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    (a) Simulated phase difference field assuming that the uniform refractivity field of a current scan is 13.9 higher than that of the reference scans. (b) An observed phase difference between current (2332 UTC 15 May 2002) and a reference (2027 UTC 14 May 2002) scan during IHOP_2002 in Oklahoma. The S-Pol radar is located in the middle of this plot. The spottiness in coverage is due to the limited number of fixed ground targets that can be used for refractivity measurements. To avoid some noise that may occur because of local variations within a few minutes, the observed phases are averaged over four consecutive scans.

  • View in gallery

    Time evolution of the phase of three neighboring targets along the same azimuth during the disappearance of trapping conditions immediately after sunrise. The phases of the two low-level targets (dotted and dashed lines) parallel each other because similar changes in path-integrated n occur in the two low-level paths between the radar and these targets. In between these low-level targets is a higher target whose phase (solid line) does not vary as much because the change in path-integrated n along the higher-level path is smaller. This occurs as a result of an adjustment in dn/dh as we move from trapping conditions (illustrated in the inset above by the stronger reflections of faraway targets) to normal propagation conditions. During trapping conditions, the higher target only 20 km away was about 180° out of phase compared to what it would have been under normal propagation conditions. This phenomenon introduces noise in the ϕϕref field, complicating the retrieval of n between targets of different heights, and it forms the basis for (14) from Fabry (2004).

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    Dependence of the pathlength on target height and propagation conditions. (a) Illustration of the geometry of the problem. (b) Examples of the ΔR variation due to the propagation conditions and target heights relative to the radar height. The solid line indicates that the target height is the same as the radar height. The dotted and dashed lines indicate, respectively, that the heights of the target are at 10 and 60 m above the radar. (c) The changes of phase differences with respect to the reference (dn/dh = −39 ppm km−1) due to the ray length changes at D = 30 km.

  • View in gallery

    Ray heights affected by propagation conditions. (a) Illustration of ray trajectory. The dotted line indicates terrain height above MSL. The gray shaded area represents the rays going toward the ground as determined by the grazing angle between the radar and ground heights at each range pixel. (b) Ray heights toward a given target as a function of distance along the surface and propagation conditions. Here, the target height is leveled with the radar height. (c) As in (b), but for a target 10 m above the radar.

  • View in gallery

    Contour plot of the variance of phase measurements in radians as a function of (dn/dh)obs − (dn/dh)ref, and the distance along the arc surface for a target height variability of 10 m. Note how even a very modest σHtarget of 10 m results in considerable phase variance.

  • View in gallery

    The azimuthally averaged phase difference (dotted line) between the time of interest (2332 UTC 15 May 2002) and the reference (2027 UTC 14 May 2002) as a function of range. The solid line is the best fit to this observed phase difference over 40 km in range, resulting in a mean refractivity difference of about 13.7 within the 40-km domain.

  • View in gallery

    (a) Topography map generated with the National Elevation Dataset of the USGS (with a resolution of 1 arcsec, approximately 30 m in space) within 60 km of the S-Pol radar. (b) The map of the NIQ (larger than −20 dB) combined with RI (larger than 0.8) observed at 2027 UTC 14 May 2002, when the reference has been prepared for the refractivity retrieval. The radar detects well many targets near range (∼10 km) and on higher terrain (e.g., the northern west area). The Beaver River valley in the northern east area is not seen by the radar because of its lower elevation.

  • View in gallery

    (a) MDH map; the terrain height is subtracted from the lowest ray height assuming that dn/dh is 20 ppm km−1 as obtained from AERI soundings (located near AERI in Fig. 7a) at 1948∼2020 UTC 16 May 2002. (b) Probability of having radar pixels at larger than a certain height over the pixels of good targets (as determined by the NIQ and RI thresholds over the radar domain: −20 dB and 0.8, respectively): P(Htarget ≥ MDHi) as a function of target height (in square). This probability is smoothed (solid line) to avoid any negative probability. (c) Probability of having a target at a specific height, P(Htarget = h), using the smoothed result from (b). This probability is applied to the radar pixel given as 150 m in range and 1° in azimuth to assign target numbers and heights in the pixel.

  • View in gallery

    Results for case 1. (a) Phase differences between 2332 UTC 15 May and 2027 UTC 14 May (reference time). Data are azimuthally averaged up to 10 km (dotted line). The solid line is the fit to the observations; aliasing every 3–4 km indicates a uniform change of 12–14 of the mean refractivity over the domain. (b) Retrieval field of refractivity differences between the time of interest and the reference. We only plot the area beyond 1.2 km because of the low quality of the near-range data. (c) Here, dn/dh estimated from the AERI soundings at the reference time (averaged over 15-min period scans; from 2020 to 2035 UTC) and at low levels (e.g., 65, 109 m AGL). A representative value (62 ppm km−1) is obtained by extrapolating dn/dh at 65 m to the lowest level of 33 m and averaged in time. (d) As in (c), but for the time of interest.

  • View in gallery

    Comparison between the observation and the simulation for case 1. (a) Phase difference, (b) its local average, and (c) the variability over an area with 2.4 km in range by 10° in azimuth are shown. (top to bottom) Each row shows the results from the observations, simulation 1, simulation 2, and simulation 3.

  • View in gallery

    As in Fig. 9, but for case 2 between 1850 UTC 16 Jun and 0843 UTC 21 May (reference time).

  • View in gallery

    As in Fig. 10, but for case 2.

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Simulation and Interpretation of the Phase Data Used by the Radar Refractivity Retrieval Algorithm

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  • 1 Department of Atmospheric and Oceanic Sciences, McGill University, Montreal, Quebec, Canada
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Abstract

The radar refractivity retrieval algorithm applied to radar phase measurements from ground targets can provide high-resolution, near-surface moisture estimates in time and space. The reliability of the retrieval depends on the quality of the returned phase measurements, which are affected by factors such as 1) the vertical variation of the refractive index along the ray path and 2) the properties of illuminated ground targets (e.g., the height and shape of the targets intercepted by radar rays over complex terrain). These factors introduce ambiguities in the phase measurement that have not yet been considered in the refractivity algorithm and that hamper its performance.

A phase measurement simulator was designed to better understand the effect of these factors. The results from the simulation were compared with observed phase measurements for selected atmospheric propagation conditions estimated from low-level radio sounding profiles. Changes in the vertical gradient of refractivity coupled with the varying heights of targets are shown to have some influence on the variability of phase fields. However, they do not fully explain the noisiness of the real phase observations because other factors that are not included in the simulation, such as moving ground targets, affect the noisiness of phase measurements.

Corresponding author address: Shinju Park, Dept. of Atmospheric and Oceanic Sciences, 805 Sherbrooke West 945, Montreal QC H3A 2K6, Canada. Email: shinju@meteo.mcgill.ca

Abstract

The radar refractivity retrieval algorithm applied to radar phase measurements from ground targets can provide high-resolution, near-surface moisture estimates in time and space. The reliability of the retrieval depends on the quality of the returned phase measurements, which are affected by factors such as 1) the vertical variation of the refractive index along the ray path and 2) the properties of illuminated ground targets (e.g., the height and shape of the targets intercepted by radar rays over complex terrain). These factors introduce ambiguities in the phase measurement that have not yet been considered in the refractivity algorithm and that hamper its performance.

A phase measurement simulator was designed to better understand the effect of these factors. The results from the simulation were compared with observed phase measurements for selected atmospheric propagation conditions estimated from low-level radio sounding profiles. Changes in the vertical gradient of refractivity coupled with the varying heights of targets are shown to have some influence on the variability of phase fields. However, they do not fully explain the noisiness of the real phase observations because other factors that are not included in the simulation, such as moving ground targets, affect the noisiness of phase measurements.

Corresponding author address: Shinju Park, Dept. of Atmospheric and Oceanic Sciences, 805 Sherbrooke West 945, Montreal QC H3A 2K6, Canada. Email: shinju@meteo.mcgill.ca

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 H2O 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

The current radar refractivity retrieval starts with the radar phase measurement 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
i1520-0426-27-8-1286-e1
The traveling time ttravel 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).
The refractive index of air near the surface is approximately 1.0003 for a standard atmosphere (Doviak and Zrnić 1993). In reality, it varies along the ray path. It is often expressed in terms of refractivity [N = (n − 1) × 106]. 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
i1520-0426-27-8-1286-e2
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 ϕtref(r) as
    i1520-0426-27-8-1286-e3
    where the overbar indicates path-averaged values, and the subscript tref 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)tref 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 Δϕ with respect to range:
    i1520-0426-27-8-1286-e4
    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 Δϕ 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 Δϕ 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.
However, identifying and quantifying the contribution of these factors to the noisiness in phase differences is not simple. For example, 1) the information of real target properties (movement, shape, etc.) and location can hardly be resolved within a radar pixel (150 m by 1°, for instance), and 2) the small-scale variability of 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.

It is well known that refractivity generally decreases with height. The gradient of the refractive index in the vertical (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 Htarget. 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 Htarget are known, R can be computed as a function of dn/dh:
i1520-0426-27-8-1286-e5
where C = (Er + Hradar)2 + (Er + Htarget)2 − 2(Er + Hradar)(Er + Htarget)cos(D/Er) by applying the law of cosines, and Er is the earth’s radius. If the radius of a ray curvature is the same as Er (which can be approximated with 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 ΔR 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 (Δϕ) 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 ΔR 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 Δϕ 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.
In fact, to examine the dependence of target heights on Δϕ 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):
i1520-0426-27-8-1286-e6
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 Htarget 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.
Now, let us analyze the impact of propagation conditions in (6). Figure 4b shows the path of the ray intercepting a target aligned with the radar (Hradar = Htarget). 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):
i1520-0426-27-8-1286-e7
where n(r, t)h is the refractivity at a given (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 (σΔϕ) according to Bevington (1969):
i1520-0426-27-8-1286-e8
Here, the variability of target heights can be quantified as the spread of the distribution of target heights (σHtarget). Figure 5 shows σΔϕ computed from (8) using simply a spread of target heights of σHtarget = 10 m. Because from (8) the uncertainty grows linearly with σHtarget, 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

The refractivity retrieval uses relative differences between the observed phase differences with respect to those made at a reference time, rather than between the transmitted phases themselves. Hence, a phase simulator has been developed to compute phase differences Δϕ with respect to the reference phase and to test the effect of propagation conditions (characterized by dn/dh) on the noisiness of Δϕ. Based on (7), for a given target location in terms of radar range R, the phase differences can be determined as
i1520-0426-27-8-1286-e9
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)tref 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(tref)Rtref 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 (Htarget) within the radar domain.
Because it is not straightforward to measure these directly, let us describe the following approach to obtain each element based on the observations available during IHOP_2002.

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 (Htarget) 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.

We have chosen these pixels based on a quality index (QI). The QI is composed of (i) the echo strength estimated at the 0° elevation angle during the dry calibration time (from 2010 to 2040 UTC 14 May) and (ii) the reliability index (RI) used to characterize the stationarity of the target (Fabry 2004). The echo strength is determined by analyzing radar-backscattered power in terms of the norm of I and Q (NIQ) in decibels computed as
i1520-0426-27-8-1286-e10
where for M samples of the complex xi,k (I, Q) signal over the pulse width (∼1 μs) at the ith 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
i1520-0426-27-8-1286-e11
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.

By combining identified targets with MDH values, we have estimated the probability of having a target higher than Htarget within a pixel as shown in Fig. 8b:
i1520-0426-27-8-1286-e12
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 NIQmin 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:
i1520-0426-27-8-1286-e13
where
i1520-0426-27-8-1286-eq1
Here, Wm 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.

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.

Phase returns from simulated targets are computed by averaging (9) for all the heights where the target is visible (with the height resolution of 1 m) according to
i1520-0426-27-8-1286-e14
If the simulated ray intercepts multiple targets within a radar pixel, the phase returns from that pixel are averaged in a similar manner.

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.

The comparisons are performed in terms of phase differences between reference time and time of interest. Because these fields are noisy, to see their patterns better we compute the local average of phase differences (Δϕ) over areas of 2.4 km in range by 10° in azimuth around each pixel:
i1520-0426-27-8-1286-e15
Also, we compute the variability expressed as the local standard deviation σΔϕ for directional data followed by Weber (1997):
i1520-0426-27-8-1286-e16
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 ΔN) 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 dϕ)/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 σΔϕ 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.
Yet, these factors would not be sufficient to explain the noise in observed phase differences. Hence, other factors must also play a significant role. Our simulation lacked a full characterization of the complexity of ground targets, that is, the geometry and surface roughness of targets or the fact that they can move (for instance, overhead irrigators deployed in the farm fields or vegetation growth). Long-term ground observation of phase as well as echo intensity over the area of interest may help to ensure fixed ground targets and avoid moving targets. Furthermore, our simulator only includes single-ray backscattering and does not consider a full description of wave propagation. Multiple reflections of rays and diffraction behind the shadow regions have thus been ignored. All these can be factors that introduce noise in phase difference observations. Their complexity makes their inclusion in the refractivity retrieval algorithm a challenge.

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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Fig. 1.
Fig. 1.

(a) Simulated phase difference field assuming that the uniform refractivity field of a current scan is 13.9 higher than that of the reference scans. (b) An observed phase difference between current (2332 UTC 15 May 2002) and a reference (2027 UTC 14 May 2002) scan during IHOP_2002 in Oklahoma. The S-Pol radar is located in the middle of this plot. The spottiness in coverage is due to the limited number of fixed ground targets that can be used for refractivity measurements. To avoid some noise that may occur because of local variations within a few minutes, the observed phases are averaged over four consecutive scans.

Citation: Journal of Atmospheric and Oceanic Technology 27, 8; 10.1175/2010JTECHA1393.1

Fig. 2.
Fig. 2.

Time evolution of the phase of three neighboring targets along the same azimuth during the disappearance of trapping conditions immediately after sunrise. The phases of the two low-level targets (dotted and dashed lines) parallel each other because similar changes in path-integrated n occur in the two low-level paths between the radar and these targets. In between these low-level targets is a higher target whose phase (solid line) does not vary as much because the change in path-integrated n along the higher-level path is smaller. This occurs as a result of an adjustment in dn/dh as we move from trapping conditions (illustrated in the inset above by the stronger reflections of faraway targets) to normal propagation conditions. During trapping conditions, the higher target only 20 km away was about 180° out of phase compared to what it would have been under normal propagation conditions. This phenomenon introduces noise in the ϕϕref field, complicating the retrieval of n between targets of different heights, and it forms the basis for (14) from Fabry (2004).

Citation: Journal of Atmospheric and Oceanic Technology 27, 8; 10.1175/2010JTECHA1393.1

Fig. 3.
Fig. 3.

Dependence of the pathlength on target height and propagation conditions. (a) Illustration of the geometry of the problem. (b) Examples of the ΔR variation due to the propagation conditions and target heights relative to the radar height. The solid line indicates that the target height is the same as the radar height. The dotted and dashed lines indicate, respectively, that the heights of the target are at 10 and 60 m above the radar. (c) The changes of phase differences with respect to the reference (dn/dh = −39 ppm km−1) due to the ray length changes at D = 30 km.

Citation: Journal of Atmospheric and Oceanic Technology 27, 8; 10.1175/2010JTECHA1393.1

Fig. 4.
Fig. 4.

Ray heights affected by propagation conditions. (a) Illustration of ray trajectory. The dotted line indicates terrain height above MSL. The gray shaded area represents the rays going toward the ground as determined by the grazing angle between the radar and ground heights at each range pixel. (b) Ray heights toward a given target as a function of distance along the surface and propagation conditions. Here, the target height is leveled with the radar height. (c) As in (b), but for a target 10 m above the radar.

Citation: Journal of Atmospheric and Oceanic Technology 27, 8; 10.1175/2010JTECHA1393.1

Fig. 5.
Fig. 5.

Contour plot of the variance of phase measurements in radians as a function of (dn/dh)obs − (dn/dh)ref, and the distance along the arc surface for a target height variability of 10 m. Note how even a very modest σHtarget of 10 m results in considerable phase variance.

Citation: Journal of Atmospheric and Oceanic Technology 27, 8; 10.1175/2010JTECHA1393.1

Fig. 6.
Fig. 6.

The azimuthally averaged phase difference (dotted line) between the time of interest (2332 UTC 15 May 2002) and the reference (2027 UTC 14 May 2002) as a function of range. The solid line is the best fit to this observed phase difference over 40 km in range, resulting in a mean refractivity difference of about 13.7 within the 40-km domain.

Citation: Journal of Atmospheric and Oceanic Technology 27, 8; 10.1175/2010JTECHA1393.1

Fig. 7.
Fig. 7.

(a) Topography map generated with the National Elevation Dataset of the USGS (with a resolution of 1 arcsec, approximately 30 m in space) within 60 km of the S-Pol radar. (b) The map of the NIQ (larger than −20 dB) combined with RI (larger than 0.8) observed at 2027 UTC 14 May 2002, when the reference has been prepared for the refractivity retrieval. The radar detects well many targets near range (∼10 km) and on higher terrain (e.g., the northern west area). The Beaver River valley in the northern east area is not seen by the radar because of its lower elevation.

Citation: Journal of Atmospheric and Oceanic Technology 27, 8; 10.1175/2010JTECHA1393.1

Fig. 8.
Fig. 8.

(a) MDH map; the terrain height is subtracted from the lowest ray height assuming that dn/dh is 20 ppm km−1 as obtained from AERI soundings (located near AERI in Fig. 7a) at 1948∼2020 UTC 16 May 2002. (b) Probability of having radar pixels at larger than a certain height over the pixels of good targets (as determined by the NIQ and RI thresholds over the radar domain: −20 dB and 0.8, respectively): P(Htarget ≥ MDHi) as a function of target height (in square). This probability is smoothed (solid line) to avoid any negative probability. (c) Probability of having a target at a specific height, P(Htarget = h), using the smoothed result from (b). This probability is applied to the radar pixel given as 150 m in range and 1° in azimuth to assign target numbers and heights in the pixel.

Citation: Journal of Atmospheric and Oceanic Technology 27, 8; 10.1175/2010JTECHA1393.1

Fig. 9.
Fig. 9.

Results for case 1. (a) Phase differences between 2332 UTC 15 May and 2027 UTC 14 May (reference time). Data are azimuthally averaged up to 10 km (dotted line). The solid line is the fit to the observations; aliasing every 3–4 km indicates a uniform change of 12–14 of the mean refractivity over the domain. (b) Retrieval field of refractivity differences between the time of interest and the reference. We only plot the area beyond 1.2 km because of the low quality of the near-range data. (c) Here, dn/dh estimated from the AERI soundings at the reference time (averaged over 15-min period scans; from 2020 to 2035 UTC) and at low levels (e.g., 65, 109 m AGL). A representative value (62 ppm km−1) is obtained by extrapolating dn/dh at 65 m to the lowest level of 33 m and averaged in time. (d) As in (c), but for the time of interest.

Citation: Journal of Atmospheric and Oceanic Technology 27, 8; 10.1175/2010JTECHA1393.1

Fig. 10.
Fig. 10.

Comparison between the observation and the simulation for case 1. (a) Phase difference, (b) its local average, and (c) the variability over an area with 2.4 km in range by 10° in azimuth are shown. (top to bottom) Each row shows the results from the observations, simulation 1, simulation 2, and simulation 3.

Citation: Journal of Atmospheric and Oceanic Technology 27, 8; 10.1175/2010JTECHA1393.1

Fig. 11.
Fig. 11.

As in Fig. 9, but for case 2 between 1850 UTC 16 Jun and 0843 UTC 21 May (reference time).

Citation: Journal of Atmospheric and Oceanic Technology 27, 8; 10.1175/2010JTECHA1393.1

Fig. 12.
Fig. 12.

As in Fig. 10, but for case 2.

Citation: Journal of Atmospheric and Oceanic Technology 27, 8; 10.1175/2010JTECHA1393.1

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