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Projection of KNMI radar images

Iwan Holleman / Hans Beekhuis

Stereographic projection (proj.4 manual)


Contents:


Geographical positions of the radars:

The geographical positions of the two weather radars of KNMI are:

Radar Longitude Latitude Longitude Latitude
De Bilt 5o10'42.04"E 52o06'06.04"N 5.17834E 52.10168N
Den Helder 4o47'23.90"E 52o57'12.02"N 4.78997E 52.95334N

Geographical lon/lat to radar image:

The KNMI radar images are provided with a polar stereographic projection. This azimuthal projection uses a central point (origin) and a reference meridian, in our case the northpole and the Greenwich meridian. A geographical longitude (L) and latitude (B) are transformed into a distance from the origin and an angle with respect to the reference meridian (L0). A stereographical projection is conformal, indicating that angles are conserved during projection. In a polar stereographical projected image, meridian become straightlines starting in the northpole and parallels of latitude become circles centered at the northpole. The KNMI projection is performed using an ellipsoide earth model with an equator radius of Re and a polar radius of Rp. The equation for calculation of the projection distance as a function of latitude R(B) is given by:

The eccentricity e is calculated from the earth model (e=0 --> spherical):

The ellipsoide factor F(B) is given by:

The projection distance as a function of latitude R(B) is now given by:

The projection distance is scaled such that the magnification of the projection is 1 at the latitude of true scale (Bw). The following equation is valid for the scaling factor Z(Bw):

The ellipsoide scale factor G(Bw) is calculated from:

The scaling factor Z(Bw) is now equal to:

The Cartesian image coordinates are calculated in a reference frame with the y-axis pointing north, parallel to the reference meridian and a perpendiculr x-axis. The Cartesian coordinates X (distance to reference meridian) and Y (distance to northpole along reference meridian) are:

Finally, the Cartesian distances are converted to image coordinates (I and J) using the pixel sizes (X and PY) and the offset of the upper-left corner (OI and OJ):

The floating-point image coordinates can be converted to integer image coordinates by means of a truncation.

Radar image to geographical lon/lat:

In order to convert image coordinates (I and J) to geographical longitude and latitude, the above procedure has to be applied in reverse order. For one step, an iterative approach is needed to solve the equation. First the integer image coordinates have to be converted to floating-point values by adding 0.5, i.e., center of pixel. Subsequently, the Cartesian projection distances can be calculated:

The distance from the projection origin (northpole) R(I,J) and the longitude L(I,J) are found using:

The latitude can be calculated from the projection distance by a two-step iteration. During the first step, the ellipsoide part of projection distance F(B) is neglected. Using the first estimate of the latitude B1, the ellipsoide correction factor can be estimated. Subsequently, an improved estimate of the latitude can be obtained.

Projection parameters of KNMI radar image with 1.0 km x 1.0 km pixels:

The parameters of the KNMI radar image are listed in the table below. Using the above mentioned equations and these parameters, a geographical longitude and latitude can be assigned to each pixel or vice versa. These coordinates apply when using the WSG84 Earth model.


Parameter Waarde
Projection Stereographic
Origin of projection L0=0.0E en B0=90.0N
Latitude of true scale Bw=60.0N
Radius on equator and pole (WGS-84) Re=6378.137 km en Rp=6356.752 km
Offset of left-upper corner OI=0 en OJ=3650.0
Pixels size PX=1.0 km en PY=-1.0 km
Number of rows and columns 765 en 700

Using these parameters the corners of the radar images are calculated to be:

Hoekpunt Lengtegraad Breedtegraad XXX[km] YYY[km]
Northwest 0.000E 55.974N 0.000 -3650.000
Northeast 10.856E 55.389N 700.000 -3650.000
Southeast 9.009E 48.895N 700.000 -4415.000
Southwest 0.000E 49.362N 0.000 -4415.000

Projection parameters of KNMI radar image with 2.5 km x 2.5 km pixels:

(Note this is a legacy product) The earthmodel used is not stricly wsg84 but the one which was used by the dutch "rijksdriehoek coordinaten amersfoort stelsel".
The parameters of the KNMI radar image are listed in the table below. Using the above mentioned equations and these parameters, a geographical longitude and latitude can be assigned to each pixel or vice versa.


Parameter Value
Projection Stereographic
Origin of projection L0=0.0E and B0=90.0N
Latitude of true scale Bw=60.0N
Radius on equator and pole Re=6378.388 km and Rp=6356.912 km
Offset of left-upper corner OI=0 and OJ=1490.906
Pixel size PX=2.5 km and PY=-2.5 km
Number of rows and columns 256 and 256

Using these parameters the corners of the radar images are calculated to be:

Corner Longitude Latitude XXX[km] YYY[km]
Northwest 0.000E 55.296N 0.0 -3727.291
Northeast 9.743E 54.818N 640.0 -3727.291
Southeast 8.337E 49.373N 640.0 -4367.291
Southwest 0.000E 49.769N 0.0 -4367.291

Projection calculations using Proj.4:

The projection calculations can also be performed using the "proj.4" library developed by the USGS and used worldwide in numerous applications. More information on the proj.4 library can be obtained via USGS Proj.4 Projection Library

The projection of the KNMI radar images is described by the following string in the proj.4 system. This string described the complete projection.

"+proj=stere +x_0=0 +y_0=0 +lat_0=90 +lon_0=0 +lat_ts=60 +a=6378.388 +b=6356.906"


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Hans Beekhuis