LEAST-SQUARES OPTIMUM-ORDER REGIONAL GEOTHERMAL GRADIENT DETERMINATION FROM DEEP BOREHOLE TEMPERATURE DATA WITH APPLICATION TO THE WESTERN DESERT OF EGYPT # E. M. ABDELRAHMAN 1 , H. M. EL-ARABY 1 , B. EL-HAKIM 2 , A. I. BAYOUMI 1, A. M. ISMAIL 2 S u mm a r y: This paper discusses an approach to determining the least-squares optimum order of a polynomial fit to temperature-gradient data. The optimum is reached when two successive least-squares regional maps are sufficiently similar. This similarity is measured by the correlation factor between the polynomials. The approach was applied to the available deep bore-hole temperature data in the central part of the Northern Western Desert of Egypt. The fitting polynomial was found to be of the second degree. Assuming the thermal conductivity to be uniformly 2.2 Wm-IK "I for the stratigraphic section in the Western Desert, the heat-flow density ranges between 35 and 70 m W/m 2. 1. INTRODUCTION Bore-hole temperature data can be used to study regional variations of geothermal gradients [1,2]. Geothermal gradients are useful in determining terrestrial heat flow which is important for constructing tectonophysical models of the evolution of a given basin and its marginal basins, the natures of which are not fully understood [3-6]. At the same time, the geothermal gradient can be used to determine the liquid hydrocarbon window where the top surface corresponds to oil generation and the bottom surface to destruction [7]. Heat-flow anomalies at the Earth's surface can be caused by the contrast between thermal conductivity and the resulting refraction of heat, the contrast in the sources of heat production, local temperature differences, convection of groundwater, and/or topographic features [8-10]. For crustal studies, it is important to remove the effect of shallow sources from the observed geothermal data. Sharp irregularities in the heat-flow data or gradients which may be due to residual anomalies [11] or erroneous data can cause unreliable crustal interpretations. In this case, regional-residual separation techniques such as those described by Griffin [12], Agocs [13], Paul [14], Henderson and CordeU [15] and many others, can be applied not only to potential field data, but also to geothermal data, because the equations describing steady-state heat flow, gravity, and magnetics are very similar. On the other hand, two-dimensional mapping of geothermal data from deep bore-holes using the least-squares method is very useful in tectonic studies. However, the effectiveness of the method depends on choosing the proper degree of the regional polynomial to fit the observed geothermal data. As the order of fit becomes high, the regional component of the geothermal data will contain a large portion of residuals. Even when the computed regional is equal to the actual regional, some distortion in the computed regional should be expected because the regional is computed from all data points including the residual effects. The aim of the present paper is to introduce a method of estimating the order of the polynomial that would provide an acceptable # Presented at the International Meeting on Terrestrial Heat Flow and the Structure of Lithosphere, Bechyn~ Castle, Czech Republic, September 2 - 7, 1991. 1 Address: Geophysics Dept., Fac. of Science, Cairo University, Giza, Egypt 2 Address: Geological Survey of Egypt, Abbassiya, Cairo, Egypt 302 Studia geoph, et geod. 37 (1993)