The raw spectral power is a superposition of signal due to radar echoes and of noise background. The noise background would result in a permanent non-zero drop size distribution and consequently in non-zero values of liquid water content and rain rate — even in precipitation free conditions. In order to avoid this bias, the noise background is estimated and removed. The noise estimation is based on the so called Hildebrand-Sekhon method. Assuming white noise it provides one mean value for describing the noise background, which is subtracted from the power spectrum. Ideally the noise-corrected spectral power (ncsp) would be zero, if there is no signal. Due to stochastic fluctuations of the actual spectral noise some ncsp-values are positive and some are negative. In case of a symmetric stochastic distribution the probability of both signs should be 0.5. Due to finite number effects the actual estimate of the noise background is slightly biased and in addition, the stochastic power distribution is not quite symmetric in reality. Therefore the implementation of the method for the MRR provides a slight preference of positive ncsp-values. The negative sign is now kept for the calculation of negative spectral drop numbers in order to avoid bias in integral products as radar reflectivity, liquid water content and rain rate.

Consequently one would expect occasional occurrence of negative signs also for these integral parameters. This is not the case due to the following reasons:

- The reflectivity is given on a logarithmic scale, which can only represent positive numbers. Values with negative sign are replaced by blanks.
- Rain rates which do not meet certain criteria (threshold of signal to noise ratio, coherence in adjacent range gates) are replaced by exact zero. (The same is true for the corresponding liquid water content).
- Only positive velocities are calculated.

The MRR standard signal processing software is adapted to the liquid phase of precipitation. It is optimized for deriving drop size distributions and corresponding integrals as for example liquid water content, rain rate or mean Doppler velocity.

In case of snow (or graupel, hail) the standard signal processing does not provide physically meaningful results, because the relations of scattering cross section versus mass and fall velocity are very different for ice crystals than for water droplets. Moreover, certain frequency intervals of the raw spectra are discarded in order to achieve stable results (see e.g. Peters et al., 2005). Particularly in case of snow the main spectral power can be concentrated in the discarded frequency range resulting in seemingly low radar sensitivity.

Maahn and Kollias (2012) have developed a special algorithm for snow detection, which is available on the web mrr_snow. The input needed for initialization is the raw spectrum as provided by the MRR. This algorithm provides reflectivity and higher spectral moments of snow echoes with enhanced sensitivity and has been checked by the authors against simultaneous measurements with a more sensitive cloud radar.

Most variables are corrected for path integrated attenuation (PIA). If PIA exceeds 10 dB, those variables (including PIA itself) are no longer considered trustworthy and the results are replaced by blanks.

In addition to the attenuation corrected reflectivity *Z* also a non-corrected version *z* is issued, which shows relative reflectivity structures also at ranges, where the absolute reflectivity cannot be determined.

The Doppler velocity* w* is not biased by attenuation and is therefore issued for all heights. Note, that *w* is issued, even, if there is no significant signal.

Ideally there should be no signal in case of no rain and *Z* (represented on a logarithmic scale) should be -∞. In reality there are mean and stochastic noise contributions, which remain in case of no signal. In FAQ MRR#001 is explained how the mean noise level is estimated and subtracted. Due to stochastic fluctuations the noise corrected samples are distributed around zero. Negative values cannot be presented on the logarithmic scale and are replaced by blanks, positive values are issued.

The positive branch of the *Z*-distribution during precipitation-free conditions can be analyzed to determine the (height dependent) detection limit of the MRR. If we assume a Gaussian distribution of *Z* (in the linear domain), centered at zero, the mean value of the positive branch is equal to the standard deviation of the distribution. The most simple estimate of the detection limit is based on the mean value of (more precisely: the arithmetic mean calculated in the linear domaine of *Z*). A value twice (tree times) as high as the mean value indicates the presence of signal with 76% (92%) probability.

In reality the distribution is not centered at zero but at some small positive value, because the noise estimation method is slightly biased. Therefore this simple estimate is conservative. If desired, the distribution center can be determined by counting the ratio of positive and negative (blanks) values and consulting the error function. Further refinement is possible by replacing the Gaussian distribution by a Chi-Square distribution with *n*-1 degrees of freedom, where *n* is the number of averaged power spectra. (*n* ≈ 58 for “Processed Data”).

Definition:

*w* is the first moment of the noise-corrected power spectrum.

Equivalent:

*w* is the reflectivity weighted mean fall velocity.

Other weighted mean velocities (for example mass-weighted) can be calculated by post-processing on the basis of the drop size distributions.

It is assumed that *w* is always downward directed (positive sign). Upward velocities (*w*_up ) will be aliased to *w* = *w*_up + *w*_nyquist with *w*_nyquist = 12.08 m/s.

*w* is always calculated, even, if there is no significant signal present. In the latter case *w* has no physical meaning but it is helpful for diagnostic purposes in case of malfunction of the MRR.

The condition RR = 0.00 can be used for masking non-physical values of *w* at heights with PIA < 10 dB. A masking algorithm of *w* working at all ranges (including PIA ≥ 10 dB ) can be based on a (height dependent) threshold *z*_t of *z* , which has been defined on the basis of mean values of *z* (= *Z* ) in precipitation-free conditions. (See FAQ MRR#004 for details).

The radar reflectivity factor is defined as

with drop diameter and drop size distribution.

If the radar wavelength is much longer than (= valid range of Rayleigh approximation) the volume reflectivity (= backscatter cross section per volume) is related to by

where depends on the dielectric constant of water. can be derived directly from the received echo power using the radar equation.

If is not much smaller than the relation between and does not hold, and thus cannot be inferred from the received echo power. For many purposes it is nevertheless useful to convert using the above relation. The result is referred to as “equivalent radar reflectivity factor” .

approaches for .

Although this condition is not fulfilled for the MRR, the radar reflectivity factor can be determined by MRR because the drop size distribution is known.

For comparison with weather radar (with longer wave lengths than MRR wave length) is the preferable variable.

The retrieval algorithm for the Doppler velocity calculates the center of gravity in the 1/e environment of the spectral peak. Even in case of pure noise the spectrum has a peak at some random position.

Consequently, the Doppler velocity has no physical meaning, if no rain has been detected. Nevertheless, the output is not suppressed because it can contain hints on the character of potential interference sources.

For automated physical interpretation of the Doppler velocity, its values should be discarded, if no rain is detected.

The tables provide the center diameters corresponding to spectral velocities. While the spectral velocities are equidistant the diameters are not equidistant, due to the non-linear relation between fall velocity and diameter.

The height dependence of density in the (standard) atmosphere leads to a height-dependent fall velocity of drops of a given size. Therefore, the center diameter corresponding to a given spectral velocity is height dependent.

The MRR does no observe the signal of single drops but the superposition of signals from many drops in the scattering volume. Therefore, the lower size threshold, which can be observed depends on the actual number density of the corresponding size class. This is not a constant but depends on the actual rain event.

In case of MRR only drops with a fall velocity (in still air) of more than or equal to 0.75 m/s are included in the analysis. This corresponds (in still air) to a minimum diameter of 0.245 mm at ground level.

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