## Abstract

This paper explores the removal of normal ground echoes (GREs) and anomalous propagation (AP) in ground-based radars using a fuzzy logic approach. Membership functions and their weights are derived from the characteristics of radar echoes as a function of radar reflectivity. The dependence on echo intensity is shown to significantly improve the proper identification of GRE/AP. In addition, the proposed method has a better performance at lower elevation angles. The overall performance is comparable with that from a polarimetric approach and can thus be easily implemented in operational radars.

## 1. Introduction

The removal of normal ground echoes (GREs) and anomalous propagation (AP) is mandatory for improving quantitative precipitation estimates (QPE). This task is rendered more difficult by the fact that GRE/AP fluctuate with time due to the propagation of the radar beam under changing atmospheric conditions. A map of typical ground echoes under normal atmospheric conditions from the McGill S-band radar is shown in Fig. 1a. When the vertical gradient of refractivity, which changes with water vapor pressure, air pressure, and temperature, is smaller than the “normal” value, superrefraction occurs (Rinehart 1997), causing a greater downward bending of radar beams, resulting into extended ground echoes, the so-called anomalous propagation or anoprop (AP) as seen in Fig. 1b. The AP usually occurs with nocturnal inversions in clear-air conditions and also appears when sharp moisture gradient are formed following the passage of a thunderstorm. Strong AP concurrent with precipitation is frequent as exemplified in Fig. 1b and thus must be eliminated in radar QPE.

Several studies have been conducted to remove GRE/AP [An excellent review and all relevant references are given by Steiner and Smith (2002)]. The removal can be applied at three steps: 1) at the radar installation (site, hardware), 2) by data processing, and 3) by comparison with other data sources (Steiner and Smith 2002). Here we focus on a method that is based on the processing of archived data that is easily accessible to users and can be easily tuned to maximize the removal of unwanted AP. The main idea of our approach is to use the spatial and temporal information of GRE/AP such as the continuity of reflectivity and texture (radial velocity and horizontal, vertical gradient of reflectivity, etc.).

Many researchers have used the spatial and temporal information of GRE/AP as a threshold or in a probabilistic manner to identify GRE/AP (Mueller and Sims 1975; Collier et al. 1980; Smith 1990; Giuli et al. 1991; Joe 1991; Pratte et al. 1993; Grecu and Krajewski 1999, 2000; Kessinger et al. 2001; Krajewski and Vignal 2001; Moszkowicz et al. 1994; vanAndel 2001; Steiner and Smith 2002). In Steiner and Smith (2002), the spatial variability of the reflectivity field and the vertical gradient of the reflectivity calculated in polar coordinates are combined with the vertical extent of radar echoes to identify AP by means of a decision tree. Kessinger et al. (1999, 2001) adopted a fuzzy logic approach that included the texture of the reflectivity, the median radial velocity, the median spectrum width, and the standard deviation of the radial velocity. Grecu and Krajewski (2000) applied a neural network based on the local characteristics of reflectivity and of velocity fields. In addition, new parameters from polarimetric measurements (the correlation coefficient *ρ*_{hv} between horizontally and vertically polarized backscatter signals, and the local variability of the total differential phase) have been shown by Ryzhkov and Zrnic (1998) to further improve the removal of GRE/AP. In this paper, we use the detection of GRE/AP with polarimetric information as a current best reference for validation.

The purpose of this paper is to develop a method of detecting anomalous propagation and normal ground echoes based on radar echo characteristics. Our approach is similar to that of Kessinger et al. (1999, 2001). However, the membership functions for the fuzzy logic are objectively derived from the statistics of observed echo characteristics (the absolute value of radial velocity, the standard deviation of reflectivity, and the vertical gradient of reflectivity) as a function of reflectivity and proper weights for these membership functions are also obtained from these statistics to optimize the removal of GRE/AP. The data used are summarized in section 2 and the methodology is discussed in section 3. The characteristics of GRE/AP and precipitation echoes, the construction of the membership functions, and the derivation of proper weights are described in section 4. The validation methodology is described in section 5. The determination of optimal thresholds is presented in section 6. Some validations are presented in section 7 by comparison with a method that is based on polarimetric information and is thus assumed to be the best identifier of GRE/AP.

## 2. Data

The McGill S-band radar measures reflectivity, radial velocity, and polarimetric parameters (differential reflectivity, correlation coefficient, and differential phase shift). It transmits a 45° linearly polarized beam and receives horizontal and vertical components simultaneously with two receivers. It scans at the speed of 6 rotations per minute (rpm) or 36° s^{−1} (equivalent to 24 elevation angles for the complete 5-min cycle) needed to maximize the volumetric surveillance (Marshall and Ballentvne 1975) and is now part of the Canadian operational radar network. Since 1994, two volumetric scans of 12 even and 12 odd elevation angles are obtained every 5 min at a resolution of 1° by 150 m. They are stored as one raw volume scan at a resolution of 1° × 1 km. Since 2002, the standard deviation of reflectivity (in decibels) within 1 km is calculated and archived at the optional degradation stage from 150 m to 1 km. No filtering of AP or of normal GREs is applied. Since the McGill S-band radar is situated in the St. Lawrence valley, the normal ground echoes are very extensive at low elevation angles (Fig. 1a) mostly due to sidelobe contamination and they appear up to 2 km in height, especially, north and south of the radar, with an intensity of 30 dB*Z* due to the Laurentians and Adirondacks Mountains. In the southeast, at a range of 120 km, where the Green Mountains are located, the ground echoes, in excess of 30 dB*Z*, are evident up to 4 km. The AP forms in a direction where there are no nearby blockages of the lower elevation angles, typically along the St. Lawrence River valley (southwest to northeast: Fig. 1b) and appears to extend up to 3–4 km in height with an intensity of over 30 dB*Z*. The AP along the north, northwest, south, and southeast is less frequent due to the partial blocking of the lowest elevation angles.

The characteristics of radar echoes are obtained from five nonprecipitation and five precipitation events listed in Table 1. Over 460 volume scans (∼40 h) are used. The five nonprecipitation cases include hours with normal ground echoes as well as with noticeable AP. The five precipitation cases have no AP and include stratiform rain, convective lines, and isolated small convective cells. With this dataset, we intend to obtain the typical characteristics of GRE/AP and precipitation echoes. For the validation of an AP removal algorithm, a different dataset is used (Table 2). This dataset includes various meteorological situations to test the applicability of an algorithm (stratiform rain, scattered showers associated with small shallow convective cells, and strong convective lines). In some cases, strong AP occurs simultaneously with precipitation. This case selection provides a broad range of situations that can occur in an operational setup.

## 3. Methodology

### a. Identification of precipitation echoes and GRE/AP

The removal of GRE/AP is performed at a resolution of 1 km and 1°. The algorithm uses three parameters (or features) derived at that resolution: the standard deviation of reflectivity [SDZ (dB)], the vertical gradient of reflectivity [VGZ (dB°^{−1})], and the absolute value of radial velocity (*V _{r}* in m s

^{−1}):

where the indices *i* and *j* are for the elevation angles and ranges, respectively. The n is the number of reflectivity measurements within the 1-km range. During data archiving procedure, SDZ is calculated from high-resolution (150 m × 1°) raw data. This parameter tests the uniformity of the reflectivity field within an area of 1 km × 1°. Since the precipitation fields are smoother than GRE/AP fields, a large value of SDZ should be an indication of GRE/AP. The *V _{r}* is used to check whether the echoes are fixed or not. However, this parameter can identify precipitation echoes around the line of zero radial velocity as GRE/AP. VGZ is calculated at each pixel with

*i*th and (

*i*+ 2)th elevation angles instead of (

*i*+ 1)th angle. This is done in order to avoid spurious gradients caused by the time difference of 2.5 min between the even and odd scanning angles. The vertical gradient for the GRE/AP is large since they are intercepted by only the lower elevation angles. However, shallow precipitation can likewise yield large values for VGZ, resulting into an ill identification. Therefore, we combine all three parameters with proper weights to compensate for the weakness of each parameter in the context of a fuzzy logic approach. The essence of this approach is that the membership functions and their weights are objectively obtained from the radar echo characteristics (see section 4). Their derivation is simple and is easily adaptable to other radars.

The overall procedure is described in Fig. 2. (More detailed explanation on the determination of the membership functions and weights will be discussed in section 4.) First we derive the three feature parameters: VGZ, *V _{r}*, and SDZ from the raw volume scan. Then, the fuzzy membership value (MF) for a given value of

*Z*and the feature parameter is calculated with predetermined membership functions (see Fig. 7 of section 4). For each parameter, four membership functions (three for VGZ) are used as a function of echo intensity (reflectivity). We have applied the additive method to obtain the total membership value (MF

_{tot1}) with the weights [

*W*(

*Z*)] shown in Table 3:

where *k* indicates the different parameters. The weights are a function of reflectivity. The multiplicative method is another way of minimizing false classification while the above additive method maximizes the probability of detection (see section 2 of Zrnic et al. 2001). Here, we have chosen the additive method to increase the probability of detection rather than leaving GRE/AP undetected. This total value is compared with a threshold value (MF_{thresh}), which is deduced from the comparison with the polarimetric identification of GRE/AP. When MF_{tot1} > MF_{thresh}, the pixel is identified as a GRE/AP. At a far range, the classification between precipitation and GRE/AP becomes less clear due to the use of range-independent membership functions. Thus, the extension of GRE/AP to nearby pixels (±1 pixel for all directions) is applied for the ranges beyond 75 km. That is, when a pixel at (range = *r*_{1}, azimuthal angle = az_{1}) is identified as a GRE/AP and *r*_{1} > 75 km, pixels at (range = *r*_{1} − 1 km to *r*_{1} + 1 km, azimuthal angle = az_{1} − 1° to az_{1} + 1°) are likewise considered as GRE/AP. The specific range of 75 km is set by trial and error to maximize the performance of the classification. A ring-shaped artifact may be present and some precipitation areas can be eliminated. However, we intend to eliminate all possible GRE/APs instead of leaving them undetected.

When MF_{tot1} ≤ MF_{thresh}, a weighted mean (MF_{tot2}) of the two membership functions for *V _{r}* and SDZ is compared with a threshold value. This second test attempts to identify AP occurring within stratiform precipitation. For example, when AP is present at lower elevation angles, the next elevation angle can intercept a bright band (BB), resulting in a small value of VGZ and thus identifying the pixel as a precipitation echo. However, when only two parameters are combined, a value of MF

_{tot2}should be larger than MF

_{thresh}. This echo will be AP and the extension of GRE/AP is applied. If an echo fails the two tests (i.e., MF

_{tot1}≤ MF

_{thresh}and MF

_{tot2}≤ MF

_{thresh}), a further check is performed in order to determine whether this assumed precipitation echo is an isolated point. When none of the eight pixels around this echo within an area of 3 km × 3° are measurable radar echoes or the fraction of GRE/AP pixels to total data pixels is larger than 50%, this echo is considered as GRE/AP and then the extension of GRE/AP area is also applied to nearby pixels. In this way, we eliminate suspicious data rather than to keep them.

GRE/AP echoes, that are identified with the above procedures, are switched to “no precipitation” or “no detectable echo.” In this paper, we have not applied any recovering of identified GRE/AP areas by interpolating or extrapolating from nearby precipitation echoes because we first want to test the performance of this procedure. The current method is performed for entire set of 24 elevation angles except for last two because GRE/AP could only occur at lower elevation angles.

## 4. Membership functions and their weights

### a. Radar echo characteristics

Using the data in Table 1, we have derived the normalized frequency distributions of the three parameters (VGZ, *V _{r}*, and SDZ) to define the characteristics of GREs, AP, and precipitation echoes (PREs; Fig. 3). Similar studies on radar echo characteristics have been conducted by Grecu and Krajewski (2000) and Steiner et al. (1999).

We have classified the radar measurement domain into two areas (D_{G} and D_{GF}) based on the average ground echo map (*Z*_{GRE}: Fig. 1a). The subscripts “G” and “GF” refer to “ground clutter region” and “ground clutter-free region,” respectively. In Fig. 1a, D_{GF} is thus the area of *Z*_{GRE} < −5 dB*Z* and D_{G} is the region where *Z* ≥ −5 dB*Z*. These two areas are used to classify radar echoes for the cases in Table 1. For the five nonprecipitation events in Table 1, the frequency distribution of normal GRE areas is derived from all echoes within D_{G} while that of AP areas is calculated from echoes within D_{GF}. The frequency distribution of PRE areas is obtained from echoes within D_{GF} for the five precipitation events in Table 1. Figure 3a shows that the absolute value of radial velocity *V _{r}* of most ground and AP echoes is within 2 m s

^{−1}. Since biological targets are not separated from ground and AP echoes, the large values of

*V*(around 2 m s

_{r}^{−1}) may be due to the contamination from insects. These values are observed specifically at the edges of ground echoes. The absolute value of radial velocity from precipitation shows a wide distribution without a distinctive peak. The AP has a narrower

*V*distribution than that of GRE. The distribution of SDZ of AP and GRE shows a broad peak around 4.5 dB and is skewed to large values. The one for PRE is narrower but symmetric with a mode around 2 dB. The two distributions are overlapping around 2 ≤ SDZ ≤ 5 dB. The VGZ distribution for GRE reveals two peaks at 0 and at 30 dB°

_{r}^{−1}. The peak at 0 dB°

^{−1}for GRE is mainly due to weak GRE at the edge of average ground echo map (see Fig. 6a). This peak is not seen with AP. The AP shows a distinctive peak at 30 dB°

^{−1}. The distribution for PRE is almost symmetric with respect to 0 dB°

^{−1}. These features of AP and PRE confirms results from Steiner et al. (1999). In general, a fuzzy separation is present between the distributions of PRE, and that for AP and GRE. There are different amounts of overlapping for the different features. This overlapping illustrates the difficulty of properly identifying GRE/AP with a single parameter and points to the necessity of combining different parameters. However, as the overlapping increases, the identification will be more difficult.

A similar analysis is performed with results stratified by four reflectivity intervals (0–10, 10–20, 20–30, and larger than 30 dB*Z*) in order to determine the dependence of the echo characteristics on reflectivity. Only two categories (GRE/AP and PRE) are shown in Figs. 4, 5 and 6. In general, the distributions of the three parameters for PRE do not vary with reflectivity whereas significant variations for GRE/AP are noticeable. The PRE distribution of *V _{r}* is independent of reflectivity (Fig. 4). The one for GRE/AP becomes narrower with higher

*Z*, indicating the distinctive stationary characteristics of strong GRE/AP. The broader

*V*distribution at weaker reflectivity indicates some GRE/APs may be caused by insects and birds. A simple

_{r}*V*threshold would be useful only to eliminate the stronger GRE/AP at the expense of also removing a significant amount of precipitation of all intensities while failing to filter out the biological targets.

_{r}Similarly, the distribution of SDZ for PRE remains unchanged for the different intervals of reflectivity (Fig. 5). However, that of GRE/AP becomes broader and shifts to larger SDZ value with increasing *Z*, resulting in a reduction of the overlapping area. This is an indication that SDZ is a better separator of the two classes at higher reflectivity. Thus, higher AP or GRE can be eliminated more effectively than weaker echoes.

The distribution of VGZ for precipitation (PRE) does not vary with reflectivity while that for GRE/AP changes significantly. However, no separation between distributions for PRE and GRE/AP is present at the weakest reflectivity (*Z* = 0 ∼ 10 dB*Z*), indicating that VGZ at this range of *Z* cannot be used for identification. The distribution for GRE/AP gradually shifts to larger VGZ with increasing *Z* and shows two distinctive peaks. An examination of the data reveals that the contribution for the first peak is mainly from areas of normal ground echoes whereas the second peak is due mainly to AP. A prominent separation occurs for *Z* > 30 dB*Z*. Without further investigation, it is clear that VGZ is the best parameter for higher reflectivity and should thus have the largest weight.

Range dependence of echo characteristics is also investigated up to 120 km (not shown). The distribution of SDZ shifts to smaller SDZ values for PRE and to larger values for GRE/AP with increasing range, leading to a better separation of the two distributions at far range. The distribution of VGZ for PRE becomes slightly broader with increasing range. For GRE/AP, the peak around 10 to 15 dB°^{−1} becomes weaker with increasing ranges and the overall distribution skews toward larger values of VGZ. The distribution of *V _{r}* becomes narrower for GRE/AP with increasing range while no noticeable change is found in PRE. The overall features show a better separation of the two distributions of PRE and GRE/AP with increasing range. This suggests that the use of range-dependent feature characteristics should only slightly improve the performance of GRE/AP classification.

In summary, the distributions for PRE do not vary significantly with reflectivity while those for GRE/AP show a prominent shift to larger SDZ and VGZ with higher reflectivity intervals. The distribution of *V _{r}* for GRE/AP becomes narrower with higher

*Z*. In other words, the overlapping areas of distributions decrease with higher

*Z*, leading to better identification of stronger GRE/AP.

### b. Membership functions

From the characteristics of precipitation echoes and GRE/AP, we now construct membership functions that are used for the identification of GRE/AP. For a given parameter (e.g., SDZ) and reflectivity interval, we have two normalized frequencies for GRE/AP (*F*_{GRE/AP}) and for precipitation (*F*_{PRE}) echoes. The membership function MF_{GRE/AP}(SDZ, *Z*) of GRE/AP is derived from the following:

where *Z* indicates the interval of reflectivity selected for Figs. 4, 5 and 6. In addition, the membership function for the entire range of reflectivity is also derived. For an example, for SDZ = 4 dB and 0 ≤ *Z* < 10 dB*Z*, we get *F*_{GRE/AP} = 0.14 and *F*_{PRE} = 0.05, yielding MF_{GRE/AP}(SDZ, *Z*) = 0.74. An echo with these characteristics thus has a greater possibility of being classified as GRE/AP, the final decision being dependent on the value and weight of the membership function for other parameters.

The derived GRE/AP membership functions for the selected reflectivity intervals are shown in Fig. 7. The VGZ membership function for 0 ≤ *Z* < 10 dB*Z* is not derived because, as seen in Fig. 6a, the two *F*_{GRE/AP} and *F*_{PRE} are unable to separate the PRE and GRE/AP classes. When the calculated MF_{GRE/AP} reaches 0.05 (or 0.95), the value of MF_{GRE/AP} below (or above) that parameter is considered as 0.00 (or 1.00). Then, a moving average of three points is applied to obtain a smooth curve. By definition, MF_{GRE/AP} = 0.5 corresponds to the intersection of the two frequency curves where the values of *F*_{GRE/AP} and *F*_{PRE} are equal. In the case of VGZ and *Z* ≥ 30 dB*Z*, this occurs at VGZ = 12 dB°^{−1} below the smallest *F* of 0.01 shown in Fig. 6d.

It is noticeable that the membership functions vary as a function of reflectivity intervals. For example, when SDZ = 4 dB, MF_{GRE/AP} increases from 0.5 when *Z* ≥ 30 dB*Z* to 0.8 when 10 ≤ *Z* < 20 dB*Z*. Here MF_{GRE/AP}(*V _{r}*,

*Z)*becomes broader as the reflectivity decreases. The change of MF

_{GRE/AP}(VGZ,

*Z*) is rather dramatic. An echo with VGZ = 10 dB°

^{−1}is favorable to be GRE/AP when 10 ≤

*Z*< 20 dB

*Z*(MF

_{GRE/AP}= 0.85) whereas it is more likely to be precipitation when

*Z*≥ 30 dB

*Z*(MF

_{GRE/AP}= 0.33). With the reflectivity-independent membership function of VGZ, the 30-dB

*Z*echo would be favorable for GRE/AP (MF

_{GRE/AP}= 0.76). This emphasizes the importance of stratifying membership functions with different reflectivity intervals. Research in hydrometeor identification considers this reflectivity dependence by using two-dimensional membership functions (Straka 1996; Carey and Rutledge 1996; Lopez and Aubagnac 1997; Zrnic and Ryzhkov 1999; Vivekanandan et al. 1999; Liu and Chandrasekar 2000). However, membership function does not change with reflectivity in Radar Echo Classifier by Kessinger et al. (1999, 2001). In general, for a given value of parameters (SDZ,

*V*, and VGZ), a single membership function for the entire reflectivity range leads to a misidentification whereby heavy precipitation is unnecessarily removed and weak GRE/APs remain. Designing reflectivity-dependent membership functions has improved the performance of classification. It is shown in section 6c. We will show to what extent membership functions that are a function of reflectivity can improve the identification based on a single membership function for the entire range of reflectivity values.

_{r}### c. Weights of membership functions

The next step is to calculate the relative importance (or weight) of each of the three selected parameters. For an example, because VGZ can identify GRE/AP better than SDZ or *V _{r}* when

*Z*≥ 30 dB

*Z*, then the largest weight should be assigned to VGZ. The three parameters should thus be combined in a way that maximizes their effectiveness. The weights can be easily calculated by comparing overlapping areas of each parameter since they represent the ambiguity between GRE/AP and precipitation echoes. For a given reflectivity interval, the weights are calculated from the following:

where *S* = (1/*A*_{SDZ}) + (1/*A*_{VGZ}) + (1/*A _{Vr}*) and

*A*is the overlapping area between normalized frequencies of GRE/AP and of precipitation echoes (e.g., the shaded area in Fig. 5a). The calculated weights and overlapping areas are listed in Table 3. Considering the entire range of reflectivity, the

*V*is the most important parameter followed by VGZ. As reflectivity increases, the VGZ becomes more dominant while the opposite occurs for

_{r}*V*. The changing weights with reflectivity underline the necessity of classifying the membership functions according to reflectivity. The increase of the total overlapping area (the last column in Table 3) with decreasing reflectivity indicates the greater ambiguity in separating GRE/AP from precipitation, in other words, stronger GRE/APs are easier to identify than weaker echoes.

_{r}## 5. Validation methods

We evaluate the performance of the algorithm in two ways: 1) by examining the resultant rainfall accumulation maps and 2) by comparing with polarimetric identification of GRE/AP at all plan position indicators (PPIs). Although the former is a subjective validation, the continuity of a rainfall accumulation field in space can easily reveal the skill of the algorithm. A few missed pixels of AP yield large accumulation values, resulting in an abrupt rainfall accumulation gradient and spatial discontinuity. The climatological *R–Z* relationship (*Z* = 210*R*^{1.47}) in Montreal, Canada (Lee and Zawadzki 2005), is used for the transformation of the measured reflectivities.

The polarimetric identification of GRE/AP can be considered as a reliable reference for validation (Giuli et al. 1991; Ryzhkov and Zrnic 1998; Zawadzki et al. 2001). The polarimetric approach adapted by the operational McGill S-band radar is based on the standard deviations of the differential reflectivity (*Z*_{DR}), of the phase shift (*ϕ _{DP}*), and of reflectivity (

*Z*). They are computed over a 1-km range using the measurements obtained at each pulse length of 150 m. Ryzhkov and Zrnic (1998) used a

*ρ*

_{hv}threshold value of 0.7 to identify GRE/AP in addition to the local variability of

*ϕ*. Detailed investigation of

_{DP}*ρ*

_{hv}with the McGill S-band radar has revealed that a significant portion of the GRE areas have

*ρ*

_{hv}values larger than 0.85 while some precipitation areas have

*ρ*

_{hv}values smaller than 0.85. Thus, we have not included

*ρ*

_{hv}in this study. The membership functions of our polarimetric approach are almost step functions with thresholds of SD

*= 1.6 dB, SD*

_{ZDR}*= 14°, SD*

_{ϕDP}_{Z}= 3.4 dB. The same weight is applied to each parameter. Consequently, when two of these standard deviations are larger than the corresponding threshold values, that is, the total membership value exceeds 0.66, the pixel is considered as a “true” GRE/AP. This threshold has been determined subjectively by expert analysis and proper weights should be determined in a follow-on study. The classification of this approach is taken to be the truth for evaluating the skill of our proposed method outlined in Fig. 2. Four skill scores [the probability of detection (POD), the false-alarm rate (FAR), the critical success index (CSI) and equitable threat score (ETS)] are calculated as follows (Germann and Zawadzki 2002):

The *hits* (*h*) are points that are correctly identified as GRE/AP by the our proposed approach, Precipitation echoes incorrectly classified as GRE/APs are *false alarm* ( *f* ) while the converse (i.e., GRE/APs judged to be precipitation echoes) are *misses* (*m*). Pixels declared as precipitation echoes by both our proposed fuzzy logic method and polarimetric approach are counted as *correct negatives* (*z*). Furthermore, we define *u* = [(*h* + *m*)(*h* + *f* )/(*h* + *m* + *f* + *z*)]. In addition to these skill scores, rainfall accumulation maps are compared after eliminating GRE/AP with the proposed and polarimetric methods.

## 6. Determination of an optimal threshold

Figure 8 shows the measured reflectivity field (Fig. 8a) that is described as an example of AP with precipitation in Fig. 1b and the membership values for *V _{r}*, SDZ, and VGZ (Figs. 8b,c,d). The total membership values (MF

_{tot1}) from the proposed fuzzy algorithm appear in Fig. 8e while the results (MF

_{tot, pol}) from the polarimetric approach are shown in Fig. 8f. Higher membership values correspond well to GRE/AP shown in Fig. 8a. The MF for VGZ is higher than 0.9 while that for SDZ is smaller particularly at the edge of AP. The MF for

*V*in Fig. 8b shows high values (>0.7) over areas of zero radial velocity in the northwest direction where precipitation particles move perpendicular to the radar beam. By combining the membership values with their proper weights, the unwanted effects of zero radial velocity are diminished as indicated by the reduced MF

_{r}_{tot1}in Fig. 8e. In addition, the edge effect from SDZ is also reduced in MF

_{tot1}. Nevertheless, in general, MF

_{tot}

*> 0.6 over AP regions and is consistent with those from polarimetric information shown in Fig. 8f.*

_{1}Our polarimetric identification of GRE/AP requires MF_{tot,pol} ≥ 0.66. Those pixels are considered as GRE/AP and are used as a reference or truth when determining the best threshold value (MF_{thresh}). This is achieved by applying various thresholds from 0 to 1 in MF_{tot1} and calculating the resulting skill scores between the current and polarimetric approaches (Fig. 9). The identification of GRE/AP is applied only up to the first test (MF_{tot1} > MF_{thresh}) in Fig. 2 and no further tests have been applied at this stage.

Although the skill scores vary with cases, POD decreases gradually up to a threshold value of 0.6 and then drops more quickly. FAR decreases rapidly and begins to maintain a constant value at MF_{thresh} ≥ 0.6. CSI and ETS have a maximum in the 0.45–0.55 range (mostly at 0.55) where FAR is significantly low and POD is relatively high. We have thus determined the optimum threshold as 0.55. However, this threshold can vary slightly from case to case and, more significantly, for different elevation angles. (A further investigation of performance with elevation angles is discussed in section 7.) However, we have used the same threshold for all cases and elevation angles. A further application of this method with data from different radars will determine to what extent this threshold depends on different climatological regions.

## 7. Evaluation

A simple evaluation has just been shown in the previous section where determining MF_{thresh} for our proposed fuzzy logic algorithm. We now proceed to a more thorough evaluation of the proposed algorithm and examine the sensitivity of the algorithm to different elevation angles and different membership functions.

### a. Two tests: MF_{tot1} and MF_{tot2}

As explained in the section 3, VGZ for GRE/AP can be small under brightband conditions, identifying them as precipitation echoes. Thus, in order to minimize the effects of a bright band, we apply an additional test with MF_{tot2} that is a weighted sum of the membership values of SDZ and *V _{r}*. Skill scores after further identifying GRE/AP with MF

_{tot2}> MF

_{thresh}are shown in Fig. 10 for the case of 8 July 2004. When compared with Fig. 9a, POD, CSI, and ETS at MF

_{thresh}> 0.6 are significantly improved although FAR is slightly worse. Note that these skill scores are obtained after applying MF

_{tot1}and MF

_{tot2}. Lower skill scores (not shown here) would be achieved by applying only MF

_{tot2}. The additional test of MF

_{tot2}> MF

_{thresh}, slightly improves POD, CSI, and ETS scores at MF

_{thresh}= 0.55, indicating that some residual GRE/APs at stratiform rain regions have been properly identified. We have verified the same conclusion for the other cases. Thus, this result demonstrates the necessity of this test to refine the identification.

### b. Dependence on elevation angles

The AP usually appears strong at lower elevation angles and weakens with increasing elevation angles. As shown in Figs. 4, 5 and 6, and Table 3, the separation between GRE/AP and precipitation echoes is less ambiguous with increasing reflectivity. This is an indication of the poor performance of the current reflectivity-dependent fuzzy logic approach for weaker reflectivity. Since the intensity of GRE/AP decreases with elevation angles, the skill of our algorithm should decrease with higher elevation angles. Figure 11 shows the skill scores at four different elevation angles after applying only MF_{tot1} > MF_{thresh}. The skill scores for the entire elevation angles for this event are shown in Fig. 9c. The skill scores dramatically improve at lower elevation angels, particularly for MF_{thresh} > 0.55 where POD, CSI, and ETS increase by larger than 0.1 although FAR increases. CSI and ETS reaches to 0.78 and 0.62, respectively, at the elevation angle of 0.9°. In addition, the optimum threshold that provides the maximum skill scores (CSI and ETS) increases from 0.45 ∼ 0.55 at higher elevation angles to 0.7 at lower angles. This shift of the threshold is an indication that GRE/APs are less ambiguous at lower elevation angles than at higher angles. This result also suggests that the optimum threshold depends on elevation angles (higher values for lower elevation angles). However, the selected threshold (MF_{thresh} = 0.55) appears as a good compromise value at overall elevation angles.

### c. Dependence of membership functions on different intervals of reflectivity

We have used membership functions that are a function of reflectivity. As shown in Fig. 7, reflectivity-dependent membership functions are significantly different from a single membership function applied to the entire range of reflectivity. Thus, the latter can lead to an incorrect identification of GRE/APs.

Figure 12 shows the skill scores and an inset of 0.9° PPI with the two different approaches (reflectivity-dependent and reflectivity-independent membership functions). The skill scores are derived from the comparison of instantaneous reflectivity maps for the entire volume scans as described in section 5. The use of reflectivity-dependent membership functions significantly improves the skill scores (Figs. 12a,b). For example, CSI and ETS increase by about 0.07 at MF_{thresh} = 0.55. POD also increases and FAR decreases. These results are due to the misclassification of precipitation echoes as GRE/AP by the single membership function as can be deduced from Fig. 7. This can be shown in the PPI maps in Figs. 12c,d. GRE/APs are effectively removed in both maps. However, the elimination of precipitation echoes is noticeable (Fig. 12d). In general, reflectivity-dependent membership functions keep the precipitation echoes. These results suggest the importance of proper membership functions as a function of reflectivity.

### d. Comparison with the polarimetric approach

As mentioned in section 3, the polarimetric information can provide a better identification of GRE/AP. In this section, the performance of our proposed fuzzy logic method is compared with that of the polarimetric approach in terms of rainfall accumulations.

Rainfall accumulations for three cases from Table 2 are shown in Fig. 13. The first (29 August 2004) is a typical convective precipitation with normal ground echoes. Strong APs are imbedded within precipitation areas in the other two cases. On 8 July 2004, scattered showers first approach from the southwest and are then followed by stratiform rain. On 19 July 2004 case, strong convective lines and several isolated convective cells approach from the southeast and become weaker after passing by the radar site. GRE/APs are not removed on the accumulation of the first column. Proposed fuzzy logic and polarimetric methods are applied to eliminate GRE/APs for the second and third columns, respectively. Unrealistically large rainfall amounts are shown in Fig. 13a over the normal ground echo regions, north, south, and southeast of the radar map (see the map of average ground echoes in Fig. 1a). In the second and third cases, the accumulation map is further seriously contaminated by AP. The application of the proposed fuzzy logic method removes most of ground echoes and AP without affecting the precipitation echoes. The resultant accumulation map is similar to the one derived with the polarimetric approach. A quantitative analysis has revealed that polarimetric accumulation is slightly larger, indicating that our proposed fuzzy logic method eliminates some of the precipitation. However, the polarimetric approach could not completely eliminate AP over the indicated area shown in Fig. 13i south of the radar (circled area), suggesting that a further optimization of the polarimetric approach is also needed.

## 8. Conclusions

Normal ground echoes (GREs) and anomalous propagation (AP) are serious obstacles for achieving accurate precipitation estimate from radars. In this paper, we have explored the characteristics of precipitation echoes and GRE/AP and have used them in removing GRE/AP. The proposed fuzzy logic method applies reflectivity-dependent membership functions and shows improvements over using simpler, reflectivity-independent membership functions. The fuzzy membership functions and the proper weights are derived from the distribution of the characteristics of precipitation echoes and GRE/AP. In this sense, our approach is more objective and can be easily adapted to local conditions.

Since the spatial structures of radar echoes depend on geographical and climatological locations as well as radar characteristics, the separation between precipitation echoes and GRE/AP should also vary with these conditions. For example, in a climatological regime where strong convective systems are dominant, the standard deviation of reflectivity from precipitation echoes should be larger. Thus, the corresponding threshold should be larger in order to remove GRE/AP effectively while keeping precipitation echoes. However, in the climatological Montreal, Canada, environment where stratiform precipitation is dominant throughout the whole year, a smaller value is more suitable. Thus, a further investigation on the dependence of echo characteristics as a function of local environment could be useful for the proper removal of GRE/AP.

Results show that even in the Montreal environment, the characteristics of GRE/AP such as the standard deviations of reflectivity, the vertical gradient of reflectivity, and the absolute value of radial velocity vary systematically with intensity (radar reflectivity) whereas they are almost constant for the precipitation echoes. Accordingly, the membership functions and weights are made to change with echo intensity unlike what has been done in Radar Echo Classifier (Kessinger et al. 1999, 2001), which has used a single membership function for the entire range of echo intensities. The variation of the membership function with reflectivity (Fig. 7) is so significant that the use of a single membership function leads to misclassification of GRE/AP. Our evaluation has shown that the use of a single membership function leads to a significant removal of precipitation echoes since, in the heavy precipitation, the membership value is always overestimated. CSI and ETS increase by 0.07 with MF_{thresh} = 0.55 (Fig. 12).

The vertical gradient of reflectivity is a key parameter when *Z* ≥ 30 dB*Z* and its importance diminishes with decreasing intensity, leading to the radial velocity as the most effective parameter (Table 3). In general, strong echoes are less ambiguous, resulting in a clear classification. The performance also depends on the radar elevation angles due to less ambiguity of GRE/APs at lower angles where the maximum values of CSI and ETS reach to 0.78 and 0.62, respectively. The evaluation shows that the current approach has comparable performance with a polarimetric approach. The proposed fuzzy logic method removes most normal ground echoes and AP although some of precipitation fields are eliminated as shown in the comparison of accumulation maps (Fig. 13).

The proposed method is simple and can be applied to any operational radar after deriving the proper membership functions and weights suitable for the local environment.

## Acknowledgments

Two of the authors (Kim and Cho) received a financial support for this research from the Technical Development for Remote Sensing Meteorology as a part of the Meteorology and Earthquake Research and Development Programs funded by the Korean Meteorological Administration. The authors are indebted to Dr. Aldo Bellon and Alamelu Kilambi for constructive comments and for optimizing the Radar Data analysis, Processing, and Interactive Display (RAPID) system for this research. This work was partially supported by a grant from the Canadian Foundation for Climate and Atmospheric Sciences.

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## Footnotes

*Corresponding author address:* Dr. GyuWon Lee, J. S. Marshall Radar Observatory, McGill University, P.O. Box 198, Macdonald Campus, Ste-Anne de Bellevue, Montreal QC H9X 3V9, Canada. Email: gyuwon.lee@mail.mcgill.ca