Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet
Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet
2.4. ~articul'ar source fields Notation: Xn the sequel only fields with a periodic time function , -- exp(iwt) are considered. For any field quantity .A(s,t) ,+ we write A(r,w) - with the understanding that the real or imaginary part of A(g,t) exp(iwl;) is meant. Furthermore, since only the TE-mode is of interest we shall drop the subscript "E". In the last section an algorithm for the cal.culation of the $-poten- tial in a layered half-space has been given. As input occurs fe(~,K;w) = fi (Krw) exp(-KZ) , tIie representation of - the source - potential. in t.he wave-nurnber space. Frolu (2.12a) , i .e. - 13=grad (a@/az) - follows that - a$/az is the magnetic scalar potential. Between f- (K, w) and the source potential be (2,~) exist the following reci:- 0 - - -- 1 2 SS ie(r,w)e dx dy --. General field ------ - - If I$ i.s symmetrical. wi.th respect to a vertical axis, i.e. @ is a (2.35) function of r = (x" +y2)1'2 only, e4s.. (2.35) and (2.36) simplify t:o In the derivation of (2.35a) and (2.3Ga) the cartesian coordinates were replaced by circular. polar coordinates x = rcos0, y = rsin0, K = KCOS$, K~ = ~siilq >: and use was 1nad.e of the identity
2n -i~cr cos (8-$1 -iKrcU J e ~B=J e c's8d0-2.fcos (~rcos0)ci0=2n~ 0 0 0 11 (icr) 0 - 2. ) 1/2 2 1/2 Froin (2.35a), (2.3Ga) follovis that if 4 is function of r= (x2-I-Y then fo is a function of K = (I:* f K ) . In certain cases (e.cj. X Y example h! below), it is on1.y possible to obtain the potential on the clrlj-ndrical axis (r = o) in closed form. Then (2.35a) reads Eq. (2.35b) can be considered as a Laplace-Transform for which the inversion is where E is. an appropriate real constant (thus that all singill.arj.- e ties of 41 (o,z) are at the left of z = E). Simpler wou.ld be the use of a table of Laplace transforms. The spectxal representation of the source is now calculated. for simple sou-rces : ' &) Vertica.:L magnetic dipo1.e ,- Locate the dipole at r = (o,o,-h), ki > o and let i.ts moment be -0 - * M - = Mz. - When it is produced by means of a small current loop then !4 = current x area of the loop. ($5 i.s positive if the di.rect:ion of the current forms with - 2 a right-handed sys::.em, and negative else.) From the scalar potential follows - Since 4 shows cylindrical symmetry, v7e obtain front (2.3Sa) on using the resu1.t I Ver kical. magnetic dipole
- Page 1 and 2: Electromagnetic Induction in the Ea
- Page 3 and 4: 6.2. Generalized matrix inversion 6
- Page 5 and 6: A1~Lernativel.y p = 30. m, where T
- Page 7 and 8: The e1ectrica:L effect - of the cha
- Page 9 and 10: The vari.ables x,y, and t which do
- Page 11 and 12: - d IufM12 > 0 . dz - - On the othe
- Page 13 and 14: a) TM-mode From (2.251, (2.26); (2.
- Page 15: with " the abbreviation + - 1 - ? =
- Page 19 and 20: In the 1irnl.t a -+ o, I +- m, M =
- Page 21 and 22: -- 2.5. Definition -- of the transf
- Page 23 and 24: we arrive at - v The same appl-ies
- Page 25 and 26: ) Computation of ---- C for a laxez
- Page 27 and 28: The approximate interpretation of C
- Page 29 and 30: I ~ispersi-on relations I Dispersio
- Page 31 and 32: where L is a positively oriented cl
- Page 33 and 34: 1 2 3 4 CPD 1 2 3 4 CPD - g) Depend
- Page 35 and 36: The TE-mode has no vertical electri
- Page 37 and 38: i I Earth Anomalous domain 3.2. Air
- Page 39 and 40: Hence, the conductivity is to be av
- Page 41 and 42: The RHS i.s a closed line integral
- Page 43 and 44: 4. Having determined B;, the coeffi
- Page 45 and 46: 3.4. Anomalous region as basic doma
- Page 47 and 48: - 6 and 6= can be so adjusted that
- Page 49 and 50: From the generalized Green's theore
- Page 51 and 52: and y can again be so adjusted that
- Page 53 and 54: 4.2. In3ral - --- equation method L
- Page 55 and 56: The element GZx is needed for all z
- Page 57 and 58: With this knowledge of the behaviou
- Page 59 and 60: After having determined Qzr VJ,; @,
- Page 61 and 62: 4.3. The surface inteyral approach
- Page 63 and 64: F At the vertical boundaries the co
- Page 65 and 66: The four equations A A A A H = i sg
2.4. ~articul'ar source fields<br />
Notation: Xn the sequel only fields with a periodic time function<br />
, --<br />
exp(iwt) are considered. For any field quantity .A(s,t)<br />
,+<br />
we write A(r,w) - with the understanding<br />
that the real or imaginary part of A(g,t) exp(iwl;) is<br />
meant. Furthermore, since only the TE-mode is of interest<br />
we shall drop the subscript "E".<br />
In the last section an algorithm for the cal.culation of the $-poten-<br />
tial in a layered half-space has been given. As input occurs<br />
fe(~,K;w) = fi (Krw) exp(-KZ) , tIie representation of - the source - potential.<br />
in t.he wave-nurnber space. Frolu (2.12a) , i .e. - 13=grad (a@/az)<br />
-<br />
follows that - a$/az is the magnetic scalar potential. Between<br />
f- (K, w) and the source potential be (2,~) exist the following reci:-<br />
0 -<br />
- -- 1<br />
2 SS ie(r,w)e dx dy<br />
--. General field<br />
------<br />
- -<br />
If I$ i.s symmetrical. wi.th respect to a vertical axis, i.e. @ is a<br />
(2.35)<br />
function of r = (x" +y2)1'2 only, e4s.. (2.35) and (2.36) simplify t:o<br />
In the derivation of (2.35a) and (2.3Ga) the cartesian coordinates<br />
were replaced by circular. polar coordinates<br />
x = rcos0, y = rsin0, K = KCOS$, K~ = ~siilq<br />
>:<br />
and use was 1nad.e of the identity
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