1. Introduction and motivation
High-resolution near-surface moisture is crucial to pursue knowledge of convective and boundary layer processes (Weckwerth et al. 1999; Weckwerth 2000; Sherwood et al. 2010). From numerical model simulations and data analysis, convection initiation and quantitative precipitation forecasting are shown to be sensitive to accurate measurements of moisture and temperature variability at the surface and in the boundary layer (e.g., Zawadzki et al. 1981; Crook 1996; Weckwerth et al. 1999). However, moisture observations at high temporal and spatial resolutions in the lower boundary layer are not readily available. The lack of information on moisture is one of the main limitations of mesoscale short-term forecasting (Emanuel et al. 1995; Dabberdt and Schlatter 1996; Fabry and Sun 2010; Hanley et al. 2011).



Comparisons between refractivity measured by the radar and other instruments in the boundary layer show high correlations in time and space (Fabry et al. 1997; Weckwerth et al. 2005; Bodine et al. 2011). Since high-temporal- (about 5–10 min) and high-spatial- (4 km by 4 km in the horizontal after smoothing) resolution refractivity retrievals have been obtained, many studies have demonstrated the potential utility of refractivity maps for studying near-surface moisture variation associated with a variety of weather phenomena, such as convection initiation, convection evolution, and characteristics of the boundary layer (Weckwerth et al. 2005; Fabry 2006; Buban et al. 2007; Chen et al. 2007; Roberts et al. 2008; Koch et al. 2008; Besson et al. 2012; Nicol et al. 2014). Refractivity maps not only provide small-scale moisture variability particularly in those areas without a dense mesonet but also show boundaries prior to the fine line of traditional reflectivity (Weckwerth et al. 2005; Heinselman et al. 2009; Wakimoto and Murphey 2010; Bodine et al. 2011). However, the coverage range of refractivity data is about 40–60 km, dictated by the topography and radio wave propagation. The range up to which refractivity data can be collected is limited; to go beyond, a radar network is needed. Thus, a networked technique has been developed for merging multiple X-band radars to extend the coverage of refractivity observations (Hao et al. 2006; Fritz and Chandrasekar 2009).
Montmerle et al. (2002) and Sun (2005) assimilated radar refractivity information to adjust the quantity and distribution of low-level moisture. The newly added information not only modified the low-level humidity field but also changed the spatial variability of moisture, which enhanced the intensity of the storm, leading to better quantitative precipitation forecasting. As a result, the research community has been preparing to assimilate the composite refractivity data from operational radar networks to numerical models in order to improve short-term forecasting skill (Besson et al. 2012; Caumont et al. 2013; Gasperoni et al. 2013; Nicol et al. 2013; Nicol and Illingworth 2013; Nicol et al. 2014).
For such quantitative applications, the accuracy of the refractivity retrieval is important and thus it is critical to gain more knowledge about the biases and the representativeness of the retrieval. Although the quality of the retrieval has been discussed from different aspects and improved in the last decade (Fabry 2004; Park and Fabry 2010; Besson et al. 2012; Parent du Chatelet et al. 2012; Caumont et al. 2013; Nicol et al. 2013; Nicol and Illingworth 2013), the unsolved problem associated with the vertical gradient of refractivity (
The goal of this research is to rethink refractivity retrieval to obtain a more accurate near-surface 2D horizontal refractivity map at a given representative height and additional information on
2. Phase difference and refractivity
a. The basis of radar refractivity retrieval

















b. Revisiting the assumptions and unsolved problems
The accuracy of retrieved refractivity critically depends on the quality of the phase differences of reliable ground targets. Phase differences caused by reasons other than the real atmospheric refractivity variations lead to noisiness in
The simplistic assumptions that were originally made by Fabry et al. (1997) to obtain a 2D refractivity field are as follows: 1) the heights of selected targets and the radar antenna height are identical (
As a result, since targets are at different heights under varying
c. Reinterpretation of the measured 

































Additional contributions to target ranges caused by target heights
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1

Additional contributions to target ranges caused by target heights
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1
Additional contributions to target ranges caused by target heights
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1






The average change of the horizontal refractivity at the radar height (
The temporal variation of the vertical refractivity profile is a source of bias for

(a) Average refractivity bias
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1

(a) Average refractivity bias
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1
(a) Average refractivity bias
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1
d. Noisy 
and local N biases









1) N at a given height above terrain
To quantitatively interpret and apply the refractivity retrieval, one generally wants to estimate the refractivity field at a given height above the terrain. The temporal change of refractivity between targets combines the 2D refractivity change at the radar height and the change of the vertical refractivity difference between the radar and the average height of targets, which are terms (i) and (ii) in (9). However, there are some more residual terms of
2) Local N bias due to target height
The measured

(a) Local refractivity bias due to the effect of the height difference between a pair of neighboring ground targets (
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1

(a) Local refractivity bias due to the effect of the height difference between a pair of neighboring ground targets (
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1
(a) Local refractivity bias due to the effect of the height difference between a pair of neighboring ground targets (
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1
3) Local N bias due to the propagation effect
The evolving propagation condition (
4) Consequences
The biases discussed above show how the data quality of retrieved refractivity is strongly affected by the diurnal evolving
In addition, the result of N estimation is very sensitive to the size of the smoothing window or of the
Previous work uses smoothing to reduce the noisy
3. Extracting 
information from returned power

a. Concept of a pointlike target










(a) Power pattern of a selected ground target as a function of antenna elevation. The target is located at the 240° azimuth and the 228th gate. Black dots show the reflectivity measured at multiple antenna elevations. These measurements are fitted with a Gaussian function of the width of the antenna beam (red line with circles). The noisy reflectivity near
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1

(a) Power pattern of a selected ground target as a function of antenna elevation. The target is located at the 240° azimuth and the 228th gate. Black dots show the reflectivity measured at multiple antenna elevations. These measurements are fitted with a Gaussian function of the width of the antenna beam (red line with circles). The noisy reflectivity near
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1
(a) Power pattern of a selected ground target as a function of antenna elevation. The target is located at the 240° azimuth and the 228th gate. Black dots show the reflectivity measured at multiple antenna elevations. These measurements are fitted with a Gaussian function of the width of the antenna beam (red line with circles). The noisy reflectivity near
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1
A point target can be identified by fitting the received powers at successive antenna elevations
A simpler method is further proposed to effectively investigate the pointlike property of targets based on the parabolic shape of
b. Using echo power at multiple elevations







Figure 5a shows the varying

(a) Representative elevation
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1

(a) Representative elevation
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1
(a) Representative elevation
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1
1) Estimating 















2) Estimating 

If
3) Normalized 
from 




























The normalized
4. Validation of 
retrievals

a. Data
The new method of

Map of height difference (m) between the terrain and the S-Pol radar (located in the center of the range rings). The gray lines show the azimuth angles at 30° intervals relative to the S-Pol radar, and the rings are in 10-km range intervals away from the radar. The BAO tower is shown as a red dot at 229.5° in azimuth and 12.56 km away from the radar. The yellow dots are the selected ground targets for
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1

Map of height difference (m) between the terrain and the S-Pol radar (located in the center of the range rings). The gray lines show the azimuth angles at 30° intervals relative to the S-Pol radar, and the rings are in 10-km range intervals away from the radar. The BAO tower is shown as a red dot at 229.5° in azimuth and 12.56 km away from the radar. The yellow dots are the selected ground targets for
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1
Map of height difference (m) between the terrain and the S-Pol radar (located in the center of the range rings). The gray lines show the azimuth angles at 30° intervals relative to the S-Pol radar, and the rings are in 10-km range intervals away from the radar. The BAO tower is shown as a red dot at 229.5° in azimuth and 12.56 km away from the radar. The yellow dots are the selected ground targets for
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1
Ground targets are first distinguished from weather and other signals using the following criteria: The average returned power at 0.3° and 0.6° elevations during the first experiment are higher than 25 dBZ, and the standard deviation of the power at each elevation over the 4 h of the first data experiment is less than 1.5 dB to ensure the stability of power returns; the average clutter phase alignment (CPA; Hubbert et al. 2009) is higher than 0.85 and its standard deviation is smaller than 0.03. High CPA implies that phase and power are consistent within the resolved volume. In addition, the pointlike nature of the target is checked by fitting a line through the first-order derivative of
The BAO tower (NOAA 2015) collects near-surface atmospheric basic variables: temperature, relative humidity, and wind every minute at 10-, 100-, and 300-m height above the ground. Only surface pressure is measured, and the pressure at other elevations is derived from the hydrostatic equation. The refractivity value at each level is calculated based on (1). The vertical profile of refractivity between different heights is obtained as an in situ observation for comparison.
b. 
estimation from selected targets

An example of a selected pointlike target illustrates how to use echo powers to estimate

Returned power variation of the selected target of Fig. 4 for 3 days. (a) Reflectivity observed at multiple radar elevations under a variety of conditions (gray dots). The colored dots highlight two specific
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1

Returned power variation of the selected target of Fig. 4 for 3 days. (a) Reflectivity observed at multiple radar elevations under a variety of conditions (gray dots). The colored dots highlight two specific
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1
Returned power variation of the selected target of Fig. 4 for 3 days. (a) Reflectivity observed at multiple radar elevations under a variety of conditions (gray dots). The colored dots highlight two specific
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1

Illustrations of how
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1

Illustrations of how
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1
Illustrations of how
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1
An ensemble of ground targets is used to estimate an average

(a) Time series of
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1

(a) Time series of
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1
(a) Time series of
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1
c. Radar–tower comparison of 

The estimated

Correlation coefficients between the time series of
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1

Correlation coefficients between the time series of
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1
Correlation coefficients between the time series of
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0224.1
The discrepancy of
In addition, differences in measurement representativeness might explain the discrepancies in
Finally, the power of a given pixel is not only affected by the beam propagation condition (
In summary, power measurements at successive low elevations can be used to qualitatively describe the diurnal
5. Concluding remarks
Variable target heights and changing
Using a theoretical reanalysis of the equation of the returned phase of a target, the representativeness of the measured phase and of the retrieved refractivity are clarified, and the systematic refractivity biases are quantified and shown to be related to the effect of
A practical method to estimate
Using this new theoretical basis, the magnitude of systematic biases in refractivity retrievals can be reduced by including the effects of terrain and target height. To make this possible, a new step-by-step processing to retrieve N based on these results should be as follows: 1) determine
Acknowledgments
The authors thank Dr. Mike Dixon for helping to collect special scans with the NCAR S-Pol radar. We also used in situ BAO observation provided by the NOAA/OAR/ESRL PSD, and we thank NOAA for its generosity. We also thank Dr. Isztar Zawadzki for the many fruitful discussions and Mr. Jonathan Vogel for assistance with the English grammar. This work was made possible thanks to the support from the Natural Sciences and Engineering Research Council of Canada (NSERC).
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