1. Introduction

Ground-based meteorological radars are being extensively used to obtain valuable information on the spatial distribution and time evolution of a wide range of atmospheric parameters, in particular hydrometeors and cloud fields (amounts, particle size distribution, and shape) as well as wind. Electromagnetic pulses are emitted with a tilt about the horizontal plane that typically varies between 0.5° and 90° (vertically pointing), depending on the desired detection range and type of application. The propagation of electromagnetic radiation through the atmosphere at typical wavelengths used in radar meteorology (between 1 mm and 10 cm) is directly related to the spatial variations of refractivity, which is itself mainly a function of air temperature, pressure, and humidity. In particular, vertical gradients of refractivity and thus the vertical stratification of the atmosphere determine the path followed by a radar beam. In certain meteorological conditions described below, low-tilt beams emitted from ground-based radar can become trapped and can even be deflected toward the surface, leading to spurious backscattered signals and hence erroneous interpretation (e.g., false precipitation echo). This phenomenon is referred to as anomalous propagation (AP), and the layer inside which the beam bends downward is called the duct.

The use of ground-based radar data is bound to increase in the future not only for model validation but also for the production of precipitation analyses (e.g., Mahfouf et al. 2007) as well as in data assimilation. In the latter context, it is hoped that radar data can improve the quality of initial conditions for numerical weather prediction (NWP) models through nudging techniques (e.g., Macpherson 2001), diabatic initialization (e.g., Ducrocq et al. 2002), or more elaborate data assimilation procedures such as four-dimensional variational data assimilation (4DVAR; e.g., Lopez and Bauer 2007) or ensemble Kalman filter (Tong and Xue 2005). With model horizontal resolution reaching 10 km or better, the excellent sampling in both space and time of radar observations makes them particularly appealing to constrain initial conditions for mesoscale models (e.g., Sun 2005). However, radar measurements can only be profitable if data affected by nonmeteorological scatterers (including AP) can be screened out prior to the desired application.

From a climatological standpoint, it is important to assess how often a given region of the globe is likely to experience conditions favorable to AP. Despite the limitations associated with the use of model analyses and limited vertical and horizontal resolutions, such knowledge could be used for guidance by radar network developers for the siting of new instruments. It could also help to identify regions in which model verification, precipitation analysis, and data assimilation based on radar data are likely to be unreliable because of frequent AP situations. Beyond weather radar meteorology, the results of such study might also be useful in other domains such as telecommunications. In the literature, global maps of vertical refractivity gradients for 100 and 1000 m above ground level (AGL) were created by Bean and Dutton (1968) for guidance of communication systems design. Radiosonde data were used to produce the Historical Electromagnetic Propagation Condition Database by Patterson (1987), which was later included in the Advance Refractive Engineering Prediction System (AREPS) of the U.S. Naval Laboratory (Patterson 1998). In a similar way, the International Telecommunication Union (ITU) periodically publishes global maps of propagation conditions, some of which were based on European Centre for Medium-Range Weather Forecasts (ECMWF) analyses (e.g., ITU 2003). In a more recent paper, von Engeln and Teixeira (2004) proposed a 6-yr global climatology of ducting events on a 160-km horizontal grid from the prospect of radio-occultation measurements from the spaceborne global positioning system (GPS). In their study, only the first-highest duct when looking downward was considered and the coarse resolution used implied that only horizontally widespread ducts were included in the statistics. The goal of the work presented here is to introduce a 5-yr global climatology of AP events from the viewpoint of ground-based radars and derived from ECMWF operational analyses at roughly 40-km horizontal resolution, which should provide information on a much finer scale over individual regions such as Europe and the United States.

Section 2 defines the various refraction regimes and explains how ducts can be simply identified from temperature, humidity, and pressure profiles. The ECMWF data used in this study are briefly described in section 3, and some remarks about the computation of refractivity vertical gradients are expressed in section 4. The main features of the global climatology of AP conditions are presented in section 5, which also includes a more detailed analysis of ducting occurrences over Europe and the United States, in which places precipitation radar networks are already well developed. Then, the sensitivity of the climatological statistics to radar tower height is discussed in section 6. A summary and conclusions are given in section 7.

2. Definition of refraction regimes

The propagation of radar beams through the atmosphere is governed by Snell’s law, that is, by the spatial variations of the index of refraction n or, equivalent, those of refractivity, N = (n − 1) × 10⁶ (n is close to unity for the air). Refractivity N can be computed from temperature T, partial pressure of water vapor e, and atmospheric pressure p (e.g., Steiner and Smith 2002) using

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where T is in kelvins and e and p are in pascals. Typical values of N range between 140 and 450. Because vertical variations of temperature, moisture, and pressure dominate over horizontal ones, the propagation of radar beams mainly depends on the vertical gradient of refractivity. The effect of the earth’s curvature (radius R) at an altitude z can be accounted for by considering a modified refractivity M (e.g., Turton et al. 1988), defined as

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Ducting (or radar beam trapping) usually occurs for beams with small tilt angles (<1°) above the local horizon when ∂M/∂z < 0 or, equivalent, when ∂N/∂z ≤ −10⁶/R ≈ −0.157 m⁻¹, assuming R can be approximated by the earth’s radius throughout the troposphere. Based on Steiner and Smith (2002), for instance, four different propagation regimes can be defined:

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From Eq. (1), one can deduce that conditions favorable to the superrefractive and ducting regimes must involve layers exhibiting either an increase of temperature or a sharp decrease of moisture with height. As ∂M/∂z becomes negative, the electromagnetic beam gets deflected toward the surface and spurious ground echoes (or ground clutter) are returned to the radar. Figure 1 gives an illustration of the different propagation regimes. One should note that the trapping layer (TL) is defined as the layer inside which ∂M/∂z remains negative. This layer is always embedded inside the duct, which extends, by definition, from the TL top down to either the surface or the level at which M becomes equal to the value at TL top. Depending on the shape of the modified refractivity profile, three types of ducts can be identified as sketched in Figs. 2a–c: surface duct, S-shaped surface duct, and elevated duct, respectively.

The most common meteorological situations involving superrefractivity or, worse, ducting are 1) temperature inversion due to nocturnal radiative cooling inside the planetary boundary layer (PBL) over land, 2) temperature inversion over a moist PBL due to anticyclonic subsidence in the trade wind region, 3) dry-air advection over sea or wet land, 4) low-level advection of moist and cool air from the sea, and 5) outflow of low-level moist and cold air from thunderstorms. In practice, the ducts that affect ground-based radars are always confined to the lowest 3000 m of the troposphere. Note that AP ground clutter can also be observed in the superrefractive regime in the absence of ducting when the radar beam is refracted downward and impinges on surrounding orography, as shown in Fig. 1 (path labeled “SUPR”).

To efficiently deviate a radar beam toward the surface, a TL must have a minimum depth D_min that is expectedly a function of the magnitude of the vertical refractivity gradient but also of the radar signal wavelength λ. The simplified formula proposed by Turton et al. (1988) leads to the expression

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where λ and D_min are in meters and C = 400 (263) for surface (elevated) ducts. For a wavelength of 10 cm, which corresponds to the upper end of the wavelength range used in radar meteorology, the minimum depth required for a ducting layer to completely deviate the radar beam is approximately equal to 22 m (16 m) for a surface (elevated) duct. In more general terms, Eq. (3) shows that D_min becomes smaller when wavelength is reduced as well as when ∂N/∂z becomes more negative (increased duct intensity). However, one should also keep in mind that AP can already lead to nonnegligible ground clutter contamination when only a few percent of the signal power (i.e., sidelobes only) are refracted downward by a shallow or weak duct. The efficiency of duct detection from model outputs will be discussed in sections 3 and 4.

3. ECMWF operational analyses

The climatological statistics are computed from ECMWF operational analyses for the period between 1 December 2000 and 30 November 2005. ECMWF operational analyses are produced using a 4DVAR data assimilation system, as described in Courtier et al. (1994). All analysis fields are retrieved on the original T511 reduced Gaussian grid, which corresponds to a quasi-uniform horizontal resolution of about 40 km over the globe. Analyses valid at 0000, 0600, 1200, and 1800 UTC are used to somewhat crudely sample the diurnal cycle. Temperature and moisture profiles are extracted on the operational 60 vertical levels (hybrid η coordinate). As an illustration of the model vertical resolution, the thickness of each model layer up to a 10-km height is displayed in Fig. 3. Note that the actual height of each model level may vary slightly among model grid points because of horizontal variations of virtual temperature and surface pressure. The current operational T799 L91 configuration could not be used in this study because these enhanced resolutions were implemented 2 yr ago, a period of time deemed too short for building a reasonable climatology. However, von Engeln et al. (2003) showed that the T511 L60 version of the ECMWF model is capable of resolving and accurately representing the vertical thermodynamical structure of the lower troposphere, including PBL height and inversion characteristics.

It is obvious, though, that ducts that extend over distances shorter than the grid spacing of 40 km will not be properly resolved in the model analyses and that the frequencies of superrefraction occurrence reported in this study will therefore tend to underestimate real values in such situations. This limitation will particularly apply to ducts that are associated with thunderstorm outflow regions that are typically only a few kilometers across. It should also be stressed that the statistics presented here depict the atmosphere as seen in the model analyses, which may sometimes depart from the real atmosphere. Nonetheless, 40-km model analyses over 5 yr are likely to provide the best three-dimensional global representation of the atmosphere available as yet.

4. Refractivity gradient calculations and statistics

Refractivity is computed from Eq. (1), and the minimum vertical gradient is estimated over all model layers starting from the lowest model level (about 10-m height) up to a height of 3000 m. This maximum height limitation is imposed as a result of the degradation of model vertical resolution with height, as illustrated in Fig. 3, and because most ducts affecting ground-based radars are found below this level. It is worth noting that such restriction was also applied in von Engeln and Teixeira (2004). On the other hand, the gradient computations are started at the lowest model level and not directly from the surface to account for the fact that most meteorological radars are installed at least 10 m above the ground (towers). In other respects, because Eq. (3) indicated a D_min of about 20 m, Fig. 3 indicates that any TL detected in model analyses is deep enough to fully refract the radar beam downward. At the same time, Fig. 3 suggests that the minimum depth of ducts that can be identified from the analysis fields increases with height. In particular, this means that low-level ducts that are a few tens of meters thick and elevated ducts thinner than a couple of hundred meters are probably underrepresented in the statistics. One should therefore expect an underestimation of the overall occurrence of ducts in the climatology shown in this paper. The magnitude of this underestimation is likely to depend on geographical location and weather regime.

In this study, statistics are calculated not only for ducting but also for superrefractive situations. Superrefraction (ducting) is assumed to occur whenever the minimum value of ∂N/∂z is lower than −0.0787 m⁻¹ (−0.157 m⁻¹). Note that in the following, all occurrences of ducting events are also counted as superrefractive cases. For each model grid point, statistics will be presented in terms of the frequency of occurrence of superrefractive and ducting events but also in terms of the bottom height h_b of the first TL encountered from the surface.

5. Results

6. Influence of radar tower height on ducting occurrence statistics

As mentioned in section 4, the climatology thus far described (hereinafter denoted CLIM10) was obtained by starting the calculations of the refractivity vertical gradient from the lowest model level (z_min) to account for the fact that most meteorological radars are installed at least 10 m AGL. In fact, the height of the towers dedicated to these instruments can reach even greater heights. For instance, among the 154 NEXRAD radars in the United States, one-third or so are installed higher than 30 m AGL and more than 40% are at a height around 20 m AGL. Because the steepest refractivity gradients over land are often found within the lowest 50 m of the troposphere, an alternative global climatology of superrefractive conditions was produced by starting the computations from the second-lowest model level (z_min ≈ 35 m). This alternative climatology will be referred to as CLIM35 and is obviously expected to exhibit less frequent superrefractive and ducting events than CLIM10 whenever strong refractivity gradients are present below z = 35 m.

Over sea, the reduction in these frequencies from CLIM10 to CLIM35 remains small (less than a few percent on average; not shown) because superrefraction is often associated with inversion layers that lie several hundred meters above the ocean surface (Fig. 6). Over land, in contrast, frequency differences are more pronounced. As an illustration, Tables 1 and 2 compare spatial averages of ducting frequencies for each season and time of the day over Europe and the United States (land only) between CLIM10 and CLIM35.

For both areas, the largest decrease (more than 10%) in frequencies occurs in the summertime and in the evening (1800 UTC in Europe; 0000 UTC in the United States). This can be explained by the fact that temperature inversions are then confined to a very thin layer close to the surface, which makes the computations of refractivity gradients very sensitive to the value assigned to z_min. These shallow inversions associated with high occurrences of ducting (as seen in Figs. 9d, 17a) correspond to the very low TL base heights found in Fig. 13d over Europe and in Fig. 21a over the eastern half of the United States. On the other hand, ducting frequencies are less sensitive to z_min (2%–5% drop only) during summer nights because, by then, inversion layers in those same regions have risen above 35 m, as indicated in Figs. 13a and 21b. Similar remarks apply to spring and autumn, although with slightly weaker reductions. In the winter season as well as during local daytime, the impact of z_min is small in absolute terms.

The summertime reduction of ducting occurrence when z_min is changed from 10 to 35 m can reach very high values locally, as illustrated in Fig. 23. These two maps, valid for late afternoon over Europe and the United States in the summertime, suggest that in the Po Valley, the northern Balkans, and the Mississipi Valley, ducting probability, as computed from the model analyses, drops by approximately 40% (from 60% to 20%).

However, it is important to emphasize that increasing the installation height of ground-based radars is not the panacea to improve the quality of radar images. Indeed, installing radars on tall towers implies that the clutter under normal conditions can be increased dramatically as a result of undesirable contributions from the sidelobes emitted by the antenna (e.g., Smith 1972; Doviak and Zrnic 1985). Despite its relative weakness in comparison with that of the main lobe echo (down 25–30 dBz typically), the contribution from sidelobes can become nonnegligible when the returned main echo is strong, in heavy-rain situations for instance. Furthermore, different portions of the sidelobes can illuminate ground targets and thereby substantially increase the intensity of the ground clutter echo, which can degrade significantly the quality and usefulness of the radar observations. The choice of the proper radar height should eventually be a compromise between all of these competing benefits and drawbacks and should be adapted to the local terrain configuration.

7. Conclusions

A 5-yr climatology of the frequency of atmospheric superrefraction and ducting occurrences and of trapping-layer base height has been constructed by deriving refractivity gradients from ECMWF analyses at a 40-km resolution and using 60 levels in the vertical direction. This horizontal grid spacing implies that small-scale ducts are likely to remain unresolved in the model analyses and therefore real occurrences of superrefraction might be underestimated in such situations (e.g., thunderstorm outflow). It would be useful to repeat this work as soon as the volume of T799 L91 analyses available becomes large enough, because enhanced vertical and horizontal resolution would likely lead to improved detection of ducts.

The purpose of this work was to provide a relatively high resolution global database of AP that can be used in various applications of weather radar observations in NWP and data assimilation, but maybe also in the field of telecommunications. In addition, the statistics presented here might as well be helpful to the GPS satellite radio-occultation community because the occurrence of ducting is known to degrade the accuracy of refractivity retrievals obtained from bending-angle measurements.

It is thought that within a few years the assimilation of precipitation radar observations could become feasible in ECMWF’s 4DVAR system, with some potential benefit for the quality of the analyses and subsequent forecasts. However, a prerequisite is that any radar data point that is contaminated by AP should be screened out prior to assimilation. Indeed, observations that supposedly have been quality controlled can still suffer from AP contamination, especially when those data are available in quasi–real time, with limited automatic sanity checks applied to them. Despite the aforementioned limitations linked to the use of model fields, the assimilation of radar data might benefit from the application to the model background state of a screening procedure based on the type of computations described in section 2, especially in situations favorable to large-scale and intense ducts. The efficiency of such procedure should improve in the coming years as the model horizontal and vertical resolutions are increased.