[44] The ECMWF meteorological fields are provided with a horizontal resolution of 0.5° × 0.5° on 91 model levels extending from Earth's surface to 80 km. The pressure, the temperature and the specific humidity are stored at grid points, specified by longitude, latitude and height above mean sea level.

[45] At first, the refractivity is computed at each grid point in the NWM domain. Hereinafter, the one dimensional arrays Ψ_{I} and Φ_{J} with *I* = 1, ., *M*_{I} and *J* = 1, ., *M*_{J} denote the horizontal grid point coordinates of the model grid. The integers *M*_{I} and *M*_{J} denote the horizontal dimensions of the model grid. The three dimensional arrays *H*_{IJK} and *N*_{IJK} for *I* = 1, ., *M*_{I}, *J* = 1, ., *M*_{J} and *K* = 1, ., *M*_{K} store the model height and the model refractivity respectively. The integer *M*_{K} denotes the vertical dimension of the model grid. To compute the refractivity at an arbitrary point *N*(*x*, *y*, *z*) a coordinate transformation routine and an interpolation routine is needed. The coordinate transformation routine consists of transforming cartesian coordinates (*x*, *y*, *z*) to spherical coordinates (*φ*, *ϕ*, *h*). Here *φ* denotes the longitude, *ϕ* denotes the latitude and *h* denotes the height. The interpolation routine consists of the following steps:

[46] Step a) Determine the neighboring grid point indices *I* and *J* and compute the increments *X* and *Y*:

[47] Step b) Compute the bilinear interpolation coefficients *X*_{1}, *X*_{2}, *X*_{3}, *X*_{4}:

[48] Step c) Determine at the neighboring grid points the adjacent grid point indices with respect to height *A*, *B*, *C*, *D* by binary search and compute the vertical interpolation coefficients *L*_{1}, *L*_{2}, *L*_{3}, *L*_{4}:

[49] Step d) Perform logarithmic interpolation to compute the refractivity values *N*_{1}, *N*_{2}, *N*_{3}, *N*_{4}:

[50] Step e) Compute the refractivity *N*:

[51] Above/below the NWM top/bottom log-linear extrapolation is performed. Beyond NWM lateral boundaries the nearest refractivity profile is used.

[52] The computation of the partial derivative of *N* with respect to *x*, *y* or *z* is performed by rigorous application of the chain rule of differential calculus. For reasons of clarity and comprehensibility we provide the computations in the interpolation routine in detail. The computations in the coordinate transformation routine are carried out in a similar manner. Hereinafter the subscript *ξ* denotes the partial derivative with respect to *x*, *y* or *z*:

[58] For better computational efficiency a user specific grid is defined where the model height satisfies for *I* = 1, ., *M*_{I}, *J* = 1, ., *M*_{J} and *L* = 0, ., *F.* For an arbitrary point in the user specific grid and . The refractivity at each grid point of the user specific grid is determined using the coordinate transformation and interpolation routine outlined above. In order to circumvent the binary search the user specific grid is defined such that a simple invertible relation exists between the height and the grid point index with respect to height. Since the refractivity tends to decrease exponentially with height, we propose the following ansatz

for *L* = 0, ., *F.* The integer *F* denotes the vertical dimension, *τ* denotes the lapse rate and *T* denotes the top height of the user specific grid. Specifically, for the ECMWF grid we set *F* = 200, *τ* = 6 and *T* = 80 km. On the uncertainty of STDs due to the interpolation routine the reader is referred to section 3.