Journal of African Earth Sciences, Vol. 16, No. 4, pp. 413-423, 1993. 0899-5362/93 $6.00 + 0.00 Printed in Great Britain Pergamon Press

First heat flow density determinations from Southeastern Zaire (Central Africa)

M. N. Sm3A~'mcn 1~, G. VASSm-rR ~ and P. Lot~s I

ICentre G6ologique et G(mphysique, Universit~ de Montpellier II 34095 Montpellier Cedex 5, France ~,aboratoire de G&~hysique, Universit~ de Lubumbashi B. P. 1825 Lubumbashi, Zaire

(First received 15th February, 1993; revised version received 30th April, 1993)

Absla'aet - First beat flow density determinations from southeastern Za~re are presented. Sites are located in the late Proterozoic metasedimentary cover of the Pan-African belt (600 Me.). For each individual boreholes, heat flow ranges between 48 and 72 mW m 2. The average value of 62 mW m "2 for the sites is similar to that of 66 mW m 2 obseawed in Zambia. Both values are higher than what is expected for P~m-African terranes. These heat flow determinations in Shaba province of southeastern Za~e, together with gravity and seismological observations, support the hypothesis of lithospheric thinning for this area. As already suggested for Zambia, this lithospberic thinning may be associated with a southwestern extension of the East African Rift System from Tanganyika across the central African plateau.

INTRODUCTION

During the las t twen tyyears , a significant n u m b e r of t e m p e r a t u r e m e a s u r e m e n t s h a s b e e n obta ined in Africa for hea t flow dens i ty de te rmina t ions (HFD). The va lues of HFD show a subs t an t i a l con t ra s t In hea t flow b e t w e e n Archaean c ra tons an d y o u n g e r s u r r o u n d i n g t e r r anes In which HFD is typical ly a b o u t 50 % grea te r t h a n In the c ra tons (e.g. Bal lard e t a / . , 1987; Nyblade et al., 1990; Lesquer and Vasseur , 1992). In Zambia where C h a p m a n and Pol lack (1977) per formed the first geo the rma l s tud ies , the high hea t flow va lue a v e r a g e of a b o u t 6 6 m W m -2 o n P r o t e r o z o i c te r ranes , w a s In terpre ted a s an evidence, along wi th se ismici ty and p l a t eau uplift, of a th inned l i thosphere a n d incipient rifting extending from Lake T a n g a n y i k a s o u t h w e s t w a r d a c r o s s t he centra l African p la teau .

S o u t h e a s t e r n Zaire is located In sou thwes t of Lake Tangany ika (Fig. 1), where geological and seismological obse rva t ions have clearly evidenced WNW-ESE tens iona l pa t tern , a s soc ia ted with the Eas t African Rift S y s t e m (Brain, 1972; M a a s h a and Molnar, 1972). It be longs to the s a m e Proterozoic tectonic province as Zambia and is also char- acter ized b y dis tens ive s t ruc tu re s , such as the y o u n g U p e m b a and Mwero g r abens t rending NE- SW (Fig. 1 and Fig. 5).

Accord ing to F a i r h e a d a n d Gird le r (1969), Fa i rhead and H e n d e r s o n (1977) a s s u m p t i o n a b o u t Eas t Africa Rift extension, va r ious geophysical obse rva t ions s u c h a s t r end of se ismici ty (Bram,

1972; Gr lmison and Chen, 1988) or HFD da ta (Chapman and Ponack, 1977; Nyblade etaL, 1990) have sugges ted the exis tence of one or several p o s s i b l e i n c i p i e n t r i f t i ng a r m s f r o m L a k e Tanganyika cont inu ing towards SW, re la ted to an a s t h e n o s p h e r e uplift. In order to d i s c u s s the rifting a s s u m p t i o n in S o u t h e a s t e r n Zaire, fu r the r geo- logical and geophysical data, s u c h a s geothermal and gravity m e a s u r e m e n t s , are needed. Thus , t e m p e r a t u r e su rveys have b e e n c o n d u c t e d in mining explorat ion boreholes , while gravity cover- age h a s been ex tended on S h a b a province of Zaire in 1990 b y the geophys ics t e a m of the Universi ty of L ubumbash i .

In th is paper , we descr ibe the r e su l t s of the p rogram ZAIFLUX a b o u t the t empe ra t u r e and hea t flow m e a s u r e m e n t s ob ta ined dur ing the las t three y e a r s in Ka tangan Copper Arc of Zah'e. These resu l t s with those from Zambia provide s u b s t a n - tial cons t r a in t s on geodynamic mode l s of as theno- sphe r i c upl if t in S o u t h e a s t Zaire a n d North Zambia.

GEOLOGICAL SETTING

In S o u t h e a s t e m Zaire, th ree wide P recambr ian be l t s are k n o w n (Fig. 1). In the nor thwes t , the U b e n d i a n (Lower Proterozoic) a n d the Kibara (Middle Proterozoic) be l t s which en tend to Zambia and to the e a s t e m Africa are in te rpre ted as the resul t of a cont inenta l collision (e.g. Daly, 1983; K a m p u n z u et al., 1985; Tsh imanga et al., 1988; Kabengele et aL, 1991). They compr i se a variety of

413

4 1 4 M. N. SEBAGENZI, G. VASSEUR and P. Louis 2 S o

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First heat flow density determinations from Southern Zaire (Central Africa)

granites, mlgmatites, gneisses, and metasediments which const i tu te the pre-Katangan basemen t complex. In the southeast , the Katangan ('late Proterozoic) is considered as a mobile belt formed during the so-called Lufilian orogeny of Pan- African age (600 Ma.).

All the sites of temperature measurements are located in the Katangan metasediments . Many papers on stratigraphy of this belt have been published (e.g. Francois, 1974; Lefebvre, 1978). The K a t a n g a n be l t c o n s i s t s of c las t ic and calcareous sediments of the Roan supergroup, s h a l e s and c a l c a r e o u s s h a l e s of the lower Kundelungu supergroup.

During the Pan-African orogeny, the Katangan and its basement have been deformed simul- taneously. Their relat ionships and the overall tectonic character of the orogeny are clearly shown In open pits and mines of the Katangan Copper Arc. The Roan rocks are Intensively deformed and during the orogeny, they formed northwards thrus t over the rocks of the Kundelungu platform, which extends to LakeTanganyika In the north (Fig. 1). In the Katangan Arc, the main features of deform- ation are foldings, reverse faults with overthrust- ing and thrus t faults whereas In the Kundelungu platform the rocks are horizontal with mInor local deformations.

The Katangan Arc is rich in polymetallic mineral- izations and several copper, cobalt, lead, zinc and uranium mining districts are located within the fold belt In the southeas t of Zaire and in the north of Zambia.

T E C H m g U E OF M SUm NTS

Temperature measuremen t s have been obtained In 1988 and 1990, In exploration boreholes drilled by GECAMINES and SODIMIZAmining companies of S h a b a Province. Surveys were conduc ted several years after drilling had ceased. They were made In water, most of the time below 50 m, the expected depth of the water table during the dry season , with a t he rmi s to r probe equ ipmen t (relative accuracy: 0.01) combined with a Metrix MX 575 multimeter, with sampling each 10 m. In order to reach the proper s teady-state during logging, it was necessary to wait about five minutes after the probe had been submerged. For most of the boreholes, temperature was measured only in the range of 50 to 200 m, with the exception of Kinsenda site, not far from the Zambia border, where boreholes are deeper (300-450 m).

Thermal conductivity of solid rock samples were measured using a t ransient line heat source method. Samples of drill core were cut parallel to the origInal core axes. For most boreholes, cores were not available, then surface rocks were collect-

415

ed to complete samples from holes. All rocks were sa tura ted with water Into a vacuum appara tus before measurement . For the main lithologies, measured thermal conductivity values of samples are: 3.56 W m q K -1 for arkose, 4.20 W m q for dolomite, 2.65 W m ~ K" for sandstone and 2.15 W m q K q for shale.

TEMPERATURE DATA

Temperature-depth profiles measured In diffe- rent sites are shown In Fig. 2. In general the holes are located in a fiat country and no topographic corrections have been made. All curves exhibit a significant negative temperature gradient In the first hundred meters. The relative minimum tem- perature is found at a depth ranging from 100 to 200 m. Examining the temperature profiles from Kinsenda site, it appears that below this depth the temperature gradient Increases until it reaches a constant value.

As it cannot be due to conductivity contrasts or variation of the water table, the anomalous curvature which may be seen as a characteristic feature of the measurements from the southeas t of Zaire and perhaps from the north of Zambia (Chapman, 1983), is interpreted as the effect of variations In surface conditions, i.e. of surface temperature (Sebagenzi e t al., 1992).

The analysis of this phenomenon was done with an inversion technique of temperature data based on the theory of heat conduct ion in a semi-infinite homogeneous medium. This process a s sumes that ground temperature variations can be modelled as a Jump of temperature, starting at an unknown time and contInuIng until present time. These variations are computed by boreholes temperature data Inversion with algorithm of Tarantola and Valette (1982). One of the parameters of the ground temperature model is the undis turbed tempera- ture gradient, used for heat flow determination. The Inversion algorithm re turns the optimal values of the parameters ( d T / d z the undis turbed tempe- rature gradient, T o the original undis turbed sur- face t e m p e r a t u r e , T the p r e s e n t s u r f a c e temperature, a T t h e dis turbance between the inl- tial tlme and the present time, t* the time Interval during which this dis turbance occurred) and a poster/or/standard deviations. The development of this Inversion technique was recently performed (e.g. Cermak et al., 1992; Sebagenzi et al., 1992).

As shown In Sebagenzi et al., the anomalous curvature can be Interpreted as the result of a warming of the ground surface temperature by 3-4°C during the last 40-100 years. This warming can be associated with the environmental changes In connection with mIning activities and urban- ization.

416

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HEAT FLOW DETERMINATION AND DEEP STRUCTURE

The HFD have been computed as the product q = k d T / d z , where k is the average conductivity on the depth of measurement and dT/dz the undis- turbed temperature gradient. The value of K has been est imated in each borehole as the harmonic average of conductivity for each lithology weighted by their thickness. The uncertainty on this average of conductivity is est imated to 10 %. The distribu- tion of heat flow for both Zambia and Zaire sites, together with regional gravity anomaly (Sebagenzi, 1993) are depicted in Fig. 4, where some HFD results from Lakes Kivu, Tanganyika, Malawi and from Tanzania reported by Nyblade e t a/. (1990), are plotted for comparison purpose. In Table 1 where the resul ts obtained from Zaire are sum- marized, we give for each borehole, elevation, un-

dis turbed temperature gradient, harmonic ave- rage conductivity and heat flow value. In addition for each site, an average heat flow is shown in last column. In Southeas t Zaire, heat flow values for each individual borehole range between 48 and 72 mW m 2 whereas site averages vary from 53 to 69 mW m -2. The es t imated error is large, bu t the various values seem consistent. The average for the various sites is 62 mW m 2. This value is similar to that of 66 mW m 2 observed from Zambia by Chapman and Pollack (1977). Both values give an average of abou t 64 mW m 2 which might be representative of the Zaire and Zambia heat flow values. This average value which is similar to those obtained in the Mozambique belt on the flanks of the Rift east branch (Nyblade eta / . , 1990), is higher by about 9 mW m 2 than the 55 mW m 2 value calculated for the Hoggar Pan-African Province (Lesquer e toL , 1989). Thus, HFD from Zambia and

First heat flow density determinations from Southern ZaIre (Central Africa)

Zaire Proterozoic ter ranes lies within the regional heat flow pat tern of 68 ± 4 mW m -2 average value for rifted continental margin in East Africa (Nyblade et oL, 1990). MbuJtmayi site HFD (Fig. 1 andTable 1) which is presented in the west of Fig. 4 where regional gravity is not av_afiable, is not considered in our discussion because this site belongs to different tectonic province of KasaI craton. How- ever, this HFD value would be taken into account in the interpretation of the HFD contrast between Archaean craton and mobile belts area~. HFD average in Proterozoic ter ranes of Zambia and Zalre is higher by 20 mW m -2 than the 44 mW m "a obtained from MbuJimayi site of Kasai Archaean craton which agrees with Tanzania and Zimbabwe cratons HFD (Nyblade et oL, 1990).

Crusta l c o n t r i b u t i o n t o s u r f a c e h e a t f low In order to discriminate the crustal component of

heat flow and that from deeper origin in the mantle, Birch et oL (1968) proposed a linear relationship between heat flow qo and heat production Ao, expressed as qo = q* + bAo, where q ' i s reduced heat flow which represents heat flow component at lower crustal level in a province and bAo the integrated heat production. The parameter b is a lengh which characterizes the vertical distribution of heat generat ion in the crust. This relation has been applied by Ballard et oL (1987) to heat flow and heat production data from Bostwana-Namibia and Zambia which belong to the same tectonic province, to obtain a reduced heat flow q* of 50 + 3 mW m "2 and a slope b of 5.5 _+ 1.1 kin.

Under the assumpt ions tha t heat production is constant for depth range 0 < z < b with b = 5.5 ± 1.1 km and tha t reduced heat flow q* of 50 ± 3 mW m -2 calculated by Ballard et ol. corresponds to heat flow at upper crustal level, an estimation of the crustal contribution to surface heat flow has been

417

made. For this estimation, a two layers crust with different heat production, has been assumed. An upper crust formed by granite gneisses, biotite gneisses, migmatl tes and amphibolites in the amphibolite facies, and a lower crust is assumed to be formed by granulite facies rocks (Chapman and Pollack, 1977). We have no heat production deter- mlnation from heat flow sites of Southeast Zaire. However in Zambia, measurement s obtained by Chapman and Pollack (1977) on some samples of granite gnelsses, biotite gnelsses, mlcaschists and amphibolites, belonging to the pre-Katangan base- ment and Katangan metasediments , revealed an average heat production of 2 . 4 ~W m a which can also be used for Southeast Zaire. According to Hadiouche and Jobert (1988), the average crustal thickness was estimated to be 36 km for this area. Then, the upper crust with average heat produc- tion of 2.4 ttW m -a, would have a thickness of 5.5 km. The lower crust would be 30.5 km thick; its heat production value can be est imated to be 0.4 I~W m a as other granulite facies ter ranes through- out the world (Pinet and Jaupar t , 1987). For this model, the values of integrated heat production are 13.2 mW m "a for upper crus t and 12.2 mW m "~ for lower crust, which lead to a mantle heat flow of 38.6 mW m 2.

However, the validity of this result depends on the reality of the heat production model and in particular on the th ickness of the enriched crust. It has been shown that lateral heat t ransfer may significantly affect the slope of the heat flow-heat p roduc t ion re la t ionship (e.g. J a u p a r t , 1983; Vasseur and Singh, 1986; Nielsen, 1987; Furlong and Chapman, 1987). This effect has been model- led quantitatively by Vasseur and Singh (1986), assuming that the heat generation distribution can be represented horizontany by an isotropic stationary function of the horizontal distance.

Table 1. Thermal gradient, est imated conductivity and heat flow density for each borehole.

SITES BOREHOLES ELEVATION TIIERMAL GRAD. CONDUCTIVITY IIEAT FLOW DENSITY (m) ('Ckm "1) (Wm"'C I) (mWm "2)

KINSENDA KIN 80B 1295 24.0 ± 1.3 2.94 ± 0.3 71 ± 11 KIN 122 1285 21.3 + 2.7 3.03 ± 0.3 65 ± 14 KIN 144 1271 24.0± 1.8 3.01 ±0.3 72± 12 KIN 149 1274 19.4 ± 2.0 2.97 ± 0.3 58 ± 12

66.5 ± 12

KIPUStll MW2 1302 21.0 + 6.0 3.30 ± 0.3 69 + 26 69.0 ± 26

I.IKASI IJKASI3 1320 17.0 + 10.0 2.82 ± 0.28 I.IKASI4 1320 23.0 ± 8.0 2.82 ± 0.28

KAMOYA 1490 17.0+ 9.0 2.801+ 0.28

48 + 32 65+29 48+ 30

53.0 + 30

MBUJIMAYI 18 595 15.0 + 6.0 2.67 + 0.3 40 + 20 KI0 590 18.0+ 5.0 2.69+0.3 48+19 44.0+20

418

As shown in the appendix, the apparent thick- ness Da of the radioactive layer deduced from heat flow-heat product ion relation is reduced with respect to its actual th ickness D by a value which depends upon k / D ratio, k being the characteristic horizontal scale length of heat production varia- tions. For reasonable values of the ratio g/D, the reduction factor lies between 1 and 0.5.

Returning to the estimation of crustal radio- active source thickness, the actual thickness D would then range between 5.5 and I I kin. The latter value would imply a crustal heat flow contri- but ion of 36.2 mW m 2 a n d therefore, a mantle heat flow of 27.8 mW m a instead of 38.6 mW m a as obtained with the former assumption.

Recent geothermal s tudies around the world have shown tha t mant le heat flow in s table Precambrian shields is close to 13 mW m "2 (e.g. Pinet and Jaupar t , 1987; Pinet et aL, 1991). In comparison with this result, it appears that mantle heat flow benea th Southeas t Zaire and North Zambia Proterozoic t e r ranes may present an excess heat flow of 14.8 mW m a.

G e o t h e r m a l i n v e s t i g a t i o n s f rom Z a m b i a (Chapman and Pollack, 1977) led to estimate similar

M. N. S~^G~czl, G. VASSAR and P. Lotns

excess mantle heat flow {mean value of 15 mW ma}. This thermal excess has already been noted and interpreted as indicator of a flux from the asthenosphere, in relation with lithospherlc thin- ning, perhaps associated with a south western ex- tension of East African Rift System {Chapman and Pollack, 1977). As Southeas t ZaZre and North Zambia belong to the same major tectonic pro- vince, we believe that our heat flow determinations support , for the Southeas t Zaire, this hypothesis which may be considered as confirmed since Southeas t Zaire and Zambia HFD agrees with the regional heat flow pat tern of Kenya and Tanzania (Nyblade etaL, 1990}, where continental rift struc- tures are well developed.

G e o t h e r m s b e n e a t h S o u t h e a s t ZaIre An estimation of temperature as a function of

depth for Southeas t Zaire and North Zambia has been attempted. Geotherms have been calculated for heat flow of 64 mW m-% the mean surface h e a t flow value for Zaire and Zambia terranes. Calcula- tions were run for the two models corresponding to the extreme assumpt ions on thevalue olD {5.5 and 11 kin). For the first model, the following para-

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First heat flow density determinations from Southera 7-.Yre (Central Africa)

meters were assumed: m e a n radiogenic heat for upper and lower crust are 2.4 and 0.4 ~tW m -s, res- pectively; thickness of 5.5 km for the upper and 30.5 km for the lower crus t and no mantle heat production contribution; uniform thermal con- ductivity of 3 W m ~ K'L For the second model, all parameters values remain the same except the upper and lower crus t thickness values which are fixed to 11 km for the upper and 25 km for the lower

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419

crust. Both geotherms presented on Fig. 3, indi- cate quite temperatures at depth. A temperature of 1250°C, which could correspond to the beginning of partial melting, is reached at depth of about 85 to 110 kin.

The predicted temperatures mus t be taken with caution because of uncertaint ies about heat gene- ration and crust layer thickness estimations, and of no sufficient constraints on thermal parameters

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Fig. 4. Regional gravity field, HFD values at the sites shown in Fig. I and at some sites reported from Kenya and Tanzania by Nyblade e t al. (1990), main faults of east central Africa associated with the East African RIll system. Gmvlty curves are in regal and HFD in mWm'~ enriched star symbol corresponds to new HFD sites reported In this study, solid triangles for Zambia (Chapman and Pollack, 1977) and solid squares for Kenya, Tanzania and Zambia HFD sites (Nyblade eta/., 1990). Thick lines display major normal faults related to main rifts l ~ means rills).

420

assumption. However, they provide a first a t tempt to estimate lithospheric temperature variations with depth in Southeas t Zaire.

D I S C U S S I O N A N D C O N C L U S I O N S

In spite of their limited n u m b e r and low accu- racy, heat flow values in the Proterozoic belts of Shaba Province of Zaire (62 m W m 2) are consistent with the resul ts from Zambia (66mW m2). These HFD values are higher than what is expected in this area of Proterozoic age and are rather similar to those observed in Rift east branch. The predict- ed high tempera tures of 1250°C at shanow level would suggest tha t mantle beneath Proterozoic terranes of Southeas t Zaire and North Zambia is anomalously hot and would indicate the presence of as thenospheric uplift at a depth ranging be-

M. N. Sm3^o~rn, G. V ^ s s ~ and P. Louxs

tween 85 and 110 kin. High heat flow in Southeas t Zaire and North

Zambia related to asthenospherlc uplift is sup- ported by other indications, such as gravity and seismological results.

Regional gravity field of Southeas t Zaire and North Zambia was derived from Bouguer anomaly map obtained using Zambia data compiled by Cowan and Pollack (1977), combined with South- east Zaire data recently acquired (Sebagenzl, 1993). Regional gravity low (amplitude < 100 rngal) trend- ing NE-SW (Fig. 4) is in agreement with Girdler predictions {Girdler, 1975}, which suggested that the regional gravity anomaly in southeas tern Africa could continue the long wavelength anomaly asso- ciated with the East African Rift System. Fig. 4 presents this regional gravity field, HFD value and principal sites location, tectonics of east and

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First heat flow density determinations

central Africa and somefauRs associated with the East African Rift System. These features show clearly a close correlation between high heat flow and regional gravity anomaly over this area. This suggests that the regional gravity low may be asso- ciated with a thermal anomaly within the upper mantle.

An important a rgument to discuss the existence of incipient riRing in this area, is the estimate of lithospheric thickness from teleseismic delay times and gravity correlation published by Fairhead and Reeves (1977}. Their pat tern for Southeast Zaire and North Zambia ter ranes of Proterozoic age, predicted a th inned lithosphere which agrees with regional gravity trending NE-SW and with high heat flow.

Seismic activity is represented by the epicenters distribution of ear thquakes with n% _> 4.0 reported by the International Seismological Center between 1965 to 1970 and some fault plane solutions are shown (Fig. 5}. The focal mechan ism of 1992 S e p t e m b e r 11 e a r t h q u a k e , S o u t h e a s t Zaire (Dorbath, personal communicat ion 1992) which Just supports our previous observations, has also been plotted. Both results could be used to con- swain the present day deformation of this zone and its relationship with the East African RIR System. Present seismic activity in Southeast Zaire and in North Zambia reveals that a broad area of activity

from Southern Zaire (Central Africa) 421

extends southwestward from Lake Tanganytka into Southeast Zaire and Zambia where events have occurred in recent years (Fairhead and Henderson, 1977; Fairhead and Stuart, 1982; Maasha and Molnar, 1972; Grimison and Chen, 1988} and are nowadays occurring (e.g. Lombe and Mubu, 1992}. Seismicity NE-SW trend is parallel to regional gravity anomaly NE-SW axis and to Upemba, Mwem, Luangwa, Lukusashi and Luano rifts (Fig. 5 and Fig. 6) and ear thquake-source mechanisms indicate a NW-SE tensional stress associated with normal faulting (Dorbath, person- al communicat ion 1992). In a recent analysis of Zambia seismicity, Lombe and Mubu (i 992) sug- gest to consider the Luangwa and Mwero rifts as already existing NE-SW trending reactivated fault zones which could indicate some weakness in the crust. In this way, the seismicity in this zone might indicate that an incipient branch of the western arm of the East African Rift System extends into Southeast Zaire and Zambia and that this branch could be traced across Lake Tanganyika and con- nects with the most active part of the Rift System (Grimison and Chen, 1988). We note that this branch coincides largely with basement s t ructures associated with the Proterozoic mobile belts which separate Archaean craton areas. It appears that the Proterozoic crustal s t ructures have a major influence on the recent crustal movements.

1.0

0.8

0.6

a

0.4

0.2

0.0 I-" 0.01

l ! i l ! i | i I ! i i ! ; i i ; I

0.1 1 i I i ! I I I i i I l ! i l i i l i

10 tO0 X/D

Fig. 6. Variation of D J D as a function of k/D.

422

A c k n o w l a d g e m e n t s - Field works were supported by the Unesco Major Project "Geology for Economic Development" Contract SC/RP 264061.8 and SC/RP 2030360.0. This project was facilitated by the following mining companies and their staff: GECAMINES (C~n~ral des Carrt~res et des Mines}, SODIMIZA (Soci~t~ de Developpement Industrtel et Mlnl~res du Zalre), MiBA (Mlnl~re de Bakwanga). H. N. Pollack of the University of Michigan (USA) and M. Q. W. Jones of the Bernard Price Insti tute of Geophysical Research, University of the Witwatersrand (RSA), are thanked for their interest in this study. We are grateful toA. Lesquer and F. Lucazeau of Centre G~log ique et C,~'ophysique CNRS, Montpellier for valuable reviews and comments on the manuscript .

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Appendix

Following Vasseur and Singh (1986), two assumptions on the vertical distribution of heat

production of the surface layer can be made : either an exponential decrease with depth as e-~°

or a c.onstant heat production for depth range 0 < z < D. In both cases the autocorrelation function

of heat production which characterizes its horizontal variations, is assumed to vary as e '"x,

where ~. is the length giving the scale of horizontal heat production variations and r is the

horizontal distance. With such hypotheses, the apparent thickness D, of heat production layer,

as deduced from heat flow-heat production relationship, is obtained in the following closed

form:

D,, :" k'dk'

"It= Jo (k'2 + l)3a (1 +13k' ) with 13 = L/D

for constant heat production for 0 < z < D depth ;

D , 1 : " 1 - e -~' , D Iz=~J0 (k '2+ l ) 3adk

for exponential heat production decrease with depth.

These results are plotted on Fig.6 as a function of the parameter L/D.

From this figure, one can see that, for a given D value, the smaller is ~., the smaller the

apparent thickness D, deduced from the. heat flow-heat production plot. Only for 13 < 1 (i.e.

lateral heterogeneities of very large horizontal extension), the apparent Do value reach the true

thickness. This corresponds to ~. values range of 1 < X < 0-.