Vol50,1,2007 93 ANNALS OF GEOPHYSICS, VOL. 50, N. 1, February 2007 Key words piezomagnetic effect – borehole mag- netic measurement – the Mogi model – Long Valley Caldera 1. Introduction Boreholes are now widely used in geophysi- cal observations, in particular for seismic and crustal strain measurements, since they greatly reduce noise resulting from artificial mechanical vibrations and increase tectonic signals as the sensor is closer to the source. Improvement of S/N ratio by the use of a borehole can be similar- ly expected in the case of tectonomagnetic obser- vations. To avoid cultural noises due to moving magnetic bodies (cars, etc.), high-powered elec- tric devices and so on, we must isolate an undis- turbed area of at least a few hundreds meters square. Under the ground, however, high-fre- quency EM noise is almost completely shielded and a moving car is not detected at a 200 m depth (Yamamoto, 1990). Furthermore, tectonomagnet- ic signals increase with depth. Drag-out effect of piezomagnetic signals due to a borehole: the Mogi source as an example Yoichi Sasai (1), Malcom J.S. Johnston (2), Yoshikazu Tanaka (3), Robert Mueller (2), Takeshi Hashimoto (4), Mitsuru Utsugi (3), Shinya Sakanaka (5), Makoto Uyeshima (6), Jacques Zlotnicki (7) and Paul Yvetot (7) (1) Disaster Prevention Division, Bureau of General Affairs, Tokyo Metropolitan Government, Tokyo, Japan (2) US Geological Survey, Menlo Park, CA, U.S.A. (3) Graduate School of Science, Kyoto University, Japan (4) Graduate School of Science, Hokkaido University, Japan (5) Faculty of Engineering and Resource Science, Akita University, Japan (6) Earthquake Research Institute, The University of Tokyo, Japan (7) Observatoire de Physique du Globe de Clermont-Ferrand, France Abstract We show that using borehole measurements in tectonomagnetic experiments allows enhancement of the ob- served signals. New magnetic dipoles, which vary with stress changes from mechanical sources, are produced on the walls of the borehole. We evaluate such an effect quantitatively. First we formulate a general expression for the borehole effect due to any arbitrary source models. This is valid everywhere above the ground surface as well as within the cylindrical hole. A first-order approximate solution is given by a line of horizontal dipoles and vertical quadrupoles along the central axis of the borehole, which is valid above the ground surface and a slight- ly away (several tens of cm) from the top of the borehole. Selecting the Mogi model as an example, we numer- ically evaluated the borehole effect. It turned out that the vertical quadrupoles produce two orders of magnitude more intense magnetic field than the horizontal dipoles. The borehole effect is very local, i.e. detectable only within a few m from its outlet, since it is of the same order or more than the case without a borehole. However, magnetic lines of force cannot reach the ground surface from a deeper portion (>10 m) of a borehole. Mailing address: Dr. Yoichi Sasai, Disaster Prevention Division, Bureau of General Affairs, Tokyo Metropolitan Government, Nishi-Shinjuku 2-8-1, Shinjuku-ku, Tokyo, 163-8001, Japan; e-mail: yosasai@zag.att.ne.jp 94 Yoichi Sasai et al. However, we need special considerations in the case of magnetic field observations. We make measurements within the source material of the magnetized crust. When we excavate a hole, ad- ditional magnetic poles are observed along its wall. When a magnetometer is placed under the ground, the effect of the shallower magnetic source is different from the on-the-ground obser- vation in which the source always lies beneath the sensor. Sasai (1994) evaluated both the effects in the case of a vertical rectangular strike-slip fault, and demonstrated that piezomagnetic signals are significantly enhanced at depth beneath the ground surface. The underground solution for the piezomag- netic field is defined as a limit of the field value when the radius of the excavated spherical cavity approaches zero (Sasai, 1991b). We assume that the drilled hole is filled up with the same mag- netic substance. In actual situations, a borehole would rarely be buried again, nor could the spher- ical cavity for a magnetometer ever be shrunk to zero volume. Usually, a casing-pipe is inserted to protect the borehole, in which a non-magnetic tube (e.g., vinyl chloride) is preferable, at least for shallow boreholes. This is because long-term al- teration of magnetic susceptibility in high µ met- als could disturb tectonomagnetic changes. When we consider such a tectonomagnetic observation system, the existence of a borehole it- self could produce an additional magnetic signal. As will be described later, we observed an inter- esting magnetic change associated with a vol- cano-tectonic event in Long Valley Caldera, Cali- fornia, U.S.A. We put two proton magnetometers only 40 m apart, one of which is located close to a buried borehole. The simple difference between the two magnetometers showed a variation more or less similar to the strain change at the observa- tion site. A possibility was that amplified tectono- magnetic signal resulted from the presence of the borehole. We will examine quantitatively how much local magnetic field will be produced on the ground simply by the existence of a borehole. 2. Theory We assume that the Earth is a homogeneous and isotropic elastic half-space with a uniformly magnetized top layer stressed by some pressure source. Suppose that there is a vertical cylindrical borehole with radius a and length L as shown in fig. 1. For the sake of simplicity, the borehole is lo- cated outside the pressure source so that no me- chanical singularity due to the source exists along the surface of the borehole. As a result of crustal stress, magnetic poles resulting from piezomag- netism of rocks appear on the wall of the borehole. Given the displacement field due to the source, we can evaluate the influence of a borehole. The piezomagnetic potential Wk ( r 0) of a stressed magnetoelastic body with its surface S and the displacement field uk(r) can be ex- pressed as (Sasai, 1983, 1991b) (2.1) where (2.2) In the above, k indicates the k-th Cartesian com- ponent, n an outward normal to the surface S, r0 the observation point, r an arbitrary point on S and . Ck is given by r r0ρ = − .div x u x u m u 2 3( )k l k k l kl2 2 2 2 δ∆ = + −c m ( ) ( ) ( ) ( ) d W C n u u n S r r m n r 3 2 2 1 1 ( ) k k k k S k 0 $ $ $ 2 2 2 2 λ µ λ µ ρ ρ ∆= − + + + + c m = F ( " 2 , ## Fig. 1. Schematic representation of a borehole, a pressure source and the magnetized crust. 95 Drag-out effect of piezomagnetic signals due to a borehole: the Mogi source as an example (2.3) in which β is the stress sensitivity, Jk k-th com- ponent of the magnetization, λ and µ Lame’s constants, and in particular µ the rigidity. Let us consider the local effect of the bore- hole above the ground surface near around its top. The piezomagnetic potential due to the boreholeWk(BH) consists of three parts (2.4) Wk (A) is the potential produced by magnetic poles along the cylindrical wall, Wk(B) by those over the circular bottom, and Wk(C) by those from the van- ishing portion of the ground surface, respectively. We will evaluate these terms separately. First we introduce the Cartesian coordinates with x-y plane on the Earth’s surface and z axis positive downward, of which origin is at the center of the borehole. We also define the cylindrical coordi- nates system which shares the origin and z-axis with the Cartesian one as shown in fig. 2. 2.1. The contribution from the cylindrical wall of the borehole: Wk(A) Let the observation point outside the mag- netoelastic body be Q(x0, y0, z0)=Q(r0, θ0, z0) and a moving point on the cylindrical surface of W W W W( ) ( ) ( ) ( )k BH k k k A B C= + − C J 2 1 3 2 k kβ µ λ µ λ µ = + + the borehole P(x, y, z) = P(r, θ, z). We find (2.5) (2.6) The integral over the cylindrical surface is giv- en by (2.7) where (2.8) (2.9) . (2.10) Since the radius of the borehole a is sufficient- ly small, the displacement at an arbitrary depth z is almost identical along the wall surface. It can be replaced with the value at the central ax- is r=0, which could be attained if the borehole were absent. FA(k), FB(k) and FC(k) are functions of z only, which are given by (2.11) (2.12) (2.13)( )F z u( )C k k r 0= =6 @ ( ) ( ) ( ) div F z y u x u u 3 2 3 2 3 3 2 2 ( ) B k k k y ky r 0 2 2 2 2 λ µ µ λ µ λ µ λ µ λ µ δ = − + − + + + + + + = < F ( ) ( ) ( ) div F z x u x u u 3 2 3 2 3 3 2 2 ( ) A k k k x kx r 0 2 2 2 2 λ µ µ λ µ λ µ λ µ λ µ δ = − + − + + + + + + = < F F u( )C k k r a= =6 @ ( ) ( ) div F u x u y u 3 2 3 2 3 3 2 2 ( )k k k k r a B y y 2 2 2 2 λ µ µ λ µ λ µ λ µ λ µ δ = − + − + + + + + + = < F ( ) ( ) div F x u x u u 3 2 3 2 3 3 2 2 ( ) A k k k kx r a x 2 2 2 2 λ µ µ λ µ λ µ λ µ λ µ δ = − + − + + + + + + = < F dF θ d dF Fθ θ d z( )W ar = ( ) cos sin cos C a r 1 ( ) ( ) ( ) ( ) k k A A k L B k C k 0 0 2 0 0 2 3 0 0 0 2 ρ θ ρ θ ρ θ θ + − + − π π π ++ ; F ## # # ( , , 0) .cos sinen r θ θ= − = − − ( ) ( )cosPQ r a z z ar20 2 2 0 2 0 0ρ θ θ= = + + − − − Fig. 2. The Cartesian and cylindrical coordinate system used in Section 2. 96 Yoichi Sasai et al. These terms are excluded from the integrals with respect to θ, and eq. (2.7) is reduced to (2.14) where (2.15) (2.16) (2.17) in which (2.18) and (2.19) I(m, n; l)’s on the righthand side of eqs. ((2.15) to (2.17)) are called the integrals of Lipschitz- Hankel type, whose characteristics are investi- gated in detail by Eason et al. (1955) . (2.20) They frequently appear in the potential problems with axial symmetry. How to reduce the integrals with respect to φ to the Lipschitz-Hankel type should be referred to Sasai (1991a). The Lipschitz-Hankel integrals are expressed with the complete elliptic integrals (Eason et al., 1955), which can be expanded into the Tailor se- ries with respect to k2=4ar0 / ρa2, where . In the present case, k2 is sufficiently small and even only the first term of the Tailor expansions works as a good approximation. Φ2 to Φ4 are approximated by (2.21) Moreover, ρa can be replaced with , , . ar ar 2 1 2 3 a a a 2 3 0 3 3 4 0 5 π ρ π ρ π ρΦ Φ Φ= = = ( ) ( )a r z za 0 2 0 2ρ = + + − ( , ; ) ( ) ( ) dI m n l J at J t e t trm ct l n 0 0= 3 −# ,cosR a ar r c z z22 0 0 2 0φ= − + = − R c2 2ρ = +φ (1, 1; 1) cos d c I4 3 0 ρ φ φ π Φ = = φ π # (0, 0; 1) cos d I c3 30 ρ φ φ π Φ = = φ π # (1, 1; 0) cos d I2 0 ρ φ φ πΦ = = φ π # 2a d z= +( )W Fr θ Φsin+ ( ) cos C F F a r 1 ( ) ( ) ( ) ( ) k k A A k B k L C k 0 0 2 0 2 0 3 0 4 θ Φ Φ Φ+ − 6 @ # because the sensor height z0 (∼ a few m) is, usu- ally, much larger than the borehole radius a (∼10 cm). Finally, eq. (2.14) can be expressed by (2.22) The first and second terms of the integrand in eq. (2.22) are horizontal dipoles in the x and y direction, while the third one a vertical quadru- pole, respectively. The vertical quadrupole is defined as the differential of a vertical dipole in the vertical direction. 2.2. The contribution from the circular bottom of the borehole: Wk(B) The moving point P(x, y, z) = P(r, θ, L) lies on the bottom circle and the outward normal is upward, and we find (2.23) . (2.24) The integral over the bottom surface is given by (2.25) where (2.26) . (2.27) Since the radius of the bottom circle is small, the displacement at the bottom can be regarded almost identical to the one at the bottom center (r = 0 , z=L). FD(k) and FE(k) are excluded from the [ ]F u( )E k k z L= = ( ) ( ) ( ) div F z u x u u 3 2 3 2 2 3 2 3 ( ) D k kz k k z z L 2 2 2 2 λ µ µ λ µ λ µ δ λ µ λ µ = − + + + + − + + + = < F ( ) d d C W r F F L z r 1 1( ) ( ) ( ) k k B D k L E k L a 0 3 0 0 2 0 ρ ρ θ= + −π ; E## (0, 0, 1)en z= − = − == ( ) 2 ( )cosPQ r r z z rrL z L 0 2 2 0 2 0 0ρ θ θ+ + − − − = aπ d z=( )W r ( ) . C F x F y F z z r 1 2 ( ) ( ) ( ) ( ) k k A A k B k L C k 0 0 3 0 0 3 0 0 0 5 0 2 0 2 ρ ρ ρ + + + − − 2 < F # ( )z zr0 0 2 0 2ρ = + − 97 Drag-out effect of piezomagnetic signals due to a borehole: the Mogi source as an example integral with respect to θ and r; eq. (2.25) is re- duced to (2.28) where (2.29) (2.30) in which (2.31) and (2.32) The Lipschitz-Hankel integral is now defined as (2.33) Further integration with respect to r in eq. (2.28) is not available analytically, and we are to find an approximate solution. Using the Tailor series expansion for I(m, n; l)’s, we obtain . (2.34) The integrals in eq. (2.28) can be easily obtained as (2.35) . (2.36) If the observation point is far from the borehole, i.e. r0 >> a, eqs. ((2.35) and (2.36)) approach zero. Even near around the borehole, i.e. r0 ∼ 0, (2.35) and (2.36) become negligibly small, because usu- ally c >> a. Thus we can neglect the contribution from the bottom circle of the borehole. ( ) d r a r c a r r c r c 1 3 a 0 0 2 2 0 0 2 2 0 2Φ = + + + − + < F# ( ) d logr r r c r a r a ca 1 0 0 0 2 2 0 0 2 2 Φ = + + + + + +# ( ) ( )r r c r r c /1 0 2 2 0 2 23 3 2 π π Φ Φ= + + = + +" , ( , ; ) ( ) ( ) .dI m n l J rt J r t e t tm n ct l 0 0 = 3 −# , .cosR r r rr c L z20 2 2 0 0φ= + − = − R cL 2 2ρ = + ( , ; ) d c I 0 0 1 L 3 0 3 ρ φ π Φ = = π # ( , ; ) d I 0 0 0 L 1 0 ρ φ πΦ = = π # ( ) ( ) ( )d d C W F r r F r r r 1 2 0 2 0 ( ) ( ) ( ) k k B D k E k a a 0 1 0 3 0 $ $ Φ Φ = = + =# # 2.3. The contribution from the circular top of the borehole: Wk(C) This term comes from the counterbalance that we have to subtract the contribution from the vanishing portion of the ground surface, which is the circular outlet of the borehole with radius a. Now the moving point P(x, y, z)=P(r, θ, 0) is on the ground z = 0 and the outward normal is up- ward, and hence (2.37) . (2.38) We follow the same procedure as in the case of Section 2.2, and easily find a solution by put- ting L = 0. Thus Wk(c) is given by (2.39) where (2.40) (2.41) Φ1 and Φ3 are defined as (2.42) (2.43) in which (2.44) and (2.45) With the aid of approximation formulas (2.34), we can evaluate the integrals on the right hand side of eq. (2.39), which are given by eqs. , ( .)cosR r r rr c z z2 0<02 2 0 0 0φ= + − = − R cS 2 2ρ = + ( , ; ) d c I 0 0 1 S 3 3 0 ρ φ π Φ = = π # ( , ; ) d I 0 0 0 S 1 0 ρ φ πΦ = = π # [ ]F u( )k k zH 0= = ( ) ( ) ( ) div F z u x u u 3 2 3 2 3 3 2 2 ( )k k k z kz z G 0 2 2 2 2 λ µ µ λ µ λ µ λ µ λ µ δ = − + − + + + + + + = < F ( ) ( ) ( )d d C W F r r F r r r 1 2 0 2 0 ( ) ( ) ( ) k k C G k H k a a 0 1 0 3 0 $ $ Φ Φ = = + =# # (0, 0, 1)en z= − = − == ( ) ( )cosPQ r r z z rr2S z 0 2 2 0 2 0 0 0 ρ θ θ+ + − − − = 98 Yoichi Sasai et al. ((2.35) and (2.36)) with c = ⎢z0⎢. We find again that Wk(C) is negligibly small because ⎢z0⎢>> a. Wk (C) may play a significant role when we observe the magnetic field very close to the surface outlet, i.e. r ∼ 0 and z0 ∼ a. In ordinary tectonomagnetic observations, however, we cannot achieve such measurements just above the ground surface be- cause of high field gradient. Hence Wk(C) can be neglected as compared with Wk(A). In conclusion, the piezomagnetic field pro- duced by the existence of a borehole can be repre- sented by the first term of the contribution Wk(A), i.e. eq. (2.22). However, we may not disregard Wk (B) and Wk(C) when we consider the effect of the casing pipe which is usually made of high µ met- als. They result in additional magnetic fields near the end of the pipe: the magnetic lines of force are absorbed into the metallic wall and they spread out from the end near the ground into the air. To evaluate this effect, the approximate solutions, i.e. eqs. ((2.35) and (36)), are no longer useful, and we must numerically conduct integrations with respect to r according to eq. (2.28) and eq. (2.39). 3. A case study: the Mogi model Suppose that there is a spherical cavity of ra- dius b at a depth of D in an elastic half-space, within which a hydrostatic pressure ∆P occurs. Such a simple model works effectively to inter- pret crustal deformation around volcanoes (Mogi, 1958). The piezomagnetic field produced by the Mogi model was investigated in detail by Sasai (1991a). The displacement field of the Mogi model is given by (3.1) (3.2) (3.3) where ( ) ( ) ( ) u C R z D z D R R z z D 2 3 1 6 z 1 3 2 3 2 5 2 µ λ µ λ µ λ µ = − + + − − + + − + ( 1 ( ) u C R R R z z Dy y y 2 3 6 y 1 3 2 3 2 5µ λ µ λ µ = + + + − + ( 2 ( ) u C R x R x R xz z D 2 3 6 x 1 3 2 3 2 5λ µ λ µ µ= + + + − + ( 2 (3.4) and . (3.5) Note that the origin of the Cartesian coordinates for eqs. ((3.1) to (3.3)) is at the ground surface right above the center of the pressure source. We should rewrite eq. (2.22), of which coordinates origin is the center of the outlet of a borehole, to the one based on the new coordinates. Let the center of the top of the borehole be (xb, yb, 0) as measured from the new origin. Then we have (3.6) where (3.7) Substituting the displacement field given by eqs. ((3.1) to (3.3)) into eqs. ((2.11) to (2.13)), we ob- tain (3.8) (3.9) (3.10) (3.11)( )F F x y( ) ( )B x bA y b*= $+=F ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) C R x z D R x z R Dx R x z z D 2 3 2 3 4 3 3 2 5 4 3 5 3 3 2 9 23 12 3 3 2 3 4 30 ( ) A z b b b b 1 5 2 5 2 2 2 5 2 7 2 $ $ µ λ µ λ µ λ µ λ µ λ µ λ µ λ µ λ µ λ λµ µ λ µ λ µ + + − + + + + + + + + + − + + + $c c c m m m ( ) ( ) C F x y R R R z z D 2 3 2 3 3 4 1 3 1 10 ( ) A y b b 1 5 2 5 2 7 µ λ µ λ µ λ µ λ µ = + + + + + + − + ( 1 $−= −F ( ) ( ) ( ) ( ) ( ) ( ) ( ) C R R x R R x z z D R R x R R z D 2 3 2 3 4 1 3 3 2 3 4 3 1 3 3 2 6 3 4 1 3 2 1 3 5 8 ( ) A x b b b 1 3 1 5 2 2 3 2 5 2 2 5 2 7 2 3 2 5 2 2 µ λ µ λ µ λ µ λ µ λ µ λ µ λ µ λ µ λ µ µ + + − + + + + − + + + + − − + − + + $ c c c c m m m m .=r= ( ) ( ) ( )r z z x x y yb b b b b2 0 2 0 2 0 2ρ + − − + − d z( ) C W a F x x F y y F r z z r 1 2 ( ) ( ) ( ) ( ) k k BH A k b b B k b b L C k b b 0 2 3 0 3 0 0 5 2 0 2 π ρ ρ ρ = − + − + + − −^ h < F # C b P 2 1 3 ∆= − == ( ) ( )R x y z D R x y z D1 2 2 2 2 2 2 2+ + − + + + 99 Drag-out effect of piezomagnetic signals due to a borehole: the Mogi source as an example (3.12) (3.13) (3.14) (3.15) (3.16) In the above, R1 and R2 should be read as . (3.17) The magnetic field components are given by differentiation of the potential (3.6) with respect to r0(x0, y0, z0) (3.18) (3.19) (3.20) The 1D integrations in the above can be achieved accurately by the double exponential formula (cf., Sasai, 1991a). ( ) ( ) ( ) ( ) ( ) ( ) . d a C Z z F x x z z F y y z z F z z z z 1 3 3 9 15 ( ) ( ) ( ) ( ) k k A k b b L B k b b C k b b 2 5 0 0 0 5 0 0 5 0 7 0 3 π ρ ρ ρ ρ ∆ = − − + + − − + − + − − < F ( 1 # ( ( ) ( ) ( ) ( ) ) ( ) d a C Y z F x x y y F F y y z z y y y y 1 3 1 3 3 15 ( ) ( ) ( ) ( ) k k A k b b b L B k b b b C k b b b b 2 5 0 0 0 3 5 0 2 5 0 7 0 0 2 π ρ ρ ρ ρ ρ ∆ = − − + + − + − + − + − − − < F ( (2 2 # ( ) ( ) ( ) ( ) ( ) ( ) d a C X z F x x F x x y y F x x x x z z 1 1 3 3 3 15 ( ) ( ) ( ) ( ) k k A k b b b L B k b b b C k b b b b 2 3 5 0 2 0 5 0 0 5 0 7 0 0 2 π ρ ρ ρ ρ ρ ∆ = − + − + + − − + − + − − − < F ( ( 2 1 # =R= ( ) ( )R x y z D x y z Db b b b1 2 2 2 2 2 2 2+ + − + + + ++=F ( ) ( ) ( ) . C R z D z D R R z z D 2 3 1 6 ( ) C z 1 3 2 3 2 5 2 µ λ µ λ µ λ µ− + − − + − + = +F ( ) C R R R z z Dy y y2 3 6 ( ) C b b by 1 3 2 3 2 5 µ λ µ λ µ + + − + = +F 6 ( ) C R x R x R x z z D2 3( )x b b b C 1 2 3 2 53 µ λ µ λ µ + + − + ( )F F x y( ) ( )B A b b z z *= ( )F F x y( ) ( )B A b b y x *= 4. The behaviour of the magnetic field above a borehole Let us investigate the behaviour of the mag- netic field above a borehole by assuming an ap- propriate model. Table I summarizes the me- chanical parameters of the Mogi model, the size and the position of a borehole, together with some magnetic parameters. As for the geomag- Table I. Parameters of the Mogi model and magnet- ic environments. Parameter name Abbr. Maximum uplift ∆h 10 cm Source depth D 3 km Rigidity µ 3.5×1010 N/m2 Average magnetization J 5 A/m Stress sensitivity β 2×10−3 MPa−1 Geomagnetic dip I0 60° Geomagnetic declination D0 N16°E Fig. 3. Total intensity changes on the ground (2.5 m above the surface) due to the Mogi model. Unit in nT. This area shows 10 km×10 km square of which center is right above the pressure source. A solid cir- cle indicates the horizontal position of a borehole (2 km north and 1 km east from the center). 100 Yoichi Sasai et al. netic environment, we assume the one in Long Valley Caldera. Figure 3 shows the total inten- sity change produced by the assumed Mogi model, in which we used Sasai’s (1991a) point source solution. The magnetic field is measured at 2.5 m above the ground surface. Suppose that there is a borehole for example at a point indicated by a solid circle in fig. 3, i.e. 0.5 km north and 1 km east from the uplift cen- ter. Figure 4a-d shows the four components of the magnetic field change at a height of 2.5 m over 10 m by 10 m square with its center just above the borehole. Note that the effect of a borehole is extremely local, i.e. within several meters from its outlet. As we can easily imagine from the two hor- izontal components and particularly from the vertical component, the magnetic lines of force spread out from the ground surface and are ab- sorbed into the borehole. Such magnetic field corresponds to the third term on the right hand side of eq. (2.22), which is a line of vertical quadrupoles along the borehole axis. The mag- netic lines of force of vertical quadrupoles ef- Fig. 4a-d. Four components of the piezomagnetic change due to a borehole: a) H, b) D, c) Z and d) F compo- nent. Each square shows 10 m×10 m area of which center is the outlet of a borehole. Unit in nT. a c b d 101 Drag-out effect of piezomagnetic signals due to a borehole: the Mogi source as an example fectively come out into the air, while those of a line of horizontal dipoles encircle within the borehole. It turned out that the contribution to the external magnetic field from the vertical quadrupoles was two orders of magnitude larg- er than that of the horizontal dipoles. However, even a line of vertical quadrupoles is successive arrangement of vertical dipoles Fig. 5a,b. Depth dependence of the contribution from the cylindrical surface of a borehole: a) 2.5 m just above the outlet of a borehole; b) 2 m north and 2.5 m above the outlet. a b 102 Yoichi Sasai et al. with opposite sign, where the magnetic field cancel with each other. The effect of deep-seat- ed magnetic sources is difficult to reach the ground surface. Figure 5a shows cumulative changes in each component of the magnetic field at a point 2.5 m just above the outlet of the bore- hole as it elongates downward. The depth de- pendence of the borehole effect is not so monot- onous as compared with fig. 5a at another point, which is represented by fig. 5b. This figure shows cumulative magnetic changes at a point 2 m north and 2.5 m above the borehole. The total intensity shows its maximum around 1 m, and then it reaches the final value around 10 m length. Generally, the magnetic field above the ground is determined by the contribution from the shallower part of a borehole from top to a 10 m length. In other words, a borehole cannot drag out the magnetic lines of force at depth. This is rather disappointing. The influence of a borehole appears very lo- cally around its outlet. We investigated the dis- tribution of total intensity changes by changing the sensor height as shown in fig. 6a at a height of 2 m and in fig. 6b 1.5 m, respectively. We may expect significant amount of enhanced magnetic signals only near around the outlet of a borehole. However, in such near field, the 1st order approximation solution eq. (2.22) may not be valid, because it gives a divergent mag- netic field at the top of a borehole. 5. Discussion Since 1998 we have conducted continuous measurements of geomagnetic total force inten- sity and area survey of Self Potential (SP) in Long Valley Caldera, California, in order to de- tect possible changes in the EM fields associat- ed with intrusive events at depth. Actually, Mueller and Johnston (1998) observed remark- able magnetic changes associated with reactiva- tion of the resurgent domes in the early 1990’s. At one observation site PLV, we installed two proton magnetometers, one close to a buried vertical pipe (borehole) and the other 40 m away in order to discriminate a sub-nanotesla change if any. An episodic event, i.e. swarm earthquakes and crustal inflation, was triggered in the caldera by the October 16, 1999, Hector Mine Earth- quake of M 7.1 in Southern California. Figure 7 shows the simple differences in the total intensi- ty between the two magnetometers at PLV and the volumetric strain changes at a different ob- servation site. Total intensity difference showed gradual increase of up to 0.3 nT from the latter half of October to early November, which is Fig. 6a,b. Changes in the total intensity above a borehole: a) 2.0 m above and b) 1.5 m above the ground sur- face. Unit in nT. a b 103 Drag-out effect of piezomagnetic signals due to a borehole: the Mogi source as an example more or less similar to the accelerated volumet- ric strain changes after the Hector Mine earth- quake. However, it should be noted that no such a step-like change occurred in the geomagnetic field as observed in the strain data. Johnston et al. (2000) obtained a source model for volumetric strain changes in Long Val- ley Caldera which were triggered by the Hector Mine earthquake. This model involved aseismic normal faulting plus inflation, about 7 km WNW from PLV site, at a depth of 7 km. The volumet- ric inflation source was required to explain the strain data. According to Utsugi et al. (2000), a nearly vertical dip-slip fault produces negligible piezomagnetic field, while a Mogi pressure source effectively generates an observable one (Sasai, 1991a). The observed magnetic changes could thus be produced mainly by the volume source. Hence the results shown in fig. 7 strong- ly motivated the present study. Magnetic field change of any tectonic mod- el is proportional to its moment or the intensity of the source. It greatly depends also on the depth of the source. The triggered normal fault- ing occurred on a slightly inclined fault of 1 km by 1 km wide with 1.4 cm dislocation at a depth of 7 km. Its seismic moment is estimated as 4.9×1019 dyne-cm. On the otherhand, the mo- ment of the Mogi source given in fig. 3 is esti- mated as C=6.3×1023 dyne-cm. This source for the strain transient triggered by Hector Mine earthquake has a moment that is too small and is too deeply located to generate any observable magnetic change even with the aid of an en- hancement effect of a borehole. Hashimoto et al. (2003) summarized the magnetic observations in Long Valley Caldera during the period from 1999 to 2001. They in- vestigated the cause of annual variations promi- nent at several stations, which were ascribed to Fig. 7. Total intensity difference between two magnetometers (Unit in nT) at PLV in the upper curve and vol- umetric strain (Unit in µ-strain) in the lower one at POP site during the period from September 28 to November 3, 1999. Total intensity and strain were measured at every 10 min interval. 104 Yoichi Sasai et al. changes in the ground temperature (Utada et al., 2000). They successfully compensated for the annual variations on the basis of ground temper- ature data. Then they removed some apparent variations in the total intensity differences caused by the vectorial differences of the local main geomagnetic field as well as by the local in- duction effect, i.e. what is called the 3-compo- nents correction or the prediction-error filter technique. They concluded that there was no sig- nificant change larger than 0.5 nT at PLV. As we have seen in fig. 7, the observed variation at PLV, if any, is at most 0.3 nT or so, which does not conflict with Hashimoto et al.’s (2003) conclu- sion. At PLV site in Long Valley Caldera, a vertical steel casing was buried in a nearby borehole. This produced an anomalously high field gradient, say 1000 nT/m or more. We thus installed the sensor within a few m of the borehole. Yamamoto (1990) conducted an experimental observation of the ge- omagnetic field at the bottom of a borehole using flux-gate magnetometers. He used a vinyl chlo- ride tube to avoid some diffculties in magnetic measurements caused by high-µ metals. Howev- er, he reported some steps in the magnetic data associated with small felt earthquakes most prob- ably caused by movement of the sensor. Although the most up-to-date Overhauser type proton mag- netometer works under a relatively high field gra- dients, only a few mm displacement of a sensor would result in an apparent variation. It would then be difficult to discriminate signals of piezo- magnetic origin. An optimum arrangement for geomagnetic observations using a borehole would be a) a bore- hole of several hundreds meters length plus; b) a spherical cavity of radius 1 to 2 m in order to avoid the high field gradient at the bottom of the borehole. The magnetometer sensor is placed at the center of the cavity, which is fixed with silica or any non-magnetic substance. We don’t use the casing pipe of high µ metal for the protection of the borehole. The magnetic field at any position in such a system, even inside the borehole, can be represented rigorously by eqs. ((2.14), (2.28) and (2.39)) with the aid of Lipschitz-Hankel integrals. The same procedure is applicable to evaluate the effect of the spherical cavity at the bottom of the borehole. Our next subject is to investigate the behaviour of the tectonomagnetic field under such an observation system. Acknowledgements We are greatly indebted to Drs. Ciro Del Ne- gro and Antonio Meloni who carefully read the manuscript and gave us useful comments to im- prove it. REFERENCES EASON, G., B. NOBLE and I.N. SNEDDON (1955): On certain integrals of Lipschitz-Hankel type involving products of Bessel functions, Phil. Trans. R. Soc. London Ser. A, 247, 529-551. HASHIMOTO, T., Y. TANAKA, M.J.S. JOHNSTON, M. UTSUGI, Y. SASAI and S. SAKANAKA (2003): On the annual variations in geomagnetic differences observed in Long Valley Caldera, California, Ann. Disas. Prev. Res. 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