## 1. Introduction

The radar refractivity retrieval developed by Fabry et al. (1997) is one way of estimating near-surface moisture using the phase measurement of the radar signal returned from ground targets such as power lines, buildings, or mountains. Because it can provide maps of near-surface moisture with high resolution in time (5 min) and space (4 km), this technique triggered high expectations in the field of quantitative forecasting of severe convective storm initiation and development (e.g., Weckwerth and Parsons 2006; Wilson and Roberts 2006).

During the last decade, the refractivity retrieval technique has been implemented on both research and operational radars and evaluated in several field experiments (Roberts et al. 2008; Cheong et al. 2008; Fritz and Chandrasekar 2009; Heinselman et al. 2009). For example, the International H_{2}O Project (IHOP_2002) had an S-band research radar (S-Pol) deployed in the southern Great Plains of the United States. One significant achievement of this experiment was the validation of the moisture radar retrievals compared with measurements from numerous conventional instruments [surface stations, aircraft, soundings, and the atmospheric emitted radiance interferometer (AERI)]. Weckwerth et al. (2005) showed that the radar moisture retrievals during IHOP_2002 were highly correlated with other moisture estimates up to 250 m above the ground. Their results suggested that radar refractivity may provide lower boundary layer moisture information for data assimilation; in this direction, Montmerle et al. (2002) assimilated moisture near the ground into the McGill short-term forecasting system. Also Wilson and Roberts (2006) suggested that moisture retrievals can possibly be useful as a precursor of convection associated with dry/convergence lines.

Despite growing interest in the use of radar refractivity, not much emphasis has been placed on the quality control of the refractivity retrieval measurements. Fabry (2005) identified the following factors as possible sources of uncertainty of the retrieval: 1) the extreme noisiness of the measured phase field and 2) simple assumptions used in the retrieval algorithm (section 2). Understanding and quantifying the noise introduced by the different sources of uncertainty affecting the refractivity retrieval will enable the development of an improved algorithm. By means of a phase simulator, we intend to assess in this study 1) which types of phase errors have a statistically predictable behavior, 2) how large those errors are, and 3) whether they account for the observed phase variability. To achieve these goals, we first review the current algorithm and speculate about its noise sources (section 2). In section 3, we describe how these noise sources are incorporated into the phase simulator. For selected cases, the simulated phase differences are validated with the observations in section 4. The conclusions follow in section 5.

## 2. Phase, phase noise, and refractivity retrieval

### a. Refractivity retrieval algorithm

*ϕ*at time

*t*and range

*r*. Here, transmit pulses are assumed to be preferably generated using a Klystron transmitter with a sufficiently stable frequency

*f*, yielding The traveling time

*t*

_{travel}in (1) is the time required for a radar ray to travel twice the pathlength (or range

*r*) to the target at the speed of propagating microwaves through the atmosphere. Here, the air reduces the speed of microwaves below the speed of light in vacuum

*c*. The ratio between the speed of light in vacuum and the speed in the atmosphere is referred to as the refractive index of air

*n*and is integrated along the ray path in (1).

*N*= (

*n*− 1) × 10

^{6}]. According to Bean and Dutton (1968), the refractivity in the lower troposphere can be empirically approximated with atmospheric pressure

*P*(hPa), temperature

*T*(K), and water vapor pressure

*e*(hPa) within an accuracy of 0.1 at microwave radar frequencies as Additionally, in warm weather conditions, spatial and temporal changes in refractivity are known to be mostly caused by changes of near-surface moisture (Fabry and Creese 1999). Therefore, the radar-measured phase can be used to first retrieve refractivity from (1) and, second, the water vapor information using (2), if the following assumptions are satisfied (Fabry et al. 1997; Fabry 2004):

- (i) Targets are rigorously stationary. Only fixed ground targets can be used in the retrieval algorithm, whereas moving targets (such as precipitation) must be avoided. This is needed to associate changes in traveling time (or phase) with changes in refractivity in the horizontal (what we want to retrieve). However, in reality, the phase returned from ground targets varies at different time scales (from a second to years) because of various phenomena such as vegetation sway and growth, propagation delay, turbulence, natural disasters, or land use changes. The current retrieval algorithm mitigates some of these factors by calibrating measured phase relative to a certain reference phase
*ϕ*_{tref}(*r*) as where the overbar indicates path-averaged values, and the subscript*t*_{ref}indicates values obtained at a reference time. - (ii) The reference for the calibration is assumed to be taken when the refractivity field is as uniform as possible. This condition is best satisfied during or immediately after stratiform rain in windy and cool conditions. Alternatively, an approximation of the standard deviation of refractivity estimates from surface weather stations can also help us to select the reference time if the number of weather stations is sufficiently dense over the radar domain. For a known
*n*(*r*)*t*_{ref}and the phase field at the selected reference time, the refractivity at the time of interest [*n*(*r*,*t*)] is obtained by computing the derivative of measured Δ*ϕ*with respect to range: Fabry (2004) used smoothing to guarantee the robustness of the retrieved refractivity field. - (iii) Phase data can be aliased. When processing (4) from (3), we must be aware that small differences of refractivity fields can result in large and ambiguous differences in phase observations. Moreover, when phase exceeds ±180°, it is still observed but is wrapped within the range ±180° (phase aliasing), which may result in some uncertainty. To minimize these errors, the algorithm smoothes Δ
*ϕ*over small regions and over short paths [i.e., using (4) twice for neighboring targets]. A key hypothesis for this to work is that all targets are on a flat terrain and at the same height as the radar (as described in Fabry et al. 1997; Fabry 2004).

### b. Noisiness of the observed phase differences

Based on (4), the quality of the retrieved refractivity is determined by the quality of phase observations (affected by instrumental or measurement errors) and by the assumptions of the algorithm itself (listed above). The latter are being investigated in this paper. If these assumptions are totally fulfilled, for the case of a uniform refractivity field, observed phase differences should result in concentric rings that only depend on range, as predicted by (3). Figure 1a shows an example of the phase differences simulated with (3), provided that the uniform *N* field at the observation time is 13.9 higher than that of the reference. Note that a 1 ppm of *N* difference, corresponding to the change of 1°C in temperature or 0.2 hPa in vapor pressure, can cause a phase change of 6.7° km^{−1} for S-Pol (2.8 GHz) according to (3). As a result, multiple aliasing appears roughly every 4 km in range.

Such uniform simulated fields are not frequently observed in reality. Figure 1b shows a field of Δ*ϕ* measurements at the 0° elevation angle obtained during IHOP_2002 (Weckwerth et al. 2004). Spatially averaged *N* differences of about 13.9 were observed between the reference and the observation times (i.e., 277.3 at the observation time and 263.4 at the reference time within the first 10 km in range). Concentric circles in the observation become less obvious with increasing range, indeed indicating the presence of horizontal refractivity gradients in this region. Moreover, the observed field is overall significantly noisier than the one simulated. This suggests that propagation delays are not only due to the horizontal variability of *N* (what we want to retrieve for moisture extraction) but also to other factors that are not taken into account by the current algorithm, namely,

- (i) ground targets may not be fully stationary,
- (ii) the reference
*N*field may not be horizontally uniform at small scales, and - (iii) the heights of targets may not be at the same height of the radar due to their different heights or complex terrain.

*N*at the reference time cannot be obtained unless more station measurements are available over the domain. The first two factors [(i) and (ii)] are thus not included in the simulation. Instead, we focused on the third factor (iii) inspired by Fabry (2004). As shown in Fig. 2, he observed that phase differences were sensitive to different target heights and the temporal change of propagation conditions. Hence, we have explored further how the lack of alignment between the radar and ground targets would affect the ray paths and result in phase noisiness contributing errors in

*N*retrieval.

*dn*/

*dh*; approximately spherically stratified in the lower atmosphere; Doviak and Zrnić 1993) determines the propagation conditions of microwaves—when the vertical profile of refractivity is constant (

*dn*/

*dh*= 0 ppm km

^{−1}), the ray will travel in a straight line. Otherwise, the ray will bend upward (downward) for propagation conditions of

*dn*/

*dh*greater (or less) than 0 ppm km

^{−1}(Bean and Dutton 1968; Sauvageot 1992; Steiner and Smith 2002). Consequently, the amount of bending determines the areas of ground (or targets above the ground) detectable at a certain distance. Similarly, for given propagation conditions the topography within the radar domain and the distribution and height of targets may also result in areas with more or less ground target measurements. Hence, as shown in Fig. 3, to determine the sensitivity of the propagation conditions (

*dn*/

*dh*) to the phase return, let us consider a given ground target with a certain height

*H*

_{target}. First, we express the location of the ray path as the incremental variables of range

*r*along the earth’s surface

*d*(Fig. 3a). The range and the distance of the target from the radar are expressed as the fixed variables of range

*R*and

*D*, respectively. Because we assume that

*D*and

*H*

_{target}are known,

*R*can be computed as a function of

*dn*/

*dh*: where

*C*= (

*E*+

_{r}*H*

_{radar})

^{2}+ (

*E*+

_{r}*H*

_{target})

^{2}− 2(

*E*+

_{r}*H*

_{radar})(

*E*+

_{r}*H*

_{target})cos(

*D*/

*E*) by applying the law of cosines, and

_{r}*E*is the earth’s radius. If the radius of a ray curvature is the same as

_{r}*E*(which can be approximated with

_{r}*dn*/

*dh*= −157 ppm km

^{−1}),

*R*is the distance following a line of constant height and is the same as

*D*. For a target at

*D*= 30 km, this can be shown in Fig. 3b by plotting Δ

*R*with respect to

*D*as a function of

*dn*/

*dh*. It is seen that the range is slightly longer 1) for a larger negative

*dn*/

*dh*(yielding more bending) and 2) for a higher target height. Although this change in range due to

*dn*/

*dh*or target height is relatively small, from millimeters to centimeters, it can trigger large changes in phase differences (Δ

*ϕ*) between reference and observation times owing to its aliasing behavior. If

*dn*/

*dh*= −39 ppm km

^{−1}at the reference time and

*N*= 300 for both the reference and observation times, the phase changes due to Δ

*R*can be shown in Fig. 3c as a function of

*dn*/

*dh*. No phase difference is observed at

*dn*/

*dh*= −39 ppm km

^{−1}and the aliasing occurs in superrefraction conditions, that is, for large negative

*dn*/

*dh*. Here, no height dependence on Δ

*ϕ*is shown because the phase changes are plotted relative to the reference propagation conditions at each fixed target height. Hence, the resulting change in pathlength due to the changes in a constant

*dn*/

*dh*between the reference and observation times should not contribute to the noisiness in the phase but simply to a bias in

*N*. Note that we still consider a constant

*dn*/

*dh*over the radar domain because the spatial variation of

*dn*/

*dh*from available measurements (e.g., soundings) cannot be resolved in the radar pixel resolution.

*ϕ*associated with changes of a

*dn*/

*dh*, we should consider the height change of the ray trajectory intercepting a given target. Hence, similarly assuming the trajectory is parabola, we can now compute the height of the ray along range

*h*(

*r*): This formula is based on Fabry (2004) [Eq. (9)] and is practically equivalent to the one in Doviak and Zrnić (1993). It has the advantage of explicitly showing the effect of propagation conditions on the path of a ray intercepting a given target. Let us first compute the ray height with

*dn*/

*dh*= −39 ppm km

^{−1}for a given target height

*H*

_{target}and the terrain (Fig. 4a). Typically, ground targets intercept the lower part of the main lobe and, at times, the sidelobes (especially at close ranges). Radar rays heading toward the surface are plotted as a gray shaded area in Fig. 4a. Note that we consider beam blockage to be caused only by the terrain and not by any structured target. In other words, we assume that the signal may be reflected by any possible structured target as well as pass around it. More importantly, the height of the lowest nonblocked ray above the terrain at a given range and azimuth can be interpreted as the minimum detectable height (MDH) for ground targets. Therefore, in clear-air conditions, strong echoes at a given location identify the presence of at least one target higher than the minimum detectable height.

*H*

_{radar}=

*H*

_{target}). For ducting conditions (

*dn*/

*dh*≤ −157 ppm km

^{−1}), rays follow a convex path to reach the targets. This implies that rays are less blocked by terrain and thus better able to detect the ground at further ranges. On the other hand, in subrefractive conditions (

*dn*/

*dh*> 0 ppm km

^{−1}), ray bends upward (concavely) and are thus likely to miss ground targets. Note in Fig. 4b that the ray trajectory is as much as 20 m below the radar height for

*dn*/

*dh*= 30 ppm km

^{−1}and would be blocked at near range before reaching the target. Consequently, targets at far range can be detected only in ducting conditions unless the ground target is taller than the minimum detectable ray height. Of course, if we consider target heights being different from the radar height, the interpretation of the phase measurement can be more complicated because the ray may hit or miss the ground targets. To include such a complication introduced by changes of propagation conditions as well as target heights on phase, we can rewrite (1) for a radar scan observed at time

*t*by substituting

*h*(

*r*) from (6): where

*n*(

*r*,

*t*)

*is the refractivity at a given (*

_{h}*r*,

*t*) and height

*h*[if not explicitly mentioned,

*n*(

*r*,

*t*) is at the radar height]. Here, the vertical gradient of refractivity

*dn*/

*dh*is assumed to be a constant. As we can see, two terms on the right of (7) are affected by

*dn*/

*dh*and the height of the target. We have also substituted

*R*with (5) even though it is small. It was added for completeness sake. To quantify their impact on phase differences from (3), we compute the propagation error of phase difference measurements (

*σ*

_{Δϕ}) according to Bevington (1969): Here, the variability of target heights can be quantified as the spread of the distribution of target heights (

*σ*

_{Htarget}). Figure 5 shows

*σ*

_{Δϕ}computed from (8) using simply a spread of target heights of

*σ*

_{Htarget}= 10 m. Because from (8) the uncertainty grows linearly with

*σ*

_{Htarget}, the variability of phase differences here is only a result of the changes in

*dn*/

*dh*between the observed and the reference time. The phase variability increases as 1)

*dn*/

*dh*departs from that of the reference time and 2) targets are located at farther ranges.

This sensitivity of the phase data to propagation conditions encouraged us to predict expected phase noise by carrying a more rigorous analysis. If all of the noise is predictable by these factors, then we can correct it to improve refractivity retrieval. Hence, we designed a phase simulator based on the equations derived above coupled with the determination of a target height distribution and the estimation of different *dn*/*dh* values. Predictions of phase noise made by the simulator will be compared with the observed phase noise, allowing us to evaluate its skill.

## 3. Phase simulator

*ϕ*with respect to the reference phase and to test the effect of propagation conditions (characterized by

*dn*/

*dh*) on the noisiness of Δ

*ϕ*. Based on (7), for a given target location in terms of radar range

*R*, the phase differences can be determined as As we see on the right-hand-side (rhs) terms of (9), phase differences depend on three factors: i) the radial (horizontal) change of refractivity (the first term), ii) the target alignment with respect to the radar associated with a constant

*dn*/

*dh*(the second term), and iii) the ray curvature relative to the curvature of the earth (the last term). The three terms depend on (

*dn*/

*dh*)

*t*

_{ref}at the reference time and (

*dn*/

*dh*) at the time of interest. In practice, to compute each term, we require the following information:

- the path-averaged refractivity
*n*(*t*)*R*for observation and*n*(*t*_{ref})*R*_{tref}for reference times between the radar and given targets; - the vertical gradient of refractivity (
*dn*/*dh*) for both observation and reference times; and - the location and height of targets (
*H*_{target}) within the radar domain.

### a. Path-averaged refractivity

One should remember that the path-averaged refractivity at an observed time *n*(*t*)*R* of the first term on the rhs of (9) is the variable to be retrieved with the refractivity retrieval algorithm. Thus, the true value of *n*(*t*)*R* is not known in advance. Instead, what we know is the average aliasing rate of measured phase differences due to the spatial average of the refractivity difference over the radar domain. This means that the spatially averaged refractivity difference will be a good estimate of the path-averaged refractivity as long as the refractivity fields are uniform at both reference and observation times (as in the example of Fig. 1). Otherwise, we should include the spatial variability of differences in *N*. Figure 6 shows an example of the observed aliasing pattern of azimuthally averaged phase differences within a 40-km range (dotted line) for the same time as in Fig. 1. The fit aliasing rate (solid line) results in a difference of about 14 between the observation and the reference times. However, although some ranges seem to have aliasing rates similar to the fit, others are totally mismatched. Because our goal is to simulate the phase field as close as possible to reality, the variability of the refractivity field should also be considered in the simulation. Hence, we used the retrieved refractivity fields to characterize the local departures from the spatial mean. It is not a desired approach to reuse the retrieved fields, but this is a realistic way to consider the spatial variability of differences in *N*.

### b. Sources of the vertical gradient of refractivity (dn/dh)

Sounding measurements are used to characterize propagation conditions by estimating *dn*/*dh* from pressure, temperature, and humidity. During IHOP_2002, several radio soundings were available from different instruments. The Homestead site, located 16 km away from the S-Pol radar, had an Integrated Sounding System (ISS) and a mobile research vehicle AERIBAGO equipped with an atmospheric emitted radiance interferometer (AERI) instrument; ceilometer; surface stations; radiosonde; and GPS antenna measuring total precipitable water. Mobile facilities and aircraft launched soundings were also available within the radar domain (Weckwerth et al. 2004).

For the simulator, the values of *dn*/*dh* were assumed to be constant for both the reference time and the time of interest. The reference times during IHOP_2002 are those used for calibration in the refractivity retrieval algorithm: between 2010 and 2040 UTC 14 May 2002 for dry conditions, and between 0830 and 0900 UTC 21 May 2002 for wet conditions. Because no radiosonde sounding is available at these times, we have used the retrieved soundings from AERI observations. The AERI retrieval has been obtained through inversion of the infrared transfer equation (Feltz et al. 2003) and derived with a high temporal resolution (less than 10 min) at discrete heights (e.g., around 44, 87, 130 m, etc.) Because we are interested in propagation conditions near the ground, a representative value of *dn*/*dh* at 65 m has been computed with *N* at the level between 44 and 87 m. Then, we have extrapolated this estimated *dn*/*dh* to a value at the level of 33 m AGL. We have used such low-level estimates because conditions near the surface have the most effect on phase measurements from ground targets.

### c. Target height simulation

The simulation of phase differences requires the location, heights, and number of ground targets that are neither moving nor changing their apparent shape. Note that the target height (*H*_{target}) in (9) includes the terrain height above MSL and the target height above the ground. Terrain height can be easily obtained from a digital elevation model (Fig. 7a around the S-Pol radar during IHOP_2002). Although it is difficult to know the exact location and height of targets within the radar domain, it is known that the area of the Great Plains has targets such as farm barns, water towers, and power poles that are generally lower than 30 m tall. We hence need to determine the height distribution of targets within a typical radar pixel that only contain fixed ground targets.

*I*and

*Q*(

*NIQ*) in decibels computed as where

*M*samples of the complex

*x*

_{i,k}(

*I*,

*Q*) signal over the pulse width (∼1

*μ*s) at the

*i*th range gate. Higher values indicate strong echoes likely from ground targets. Hence, we have first established that values of

*NIQ*exceeding −20 dB are returned from fixed ground targets. Note that

*NIQ*is only the instantaneous signal strength and so does not provide target reliability. Therefore, the RI between 0 (bad) and 1 (good) has been also obtained as a measure of the coherence of

*NIQ*as from

*S*scans at the 0° elevation during a period of frequent scan every 1 min instead of the usual complete volume scan every 5 min (e.g., from 1849 to 2027 UTC 16 May) during IHOP_2002. As a result, Fig. 7b shows the

*NIQ*field (larger than −20 dB) combined with RI (larger than 0.8) at the reference time used for the refractivity retrieval. The selected area of

*NIQ*corresponds well to the area of higher terrain around the radar (Fig. 7a).

From this information, we have inferred the distribution of the heights of solid ground targets within the radar domain as follows. First, solid targets are identified based on QI. In parallel, we have simulated the minimum detectable height (see section 2b and Fig. 4a) over the radar domain; if a target is observed at a radar pixel (150 m in range by 1° in azimuth) with a given MDH, that pixel must contain a target higher than the MDH.

For consistency, the simulation has been performed using the propagation conditions estimated for the fast-scanning period (during which RI was obtained). Hence, we have used a value of *dn*/*dh* = 20 ppm km^{−1}, which is obtained using AERI measurements from 1948 to 1957 UTC 16 May 2002. In the simulation, the height of the radar and the propagation conditions plays an important role. Considering the size of the antenna dish of S-Pol (∼10 m) and of its supporting structure, the radar height is estimated to be 15 m above the ground (893 m MSL at the S-Pol site). Figure 8a shows the result of the simulated MDH map. Areas in black (0 m) indicate where the lowest ray hits the ground, which corresponds well to areas with the high ground echo intensity of Fig. 7b.

*H*

_{target}within a pixel as shown in Fig. 8b: where

*i*indicates the split of a target with the height interval of Δ = 1 m from 0 m to the maximum value of MDH above the ground, and

*NIQ*

_{min}is the threshold on

*NIQ*used to identify solid targets. We estimated the probability of having a target of a certain height in a pixel of the domain by differentiating (12) with respect to the height interval as: where Here,

*W*is a correction term that considers the possibility of having multiple targets within a single radar pixel. Figure 8c shows the histogram of target heights obtained over the S-Pol domain (up to 60 km in range) as a function of target height. As expected in the Great Central Plains (and mostly everywhere else), low targets are much more frequent within the radar domain.

_{m}Note that the distribution of Fig. 8c is used to simulate the location and height of targets within the radar domain under the assumption that targets are uniformly distributed in space. However, in reality they are quite randomly distributed. To compensate for this, we consider a possibility of having more than one target per pixel. So, the number of targets of a certain height in each pixel is randomly generated based on a Poisson distribution (Kalbfleisch 1985). The expected value of the distribution for each height is set according to the frequency obtained in Fig. 8c. In our case, the simulated number of targets resulted in mostly one and rarely two per pixel from continuous (in space) targets detected near the radar seen in Fig. 4a.

## 4. Validation of the phase simulator

The validation of phase differences simulated with (9) can be done by comparison with phase differences from real observations. This section presents two cases chosen because of the availability of i) the values of mean refractivity difference and ii) *dn*/*dh* soundings. All radar measurements (i.e., phase and retrieved refractivity) are obtained at 0° elevation angle and averaged over 15 min. This can help mitigate measurement noise that may introduce additional complexity in the comparison. Also, only radar pixels with high QI are considered (as described in section 3c), which guarantees that only solid targets have been used. Because ground targets are expected to be better observed at near range than far range, we present results up to 10 km in range based on the density of high NIQ seen in Fig. 7b; we also found that results obtained beyond 10 km in range did not add any insight to the analysis to follow.

*ϕ*

*σ*

_{Δϕ}for directional data followed by Weber (1997): This formula is similarly used to estimate Doppler spectrum width (Lhermitte 2002). The size of the area considered (2.4 km by 10°) was chosen to be large enough to obtain proper statistics in (16) while remaining small enough not to be influenced unduly by changes of

*N*in space.

### a. Case 1: 2332 UTC 15 May 2002

This is the case already analyzed throughout the paper (see section 2). The reference refractivity field is the one used for dry conditions at 2027 UTC 14 May 2002. In Fig. 1, we could identify ringlike patterns in the phase difference data. This pattern corresponds to a spatial mean difference of refractivity of about 13.9, as shown in the fitting exercise of Fig. 9a. The fit matches well with the observation up to 6 km but varies beyond that range. This inhomogeneity is partly reflected on the retrieved *N* difference field showing a west–east gradient and some small-scale variability at ranges beyond 6 km (Fig. 9b). For the propagation conditions, we have used the values of near-surface *dn*/*dh* estimated from AERI soundings (see its location in Fig. 7a): 62 ppm km^{−1} for the reference time and 25 ppm km^{−1} for the time of interest (Figs. 8c and 8d, respectively). Both periods had subrefractive conditions and show some variability of *dn*/*dh* in the vertical.

With the estimated propagation conditions at the reference and observation times, the simulations of phase differences are presented considering the three terms of (9) additively. In other words, simulation 1 includes only the radial change of refractivity, and simulation 2 adds the influence of target heights associated with a given *dn*/*dh* at observation time. Finally, simulation 3 gathers simulations 1 and 2 as well as the effect of ray curvature depending on *dn*/*dh*. Figure 10 shows the comparisons between the three simulations and the observations in terms of phase difference (Fig. 10a), spatially averaged phase differences (Fig. 10b), and local phase variability (Fig. 10c). First of all, the coverage of targets visible in the simulated fields of phase differences is in reasonable agreement with the coverage of targets in the observations. Although the patterns in the observations are patchier than in the simulations, their aliasing patterns resemble each other. For example, the ring patterns are skewed toward the west because of the presence of the east–west gradient of refractivity mentioned earlier. Hence, the simulator has produced more realistic results than those presented in Fig. 1a, where rings are purely concentric because the *N* difference field is considered uniform. In terms of the noisiness of the phase difference fields, the smooth simulated fields show some small-scale wavy patterns and similar values of standard deviation as the observations (Figs. 10b and 10c, respectively). If we focus on the simulations in Fig. 10c, the variability becomes slightly larger in simulation 2, which includes target information, than in simulation 1 where only the effect of the horizontal refractivity field is considered. Finally, simulation 3 is almost identical to that in simulation 2. This is not surprising at this near range because the third term of (9) is only significant at far ranges. From this case, the simulation seems to show some skill in reproducing noisiness. However, note that this case showed relatively large mean refractivity differences (of around 13.9) between the reference and the observation times. This might be responsible for a significant part of the variability of the phase difference fields. Hence, we have chosen another case with an observed mean refractivity similar to that at the reference time to better illustrate the impact of the propagation factors on the simulated phase differences.

### b. Case 2: 1850 UTC 16 June 2002

The time of interest (1850 UTC 16 June) is selected because the phase aliasing rate (or the mean of Δ*N*) is very low, that is, the average refractivity is very similar to that of the reference time for wet conditions (0843 UTC 21 May). Figure 11a shows that the mean *d*(Δ*ϕ*)/*dr* is much smaller than in the previous example (Fig. 9a), and its best fit yields a mean refractivity difference of about 0.56 up to 10 km in range. This small value is in good agreement with the overall refractivity differences of the retrieved fields (as seen in Fig. 11b). Unlike the dry reference time used in case 1, propagation conditions for this wet case are characterized by an almost constant vertical profile of *dn*/*dh* near the ground. The estimates of *dn*/*dh* are similar at each level of 33, 65, and 109 m, but slightly different between the reference and observation times (−37 versus −52 ppm km^{−1} shown as Figs. 11c and 11d, respectively).

Observations and simulations for this case are presented in Fig. 12. The observed phase difference field (Fig. 12a) shows almost no aliasing pattern within 10 km in range. This is well reproduced in the three stages of simulation. The smoothed fields of Fig. 12b show that the simulated phase differences resemble the observations in general. The effect of target height and propagation conditions is also shown in Fig. 12c; simulations 2 and 3 are slightly noisier than simulation 1 and seem to have more impact at farther ranges (see around 10 km in range in the northeast area) than simulation 1. If the observation time had been more superrefractive, the simulated results would have been better obtained because the difference in *dn*/*dh* between the reference and observation times may play more in the phase simulator. However, as seen in Fig. 12c, the variances of the simulated fields remain much smaller than those of the observations. For example, in terms of the root-mean-square error of *σ*_{Δϕ} over the domain of 10-km range, we have obtained about 10° (70°) from the simulated (observed) fields.

From the simulation, therefore, we have learned that higher variability appears 1) with larger differences in the propagation conditions between the reference time and the time of interest and 2) at farther range. Nevertheless, we have not been able to approach the variability of observed fields that can be 7 times larger than those of simulated fields, especially for case 2. These large differences are observed at near range, where none of the terms of (9) are significant. Hence, the explanation for the noisiness in the phase data must lie elsewhere.

## 5. Conclusions

The phase measurements of ground targets used in the radar refractivity retrieval algorithm are often noisy, yielding ambiguous retrieval results. This paper has attempted to reproduce the noisiness of the phase measurements by rewriting the equations of the algorithm to include the change of ray trajectories to ground targets over complex terrain as a function of the propagation conditions. Observed phase differences were used to validate our simulations. From the analysis of two selected cases during IHOP_2002, we have seen that phase difference simulations are sensitive to propagation conditions. This effect would also be more significant at far range. However, the simulated results at near range where ground targets are denser and of “better quality” than those at far range suggest that the factor of *dn*/*dh* and the target height variability cannot fully explain the noisiness of observed phase differences. The reasons for the discrepancy could be due to factors not accounted in the simulation such as the following:

- (i) Here,
*dn*/*dh*obtained at a single location is used for the entire radar domain. - (ii) The small-scale horizontal variability of refractivity at the reference time (supposed to be uniform in the simulator) remains still unresolved. Moreover, calibration times should be carefully selected on the basis of not only a horizontally uniform
*N*but also on a uniform*dn*/*dh*near the ground.

## Acknowledgments

We thank Prof. Isztar Zawadzki for many valuable comments. The authors are grateful to Dr. Marc Berenguer for his early review of the manuscript and numerous advices. We also thank Dr. Aldo Bellon for his insightful review. This work was completed and made possible with thanks to the Canadian Foundation for Climate and Atmospheric Sciences.

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