Map projections and topography in atmospheric mesoscale modeling

Digital Elevation Models (DEMs) of the Earth surface are given by a data set {lambda(k), phi(k), h(k)}(k=1)(K), where h(k), denotes the terrain height on a point (lambda(k), phi(k)) of an ellipsoidal Earth model epsilon. Map projections have been widely used in mesoscale modeling to get a DEM on a model domain D of a plane surface P. The procedure consists in applying a map projection to each point (lambda(k), phi(k)) to get a point (X-k(p), Y-k(p)) on D for which the correct terrain height Z(k)(p) is taken as h(k). This implies that P coincides with the tangent plane T to epsilon at a point (lambda(c), phi(c)) which can be located at the centre of D. In this work a method is proposed to get a DEM {X-k, Y-k, Z(k)}(k=1)(K) on T whose accuracy is similar to that of the original DEM {lambda(k), phi(k), h(k)}(k=1)(K); the method is extended to get a DEM on a spherical Earth model, The method is based on the transformation T that yields the point (X-k, Y-k) on T projected by a point P-lambdaphih in the tridimesional space defined by the data lambda(k), phi(k), h(k). T also yields the correct height Z(k) of P-lambdaphih with respect to the plane T. By construction (X-k, Y-k) is the unique point in 7 for which the correct terrain height can be computed with the data lambda(k), phi(k), h(k) alone. It is shown that the points (X-k(p), Y-k(p)) from map projections do not coincide with the corresponding one (X-k, Y-k), hence the correct height Z(k)(p) on each (X-k(p), Y-k(p)) cannot be computed with the data lambda(k), phi(k), h(k) alone. The two immediate approximations Z(k)(p) similar to Z(k) and Z(k)(p) similar to h(k) are studied, The uncertainty Deltah(k) = +/-30 m reported for a DEM is used to show that the estimations Z(k)(p) similar to Z(k) and Z(k)(p) similar to h(k) are valid on regions of 500 x 500 and 60 x 60 km(2) respectively. It is shown that if the point (X-k(p),Y-k(p)) is near the boundary of a model domain of 1300 x 1300 km(2) the estimation Z(k)(p) similar to h(k) has an error of approximately 30 km. and for a domain of 3300 x 3300 km(2) the corresponding error may be 200 km. It is shown by means of analytic solutions of mesoscale meteorological equations that the innacuracy of the DEM {X-k(p),Y-k(p), h(k)}(k=1)(K) can yield a wrong wind field.