Quality control algorithms developed for operational use in the Polish radar network are listed in Table 2. Each particular algorithm is discussed in separate sections that comprise a description of the algorithm for data correction and quality characterization, and then an example is demonstrated. Since only the measurement points in 3-D data space, so called gates, are considered here, the data continuity problem is not addressed in the paper.

#### 2.1. Technical radar parameters

This algorithm aims to provide a data quality metric without introducing any correction. As quality factors the set of technical radar parameters that impact on data quality is selected.

A set of such factors, listed in Table 3, is based on EUMETNET OPERA Programme findings (Holleman *et al.*, 2006) and complemented with other parameters. All the quality factors are static within the whole radar range as well as in time (apart from the date of last electronic calibration). The threshold values for which *QI*_{SYSi} become lower than 1 have been set according to the authors' experience.

Table 3. Quality factors related to the radar system technical parametersIndividual quality factor | Importance | Unit | Individual *QI*_{SYSi} |
---|

Operating frequency | Affects attenuation in precipitation | GHz | = 0.9 if X-Band |

| | | = 1.0 otherwise (C or S) |

Beam width | Affects area of measurement averaging | ° | = 0.9 if > 1° |

| | | = 1.0 otherwise |

Pointing accuracy in elevation | Affects measurement precision | ° | = 0.9 if > 0.1° |

| | | = 1.0 otherwise |

Pointing accuracy in azimuth | Affects measurement precision | ° | = 0.9 if > 0.1° |

| | | = 1.0 otherwise |

Are statistical clutter map or Doppler-filtered signal used? | Allows clutter removal | (Flag) | = 0.5 if ‘no’ |

| | | = 1.0 if ‘yes’ |

Minimal detectable signal at 1000 m | Affects sensitivity to snow and drizzle detection | dBZ | = 0.9 if > − 40 dBZ |

| | | = 1.0 otherwise |

Antenna speed (in azimuth) | Affects measurement precision | ° s^{−1} | = 0.9 if > 15°s^{−1} |

| | | = 1.0 otherwise |

Is radome attenuation corrected? | Causes measurement overestimation | (Flag) | = 0.9 if ‘no’ |

| | | = 1.0 if ‘yes’ |

Date of last electronic calibration | Affects measurement precision | (Date) | = 0.9 if more than 180 days |

| | | = 1.0 otherwise |

Time sampling | Affects measurement stability | (Numeric) | = 0.9 if < 30 |

| | | = 1.0 otherwise |

Range sampling | Affects measurement stability | (Numeric) | = 0.9 if < 5 |

| | | = 1.0 otherwise |

All the factors affect individual quality index, *QI*_{SYS}, connected with radar system parameters according to formula:

- (4)

In the cases when some technical parameters employed in the algorithm are missing, the quality of analysed data should be arbitrary decreased. It is assumed that lack of one set of information from Table 3 still allows generation of the *QI*_{SYS} value, but the individual *QI*_{SYSi} connected with the missing information is set as equal to 1.0. If two or more quality factors are missing then the total *QI*_{SYS} is treated as ‘no data’ (as a consequence the total quality index of the analysed data is also set to *QI* = ‘no data’).

#### 2.2. Horizontal and vertical broadening of a radar beam

Radar measurements are performed along each radar beam at successive gates, which represent certain surrounding areas determined by the beam width and pulse length. Since the radar beam broadens with the distance to the radar site, the measurement comes from a larger volume and related averaging errors increase as well (Figure 3). There is no possibility of correcting this effect. However, it can be determined quantitatively and taken into account in the related quality index *QI*_{BROAD}.

The horizontal and vertical broadening of the radar beam for each gate can be computed when its polar co-ordinates are known: ε (elevation angle), α (azimuth angle) and *l* (radial distance to radar site in km), and the two parameters of the radar beam: ϕ (beam width) and *P*_{L} (measurement gate length in km) calculated from the radar pulse length. The horizontal and vertical broadenings *L*_{H} and *L*_{V} are described by the following formulae, calculated from vertical cross-section through radar beam (Figure 3):

- (5)

The starting point to the investigation of relationships between the quality indicators *L*_{H} and *L*_{V} and related quality indices *QI*_{LH} and *QI*_{LV} is the finding of Szturc *et al.* (2009) that radar data near the ground are not burdened with errors when the distance to the radar site is not longer than 89 km, whereas it is quite wrong when the distance is over 195 km. Threshold values for *L*_{H} and *L*_{V} have been estimated from Equation (5) for the two boundary distances assuming *P*_{L} = 0.3 km, which relates to a standard radar long pulse of 2 µs.

Assuming that the linear shape of Equation (2) can be applied for *QI*_{LH} and *QI*_{LV} determination, from which *QI*_{BROAD} is calculated, boundary values can be estimated for the both quality indicators *L*_{H} and *L*_{V} as:

- (6)

and finally:

- (7)

where *a*_{LH} = 1.1 km, *b*_{LH} = 2.5 km, and *a*_{LV} = 1.5 km, *b*_{LV} = 3.2 km are the boundary values of the radar beam broadening.

In Figure 4 quality index fields related to the indicators for three selected elevations are displayed. It is obvious that the broadening becomes bigger with distance to the radar site and beam elevation. The beam cross-section area is related to spatial averaging of measurement. It can be observed that differences between particular elevations are very small. That is a result of radar volume geometry: the height of the volume is very small in comparison to the horizontal radius (about 20 km against 250 km).

#### 2.3. Ground clutter removal

The correction of radar images due to contamination by ground clutter is commonly made at a level of radar system software which uses statistical or Doppler filtering (e.g. Selex, 2010). In such a situation the information about the correction is not available so an indirect method of generation of a ground clutter map for the lowest (and higher if necessary) scan elevation must be employed, e.g. by using a digital terrain map (DTM).

In order to determine areas contaminated by ground clutter, a diagram of partial beam blockage values (PBB) is analysed. The PBB is defined as a ratio of blocked beam cross-section area to the whole one. A given gate is considered a ground clutter if increase in PBB along the radar beam exceeds 0.005, i.e. 0.5% (Figure 5).

Gates where ground clutter was detected should be characterized by lowered quality index. A simple formula for quality index *QI*_{GC} related to the clutter presence can be written as:

- (8)

where *a*_{GC} = 0.5 (Fornasiero *et al.*, 2005). The quality index is decreased in each gate in which ground clutter was detected even if it was removed.

An example of the ground clutter field and related quality index, *QI*_{GC}, is presented in Figure 6. Areas closer to the radar site, especially, are contaminated by such clutter due to the Earth's curvature, even if higher mountains are noticeable on the DTM at a greater distance.

#### 2.4. Removal of geometrically-shaped non-meteorological echoes

Apart from ground clutter, other phenomena such as specks, external interference signals (e.g. from the Sun and Wi-Fi emitters), biometeors (flocks of birds, swarms of insects), anomalous propagation echoes (so called anaprop) and sea clutter, are considered as non-meteorological clutter. Effective removal of such echoes is possible while using dual-polarization radars, however correction algorithms must be employed in single-polarization radars as well.

The spatial pattern of the precipitation field is considered as the most essential criterion while developing algorithms for removal of such spurious echoes. In a simpler approach, features of the 2-D radar reflectivity pattern, i.e. considering each elevation, ε, separately, are investigated. Because various types of non-precipitation echoes can be found in radar observations, in practice individual sub-algorithms must be developed to address each of them.

##### 2.4.1. Removal of external interference signals

Signals coming from external sources that interfere with the radar signal have become sources of non-meteorological echoes in radar images more and more often. Their effect is similar to a spike generated by the Sun, but they are observed at all azimuths in any time and mainly at lower elevations and may reach very high reflectivity.

The spurious echoes from the Sun and external interference, so called spikes, are characterized by their spatial structure that clearly differs from the precipitation field pattern. The shape of such echo is very specific: it is similar to a spike along the whole or large part of a single or a few neighbouring radar beams. Commonly, the reflectivity field structure is investigated to detect such echo on a radar image (Zejdlik and Novak, 2010). Recognition of this type of echo is not very difficult unless it interferes with a precipitation field. An algorithm developed for the removal of such artefacts should identify spikes, cut them out from the precipitation field, and replace them with proper reflectivity values.

In the proposed algorithm, three stages of spike removal are introduced: for “wide', ‘narrow’, and ‘high’ types of spikes.

For ‘wide’ spikes an algorithm based on analysis of spatial structure of radar echo is employed. The variability of the echo across and along the radar beam is examined using locally determined reflectivity variances, so a given echo is classified as a potential spike if the first variance is high whereas the latter is low:

- (9)

where var_{across} is the variance for a given gate calculated across the radar beam at a distance up to ± 3° in azimuth; var_{along} is the variance in a given gate calculated along the radar beam at a distance up to ± 15 km; *Z* is the radar reflectivity expressed in dBZ or mm^{6} m^{−3}.

If more than 45% of the gates along a radar beam are classified as potential spikes then (1) all the gates with spikes are replaced by reflectivities interpolated from neighbouring beams, and (2) the whole beam is treated as burdened with spike and quality index of all the gates on the beam is decreased.

The next algorithm is employed to recognize and remove ‘narrow’ spikes, i.e. not wider than 7° in azimuth. In the first step, gates with potential ‘narrow’ spikes are detected. This algorithm is applied to each gate (α,*l*) with echo detected, that means *Z*(α,*l*) > − 32 dBZ (which is the lowest reflectivity measured by radar). For these gates, reflectivities at neighbouring beams of azimuth α ± *d* (where *d* = 3°) in the same distance to radar *l* are checked:

- (10)

The procedure is repeated for *d* = 2 ° and 1 °. Next, the number of potential gates along a given radar beam is computed and if the number is higher than 25% then (1) all the gates with spikes are replaced by reflectivities interpolated from neighbouring beams, and (2) the quality index of all the gates on the beam is decreased.

‘High’ spurious echoes are all echoes detected at altitudes higher than 20 km where it is not possible for any meteorological echo to exist. All the ‘high’ echoes are removed.

Quality index *Q*_{SPIKE} for the whole radar beam in which spike is detected is given by:

- (11)

where *a*_{SPIKE} = 0.5.

##### 2.4.2. Speck removal

Generally, specks are isolated radar gates with or without echo. So called reverse specks are isolated gates without radar echo (*Z* = − 32 dBZ) surrounded by an echo field. The introduced algorithm is employed to each elevation scan separately. A grid of 3 × 3 gates around a given gate (α,*l*) is considered. The number of non-echo gates *s*_{rspeck}(α,*l*) in the grid is calculated from:

- (12)

where *m*, *n* are the polar co-ordinates of gates inside a 3 × 3 grid; *Z*(*m*, *n*) is the radar reflectivity in the gate (*m*, *n*) (in dBZ).

The parameter of the algorithm is threshold *t* for *s*_{rspeck}(α,*l*) value. If the threshold for (α,*l*) gate is not achieved then the gate is classified as a reverse speck and its reflectivity is assigned to an average from all precipitation gates inside the grid. The threshold, *t*, is set to 3.

As opposed to reverse specks, the ordinary ones are gates in which isolated echoes, which can be considered as measurement noise, are observed. The algorithm of speck removal is analogous to the one used for reverse specks. A grid of 3 × 3 gates around a given gate (α,*l*) is also considered. The number of echo gates *s*_{speck}(α,*l*) in the grid can be calculated from (Jurczyk *et al.*, 2008):

- (13)

where denotation is the same as for Equation (12).

If the threshold *t* for a number of surrounding precipitation gates *s*_{speck}(α,*l*) is not achieved then (α,*l*) gate is classified as a speck echo and the echo is removed, i.e. reflectivity *Z* = − 32 dBZ is set for the gate. The threshold is set as equal to 3. This algorithm is launched twice to clean the data more thoroughly.

The related quality index *QI*_{SPECK} depends on the presence of a removed speck or reverse speck in the given gate:

- (14)

where *a*_{SPECK} = 0.9 (specks are less corrupting of a radar image than spikes because of their isolated character).

Figure 7 shows an example of the performance of the two sets of algorithms launched consecutively: SPIKE (removal of spikes from external interference signals) and SPECK (removal of reverse specks and specks), is depicted for the Legionowo radar, which is the most strongly contaminated with spike echoes within POLRAD network.

#### 2.5. Beam blockage

Radar beams can be blocked by ground targets, i.e. places where the beam hits terrain. A geometrical approach is applied to calculate the degree of the beam blockage. This approach is based on calculation of which part of radar beam cross-section is blocked by any topographical object. For this purpose a degree of partial beam blocking (PBB) is computed from a digital terrain map (DTM) taking into account the highest blocked point in the given beam cross-section.

According to Bech *et al.* (2003, 2007), the partial beam blockage PBB may be calculated using the formula:

- (15)

where *r*_{b} is the radius of radar beam cross-section at the given distance from radar, and *y* is the difference between the height of the terrain and the height of the radar beam centre. The partial blockage takes place when − *r*_{b} < *y* < *r*_{b}, and varies from 0 to 1 (see Figure 8).

The quantity *y* in Equation (15) and Figure 8 is calculated as an altitude obtained from DTM for the radar gate located in the beam centre, reduced by quantity *h* involving: (1) altitude of radar antenna, *h*_{0}, (2) difference of altitude due to the Earth curvature, and (3) difference of altitude due to antenna elevation, ε:

- (16)

where *r*_{e} is the effective Earth's radius (8493 km) and *l* is the distance to the radar site.

Correction of partial beam blocking on precipitation rate is made by applying a multiplicative correction factor (Bech *et al.*, 2007):

- (17)

where *b* = 1.6 is the exponent in the Marshall-Palmer *Z*–*R* relationship, and for radar reflectivity (in dBZ):

- (18)

The correction is introduced if the PBB value is smaller than 0.7. For higher PBB Bech *et al.* (2007) propose to mark data in such gate as ‘no data’, however in the present paper it is proposed to take reflectivity from neighbouring higher elevations. Only when such data are not available is the ‘no data’ mark is assigned.

A quality of measurement burdened by beam blockage dramatically decreases. The quality index *QI*_{PBB} of gates where the radar beam is considered as blocked is expressed by the formula:

- (19)

where the co-efficient *a*_{PBB} can be set as 0.5 (Fornasiero *et al.*, 2005), or 0.7 (Bech *et al.*, 2007); here the value *a*_{PBB} = 0.7 is also applied.

If reflectivity in a specific gate has been replaced by reflectivity from higher elevation then *QI*_{PBB} is taken also from the higher elevation, but multiplied by a factor (1 − *a*_{PBB}). Finally:

- (20)

where *QI*_{PBB}(*el* + 1) means the *QI*_{PBB} calculated for the relevant gate in the elevation *el* + 1, *el* is the number of elevation (numbered from the lowest to the highest).

In Figure 9 an example for the Pastewnik radar is presented. The Sudety Mountains lying south of the radar site are blocking the southwest sector. The DTM for the Pastewnik radar is depicted with related ground clutter and blocked areas for the lowest elevation 0.5°.

#### 2.6. Attenuation in rain

Attenuation is generally defined as a decrease in radar signal power after passing a meteorological object, which results in an underestimation of the measured rain:

- (21)

where *A* is the attenuation (in dB); *Z*_{cor} is the non-attenuated reflectivity (in mm^{6} m^{−3}) and *Z* is the measured one.

The aim of the algorithm is to calculate the non-attenuated rain. Empirical formulae for determination of specific attenuation can be found in the literature. Using 5.7 cm radar wavelength (C-Band radar) the two-way specific attenuation *A*_{(i−1, i)} (in dB km^{−1}) in precipitation between measurement gates *i* − 1 and *i* at 18 °C can be estimated assuming the Marshall-Palmer *Z*–*R* relationship (Battan, 1973):

- (22)

where *R*_{(i)} is the precipitation rate in a measurement gate *i* (in mm h^{−1}); *Z*_{(i)} is the radar reflectivity in the gate *i* (in mm^{6} m^{−3}).

A reflectivity-based correction made iteratively (‘gate by gate’) is a common technique for correction of attenuation in rain. However, ‘gate by gate schemes are notoriously unstable and very sensitive to small calibration errors’ (Illingworth, 2004). Dual-polarization measurement seems more effective way to estimate the attenuation operationally (e.g. Bringi *et al.*, 2007), but in the case of a single-polarization radar the reflectivity-based algorithm is the sole solution.

The iterative algorithm is also employed in the RADVOL-QC model (Figure 10). Assuming that in the previous algorithm steps the specific attenuations *A*_{(0, 1)}, …, *A*_{(i−2, i−1)} and their sums PIA_{(0, 1)}, …, PIA_{(0, i−1)}, called the path-integrated attenuations, were calculated:

- (23)

From reflectivity *Z*_{(i)} measured in *i*-gate, PIA_{(0, i−1)} integrated from the radar site to the *i* − 1-gate, and first guess of attenuation *A*′_{(i−1, i)} between *i* − 1 and *i*-gates calculated from Equation (22), the non-attenuated reflectivity *Z*_{cor(i)} can be obtained according to the formula:

- (24)

The attenuation *A*_{(i−1, i)} for the distance between the two neighbouring gates *i* − 1 and *i* can then be calculated from Equation (22), and consequently the path-integrated attenuation along the whole distance from the radar site to the given *i*-gate can be obtained from:

- (25)

This attenuation value will be applied in the next step of the iterative algorithm.

In order to avoid instability in the algorithm, certain threshold values have to be set to limit the corrections for both specific attenuation *A*_{(i−1, i)} and path-integrated attenuation *PIA*_{(0, i)}.

The described algorithm is applied only for the gates where echo was detected, otherwise:

- (26)

The magnitude of the attenuation in precipitation PIA (in dB) (Equation (25)) can be considered as a quality factor for the given measurement gate *i*. Therefore the relevant quality index *QI*_{ATT} is calculated from the formula:

- (27)

where parameters *a*_{ATT} = 10 dB and *b*_{ATT} = 5.0 dB are empirically determined.

In Figure 11 an example of the attenuation field for the lowest elevation is presented for the data shown in Figure 2. The corrections are employed only if any echo is observed in the given gate, however for quality characterization the data quality decreases behind any echo area for both echo and non-echo gates.