(Slant delay mesoscale functions)

M. Figurski, M. Gałuszkiewicz, P. Kamiński, K. Kroszczyński Mesoscale mapping functions (Slant delay mesoscale functions)

M. Figurski, M. Gałuszkiewicz, P. Kamiński, K. Kroszczyński We present a prototype of computer module for GPS slant delay determination using data from COAMPS (Coupled Ocean/Atmosphere Mesoscale Prediction System) NRL (Naval Research Laboratory). COAMPS is a mesoscale non-hydrostatic model of the atmosphere which is run on IA64 Feniks computer cluster in the Department of Civil Engineering and Geodesy of the Military University of Technology. The slant delay is the result of integrating the ray (eikonal) equation for the spatial function of tropospheric refraction along the GPS wave propagation path. The work is a phase of research concerning the impact of mesoscale atmospheric phenomena on the tropospheric delay of the GPS signal.

M. Figurski, M. Gałuszkiewicz, P. Kamiński, K. Kroszczyński Development (some facts) Neil A. E. (). Preliminary evaluation of atmospheric mapping functions based on numerical weather models. Physics an Chemistry of the Earth A(6), pp. 475-48. Boehm, J., A.E. Niell, P. Tregoning, H. Schuh (6), Global Mapping Functions (GMF): A new empirical mapping function based on numerical weather model data, Geoph. Res. Letters, Vol. 33 Boehm, J., B. Werl, and H. Schuh (6), Troposphere mapping functions for GPS and very long baseline interferometry from European Centre for Medium-Range Weather Forecasts operational analysis data, J. Geophys. Res.,. R. Eresmaa, H. Järvinen, An observation operator for Ground-based GPS slant delays, Tellus (6), 58A, 3-4. R. Eresmaa, H. Järvinen, S. Niemelä, K. Salonen, Azimuthal asymmetry in groundbased GPS slant delay observations and their NWP model counterparts Atmospheric Chemistry and Physics Discussions, Vol. 7, pp 379-3, 7--7

M. Figurski, M. Gałuszkiewicz, P. Kamiński, K. Kroszczyński Slant delay module Mesoscale model - COAMPS Subroutine: Refraction Subroutine: Refraction approximation Subroutine: Ray path and slant delay integration Slant delay τ τ = ( ) = 6 n ds Nds p d e e N = k Zd + k Zw + k3 Z w T T T n refraction (refractive index), N refractivity, p d and e partial pressure of dry air and of water vapor,t temperature, Z d i Z w compressibility of dry air and of water vapor,k, k, k 3 constants determined experimetally.

M. Figurski, M. Gałuszkiewicz, P. Kamiński, K. Kroszczyński COAMPS Mesoscale model domain Atmospheric model vertical grids (Terrain following sigma Z system ) ( )/( ) top s top s z z z z σ = z top = H depth of the model domain (3.5 km ~ hpa), z s height of topograhy, kka number of levels (kka = 3). Horizontal grids - Lambert Conformal projection.

M. Figurski, P. Kamiński, K. Kroszczyński Prognostic parameters: π Exner pressure (function), θ potential temperature, q specific humidity,. u, v, w velocity components Diagnostic parameters: T temperature, e partial pressure of water vapor p d partial pressure of dry air

M. Figurski, P. Kamiński, K. Kroszczyński Refractivity approximation N = N + w ( N - N ), w = ( z - z )/( z - z ) * * i, j, m i, j, k- i, j, k i, j, k- m i, j, k- i, j, k i, j, k- Linear aproximation z i, j, k * z m * z m h * m * h m z i, j, k z i, j, k+ + * z m * h m N = N ( x)( y)( z) + N x( y)( z) + N ( x) y( z) + N ( x)( y) z+ xyz Trilinear interpolation N x ( y ) z + N ( x ) yz + n xy ( z ) + N xyz x = x / dx, y = y / dy, z = z / dh Polynomial structure of N makes easy gradient N calculation *

M. Figurski, P. Kamiński, K. Kroszczyński Ray tracing in the atmosphere. Ray definition. Propagation of rays are determined by the eikonal equation: k! ϕ ( r ) r ( ) n ( r), k 3 i = i! = = where the gradient of ϕ ( r! ) gives the direction of the ray, index of atmospheric refraction. The ϕ () r! nr (! ) function is often called eikonal.

M. Figurski, P. Kamiński, K. Kroszczyński The ray equation is the eikonal equation in a new ray based coordinate! d! dr! nr ( ) = nr ( ) ds ds where s denotes the ray path. The equatio nabove can be given by following two coupled differential equations! dr! =! v ds n ( r )! dv! = nr ( ) ds! v dx t ds dy t ds dz t ds α β γ o = ( ( )/, ( )/, ( )/ ) = (cos, cos,sin(9 ))!!! r = [ x( t), y ( t), z( t)] - point of trajectory, v - tangent vector (to r ). These coupled differential equations, that determine the! ray propagation in a medium where the index of refraction as a function of position r! r! is given by, n( ), can for example be solved using the Runge-Kutta technique.

M. Figurski, P. Kamiński, K. Kroszczyński

4 45 5 55 6 65 M. Figurski, P. Kamiński, K. Kroszczyński 6 5 3 3 3 3 4 8 5 5 8 - -3 5 6 3-4 -4 35 7 x -3 4 - - 3 x 7 5 5-4 -4-3 - - 4 3 4 Refraction Module - - -4-4 4-6 Slant delay o Elevation angle [ ] eci 8 7 x 4 4

35 4 45 5 55 6 65 7 M. Figurski, P. Kamiński, K. Kroszczyński 4 6 8 (44,55) 4 (7,66) 6 8 - -5 - -5 5 5 - -8-6 -4-4 6 8 Model domains width 33 km and 3 km resolutions

75 7 3 7 6 5 7 7-5 65-5 6 3 8 3 8 6 65 7 65 6 55 6 4 3 75-8 8 45 9 3 38 3 8 3 85 6 33 6 5 3 4 75 75 5-5 6 65 85 3 6 8 7 33 6 6 5 3 45 5 5 55 6 3 6 5 8 55 7 38 3 4 45 8 8 6 3 3 4 5 5 4 5 Pole współczynnika refrakcji N. 3 sigma powierzchnia modelu COAMPS 4 7 6 9 34 7 3 3 8 56 3 6 9 6 3 6 35 5 M. Figurski, P. Kamiński, K. Kroszczyński 3 8 6 6 5 3 4 4 8 3 3 4 34 4 38 4 5 5 45 3 6 4 3 4 8 3 5 5 45 5 55 3 6 45 5 5.86.88.85 6 3 5.84.89.88.86.88 55 5.8 8 3.83 3 6 3 Refractivity fields.89 3 5.88.9.89 6 3

6 55 5 45 M. Figurski, P. Kamiński, K. Kroszczyński Evaluation slant delay functions 6 4 i 8 (74,87).8.6.4 6 8 4 6 j 5 5 3 Scaning domain 5 5 8. Reference point Scaning process for various 5 elevation ξ, φ -azimuth. 3-5 -5 5

4 3.5 9.5 τ(ξ,φ) 9.5 - - -3 4 τ max 3-4 -4-4 4 - ξ = 3,4,...,9, φ = 8,..,8 3-4 -4-4 3 d and 3d anisotropic distribution of slant delay τ(ξ,φ), ξ elevation, φ azimuth. 4

M. Figurski, P. Kamiński, K. Kroszczyński.5 τ R (ξ,φ) 9.5.5 -.5.5-3 -.5 - - -.5 - -.5 - -.5.5-3 - - - τ ( ξ, φ) = τ( ξ, φ) τ( ξ, φ), ξ = 3, 4,...,9, φ = 8,..,8 R Reduced anisotropic distribution of slant delay.

Satellite MSG image (Visible channel) 55 COAMPS working area 55 τ 9 9 τ(ξ,φ) (67,64) 3 5 5 τ 6 9 3-6 - -3-4 -4-3 - - 3 5 4 3 τ d and 3d anisotropic distribution of slant delay τ(ξ,φ), ξ elevation, φ azimuth - - -3 τ 3-3 - 5 4 3-3

3 3 4 5 6-9 τ R (ξ,φ) - - -5 - -5 5 5.5.5.5.5 -.5 3 - -.5 - -.5 - -.5 - -.5.5.5 Reduced anisotropic distribution of slant delay.5.5 -.5 - -.5 - -

M. Figurski, P. Kamiński, K. Kroszczyński Module possibilities Temporary and spatial analysis atmospheric refraction GPS slant delay.