## 1 Introduction

[2] Regional models of the Earth's lithosphere are frequently setup in planar approximation using Cartesian coordinates. Specifically, the universal transverse mercator (UTM) projection is commonly used for the horizontal coordinates, and heights and depths are given with respect to the associated plane. The regional models are usually based on seismic and/or magnetic and gravitational potential data acquired on or close to the Earth's surface [e.g., *Zeyen and Fernàndez*, 1994; *Tašárová et al*., 2006; *Popowski et al*., 2009; *Barrère et al*., 2011]. Because of the limited height of data acquisition, and as long as the extension of the study area is limited, planar approximation may be sufficient and the relation between the model reference frame (MRF) and measurement reference frame is straightforward. In March 2009, however, the Gravity field and steady state Ocean Circulation Explorer (GOCE) satellite was launched, which gravitational gradient measurements have found application in lithospheric modeling [*Bouman et al*., 2011a; *Braitenberg et al*., 2011; *Álvarez et al*., 2012; *Hirt et al*., 2012]. Because of the height of the satellite—GOCE has a perigee height of 255 km above the Earth's surface—planar approximation may no longer be adequate.

[3] The GOCE observed gradients are given in geocentric coordinates, which can easily be referred to the reference ellipsoid because the transformation from spherical coordinates to ellipsoidal coordinates is straightforward. The MRF commonly uses the UTM-projection, which is a transversal cylindrical projection and at the central meridian the ellipsoid and cylinder coincide. Also, the relation between the ellipsoid and the plane is known, at least when the height is zero. For GOCE gradients at satellite altitude the height is obviously not zero and the exact relation needs to be derived. With respect to the vertical direction the ellipsoidal normal coincides with the planar normal for the central meridian. For other meridians there is an angle between the ellipsoidal and planar normal. In the plane the *x*-axis points east (easting), the *z*-axis is along the normal and the *y*-axis points north (northing). Also, map north and east in the plane do not coincide with geographical north and east of the ellipsoid except for the central meridian. In addition, it is not directly clear how to connect the planar Cartesian frame from a UTM projection to spherical or ellipsoidal coordinates at satellite altitude.

[4] In this paper, we will derive the relation between UTM coordinates and associated height, and their geocentric counterparts, as well as the relation between the orientation in the model and measurement reference frames. The reference frames are defined in section 2 and the relations between coordinates and orientations are derived in section 3. The relations are illustrated by examples derived from GOCE gravity field models as well as synthetic airborne gradiometer data. Section 4 addresses the validity of planar versus spherical approximation as far as topographic reductions are concerned.