
I,
.Q
1.
1.1
dmv
FIG. 8. (a) 94
Journal of the Franklin Institute Pergamon Press Ltd.
Thickness of Coal or Lignite Seam
(b)
FIG. 8. (a) The coefficient
a, as a function of the layer thickness d. (b) The coefficient function of the layer thickness d.
b, as a
By using Eqs. (25), (27) and (29), we can find that
(31) and
AK=
AR, -x,Ad Yl
.
Equations (30)-(32) give the “exact” values for the thickness contrast K. Vol. 320, No. 2, pp. X3-101, August Printed in Great Britain
(32)
d and the conductivity
1985
95
J. N. Sahalos and G. Kyriacou hzo.1 Cm> -*-.-
0,=0.5x10-3
mho/m
-_---0,=10-3 CT,-1.5 x10-3
/
I /’
dd.1
IRrl t
Cm)
d=O.6("')
c
4
I
7
FIG.
Example 1 We suppose
96
9
11
13
I
15
K = q/a,
9. The ratio R vsK for h = 0.1 m and different conductivities cl.
that we are on a two-layer
earth and measure
R, =
6.18
for h, = 0.1 m
R, =
8.12
for h, = 0.2 m
R3 = 10.33
for h3 = 0.3 m.
the following
R:
Journalof the FranklinInstitute PergamonPressLtd.
Thickness of Coal or Lignite Seam
9
-25-
k0.1
m
500
KHZ
Ul=
10 -3
U/m
8
7 I?ii
1 6
51-I
4I-
S-
I 5
dI
6
7
8
9
I 10
K=+
FIG. 10. Ratio R as a function of contrast conductivity K for d = 0.1-1.1 m and h = 0.1 m.
We use Figs. l&12
and find the following
pairs :
for h = 0.1 m
(d,,K,)
= (1.1,7.75),(1.0,9.5)
for h = 0.2 m
&,I&)
= (1.1,7.65),(1.0,8.5),(0.9,9.25)
for h = 0.3 m
(d,,K,)
= (1.1,7.5),(1.0,8),(0.8,8.25).
Vol. 320, No. 2, pp. 83-101, Printed in Great Britain
August
1985
97
J. N. Sahalos and G. Kyriacou
h=
0.2
m
500
KHz
Q1=10-3
u/,
9
I-
;s II a
/d=.z
52d=J 4-
3
4
5
6
7
a
9
I 10
K=2 01
FIG. 11. Ratio R as a function
of contrast
conductivity
From the above pair of values we get an initial computation of Ad and AK gives at last that d = 1.03 m The above method
and
K for d = 0.1-1.1 m and h = 0.2 m.
estimate
(d,, K,) = (1.1,7.63). A
K = 7.14.
is based on the knowledge
of the coal or lignite conductivity Journal
98
of the Franklin Institute Pergamon Press Ltd.
Thickness of Coal or Lignite Seam
h=o.s m
500 KHz
al=
1o-3
u/,
ia
8
7
6
-Y Jd=*’ I”“” 5
6
7
8
* 9
.
I 10
K,a2 01
FIG. 12. Ratio R as a function of contrast conductivity K for d = 0.1-1.1 m and h = 0.3 m.
crl. In most cases we do not have exactly the same o1 in different parts of the mine and for each place we need to know the conductivity. This is very difficult and maybe impossible. So, one method is to use a mean value of the mineral conductivity. The error in the thickness detection will increase with the error in the conductivity expectation. Another method independent of the conductivity can overcome all of these difficulties. Vol. 320, No. 2, pp. 83-101, August 1985 Printed in Great Britain
99
J. N. &halos and G. Kyriacou
We have the three ratios R,, R,, R, for h, = 0.1 m, hz = 0.2 m and h, = 0.3 m correspondingly. Dividing R, and R, by R3, we obtain from (26) that RI/R, and R,/R, are independent of the conductivity. So, by using the initial values d, K taken from Figs. 10-12 we apply an iteration procedure as follows. We apply the initial values to the ratio I1 = RI/R, and from Eqs. (27), we have (al-R,a,)+(B1-~1P3)d0+(Y1-alY3)KO+(81-~1BJ)doK0 +(sl-~i&j)d~+(rl-;llr3)KOd~
= 11. (33)
If d, and Kc, are the exact values X1must be zero. In general X1 # 0. We suppose that de is exact and Ke is not. If the exact K is K = K, + AK then we must have that AK=
n; (Y1-alY3)+(81-~183)d0+(rl-jllr3)d~’
(34)
From I, = R,fR,, we obtain (Q -&a,)
+ (PZ- Ms)de + (yZ- J&K
+ (6, - M&K + (sZ- &,)d;
+ (Cz- &)Kd;
= nl, ;
(35)
X2must be equal to zero if K and d, are exact. We suppose that the exact d is d = do + Ad and from (35) we have Ad = (Bz -&L)
-(a, - U,)K
n; + 2(s2 -&s&0
+ 2(& -U&K&
(36)
We use as d,, the value do + Ad and we find from (33) and (34) a new AK. The new AK from (35) gives a new Ad from (36). The iteration is continued until Ad and AK have the minimum desired values. If the first value of AK is large compared to K, we can start from Ad by using (36). Example 2
Suppose that we have R, =
6.2
for h, = 0.1 m
R, =
8.3
for hz = 0.2 m
R, = 10.4
for h3 = 0.3 m.
We start from the initial values do = 1.1 and K, = 7.63. By using Eq. (36) first, after four iterations, we found that d = 1.189 m
and
K = 8.634.
In this study a simple method for determining the thickness of a two-layer earth structure is given. The scheme is based on the mutual coupling of two small current loops. From the zero secondary coupling in two different heights we measure two Journal
100
of the Franklin Institute Pergamon Press Ltd.
Thickness
of Coal or Lignite Seam
different values of R. We generate the initial estimates do and K, which are improved by two different procedures. The whole method is very simple, and the lengthy and nontrivial computations using Sommerfeld integrals can be omitted. References (1) A. Sommerfeld, “Euber die Ausbreitung der wellen in der drahtlosen Telegraphie”, Ann. Phys., Vol. 81, p. 1135, 1909. (2) A. Baiios, “Dipole Radiation in the Presence of a Conducting Hall-Space”, Pergamon Press, New York, 1966. (3) J. R. Wait, “Induction by an oscillating magnetic dipole over a two-layer ground”, Appl. Sci. Res., Vol. B-7, pp. 73-80, 1958. (4) J. R. Wait, “Electromagnetic Probing in Geophysics”, Golem Press, Boulder, Colorado, 1971. (5) S. Maley, “A method for the measurement of a two-layer stratified earth”, Trans. IEEE, AP-11, pp. 366369,1963. (6) D. Chang, “A new EM sensor for the detection of roof thickness in a coal mine”, Scientific Report No. 26, Electromagnetics Laboratory, University of Colorado, Boulder, Colorado. (7) D. Chang and J. R. Wait, “An analysis of a resonant loop as an electromagnetic sensor of coal seam thickness”, Proc. of URSI Conference on Propagation in Non-ionized Media, LaBaule, France, pp. 141-146, 1977. (8) M. Ralston and J. R. Wait (1977), “Theory of low frequency ground conductivity measurement with application to probing of roof structures in coal mines”, EM Report No. 2, CIRES, University of Colorado, 1977. (9) P. Wilson, “A remote sensing scheme for detecting the thickness of a coal seam”, Masters Thesis, Department of Electrical Engineering, University of Colorado, Boulder, Colorado, 1978.
Vol. 320, No. 2, pp. 83-101, Augusr Printed in Great Britain
i985
101
and G.KYRIACOU
of Electrical
Engineering,
University of Thrace,
Xanthi,
Greece
ABSTRACT : The use of two small current loops for probing the thickness of a two-layer earth structure is discussed. The thickness and the conductivity contrast can be found by zeroing the secondary magnetic3eld on the axis of the receiving loop. The inversionfrom the measured data to the layer thickness proves that our procedure is simple. The whole method can be incorporated into an automated mining machine for sensing a coal or lignite seam with a slate backing.
I. Zntroduction In the development of better mining machines, it is important to have sensors which are capable of detecting the thickness of a mineral. In our case, we are talking about sensing a thin slab of coal or lignite, say a maximum of 1 m in thickness, backed by a reflecting material such as draw slate. Most of the methods previously reported are based upon radiation in front of the coal or lignite surface. The radiation problem of an electric or magnetic dipole in the presence of the earth is not new. Sommerfeld (1) in his classic 1909 paper presented the problem of a vertical electric dipole (VED) located at the interface separating two regions. Various investigators wrote papers on VED, horizontal electric dipole (HED), vertical magnetic dipole (VMD) or horizontal magnetic dipole (HMD) located in a multiple-layered earth, Bafios (2) presented a complete and systematic investigation for VED, HED, VMD and HMD by using the saddle-point method of integration and by deriving many expressions for a variety of ranges and parameters. An excellent extension on multiple-layered earth was produced by Wait (3, 4). Wait’s work can be applied to the detection of fragmented rock over a sedimentary rock base, underground water and mineral strips. Electric and magnetic dipoles are used as models of small current elements and loops. The input impedance of one dipole or the mutual coupling between pairs of dipoles can give the electric characteristics and the thickness of the ground. The induction methods, characterized by the use of dipoles in geophysical prospecting, have been given by Wait (4) in his overview of the most actively used techniques. These techniques are separated in two categories : the surface and the subsurface. In subsurface techniques the electrodes are placed below the surface of the ground and resistivity in the vicinity of the hole is measured.
OThe Franklin Institute 00164032/85$3.00+0.00
83
J. N. Sahalos and G. Kyriacou
Surface methods include magnetotelluric resistivity, induced polarization, wave tilt and radio wave methods. This study deals with the surface methods and discusses the measurement of the two small current loops in a two-layer earth. Various authors have dealt with this problem : see Maley (5), Chang (6) and Chang and Wait (7). The mutual coupling of two small current loops is well detailed by Ralston and Wait (8) and Wilson (9). However, these works could not lend themselves to a simple automated mining machine. Wilson showed a more desirable approach to finding the layer thickness by using a mini-computer incorporated into a mining device. In his inversion scheme, Wilson tried to find the layer thickness. However, we believe that the high cost of the method made it unfeasible. Our procedure generates theoretical curves, and matches the zero coupling data with the thickness of the layer. In order to make the method more compatible with an automated mining machine, we use a least square method which gives the thickness and the conductivity contrast of the layers. II. Formulation
A small horizontal current loop is located at a height h above an air-earth interface. The current distribution is essentially uniform and thus can be represented as a vertical magnetic dipole (VMD) with moment, i.e. m = ilds.
(1)
The configuration to be analyzed is shown in Fig. 1. The wavenumbers of a plane wave in a region are kj = wJ(P&J
(2)
d 1
I
(~zlQ*Jo)
FIG. 1. A VMD above a two-layer half-space model. Journal
84
of the Franklin Institute Pergamon Press Ltd.
Thickness of Coal or Lignite Seam where (3) E, is the relative dielectric constant and .sOis the free space permittivity. The electric and magnetic fields are formulated in terms of the magnetic Hertzian vector potential II*. The total Hertzian vector fjr* is the sum of the primary field If*P and the secondary field i=f*’ composed of reflections and transmissions due to the presence of loose earth. i’i*j’ has the i direction and satisfies the following equation : (V” + k;)ll*P = -Ids@ The solution
of Eq. (4) has the following n*P=
_._
Ids
m
J47L s 0
-X0). :
expression
e-juolz-hl J,(ap)c
Using Eq. (5) as a guide, we seek the general solution Ids -jG
s
.Ids
HI* = I
-J471
(4)
z> 0
> z > -d
s0
ada T(a) @2zJ0(ap) -,-d>z. uo uj are the transverse
wavenumbers
uj =
(6)
defined by
(ky -a2)1’2 -j(a’
(5)
of the form (9) :
m o Ce- bolz-hl +R(a) e- jub +“1 JO(ap) $, co [a(a)~“+B(a)e-‘““1J,(ap)~,O
da_
R,(k$ < a2
- kT)l12 R,(kT) < a2
From the continuity of the tangential interfaces, we can find that
electric
j = o 1 2 ’ ’ ’
and magnetic
(7) fields at the two
(8) A(a) =
B(a)
(l+~)+(l-~)R(a)e~juo”
= (l-~)+(l+f+(a)e-‘oh
and qa)
=
e
-juoh
. e_iwd
{[(l+~)+(l-$)R(a)]eplY1d
Vol. 320, No. 2, pp. 83-101, August 1985 Printed in Great Britain
85
J. N. Sahalos and G. Kyriacou
The magnetic field can be found from II* via the following equation : If = V(V - II*) + k%* J *
(12)
We know that the primary field II*p is the free space field due to a magnetic dipole, and can be expressed by n*p = E
eeikoRO
4n
(13)
R,
where R, = [p2+(z-h)2]1/2. From (12) and (13) the total magnetic field can be expressed by
+p p OS
omR(a)e- iuo(zfhfJ,(ap)a2 da
H,=g(z-h)y
3jk, e-ikoRo -!- +--R; R$
+z p OS
om R(a)e-
kg R; )
>
0
R(a) e-iuo(z+h)Jo(ap)$ s 0
da
(14)
1 _!_+!!k+ 1
iua(z+h)J,(ap)a2 da
co -.i
1
1
0
.
I
Our particular scheme is based on the definition of the mutual coupling between two small current loops. Suppose that we have one horizontal current loop and one loop in an arbitrary direction (Fig. 2). Both loops are placed at a height h above the air-earth interface and the separation between the loops is po. To have a zero secondary coupling between the two loops we must have a zero secondary magnetic field on the axis of the current loops. If we assume that the horizontal is the transmitting loop then the components of the total transmitting to the other loop are H”z .
H; Hz=:
(15) a3'l
J,(ap,) u. da R(a) e- jZuoh
and
f&=H”,=~
s cc
R(a) e- j2”OhJ,(apo)a2 da.
0
From H, we can subtract Hfl because this shows the primary field. The subtraction in practice will be by zeroing the ammeter ; an indication that the two loops are parallel 86
Journal
of the Franklin Institute Pergamon Press Ltd.
Thickness of Coal or Lignite Seam
-
-f h
FIG. 2. The nonparallel
loop configuration.
on the same height and very far from the earth. When we put the loops near the earth, the ammeter will give an indication other than zero. This creates the H”, field. By swivelling the receiving loop until we find a null position there results l? = Hi cos 8 + Hi sin 8 = 0
(17)
or R = -tan
8 = H”,/H;.
(18)
It is easy to measure the angle 0 and as we know we can find the ratio of H”,/H”, as a function of the thickness d and the contrast conductivity. Equations (15Hl8) agree with the results given by Wilson (9). For the quasi-static case we suppose that k, -+ 0, and as we can see in Refs. (8) and (9), R is given in the following form : (Gl)(A2+1)-1’2+(B*+l)-i/2 R = (l-K)A(A2+1)-1/2-B(B2+1)-1/2+K where K is the conductivity
contrast
A = 2(h+d)/p
(1%
~,/a,, and
B = 2h/p.
(20)
Equation (19) is used in (9) as the first approximation of the value of R. We must say that for low frequencies R converges nicely to its quasi-static limit. In this problem we use the maximum “low frequency” because the magnitude of the field components are proportional to the frequency. Hence, the lower frequency needs more sensitive equipment. We also found that R is “independent” of frequency for loop height h = O-O.4 m and frequency f = &500 kHz. This is shown in Figs. 3 and 4 where we have the amplitude of R as a function of frequency for various loop heights and two different Vol. 320, No. 2, pp. 83-101, August 1985 Printed in Chat Britain
87
J. N. Sahalos and G. Kyriacou
layer thicknesses. From Figs. 3 and 4 we can see that a frequency of 500 kHz would be a. practical choice. To find the thickness d we would start from the quasi-static value of R. By using the inverse method given in (9) we can find d. The only problem was the high cost of computer time which made the whole method impractical. An easy method to use can be made if we get analytic results of R as a function of d,
D=0.6
m
-17.
3a
25
15
10
5
1
,
3
I
4
I
I
I
I
7
8
9
10
FREQ
(HZ)
FIG. 3. Ratio R as a function of frequency for h = 0.1,0.4,0.6 and 1.1 m for D = 0.6 m.
88
Journal of the Franklin
Institute Pergamon Press Ltd.
Thickness of Coal or Lignite Seam
35
KHz
30
25
c 5 II Dz 20
F-
L
h=o.4
15
10
h=o.lm
5
1
I
-r 2
3
4
5
II
I
,
I
I
I
6
7
6
9
hii (f)
FREQ
(HZ)
FIG. 4. Ratio R as a function of frequency for h = 0.1,0.4,0.6 and 1.1 m for D = 1.1 m. h and K. These results, with the help of the least square simple expression of R.
method,
can produce
a
III. Numerical Results One of the main decisions to be made before starting the computation is. the definition of the separation distance (po) between the two loops. We calculate the Vol. 320, No. 2, PP. 83-101, August 1985 Printed in Great Britain
89
J. N. Sahalos and G. Kyriacou ratio R and the angle 6 of the receiving loop for the quasistatic case vs the layer thickness (d) and the height (h). From the results, we found that for h < 0.4 m ad/ah and M/ad increase as the separation distance decreases. From a practical point of view we must have the maximum possible difference in the angle f3for two different heights. To achieve our purpose we choose a separation distance equal to 0.2 m. This distance gives the best combination of discrimination and construction ability. For coal and lignite seams we would expect conductivity ai in the neighborhood of 1O-3 mhos/m. We use at first a conductivity equal to 10e3 mhos/m and we make all the calculations for the ratio R. In Fig. 5 we can see the ratio R as a function of K for h = 0.1-1.1. Figure 5 is given for a layer thickness equal to d = 0.1 m. Similar figures can be taken for any value of d. From a series of results we can see that there is a linear relation between R and K for h = 0.1-0.4 m. We suppose that R(K) = A+B.K (21) where A, B must be functions of d and h. To find the form of those functions we make plots of A(h) and B(h) vs h for given values of d. Figure 6 shows the above plots. As we can see A(h) and B(h) are of the following form : A(h) = a, + a,h
(22)
B(h) = bl + b,h + b,h.
The coefficients ai and bi are functions of d and are given in Figs. 3 and 8. Both figures show that ai and hi(d) must be of the form : f(d) = cl +c,d+c,d’.
(23)
We note that a, and b, are related with d in a third degree polynomial expression. For the approximation (23) and the third degree polynomial we find that the accuracy is approximately the same. From the above discussion the expression R(d, K, h) is of the form R(d, K, h) = a,(d) + a,(d) * h + [b,(d) + b,(d) * h + b,(d). h’]K. By the least square method
we obtain
R(d, K, h) = 1.398 -4.079d
(24)
values of a, bi and we have
+ 4.837d’ + (17.848 + 23.278d
- 22.982d2)h2 + [O.11+ 0.0054d - 0.063d2 + (- 0.654 + 4.845d - 2.867d’)h + (0.753 - 8.959d + 6.685d2)h2]K.
(25)
Equation (25) gives the ratio R as a function of d, h and K for a given conductivity ol. Since the conductivity is in the neighborhood of 10m3 mhos/m we can make calculations for the ratio R in the two boundary values of the conductivity. Thus in Fig. 9 we have R vs K for a given h and different d. From a series of results similar to those of Fig. 9, by using the least square method, we can find that the real ratio R, R,, is given by R,(d, K,h,o,) 90
= [l +62.64(10-3-~a,)]R(d,
K, h).
(26)
Journal of the Franklin Institute Pergamon Press Ltd.
Thickness
of Coal or Lignite Seam
h=l .I
hz.9 h= .a h=
.7
-h=.6
hz.5 h=
.4
h=,s
hz.2
5---
hz.1
0,
4
5
6
7
6
9
10
KZ=
02
FIG. 5. Ratio R as a function of contrast conductivity K for h = 0.1-1.1 m and D = 0.1 m.
IV. A Scheme to Extract the Layer Thickness The definition of d and K is given if we know two different different heights h and the exact value of ol. Vol. 320, No. 2, pp. 83-101, August Printed in Great Britain
1985
values of R for two
91
J. N. Sahalos and G. Kyriacou
, dzl.1
t
1.
d= d-.9
BP
A(h:
dz.8 dx.3 d= .2
.:
d= .7
dz1.l d= .I d=
8
.:
.6
d=s
d=
6
.4
dz.3
.l
\ dz.2
\ 1
I
3
8
.l
.2
.3
.4
.a h(m)
.l
d=.l .2
.3
.4
h(m)
FIG. 6. A(h) and B(h) coefficients vs h for different d.
As we said before we want a simple procedure which can be applied in a microcomputer. If we try to solve for d and K a system of two equations as in (26), there results a fourth degree equation which can only be solved numerically. To overcome this difficulty we use Figs. 10-12 which represent R(K) vs K for h = 0.1,0.2 and 0.3 m with d from 0.1 to 1.1 m. These figures give the ratio R for (rr = 10e3 mhos/m. We measured the values of R, for h = 0.1,0.2 and 0.3 m. In dividing those values by [l +62.64(10e3 -a,)], we find the value of R which must correspond to c1 = 10m3 mhos/m. With the help of Figs. l&12 we found an initial pair of d, and K,. We define as R, and R, the divided values for hl = 0.1 m and h, = 0.2 m. By using the initial values da, K, from Eq. (25) we can find two theoretical ratios R which for 92
Journal of the Franklin Institute Pergamon Press L.td.
Thickness
of Coal or Lignite Seam
2.5
1.5
al,bl,bz
k t 1
.5
.9 d (ml
FIG. 7. The coefficients a,, b,, b, as functions of the layer thickness d. h, and h, will be R; and R;. The ratios will be given from (25) in the following form :
, R; = “l+Bldo+y1K,+61do~io+e,d~+r,K,d~ R’, = a2 +&do +y2K, + S,d,K,, + E2d? + c2K0d$
I
(27)
The coefficients cli,fib yi, 6, q, ci are functions of the height hi and are given by Eli= 1.3998 + 17.848 hi /Ii = - 4.0794 + 23.278 hi yi = 0.11-0.6544
hi + 0.754 hf
6i = 0.0542 + 4.8454 hi- 8.9592 hi2 ei = 4.8368 -22.982
I
(28)
hi
li = - 0.0626 - 2.867 hi + 6.685 hf. Vol. 320, No. 2, pp. 83-101, August 1985 Printed in Great Britain
93
J. N. Sahalos and G. Kyriacou We write that Xi = pi + 6iK, + 2&id, + 21;id,K, yi = yj + 6id, + [id; (29)
ARi = R,-R; i = 1,2. For a first degree approximation
I the exact values of d and K are
d = d,+Ad
K = K,+AK.
and
(30)
(a)
24
2:
2:
a2 2’
I
I
I
I
I
I
.2
,3
9
,5
.6
,7
I
,S
1
I,
.Q
1.
1.1
dmv
FIG. 8. (a) 94
Journal of the Franklin Institute Pergamon Press Ltd.
Thickness of Coal or Lignite Seam
(b)
FIG. 8. (a) The coefficient
a, as a function of the layer thickness d. (b) The coefficient function of the layer thickness d.
b, as a
By using Eqs. (25), (27) and (29), we can find that
(31) and
AK=
AR, -x,Ad Yl
.
Equations (30)-(32) give the “exact” values for the thickness contrast K. Vol. 320, No. 2, pp. X3-101, August Printed in Great Britain
(32)
d and the conductivity
1985
95
J. N. Sahalos and G. Kyriacou hzo.1 Cm> -*-.-
0,=0.5x10-3
mho/m
-_---0,=10-3 CT,-1.5 x10-3
/
I /’
dd.1
IRrl t
Cm)
d=O.6("')
c
4
I
7
FIG.
Example 1 We suppose
96
9
11
13
I
15
K = q/a,
9. The ratio R vsK for h = 0.1 m and different conductivities cl.
that we are on a two-layer
earth and measure
R, =
6.18
for h, = 0.1 m
R, =
8.12
for h, = 0.2 m
R3 = 10.33
for h3 = 0.3 m.
the following
R:
Journalof the FranklinInstitute PergamonPressLtd.
Thickness of Coal or Lignite Seam
9
-25-
k0.1
m
500
KHZ
Ul=
10 -3
U/m
8
7 I?ii
1 6
51-I
4I-
S-
I 5
dI
6
7
8
9
I 10
K=+
FIG. 10. Ratio R as a function of contrast conductivity K for d = 0.1-1.1 m and h = 0.1 m.
We use Figs. l&12
and find the following
pairs :
for h = 0.1 m
(d,,K,)
= (1.1,7.75),(1.0,9.5)
for h = 0.2 m
&,I&)
= (1.1,7.65),(1.0,8.5),(0.9,9.25)
for h = 0.3 m
(d,,K,)
= (1.1,7.5),(1.0,8),(0.8,8.25).
Vol. 320, No. 2, pp. 83-101, Printed in Great Britain
August
1985
97
J. N. Sahalos and G. Kyriacou
h=
0.2
m
500
KHz
Q1=10-3
u/,
9
I-
;s II a
/d=.z
52d=J 4-
3
4
5
6
7
a
9
I 10
K=2 01
FIG. 11. Ratio R as a function
of contrast
conductivity
From the above pair of values we get an initial computation of Ad and AK gives at last that d = 1.03 m The above method
and
K for d = 0.1-1.1 m and h = 0.2 m.
estimate
(d,, K,) = (1.1,7.63). A
K = 7.14.
is based on the knowledge
of the coal or lignite conductivity Journal
98
of the Franklin Institute Pergamon Press Ltd.
Thickness of Coal or Lignite Seam
h=o.s m
500 KHz
al=
1o-3
u/,
ia
8
7
6
-Y Jd=*’ I”“” 5
6
7
8
* 9
.
I 10
K,a2 01
FIG. 12. Ratio R as a function of contrast conductivity K for d = 0.1-1.1 m and h = 0.3 m.
crl. In most cases we do not have exactly the same o1 in different parts of the mine and for each place we need to know the conductivity. This is very difficult and maybe impossible. So, one method is to use a mean value of the mineral conductivity. The error in the thickness detection will increase with the error in the conductivity expectation. Another method independent of the conductivity can overcome all of these difficulties. Vol. 320, No. 2, pp. 83-101, August 1985 Printed in Great Britain
99
J. N. &halos and G. Kyriacou
We have the three ratios R,, R,, R, for h, = 0.1 m, hz = 0.2 m and h, = 0.3 m correspondingly. Dividing R, and R, by R3, we obtain from (26) that RI/R, and R,/R, are independent of the conductivity. So, by using the initial values d, K taken from Figs. 10-12 we apply an iteration procedure as follows. We apply the initial values to the ratio I1 = RI/R, and from Eqs. (27), we have (al-R,a,)+(B1-~1P3)d0+(Y1-alY3)KO+(81-~1BJ)doK0 +(sl-~i&j)d~+(rl-;llr3)KOd~
= 11. (33)
If d, and Kc, are the exact values X1must be zero. In general X1 # 0. We suppose that de is exact and Ke is not. If the exact K is K = K, + AK then we must have that AK=
n; (Y1-alY3)+(81-~183)d0+(rl-jllr3)d~’
(34)
From I, = R,fR,, we obtain (Q -&a,)
+ (PZ- Ms)de + (yZ- J&K
+ (6, - M&K + (sZ- &,)d;
+ (Cz- &)Kd;
= nl, ;
(35)
X2must be equal to zero if K and d, are exact. We suppose that the exact d is d = do + Ad and from (35) we have Ad = (Bz -&L)
-(a, - U,)K
n; + 2(s2 -&s&0
+ 2(& -U&K&
(36)
We use as d,, the value do + Ad and we find from (33) and (34) a new AK. The new AK from (35) gives a new Ad from (36). The iteration is continued until Ad and AK have the minimum desired values. If the first value of AK is large compared to K, we can start from Ad by using (36). Example 2
Suppose that we have R, =
6.2
for h, = 0.1 m
R, =
8.3
for hz = 0.2 m
R, = 10.4
for h3 = 0.3 m.
We start from the initial values do = 1.1 and K, = 7.63. By using Eq. (36) first, after four iterations, we found that d = 1.189 m
and
K = 8.634.
In this study a simple method for determining the thickness of a two-layer earth structure is given. The scheme is based on the mutual coupling of two small current loops. From the zero secondary coupling in two different heights we measure two Journal
100
of the Franklin Institute Pergamon Press Ltd.
Thickness
of Coal or Lignite Seam
different values of R. We generate the initial estimates do and K, which are improved by two different procedures. The whole method is very simple, and the lengthy and nontrivial computations using Sommerfeld integrals can be omitted. References (1) A. Sommerfeld, “Euber die Ausbreitung der wellen in der drahtlosen Telegraphie”, Ann. Phys., Vol. 81, p. 1135, 1909. (2) A. Baiios, “Dipole Radiation in the Presence of a Conducting Hall-Space”, Pergamon Press, New York, 1966. (3) J. R. Wait, “Induction by an oscillating magnetic dipole over a two-layer ground”, Appl. Sci. Res., Vol. B-7, pp. 73-80, 1958. (4) J. R. Wait, “Electromagnetic Probing in Geophysics”, Golem Press, Boulder, Colorado, 1971. (5) S. Maley, “A method for the measurement of a two-layer stratified earth”, Trans. IEEE, AP-11, pp. 366369,1963. (6) D. Chang, “A new EM sensor for the detection of roof thickness in a coal mine”, Scientific Report No. 26, Electromagnetics Laboratory, University of Colorado, Boulder, Colorado. (7) D. Chang and J. R. Wait, “An analysis of a resonant loop as an electromagnetic sensor of coal seam thickness”, Proc. of URSI Conference on Propagation in Non-ionized Media, LaBaule, France, pp. 141-146, 1977. (8) M. Ralston and J. R. Wait (1977), “Theory of low frequency ground conductivity measurement with application to probing of roof structures in coal mines”, EM Report No. 2, CIRES, University of Colorado, 1977. (9) P. Wilson, “A remote sensing scheme for detecting the thickness of a coal seam”, Masters Thesis, Department of Electrical Engineering, University of Colorado, Boulder, Colorado, 1978.
Vol. 320, No. 2, pp. 83-101, Augusr Printed in Great Britain
i985
101