wiki:cookbook/DEALIAS
Last modified 4 years ago Last modified on 01/08/14 09:50:45

1. Algorithm name

Dealiasing of radial velocities in polar space (DEALIAS)

2. Basic description

a) Physical basis of the algorithm

The algorithm detects abrupt velocity changes between neighboring measurements and eliminates (multiple) folding.

b) Amount of validation performed so far

The dealiasing and wind profile algorithms have been been validated for two Swedish radar sites using radiosonde observations over a 18-month-period.

Haase, G., 2013: Validating Baltrad and Nordrad wind profile retrievals using radiosonde observations, 21 pages.

c) References (names and contact information of all developers during the evolutionary history, scientific papers)

The software has been developed by Günther Haase and Tomas Landelius (SMHI).

Haase G. and Landelius T., 2004: De-aliasing of Doppler Radar Velocities Using a Torus Mapping. J. Atmos. Oceanic Technol., 21, 1566-1573.

3. ODIM metadata requirements for I/O

nbins, nrays, gain, offset, nodata, undetect

4. Input data

a) What kind of radar data (including the list of previous algorithms and quality flags applied)

Polar volume or scan

b) Other data (optional and mandatory, applying “universally” agreed formats, geometry)

5. Logical steps

The polar coordinates r (the radial coordinate) and φ (the angular coordinate) are defined in terms of Cartesian coordinates by

\begin{eqnarray*}
 x & = & r \cos \phi \\
 y & = & r \sin \phi \;.
\end{eqnarray*}

Thus the unambiguous wind velocity Vu can be mapped onto the circumference of a cylinder with radius r:

\begin{equation*}
  2 \pi r = 2 V_u \; \Leftrightarrow \; r = \frac{V_u}{\pi} \;.
\end{equation*}

Assuming that φ equals π for Vo=Vu results in

\begin{equation*}
  \phi = V_o \; \frac{\pi}{V_u}
\end{equation*}

where Vo is the observed radial wind. Hence, the corresponding Cartesian coordinates are given by

\begin{eqnarray*}
  x_o & = & \frac{V_u}{\pi} \cos \left( V_o \; \frac{\pi}{V_u} \right) \\
  y_o & = & \frac{V_u}{\pi} \sin \left( V_o \; \frac{\pi}{V_u} \right) \;.
\end{eqnarray*}

Next, a set of M x N radial wind test functions (Vt) with M different wind speeds (f) and N different wind directions (d) is created. The zonal (u) and meridional wind components (v) are defined as:

\begin{eqnarray*}
  u_{ij} & = & f_i \; \sin d_j \;\;\; i=1, ..., M \mbox{ and } j=1, ..., N \\
  v_{ij} & = & f_i \; \cos d_j \;\;\; i=1, ..., M \mbox{ and } j=1, ..., N \;.
  \end{eqnarray*}

Assuming a uniform wind the most elementary equation for the radial velocity is

\begin{equation*}
  (V_t)_{ij} = (w_{ij} + W_f) \sin \theta + u_{ij} \cos \theta \sin \phi + v_{ij} \cos \theta \cos \phi \hspace{10mm} (1)
\end{equation*}

where Wf is the terminal fall velocity of the hydrometeors, w is the vertical wind component, and θ is the elevation angle of the radar scan. For small elevation angles Eq. (1) becomes

\begin{equation*}
  (V_t)_{ij} = u_{ij} \sin\phi + v_{ij} \cos\phi \;.
\end{equation*}

Hence, the Cartesian coordinates of the test functions are given by

\begin{eqnarray*}
  (x_t)_{ij} & = & \frac{V_u}{\pi} \cos \left( (V_t)_{ij} \; \frac{\pi}{V_u} \right) \\
  (y_t)_{ij} & = & \frac{V_u}{\pi} \sin \left( (V_t)_{ij} \; \frac{\pi}{V_u} \right) \;.
\end{eqnarray*}

Then the radial wind test function V*t closest to the observed radial winds is derived using the minimum distance approach:

\begin{eqnarray*}
  x_t^*, y_t^* & = & \min\limits_{ij} \left\{ \left|(x_t)_{ij}-x_o\right| + \left|(y_t)_{ij}-y_o\right| \right\} \\
  V_t^* & = & u^* \sin\phi + v^* \cos\phi \;.
\end{eqnarray*}

Finally, the dealiased radial winds (Vd) are calculated by adding multiples of the unambiguous velocity interval to the observed radial winds:

\begin{eqnarray*}
  k^* & = & \min\limits_{k} \left\{ \left|2\,k\,V_u - (V_t^* -V_o)\right| \right\} \;\;\; k \in \{..., -2, -1, 0, 1, 2, ...\} \\
  V_d & = & V_o + 2\,k^*\,V_u \;.
\end{eqnarray*}

For each scan radial winds with constant range are dealiased at the same time. Generally, dealiasing should not be performed on data with large gaps.

6. Output.

a) Data type using ODIM notation where possible, e.g. DBZH

VRAD

b) Added quality indicators

If no output file is specified data, gain, and offset will be overwritten.

7. Outline of a test concept exemplifying the algorithm, as a suggestion for checking that an implementation has been successful.