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Journal topic
**Atmospheric Measurement Techniques**
An interactive open-access journal of the European Geosciences Union

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- About
- Editorial board
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- About
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- Abstract
- Copyright statement
- Introduction
- Measurement basis and method
- Boundary-layer measurements and analysis
- Conclusions
- Data availability
- Appendix A: Error in detected power
- Appendix B: Monte Carlo Analysis
- Appendix C: Retrieval dependence on assumed pressure and temperature values
- Competing interests
- Acknowledgements
- References

**Research article**
06 Dec 2018

**Research article** | 06 Dec 2018

Boundary-layer water vapor profiling using differential absorption radar

- Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, USA

- Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, USA

**Correspondence**: Richard J. Roy (richard.j.roy@jpl.nasa.gov)

**Correspondence**: Richard J. Roy (richard.j.roy@jpl.nasa.gov)

Abstract

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Remote sensing of water vapor in the presence of clouds and precipitation
constitutes an important observational gap in the global observing system. We
present ground-based measurements using a new radar instrument operating near
the 183 GHz H_{2}O line for profiling water vapor inside of
planetary-boundary-layer clouds, and develop an error model and inversion
algorithm for the profile retrieval. The measurement technique exploits the
strong frequency dependence of the radar beam attenuation, or differential
absorption, on the low-frequency flank of the water line in conjunction with
the radar's ranging capability to acquire range-resolved humidity
information. By comparing the measured differential absorption coefficient
with a millimeter-wave propagation model, we retrieve humidity profiles with
200 m resolution and typical statistical uncertainty of 0.6 g m^{−3} out
to around 2 km. This value for humidity uncertainty corresponds to
measurements in the high-SNR (signal-to-noise ratio) limit, and is specific to the frequency band
used. The measured spectral variation of the differential absorption
coefficient shows good agreement with the model, supporting both the
measurement method assumptions and the measurement error model. By performing
the retrieval analysis on statistically independent data sets corresponding
to the same observed scene, we demonstrate the reproducibility of the
measurement. An important trade-off inherent to the measurement method
between retrieved humidity precision and profile resolution is discussed.

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How to cite.

Roy, R. J., Lebsock, M., Millán, L., Dengler, R., Rodriguez Monje, R., Siles, J. V., and Cooper, K. B.: Boundary-layer water vapor profiling using differential absorption radar, Atmos. Meas. Tech., 11, 6511–6523, https://doi.org/10.5194/amt-11-6511-2018, 2018.

Copyright statement

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Copyright statement.

^{©} 2018 California Institute of Technology. Government sponsorship acknowledged.

1 Introduction

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In this work, we discuss the implementation of differential absorption radar
(DAR) for measuring humidity profiles inside of boundary-layer clouds
(Lebsock et al., 2015; Millán et al., 2016). The DAR method, which is the microwave analog of the
mature differential absorption lidar (DIAL) method (Browell et al., 1979), combines
the range-resolving capabilities of radar with the strong frequency
dependence of atmospheric attenuation near a molecular rotational absorption
line to retrieve density profiles of the absorbing gas along the line of
sight. Recently, there was a demonstration of microwave integrated path
differential absorption in airborne measurements of sea surface air pressure
without range resolution (Lawrence et al., 2011), utilizing the 60 GHz O_{2}
line to measure the total oxygen column. More recently, our group
demonstrated a ground-based DAR for humidity sounding operating between 183
and 193 GHz (Cooper et al., 2018), with primary sensitivity to upper tropospheric
water vapor due to significant attenuation in the lower troposphere at these
frequencies. That work included a comparison of differential absorption
measurements with a millimeter-wave propagation model showing good agreement,
and left the topics of error analysis and profile inversion for future
investigation. While the 183 to 193 GHz band is attractive for DAR
measurements because of the large differential absorption values achievable,
transmission at frequencies between 174.8 and 191.8 GHz is prohibited due to
reservation for passive-only remote sensing (NTIA, 2015). On the
other hand, the 167 to 174.8 GHz band offers fewer transmission
restrictions, and features lower absolute absorption, thus enabling
penetration into the boundary layer from an airborne or spaceborne platform.
Of course, the smaller absolute absorption is accompanied by decreased
differential absorption, making the profiling capabilities of this radar
coarser than the 183 to 193 GHz DAR. Furthermore, the surface returns in
both cloudy and clear-sky areas make a DAR measurement of the total
water column possible.

The DAR approach has two unique aspects that complement existing methods for remotely sensing water vapor. First, because of its ranging capabilities it has precise height registration, unlike passive sounding whereby weighting functions can encompass broad swaths of the atmosphere. Second, in contrast with other methods the DAR signal increases with increasing cloud water content and precipitation, with the obvious caveat that the radar signal-to-noise ratio (SNR) will decrease from attenuation as the beam penetrates into the volume. The DAR therefore nicely complements the infrared and microwave sounding techniques, as well as differential absorption and Raman lidar techniques that are commonly used to remotely sense water vapor from the ground (Spuler et al., 2015; Whiteman et al., 1992; Wulfmeyer and Bösenberg, 1998), with a notable airborne DIAL system being the Lidar Atmospheric Sensing Experiment (LASE) (Browell et al., 1998). Importantly, millimeter-wave transparency in clouds allows for airborne or spaceborne measurements of lower tropospheric humidity in cloudy scenes, while DIAL systems typically cannot measure inside boundary-layer clouds due to high optical thickness.

In addition to primary applications in profiling water vapor within clouds, the instrument architecture discussed here represents an important application of recent advances in solid-state G-band technology to meteorological radar. Indeed, there has been lingering interest within the atmospheric remote sensing community for decades in utilizing G-band radar for cloud and precipitation studies, with earlier attempts hampered by limited sensitivity due to available technology (Battaglia et al., 2014). The addition of G-band reflectivity measurements to multi-frequency radar systems, for example a dual-frequency W- and Ka-band system, could provide significantly more information than additional measurements at a lower frequency because the scattering properties at G band for typical cloud particle sizes are not of Rayleigh character.

Here we present ground-based measurements using a 167 to 174.8 GHz DAR, provide in-depth measurement error analysis with emphasis on the role of background noise power, and develop a retrieval algorithm based on performing least squares fits of a spectroscopic model to the data. The retrieved profiles constitute the first active remote sensing measurements of water vapor profiles inside of clouds, and open up possibilities for a variety of scientific studies, including investigation of in-cloud humidity heterogeneity and the coupled relationship between boundary-layer clouds and thermodynamic profiles.

2 Measurement basis and method

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The DAR technique (Cooper et al., 2018; Lawrence et al., 2011; Millán et al., 2014) utilizes range-resolved radar echoes at multiple carrier frequencies in the vicinity of a gaseous absorption line to probe the frequency-dependent optical depth between two points along the radar line-of-sight. The radar echoes, or returns, may originate from cloud hydrometeors or, in the case of an airborne system, from the Earth's surface as well, enabling total column optical depth measurements. For closely spaced transmission frequencies near the absorption line center, the hydrometeor scattering properties vary little, while the gaseous absorption exhibits strong frequency dependence. By comparing with a known propagation model, these measurements can be employed to retrieve range-resolved density profiles of the absorbing molecule. Furthermore, because of the differential nature of the measurement, one does not require absolute calibration of the radar receiver in order to obtain absolute density values for the absorbing molecule. In the case of a calibrated receiver, both range-resolved density profiles of the absorbing molecule and microphysical properties of the reflecting medium can be retrieved.

Assuming negligible multiple scattering, the radar echo power received from a
collection of scatterers filling the beam at a distance *r* is

$$\begin{array}{}\text{(1)}& {\displaystyle}{P}_{\mathrm{e}}(r,f)=C\left(f\right)Z(r,f){r}^{-\mathrm{2}}{e}^{-\mathrm{2}\mathit{\tau}(r,f)},\end{array}$$

where *C*(*f*) includes the frequency dependence of the radar hardware (e.g.,
transmit power and gain), *Z*(*r*,*f*) is the (unattenuated) reflectivity, and
*τ*(*r*,*f*) is the one-way optical depth including contributions from gaseous
and particulate extinction. Taking the ratio of powers for two different
ranges *r*_{1} and ${r}_{\mathrm{2}}={r}_{\mathrm{1}}+R$ and assuming frequency independence of the
reflectivity and particulate extinction, we find

$$\begin{array}{}\text{(2)}& {\displaystyle \frac{{P}_{\mathrm{e}}({r}_{\mathrm{2}},f)}{{P}_{\mathrm{e}}({r}_{\mathrm{1}},f)}}={\displaystyle \frac{Z\left({r}_{\mathrm{2}}\right)}{Z\left({r}_{\mathrm{1}}\right)}}{\left({\displaystyle \frac{{r}_{\mathrm{1}}}{{r}_{\mathrm{2}}}}\right)}^{\mathrm{2}}{e}^{-\mathrm{2}\mathit{\beta}({r}_{\mathrm{1}},{r}_{\mathrm{2}},f)R},\end{array}$$

where

$$\begin{array}{ll}{\displaystyle}\mathit{\beta}({r}_{\mathrm{1}},{r}_{\mathrm{2}},f)& {\displaystyle}={\displaystyle \frac{\mathit{\tau}({r}_{\mathrm{2}},f)-\mathit{\tau}({r}_{\mathrm{1}},f)}{R}}\\ \text{(3)}& {\displaystyle}& {\displaystyle}={\displaystyle \frac{\mathrm{1}}{R}}\underset{{r}_{\mathrm{1}}}{\overset{{r}_{\mathrm{2}}}{\int}}\left[\sum _{j}{\mathit{\rho}}_{j}\left(r\right){\mathit{\kappa}}_{j}(r,f)+{\mathit{\beta}}_{\text{part}}\left(r\right)\right]\mathrm{d}r\end{array}$$

is the average absorption coefficient between *r*_{1} and *r*_{2}, *ρ*_{j}(*r*) is
the density of the gas component with label *j*, *κ*_{j}(*r*,*f*) is the
corresponding mass extinction cross section, which varies with *r* due to
pressure and temperature, and *β*_{part}(*r*) is the particulate
extinction coefficient integrated over local drop size distributions (DSDs).

Restricting our analysis to millimeter-wave propagation near the 183 GHz
water vapor absorption line, the sum over gaseous absorption terms can be
replaced by ${\mathit{\rho}}_{\mathrm{v}}\left(r\right){\mathit{\kappa}}_{\mathrm{v}}(r,f)+{\mathit{\beta}}_{\text{gas,bg}}\left(r\right)$, where the
subscript v corresponds to water vapor and *β*_{gas,bg} is the
background gas absorption coefficient due to all other components, which is
assumed to be frequency-independent. Assuming that pressure and temperature
vary slowly compared to the length scale *R*, we can therefore write
Eq. (3) as

$$\begin{array}{ll}{\displaystyle}\mathit{\beta}({r}_{\mathrm{1}},{r}_{\mathrm{2}},f)& {\displaystyle}={\stackrel{\mathrm{\u203e}}{\mathit{\rho}}}_{\mathrm{v}}({r}_{\mathrm{1}},{r}_{\mathrm{2}}){\mathit{\kappa}}_{\mathrm{v}}\left(f\right)+{\stackrel{\mathrm{\u203e}}{\mathit{\beta}}}_{\text{gas,bg}}({r}_{\mathrm{1}},{r}_{\mathrm{2}})\\ \text{(4)}& {\displaystyle}& {\displaystyle}+{\stackrel{\mathrm{\u203e}}{\mathit{\beta}}}_{\text{part}}({r}_{\mathrm{1}},{r}_{\mathrm{2}}),\end{array}$$

where the overbar symbol implies taking the mean value between *r*_{1} and
*r*_{2}. Thus, we see that measuring the frequency-dependent contribution to
the optical depth between *r*_{1} and *r*_{2} reveals the average water vapor
density given the known absorption line shape *κ*_{v}(*f*).
Figure 1 shows the frequency dependence of the gaseous
absorption coefficient *ρ*_{v}*κ*_{v}(*f*)+*β*_{gas,bg} in the vicinity of the 183 GHz water vapor line for
*P*=1000 mbar, *T*=285 K, and *ρ*_{v}=10 g m^{−3}. For
this work, we utilize the millimeter-wave propagation model from the EOS
Microwave Limb Sounder (Read et al., 2004). The 167 to 174.8 GHz transmission
band is highlighted in green, as well as shown in the inset of
Fig. 1, revealing a differential absorption coefficient of
3 dB km^{−1} for 10 g m^{−3} of water vapor.

Important to the validity of this DAR method is the dominance of gaseous
differential absorption over particulate differential absorption, since we
assume that *β*_{part} is frequency-independent. To investigate this
for boundary-layer clouds, we perform Mie scattering calculations for liquid
spheres and integrate the scattering parameters over DSDs corresponding to
clouds and rain for a range of mean diameters. For the DSD, we use a modified
gamma distribution of the form

$$\begin{array}{}\text{(5)}& {\displaystyle}N\left(D\right)={\displaystyle \frac{{N}_{\mathrm{0}}}{\mathrm{\Gamma}\left(\mathit{\nu}\right)}}{\left({\displaystyle \frac{D}{{D}_{n}}}\right)}^{\mathit{\nu}-\mathrm{1}}{\displaystyle \frac{\mathrm{1}}{{D}_{n}}}{e}^{-D/{D}_{n}},\end{array}$$

where *N*_{0} is a normalization factor with units of particle number per
volume that fixes the total liquid water content ℒ, *ν* is the
shape parameter, which is set to 4 for clouds and 1 for rain, and *D*_{n} is
the characteristic diameter. For rain we enforce an additional constraint
that ${N}_{\mathrm{0}}={x}_{\mathrm{1}}{D}_{n}^{\mathrm{1}-{x}_{\mathrm{2}}}$, where *x*_{1}=26.2 m${}^{{x}_{\mathrm{2}}-\mathrm{4}}$ and *x*_{2}=1.57 have been determined in previous studies by comparing to observations
(Abel and Boutle, 2012). This allows the entire rain distribution to be determined by
the liquid water content. The rain rate is calculated from this distribution
by using the terminal velocity relation from Beard (1976).

The results are shown in Fig. 2, where we plot the
differential particulate extinction, $\mathrm{\Delta}{\mathit{\beta}}_{\text{part}}\left(f\right)={\mathit{\beta}}_{\text{part}}\left(f\right)-{\mathit{\beta}}_{\text{part}}\left({f}_{\mathrm{0}}\right)$, as a function of *D*_{n}. Here
*f*_{0}=167 GHz corresponds to the low-frequency end of the transmission
band. In Fig. 2a, the corresponding rain rate is
displayed on the upper horizontal axis. For the cloud species, the
normalization parameter *N*_{0} is not fixed by any additional constraint, and
is therefore determined at each *D*_{n} to fix ℒ, which is set here
to 500 mg m^{−3}. To find the differential particulate extinction for
other values of ℒ, one can linearly scale the values in
Fig. 2b. Clearly for precipitation scenarios, the
differential extinction from rain is more than 2 orders of magnitude smaller
than that from water vapor. For clouds in the limit of small
diameter, the differential particulate extinction asymptotes to the Rayleigh
value of $\mathrm{\Delta}{\mathit{\beta}}_{\text{part}}^{\text{Rayleigh}}=\mathrm{6}\mathit{\pi}\mathcal{L}\phantom{\rule{0.125em}{0ex}}\text{Im}\left({K}_{\mathrm{w}}\right)(f-{f}_{\mathrm{0}})/\left({\mathit{\rho}}_{\mathrm{w}}c\right)=\mathrm{0.2}$ dB km^{−1} for *f*=174.8 GHz, where *ρ*_{w} is the
density of liquid water, ${K}_{\mathrm{w}}=({m}_{\mathrm{w}}^{\mathrm{2}}-\mathrm{1})/({m}_{\mathrm{w}}^{\mathrm{2}}+\mathrm{2})$, *m*_{w} is the complex
refractive index of water, and *c* is the speed of light. For larger values
of *D*_{n}, the differential extinction is enhanced by a resonant feature
characteristic of Mie scattering. Thus, for thick clouds with ℒ
as large as 500 mg m^{−3}, especially those that contain drizzle drops
which tend to lie near this resonant size, there are important bias
considerations that warrant future study in order to establish the
application of DAR in these particular scenarios. Specifically, to mitigate
the potential biases stemming from scattering by hydrometeors, the
unattenuated reflectivity can be used to distinguish clouds from
precipitation, and the frequency-dependent scattering effects can be modeled
and incorporated in the retrieval.

Due to the lower transmit power as compared to conventional radar systems at
lower frequencies, the 170 GHz radar is operated in a frequency-modulated
continuous-wave (FMCW) mode, which can offer increased sensitivity relative
to a pulsed system with the same power because the transmitter is always on.
The basic principle of FMCW radar is outlined in Fig. 3. The
transmitted signal is frequency-modulated with a linear chirp waveform of
bandwidth Δ*F*_{chirp} and duration *T*_{chirp}. After
scattering off of a target at a distance *r* from the radar, the received
chirp is delayed in time by an amount 2*r*∕*c*, leading to a fixed frequency
offset of $\mathit{\delta}f=\mathrm{2}\mathrm{\Delta}{F}_{\text{chirp}}r/\left(c{T}_{\text{chirp}}\right)$ relative to
the transmitted frequency chirp. By downconverting the received signal using
the transmitted frequency *f*(*t*) shifted by 5 MHz for convenient
amplification and detection, the fixed frequency offset between transmitted
and received chirps is converted into a constant frequency signal in the
intermediate frequency (IF) stage. Signal processing techniques are then used
to convert the IF time-domain signal to a range-resolved power spectrum. In
the IF power spectrum, the zero-range point is located at 5 MHz and the echo
power from a range *R* is located at *f*_{IF}(*r*)=5 MHz ± *δ**f*(*r*), where the positive(negative) sign applies for
decreasing(increasing) frequency chirps.

Our system utilizes state-of-the-art millimeter-wave components designed at
the Jet Propulsion Laboratory (JPL) and builds on years of FMCW radar
development for security and planetary science applications
(Cooper et al., 2011, 2017). The architecture is similar to that presented in an
earlier work (Cooper et al., 2018) which demonstrated the DAR technique between 183
and 193 GHz, but modified to transmit in the 167 to 174.8 GHz band, in which
transmission is not prohibited by international regulations
(NTIA, 2015), to perform narrow-bandwidth frequency chirps, and
to provide a 5 MHz offset of the zero-range radar signal from zero frequency
within the IF band. The IF offset is helpful for future calibrated power
measurements because of various effects that inhibit accurate power
estimation near zero frequency. The radar has an average transmit power of
140 mW, is outfitted with a 6 cm primary aperture with corresponding gain
of 40 dB, and uses a frequency chirp of bandwidth Δ*F*_{chirp}=60 MHz and duration *T*_{chirp}=1 ms, resulting in a range
resolution of $\mathrm{\Delta}r=c/\mathrm{2}\mathrm{\Delta}{F}_{\text{chirp}}=\mathrm{2.5}$ m. In general, the
choice of radar range resolution involves a compromise between acquiring more
statistically independent samples within a given target volume to reduce
uncertainty for bright targets and having a longer integration time to
reduce noise power and thus increase the SNR for weak targets. The choice of
2.5 m allows us to downsample the range dimension by a factor of 11 to
realize our desired profile resolution of 27.5 m, with decreased uncertainty
for the bright clouds measured in this work. A summary of relevant radar
hardware parameters is given in Table 1.

To process the downconverted radar signal, we first sample it using an
analog-to-digital converter (ADC) with a sampling frequency of 20 MHz for
the 1 ms duration of the chirp. Then we apply a Hanning window in the time
domain before performing a fast Fourier transform (FFT) to obtain the
range-resolved power spectrum. Application of the Hanning window reduces
side lobes from bright targets as well as the large transmit–receive leakage
signal that is always present at zero range. For the radar parameters listed
above, the corresponding conversion factor from IF frequency to the target range
is $\mathit{\delta}f\left(r\right)/r=\mathrm{400}$ kHz km^{−1}.

The starting point for assessing the achievable precision in humidity using
DAR measurements is the statistical uncertainty of the radar power
measurements themselves. Until this point, we have ignored the role of
background noise power in the radar spectrum, which is an important factor in
any realistic receiver. In general, the noise power within a given radar
range bin *P*_{n} is proportional to the sum of the receiver noise
temperature and the antenna temperature, which itself is proportional to the
scene brightness temperature. By considering the simultaneous coherent
detection of noise (*P*_{n}) and radar echo (*P*_{e}) power,
one can show that the statistical uncertainty of the *detected* power,
${P}_{\mathrm{d}}={P}_{\mathrm{e}}+{P}_{\mathrm{n}}$, is given by (see
Appendix A)

$$\begin{array}{}\text{(6)}& {\displaystyle}{\mathit{\sigma}}_{\mathrm{d}}={\displaystyle \frac{\mathrm{1}}{\sqrt{{N}_{\mathrm{p}}}}}{\left({P}_{\mathrm{e}}^{\mathrm{2}}+\mathrm{2}{P}_{\mathrm{e}}{P}_{\mathrm{n}}+{P}_{\mathrm{n}}^{\mathrm{2}}\right)}^{\mathrm{1}/\mathrm{2}},\end{array}$$

where *N*_{p} is the number of radar pulses transmitted.

In order to accurately determine the frequency-dependent optical depth
between two range bins, it is critical to obtain a separate measurement of
the background noise power in the absence of radar echoes and subtract this
off of *P*_{d}. To see why this is, consider
Eq. (2) with the left-hand side replaced by
${P}_{\mathrm{d}}({r}_{\mathrm{2}},f)/{P}_{\mathrm{d}}({r}_{\mathrm{1}},f)$, which is equivalent to
interpreting the detected power as the true echo power, set *Z*(*r*_{2})=*Z*(*r*_{1})
for simplicity, and consider the limit *P*_{e}≪*P*_{n} (i.e.,
*P*_{d}→*P*_{n}). In this case we would find that
$\mathrm{exp}(-\mathrm{2}\mathit{\beta}({r}_{\mathrm{1}},{r}_{\mathrm{2}},f\left)R\right)\to \mathrm{1}$ regardless of the actual value of
*P*_{e}, and thus would incorrectly estimate a vanishing water vapor
density, when in fact it is the echo power which has vanished. Similarly, for
modest values of the $\text{SNR}\equiv {P}_{\mathrm{e}}/{P}_{\mathrm{n}}$, this would lead to a systematic underestimate of
the true humidity. Therefore, after subtracting the separate noise power
measurement from *P*_{d} we obtain a measurement of *P*_{e}
with total uncertainty ${\mathit{\sigma}}_{\mathrm{e}}=({\mathit{\sigma}}_{\mathrm{d}}^{\mathrm{2}}+{\mathit{\sigma}}_{\mathrm{n}}^{\mathrm{2}}{)}^{\mathrm{1}/\mathrm{2}}$,
where ${\mathit{\sigma}}_{\mathrm{n}}={P}_{\mathrm{n}}/\sqrt{{N}_{\mathrm{p}}}$ is the noise power
measurement uncertainty (see Eq. 6 with
*P*_{e}=0). The relative uncertainty in the measured echo power is
therefore

$$\begin{array}{}\text{(7)}& {\displaystyle \frac{{\mathit{\sigma}}_{\mathrm{e}}}{{P}_{\mathrm{e}}}}={\displaystyle \frac{\mathrm{1}}{\sqrt{{N}_{\mathrm{p}}}}}{\left(\mathrm{1}+{\displaystyle \frac{\mathrm{2}}{\text{SNR}}}+{\displaystyle \frac{\mathrm{2}}{{\text{SNR}}^{\mathrm{2}}}}\right)}^{\mathrm{1}/\mathrm{2}}.\end{array}$$

As will be discussed in Sect. 3, the range
dimension is purposefully oversampled in our measurements, allowing us to
decrease the statistical power uncertainty at a given range by averaging
*N*_{b} adjacent range bins. The resulting relative power uncertainty
is given by

$$\begin{array}{}\text{(8)}& {\displaystyle \frac{{\mathit{\sigma}}_{\mathrm{e}}}{{P}_{\mathrm{e}}}}={\displaystyle \frac{\mathit{\xi}\left({N}_{\mathrm{b}}\right)}{\sqrt{{N}_{\mathrm{p}}{N}_{\mathrm{b}}}}}{\left(\mathrm{1}+{\displaystyle \frac{\mathrm{2}}{\text{SNR}}}+{\displaystyle \frac{\mathrm{2}}{{\text{SNR}}^{\mathrm{2}}}}\right)}^{\mathrm{1}/\mathrm{2}},\end{array}$$

where *ξ*(*N*_{b})≥1 is a factor of order unity accounting for
covariances between adjacent range bins that arise due to applying a window
function to the time-domain radar signal before transforming to Fourier
space. For the Hanning window used in this work, this function is given by
$\mathit{\xi}\left({N}_{\mathrm{b}}\right)={\left(\mathrm{1}+\frac{{N}_{\mathrm{b}}-\mathrm{1}}{{N}_{\mathrm{b}}}\frac{\mathrm{8}}{\mathrm{9}}\right)}^{\mathrm{1}/\mathrm{2}}$.

Under the simplifying assumptions introduced in Sect. 2.1, and assuming that pressure and
temperature are known as a function of range, the inverse problem to retrieve
humidity can be solved directly. The implications of the latter assumption
are explored in Appendix C. To invert the radar
spectra, we consider a set of measured echo powers *P*_{e}(*r*_{i},*f*_{j})
for ranges $\mathit{\{}{r}_{\mathrm{1}},{r}_{\mathrm{2}},\mathrm{\dots},{r}_{m}\mathit{\}}$ and transmission frequencies
$\mathit{\{}{f}_{\mathrm{1}},{f}_{\mathrm{2}},\mathrm{\dots},{f}_{{N}_{f}}\mathit{\}}$, where ${r}_{i+\mathrm{1}}-{r}_{i}=\mathrm{\Delta}r$ is the radar
range resolution. We note that in most circumstances we employ a retrieval
step size *R* that is larger than Δ*r*, since, as we will show below, the
precision in our retrieved humidity scales favorably with total optical depth
and hence with increasing *R*. Then, given a step size such that $R={r}_{i+S}-{r}_{i}$ for some integer *S*, we form the frequency-dependent measured
quantity

$$\begin{array}{}\text{(9)}& {\displaystyle}{\mathit{\gamma}}_{i}\left({f}_{j}\right)=-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}R}}\mathrm{ln}\left[{\left({\displaystyle \frac{{r}_{i+S}}{{r}_{i}}}\right)}^{\mathrm{2}}{\displaystyle \frac{{P}_{\mathrm{e}}({r}_{i+S},{f}_{j})}{{P}_{\mathrm{e}}({r}_{i},{f}_{j})}}\right]\end{array}$$

for each starting range *r*_{i}. From Eq. (2), we see that
we can extract the average humidity between *r*_{i} and *r*_{i+S} by performing
a least squares fit of the function

$$\begin{array}{}\text{(10)}& {\displaystyle}\widehat{\mathit{\gamma}}\left(f\right)=\stackrel{\mathrm{\u203e}}{\mathit{\rho}}{\mathit{\kappa}}_{\mathrm{v}}\left(f\right)+B\end{array}$$

to the measurements for each *i*, where *B* is a frequency-independent offset
containing information about dry air gaseous absorption, particulate
extinction, and the relative reflectivity of the two ranges in question. We
drop the *v* subscript on the water vapor density in the above equation for
simplicity of notation. The resulting humidity estimates
$\mathit{\{}{\stackrel{\mathrm{\u203e}}{\mathit{\rho}}}_{\mathrm{1}},{\stackrel{\mathrm{\u203e}}{\mathit{\rho}}}_{\mathrm{2}},\mathrm{\dots},{\stackrel{\mathrm{\u203e}}{\mathit{\rho}}}_{m-S}\mathit{\}}$ have
a corresponding range axis
$\mathit{\{}{\stackrel{\mathrm{\u203e}}{r}}_{\mathrm{1}},{\stackrel{\mathrm{\u203e}}{r}}_{\mathrm{2}},\mathrm{\dots},{\stackrel{\mathrm{\u203e}}{r}}_{m-S}\mathit{\}}$, where
${\stackrel{\mathrm{\u203e}}{r}}_{i}=({r}_{i}+{r}_{i+S})/\mathrm{2}$, and have associated uncertainties
determined from the fitting procedure.

Using standard error propagation, the estimated uncertainty in the measured
quantity *γ*_{i}(*f*_{j}) defined in Eq. (9) is

$$\begin{array}{}\text{(11)}& {\displaystyle}{\mathit{\sigma}}_{{\mathit{\gamma}}_{i}}\left({f}_{j}\right)={\displaystyle \frac{\mathrm{1}}{\mathrm{2}R}}{\left[{\left({\left.{\displaystyle \frac{{\mathit{\sigma}}_{\mathrm{e}}}{{P}_{\mathrm{e}}}}\right|}_{{r}_{i+S},{f}_{j}}\right)}^{\mathrm{2}}+{\left({\left.{\displaystyle \frac{{\mathit{\sigma}}_{\mathrm{e}}}{{P}_{\mathrm{e}}}}\right|}_{{r}_{i},{f}_{j}}\right)}^{\mathrm{2}}\right]}^{\mathrm{1}/\mathrm{2}}.\end{array}$$

In order to derive a simple analytical expression for the relative
uncertainty in the retrieved humidity, we restrict ourselves for the moment
to considering two transmission frequencies, *f*_{1} and *f*_{2}. In this case,
we can combine Eqs. (2), (4), and
(9) to obtain the humidity directly,

$$\begin{array}{}\text{(12)}& {\displaystyle}\stackrel{\mathrm{\u203e}}{\mathit{\rho}}\left({\stackrel{\mathrm{\u203e}}{r}}_{i}\right)={\left[{\mathit{\kappa}}_{\mathrm{v}}\left({f}_{\mathrm{2}}\right)-{\mathit{\kappa}}_{\mathrm{v}}\left({f}_{\mathrm{1}}\right)\right]}^{-\mathrm{1}}\left[{\mathit{\gamma}}_{i}\left({f}_{\mathrm{2}}\right)-{\mathit{\gamma}}_{i}\left({f}_{\mathrm{1}}\right)\right],\end{array}$$

with the associated relative uncertainty

$$\begin{array}{ll}{\displaystyle}{\left.{\displaystyle \frac{{\mathit{\sigma}}_{\stackrel{\mathrm{\u203e}}{\mathit{\rho}}}}{\stackrel{\mathrm{\u203e}}{\mathit{\rho}}}}\right|}_{{\stackrel{\mathrm{\u203e}}{r}}_{i}}& {\displaystyle}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}\mathrm{\Delta}\mathit{\tau}}}\left[\sum _{j=\mathrm{1},\mathrm{2}}\left({\left({\left.{\displaystyle \frac{{\mathit{\sigma}}_{\mathrm{e}}}{{P}_{\mathrm{e}}}}\right|}_{{r}_{i+S},{f}_{j}}\right)}^{\mathrm{2}}\right.\right.\\ \text{(13)}& {\displaystyle}& {\displaystyle}{\left.\left.+{\left({\left.{\displaystyle \frac{{\mathit{\sigma}}_{\mathrm{e}}}{{P}_{\mathrm{e}}}}\right|}_{{r}_{i},{f}_{j}}\right)}^{\mathrm{2}}\right)\right]}^{\mathrm{1}/\mathrm{2}},\end{array}$$

where $\mathrm{\Delta}\mathit{\tau}=\left[{\mathit{\kappa}}_{\mathrm{v}}\left({f}_{\mathrm{2}}\right)-{\mathit{\kappa}}_{\mathrm{v}}\left({f}_{\mathrm{1}}\right)\right]\stackrel{\mathrm{\u203e}}{\mathit{\rho}}\left({\stackrel{\mathrm{\u203e}}{r}}_{i}\right)R$ is
the differential optical depth for *f*_{1} and *f*_{2} between range bins *r*_{i}
and *r*_{i+S}. Equation (13) reveals that there
are three linked quantities determining the sensitivity of the system:
(1) the magnitude of the DAR signal quantified by Δ*τ*, (2) the
statistical uncertainty of the power measurements given by the quadrature sum
of relative errors in Eq. (13), and (3) the
relative uncertainty in the derived value for the humidity. Thus, given a set
of measured echo powers and a specific value for the humidity, there is a
trade-off between spatial resolution of the retrieval and relative
uncertainty in the humidity estimate.

An important and subtle point regarding the uncertainty in the measured
quantity *γ*_{i}(*f*_{j}) is that Eq. (11) relies on a
Taylor expansion in the relative error *σ*_{e}∕*P*_{e}, and
therefore is only valid for measurements with SNR above some
critical value that depends on the number of measurements *N*_{p}.
Because there is no closed-form expression for the probability distribution
function (PDF) of *γ*_{i}(*f*_{j}), we resort to a Monte Carlo analysis, which
is described in Appendix B, to generate the relevant PDFs for the parameters used in this work numerically. From this
analysis, we find that for *N*_{p}=2000 pulses and *N*_{b}=11 averaged bins, the Taylor expansion method is accurate for measurements
with $\text{SNR}>-\mathrm{10}$ dB.

We note here that it is typical of differential absorption systems to utilize
only two frequencies: one online and one offline. However, in this work we
are concerned with validating both the spectroscopic model used and the radar
hardware itself, which could be subject to unknown frequency-dependent
systematic effects. The regression approach discussed above thus provides for
a robust comparison of the measured frequency dependence *γ*_{i}(*f*_{j}) with
the model $\widehat{\mathit{\gamma}}\left(f\right)$, while a two-frequency approach would mask
inconsistencies between measurements and model, or systematic hardware
effects, since the two free parameters $\stackrel{\mathrm{\u203e}}{\mathit{\rho}}$ and *B* are fully
determined given two frequency points. Furthermore, a distributed set of
frequencies allows for the possibility of extending retrievals deeper in
range for moist atmospheres, as frequencies closer to the line center will be
attenuated more strongly, and can be excluded from the fits described above
when the critical SNR value is reached.

3 Boundary-layer measurements and analysis

Back to toptop
In this section we report on measurements performed at JPL on 15 March 2018 using the proof-of-concept differential absorption radar described in Sect. 2.2. For these measurements, we implement a new signal processing technique for real-time noise floor characterization, utilizing a triangle-wave frequency chirp (i.e., bidirectional) instead of a sawtooth-wave chirp (i.e., unidirectional). According to FMCW radar principles, the echo spectrum switches from residing on the low- to the high-frequency side of the zero-range signal (i.e., 5 MHz) for increasing and decreasing linear frequency chirps, respectively. As shown in Fig. 4a, this fast switching of the chirp direction alternately exposes the noise floor on each side of the zero-range point within the IF band, and provides accurate and nearly continuous estimation of the system noise power and the passive signal corresponding to the scene brightness temperature at each frequency bin. This technique is especially advantageous for airborne/spaceborne applications, as the brightness temperature of the observed scene can change on fast timescales due to different surface types (e.g., ocean versus land) and from the presence or absence of clouds.

Figure 4 showcases a few aspects of a single
ground-based DAR measurement, for which the conditions were light drizzle and
a cloud located a few hundred meters off the ground. For all the field
measurements discussed in this work, we acquire *N*_{p}=2000 pulses
for each of 12 frequencies equally spaced between 167 and 174.8 GHz, with
the radar positioned just inside a building, pointing at 30^{∘}
elevation. The experimental sequence is as follows: first, we perform 40
frequency chirps at a given transmission frequency before switching to
another frequency, which takes 1 ms. The received signal is downconverted to
baseband, digitized in an ADC, and processed in
real time as described in Sect. 2.2.
We achieve a system duty cycle of >90 %, resulting in a total
measurement/observation time of ≈25 s.

By subtracting the respective noise floors from the increasing and decreasing
frequency chirp measurements (Fig. 4a), and subsequently
combining the mirrored spectra, we obtain our estimate of the echo power
spectra. In Fig. 4b, we plot the echo power spectra
scaled by *r*^{2} for the 12 transmission frequencies before bin averaging,
which reveals the range dependence of the quantity $Z\left(r\right)\mathrm{exp}(-\mathrm{2}\mathit{\tau}(r,f\left)\right)$.
Each spectrum is normalized to its value at 100 m. Thus, we observe the
differential absorption due to water vapor directly from the spreading of the
spectra with increasing range, whereby for a particular range, the plotted
values increase monotonically with decreasing transmit frequency. After
averaging the quantity ${r}_{i}^{\mathrm{2}}{P}_{\mathrm{e}}({r}_{i},{f}_{j})$ within a swath of size
*N*_{b}=11, we filter the spectra based on the Monte Carlo analysis
in Appendix B, keeping only those points with
$\text{SNR}>-\mathrm{10}$ dB, and are left with the smoothed profiles shown in
Fig. 4c. Figure 4d shows the
relative error in the binned (*N*_{b}=11) echo power measurement (blue
circles) plotted against the measured SNR for all 12 frequencies.
The measured values agree very well with those predicted by
Eq. (8) (black dashed line), indicating that our
statistical model based on speckle noise, which underlies the Monte Carlo
simulations implemented in this work, is accurate.

Using the averaged, filtered spectra in Fig. 4c, we
proceed towards retrieving the water vapor density profile using the
procedure outlined in Sect. 2.4. For the profiles
presented in this section, we utilize a retrieval step size of *R*=200 m.
Beginning with an initial range of *r*_{1}=100 m, we form the 12 quantities
*γ*_{i}(*f*_{j}) for each starting *r*_{i} in the set $\mathit{\{}{r}_{\mathrm{1}},{r}_{\mathrm{2}},\mathrm{\dots},{r}_{m-S}\mathit{\}}$, and perform a least-squares fit of the function
$\widehat{\mathit{\gamma}}\left(f\right)$ to the data at each range point. Note that the retrieved
water vapor density ${\stackrel{\mathrm{\u203e}}{\mathit{\rho}}}_{i}$ is related only to the difference
between the value of the fitted function at 174.8 and 167 GHz, while the
offset is related to particulate extinction and hydrometeor reflectivity, and
is disregarded in this work. The pressure and temperature dependence of the
absorption line shape is included in the fitting model using reported values
at the surface from a nearby weather station, and assuming an exponential
pressure profile with a scale height of 7.5 km and a temperature lapse rate
of 6 ^{∘}C km^{−1}. We note that for the relatively short vertical
extents of the profiles from these ground measurements (e.g., 1.4sin30^{∘} km for Figs. 4 and
5), the retrieved $\stackrel{\mathrm{\u203e}}{\mathit{\rho}}$ values are quite
insensitive to the assumed thermodynamic profiles (see
Appendix C).

An important element of the DAR technique in general is utilizing an accurate
model for the absorption line shape. Examples of line shape fits to the data
are shown in Fig. 5a for three different values of
SNR, with arbitrary offsets imposed on the three traces to permit
simultaneous plotting. To assign SNR values to these points, we
compute the mean SNR for the 12 frequencies at *r*_{i} and *r*_{i+S},
and use the smaller of the two. Clearly the millimeter-wave model accurately
captures the frequency dependence of the measurements, which is supported
quantitatively by the typical reduced chi-square values of ${\mathit{\chi}}_{\text{red}}^{\mathrm{2}}\approx \mathrm{1}$ for these fits. The retrieved water vapor density profile is shown
in Fig. 5b, where the range ${\stackrel{\mathrm{\u203e}}{r}}_{i}$ assigned
to each fitted value ${\stackrel{\mathrm{\u203e}}{\mathit{\rho}}}_{i}$ is the midpoint of *r*_{i} and
*r*_{i+S}. Also plotted here is an estimate of the saturation vapor density
given our lapse rate assumption. This profile is consistent with a cloud base
between 400 and 600 m and shows qualitatively good agreement with the
expectation that the relative humidity is approximately 100 % in liquid
cloud layers. Note that because the retrieved values correspond to the mean
humidity between *r*_{i} and *r*_{i+S}, we effectively retrieve the profile
convolved with a box of size *R* (200 m here). For this retrieval, the
absolute humidity errors lie between 0.55 and 0.60 g m^{−3} until around
1 km (SNR≈10 dB), where the error steadily increases until
the final retrieval point at 1.25 km with ${\mathit{\sigma}}_{\stackrel{\mathrm{\u203e}}{\mathit{\rho}}}=\mathrm{2.9}$ g m^{−3}. The value of ${\mathit{\sigma}}_{\stackrel{\mathrm{\u203e}}{\mathit{\rho}}}$ in the
high-SNR regime (i.e., the first 1 km) remains roughly constant, even
though $\stackrel{\mathrm{\u203e}}{\mathit{\rho}}$ varies by a factor of 3, since the absolute humidity
error is independent of the humidity itself, and depends only on the
differential mass extinction cross section
*κ*_{v}(174.8 GHz) − *κ*_{v}(167 GHz), the retrieval step size
*R*, and the power measurement uncertainty (see
Eq. 13).

Though we do not have independent, coincident water vapor profile
measurements with which to validate the accuracy of the retrieval, we can
investigate repeatability of this DAR method by performing the retrieval on
coincident, independent DAR measurements of the same exact scene. To do so,
we acquire *N*_{p}=4000 pulses at each frequency with a total
measurement time of 50 s, and parse the data into two groups of
*N*_{p}=2000 pulses both spanning the full 50 s. The results are
shown in Fig. 6, where we also present measurements of
different cloud and precipitation scenarios than that presented in
Figs. 4 and 5. In
Fig. 6, panels (a) and (b) correspond to light rain at
the surface with a cloud boundary at 1 km range, and panels (c) and (d) to
heavy rain at the surface with strong particulate extinction. The retrievals
from the two independent sample sets in both cases agree quite well, which
showcases the reproducibility of the measurement and indicates that the
estimated humidity error accurately captures the sample scatter.

Given a measured range-resolved echo power spectrum, what retrieval range
resolution can we achieve for a specified minimum retrieval precision? As
discussed briefly in Sect. 2.4, the relative
error in the retrieved humidity ${\mathit{\sigma}}_{\stackrel{\mathrm{\u203e}}{\mathit{\rho}}}/\stackrel{\mathrm{\u203e}}{\mathit{\rho}}$
(see Eq. 13) for a given power measurement
uncertainty varies inversely with the differential optical depth, and thus
depends on both the retrieval step size *R* (i.e., retrieval resolution) used
*and* the absolute value of the humidity $\stackrel{\mathrm{\u203e}}{\mathit{\rho}}$.
Alternatively, one can look at the absolute error and rearrange
Eq. (13) to find that, for a given pair of
frequencies and power measurements at two ranges, the product of
${\mathit{\sigma}}_{\stackrel{\mathrm{\u203e}}{\mathit{\rho}}}$ and *R* is constant. Hence, reducing the retrieval
step size by some factor increases the absolute humidity error by the same
factor. In future work we will implement a retrieval algorithm that has
adaptive range resolution based on both the inherent signal (i.e., humidity)
and the measurement noise.

4 Conclusions

Back to toptop
A proof-of-concept humidity-profiling DAR operating between 167 and 174.8 GHz has been constructed and tested from the ground. The instrument builds on progress made in an earlier version operating between 183 and 193 GHz (Cooper et al., 2018), and employs a new signal processing technique for performing real-time noise power spectrum characterization and subtraction, providing for higher accuracy measurements of the radar echo power. A new direct inversion algorithm for retrieving humidity based on least squares fits to a spectroscopic model is applied to the measured echo power spectra, showing close agreement between the measurement and model frequency dependence. The humidity profiles retrieved from two statistically independent measurement sets of the exact same scene are in close agreement, highlighting the reproducibility of the method. The uncertainties in the power measurements, which in part determine the retrieved humidity uncertainty, agree very well with a statistical model based on radar speckle noise that incorporates the effects of background noise subtraction and downsampling, or binning, of the measured spectra.

Development of an operational airborne 167–174.8 GHz DAR is currently in progress, which will include an additional 20 dB of antenna gain and a factor of 4 increase in transmit power. Important future steps for this instrument include validation of the measurement accuracy using coincident measurements of humidity, pressure, and temperature (e.g., from radiosondes), and eventually testing from an airborne platform. Specifically, the surface returns while measuring from an airborne platform will be investigated for the retrieval of total column water within the boundary layer. A more significant augmentation of the system could include the addition of passive radiometric channels near the 183 GHz line. This would allow for continuous measurement of vertical humidity profiles when transitioning between clear-sky and cloudy areas, and opens the possibility to study biases in the humidity retrieved from radiometric measurements that are caused by scattering and emission from clouds.

Data availability

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Data availability.

All of the data used in this paper are presented in the figures.

Appendix A: Error in detected power

Back to toptop
In this appendix, we derive the expression for the detected power uncertainty
within a single radar range bin in the presence of background noise
(Eq. 6). To do so, we begin by assuming that all
targets within the scattering volume are randomly distributed, leading to the
well-known Rayleigh fading model for the received echo signal (Ulaby et al., 1982).
In the context of FMCW radar, we then consider the received complex electric
field amplitude *E*_{i} corresponding to the *i*th frequency bin in the FFT
spectrum, where we only consider the polarization direction that couples into
the radar receiver. Within the Rayleigh fading model, it is shown that for
${E}_{i}={E}_{\mathrm{0},i}{e}^{i{\mathit{\varphi}}_{i}}$, the modulus of the field amplitude *E*_{0,i} is
normally distributed with zero mean and standard deviation *σ*_{E}, and
the phase *ϕ*_{i} is uniformly distributed over the interval [0,2*π*].
Alternatively, we can write the corresponding voltage in the receiver as
${V}_{e,i}={a}_{i}+i{b}_{i}$, where *a*_{i} and *b*_{i} are uncorrelated and are both
normally distributed with zero mean. Then, from the expression converting
electric field to power, ${P}_{\mathrm{e},i}=|{V}_{\mathrm{e},i}{|}^{\mathrm{2}}=\mathit{\alpha}|{E}_{i}{|}^{\mathrm{2}}$, we find
the probability distribution function for the received echo power

$$\begin{array}{}\text{(A1)}& {\displaystyle}p({P}_{\mathrm{e},i}\ge \mathrm{0})={\displaystyle \frac{\mathrm{1}}{\langle {P}_{\mathrm{e},i}\rangle}}{e}^{-{P}_{\mathrm{e},i}/\langle {P}_{\mathrm{e},i}\rangle},\end{array}$$

where the mean equals the variance and is given by $\langle {P}_{\mathrm{e},i}\rangle =\mathrm{2}\mathit{\alpha}{\mathit{\sigma}}_{E}^{\mathrm{2}}$, *α* is a field-to-voltage conversion factor for the
radar, and $p({P}_{\mathrm{e},i}<\mathrm{0})=\mathrm{0}$. Furthermore, we find that $\langle {a}_{i}^{\mathrm{2}}\rangle =\langle {b}_{i}^{\mathrm{2}}\rangle =\langle {P}_{\mathrm{e},i}\rangle /\mathrm{2}$. Though not proven here,
the Rayleigh fading model also shows that the expectation value of the
received power from *N* randomly distributed targets is the sum of the
expectation values of the individual target echo powers.

Similarly, one can show that Gaussian white noise in the radar signal, which
comes from both the scene brightness temperature and the radar electronics,
results in a noise voltage within the *i*th frequency bin of the FFT spectrum
with Fourier coefficient ${V}_{\mathrm{n},i}={c}_{i}+i{d}_{i}$, where $\langle {c}_{i}\rangle =\langle {d}_{i}\rangle =\mathrm{0}$ and $\langle {c}_{i}^{\mathrm{2}}\rangle =\langle {d}_{i}^{\mathrm{2}}\rangle =\langle {P}_{\mathrm{n},i}\rangle /\mathrm{2}$. We proceed towards deriving
Eq. (6) by considering the coherent detection of
both the radar echo and noise signals. In this case, the detected voltage
signal in the Fourier domain within the *i*th range bin is ${V}_{\mathrm{d},i}={V}_{\mathrm{e},i}+{V}_{n,i}$, and the detected power is ${P}_{\mathrm{d},i}=|{V}_{\mathrm{d},i}{|}^{\mathrm{2}}$. Using the expectation values listed above, it is easy
to show that

$$\begin{array}{}\text{(A2)}& {\displaystyle}\langle {P}_{\mathrm{d},i}\rangle =\langle {P}_{\mathrm{e},i}\rangle +\langle {P}_{\mathrm{n},i}\rangle \end{array}$$

and

$$\begin{array}{}\text{(A3)}& {\displaystyle}\text{Var}\left({P}_{\mathrm{d},i}\right)={\left(\langle {P}_{\mathrm{e},i}\rangle +\langle {P}_{\mathrm{n},i}\rangle \right)}^{\mathrm{2}}.\end{array}$$

Therefore, we recover Eq. (6) by computing the
standard error for *N* independent measurements, ${\mathit{\sigma}}_{\mathrm{d},i}^{\mathrm{2}}=\text{Var}\left({P}_{\mathrm{d},i}\right)/N$.

Appendix B: Monte Carlo Analysis

Back to toptop
As discussed in Sect. 2.3,
subtracting off the noise power contribution to the detected power
*P*_{d} is critical for accurate humidity estimation. However, for low
values of SNR, we clearly expect the result $\langle {P}_{\mathrm{e}}\rangle =\langle {P}_{\mathrm{d}}\rangle -\langle {P}_{\mathrm{n}}\rangle $ to be
negative some of the time due to finite sampling, where “〈⋯〉” denotes the sample average. This is nonphysical.
Therefore, in order to account for these finite sampling effects and the
potential breakdown of standard error propagation when
*σ*_{e}∕*P*_{e} is not small, we employ a Monte Carlo
simulation of the DAR measurement. The PDF for the echo power received from
randomly distributed hydrometeor targets within a single range bin is given
by Eq. (A1). In order to generate random samples of the radar
spectrum for transmission frequency *f*_{j}, we begin with an idealized
spectrum 〈*P*_{e}(*r*_{i},*f*_{j})〉 for which we set $C\left(f\right)=Z(r,f)=\mathrm{1}$ and *ρ*_{v}(*r*)=constant, and sample the distribution
(Eq. A1) at each range bin *r*_{i}. We then perform a fast
Fourier transform (FFT) to obtain the corresponding time-domain radar signal,
add the effects of background noise using Gaussian white noise, and apply a
Hanning window. Taking the inverse FFT thus supplies a single random
realization of a measured *P*_{d} spectrum. For the simulated spectra
used in this work, we generate *N*_{p}=2000 radar pulses with range
resolution Δ*r*=2.5 m and average them to realize a single radar
measurement. These values for *N*_{p} and Δ*r* are the same
parameters utilized in the field measurements presented in
Sect. 3. We generate 10 000 averaged spectra
for both *P*_{d} and *P*_{n}, giving 10 000 random
realizations of the echo power measurement. For these simulations, we use
*f*_{j}=167 GHz and *ρ*_{v}=7.4 g m^{−3}.

Our aim is to utilize the Monte Carlo simulations to inform where the Taylor
expansion method for error propagation breaks down in our estimation of
${\mathit{\sigma}}_{{\mathit{\gamma}}_{i}}\left({f}_{j}\right)$, and thus provide a criterion for filtering our
measurements. To do so, we fix *N*_{b}=11 and the step size *S*=10
(i.e., 275 m) and compute the mean and standard deviation of the Monte Carlo
probability distribution for the two-way transmission between *r*_{i} and
*r*_{i+S} for each *r*_{i}. Figure B1 shows the
results, where we plot the Monte Carlo mean value divided by the a priori
two-way transmission used to generate the Monte Carlo results, as a function
of the SNR at *r*_{i+S}. The gray shaded area represents the
SNR-dependent errors predicted from Eq. (8).
There are two notable deviations that arise for SNR values below
0.1: (1) the error estimated using the standard error propagation formalism
begins underestimating the true standard deviation calculated using the Monte
Carlo ensemble, and (2) the mean of the Monte Carlo-generated distribution
systematically overestimates the true two-way transmission. We note here that
this point of departure between the naive error propagation estimate and that
from the Monte Carlo distributions does not depend on *N*_{b} or *S*,
but is determined by the number of independent pulses *N*_{p} used to
realize a single radar measurement. From these simulations, we conclude that
the standard error propagation model is sufficient for $\text{SNR}>-\mathrm{10}$ dB. Therefore, after downsampling the measured spectra with
*N*_{b}=11, we eliminate all measured values with $\text{SNR}<-\mathrm{10}$ dB, as described in Sect. 3.1.

Appendix C: Retrieval dependence on assumed pressure and temperature values

Back to toptop
To assess the dependence of the retrieved humidity on temperature and
pressure, we will consider again the case of the two-frequency measurement,
using transmission frequencies *f*_{1}=167 GHz and *f*_{2}=174.8 GHz. Then,
for a given starting range *r*_{i} and step size *R*, we use the measured
quantities *γ*_{i}(*f*_{1}) and *γ*_{i}(*f*_{2}) to solve for the mean humidity
between the two ranges,

$$\begin{array}{}\text{(C1)}& {\displaystyle}{\stackrel{\mathrm{\u203e}}{\mathit{\rho}}}_{i}={\displaystyle \frac{{\mathit{\gamma}}_{i}\left({f}_{\mathrm{2}}\right)-{\mathit{\gamma}}_{i}\left({f}_{\mathrm{1}}\right)}{{\mathit{\kappa}}_{\mathrm{v}}({f}_{\mathrm{2}},P,T)-{\mathit{\kappa}}_{\mathrm{v}}({f}_{\mathrm{1}},P,T)}}={\displaystyle \frac{{\mathit{\gamma}}_{i}\left({f}_{\mathrm{2}}\right)-{\mathit{\gamma}}_{i}\left({f}_{\mathrm{1}}\right)}{\mathrm{\Delta}{\mathit{\kappa}}_{\mathrm{v}}(P,T)}},\end{array}$$

where we now explicitly write *κ*_{v} as a function of temperature *T* and
pressure *P*, and we have defined the differential mass extinction cross
section Δ*κ*_{v}(*P*,*T*) for these two frequencies. Given reference
values of *P*_{0}=1000 mbar and *T*_{0}=285 K, and corresponding retrieved
humidity ${\stackrel{\mathrm{\u203e}}{\mathit{\rho}}}_{i,\mathrm{0}}$, we calculate the error in our humidity
estimate for different conditions *P* and *T* as

$$\begin{array}{}\text{(C2)}& {\displaystyle \frac{{\stackrel{\mathrm{\u203e}}{\mathit{\rho}}}_{i}-{\stackrel{\mathrm{\u203e}}{\mathit{\rho}}}_{i,\mathrm{0}}}{{\stackrel{\mathrm{\u203e}}{\mathit{\rho}}}_{i,\mathrm{0}}}}={\displaystyle \frac{\mathrm{\Delta}{\mathit{\kappa}}_{\mathrm{v}}({P}_{\mathrm{0}},{T}_{\mathrm{0}})}{\mathrm{\Delta}{\mathit{\kappa}}_{\mathrm{v}}(P,T)}}-\mathrm{1}.\end{array}$$

Figure C1 shows the humidity error for pressure deviations of ±20 mbar and temperature deviations of ±15 K. Here we see that the retrieved humidity is very weakly dependent on the assumed pressure, and only accrues an error of 10 % for a temperature deviation of about 8 K.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

This research was supported by NASA's Earth Science Technology Office under
the Instrument Incubator Program, and was carried out at the Jet Propulsion
Laboratory (JPL), California Institute of Technology, Pasadena, CA, USA,
under contract with the National Aeronautics and Space Administration.
Richard J. Roy's research was supported by an appointment to the NASA
Postdoctoral Program at JPL, administered by Universities Space Research
Association under contract with NASA.

Edited
by: Murray Hamilton

Reviewed by: Gerald Mace and two anonymous
referees

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Short summary

The measurement of water vapor profiles inside clouds with high spatial resolution represents an outstanding problem in atmospheric remote sensing. Here we present measurements from a proof-of-concept millimeter-wave (170 GHz) cloud radar aimed at filling this observational gap, and demonstrate the ability to retrieve in-cloud water vapor profiles with high precision and resolution. This technology could meaningfully impact future satellite-based measurements of water vapor.

The measurement of water vapor profiles inside clouds with high spatial resolution represents an...

Atmospheric Measurement Techniques

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