Introduction
In this work, we discuss the implementation of differential absorption radar
(DAR) for measuring humidity profiles inside of boundary-layer clouds
. The DAR method, which is the microwave analog of the
mature differential absorption lidar (DIAL) method , 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 , utilizing the 60 GHz O2
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 , 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 (). 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 , with a notable airborne DIAL system
being the Lidar Atmospheric Sensing Experiment (LASE) .
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 . 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.
Measurement basis and method
Differential absorption radar
The DAR technique 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
Pe(r,f)=C(f)Z(r,f)r-2e-2τ(r,f),
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 r1 and r2=r1+R and assuming frequency independence of the
reflectivity and particulate extinction, we find
Pe(r2,f)Pe(r1,f)=Z(r2)Z(r1)r1r22e-2β(r1,r2,f)R,
where
β(r1,r2,f)=τ(r2,f)-τ(r1,f)R=1R∫r1r2∑jρj(r)κj(r,f)+βpart(r)dr
is the average absorption coefficient between r1 and r2, ρj(r) is
the density of the gas component with label j, κj(r,f) is the
corresponding mass extinction cross section, which varies with r due to
pressure and temperature, and βpart(r) is the particulate
extinction coefficient integrated over local drop size distributions (DSDs).
Gaseous absorption coefficient (one-way) calculated using the model
from and the parameters listed in the figure. The green shaded
region and inset highlight the 167 to 174.8 GHz transmission band for this
work.
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 ρv(r)κv(r,f)+βgas,bg(r), where the
subscript v corresponds to water vapor and βgas,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. () as
β(r1,r2,f)=ρ‾v(r1,r2)κv(f)+β‾gas,bg(r1,r2)+β‾part(r1,r2),
where the overbar symbol implies taking the mean value between r1 and
r2. Thus, we see that measuring the frequency-dependent contribution to
the optical depth between r1 and r2 reveals the average water vapor
density given the known absorption line shape κv(f).
Figure shows the frequency dependence of the gaseous
absorption coefficient ρvκv(f)+βgas,bg in the vicinity of the 183 GHz water vapor line for
P=1000 mbar, T=285 K, and ρv=10 g m-3. For
this work, we utilize the millimeter-wave propagation model from the EOS
Microwave Limb Sounder . The 167 to 174.8 GHz transmission
band is highlighted in green, as well as shown in the inset of
Fig. , 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 βpart 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
N(D)=N0Γ(ν)DDnν-11Dne-D/Dn,
where N0 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 Dn is
the characteristic diameter. For rain we enforce an additional constraint
that N0=x1Dn1-x2, where x1=26.2 mx2-4 and x2=1.57 have been determined in previous studies by comparing to observations
. 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 .
Differential particulate extinction coefficients
for (a) rain and (b) cloud. In the case of rain, the DSD
characteristic diameter Dn determines the liquid water content and hence
the rain rate, while for clouds we fix L=500 mg m-3. The
legend in (a) gives the corresponding frequencies in
GHz.
The results are shown in Fig. , where we plot the
differential particulate extinction, Δβpart(f)=βpart(f)-βpart(f0), as a function of Dn. Here
f0=167 GHz corresponds to the low-frequency end of the transmission
band. In Fig. a, the corresponding rain rate is
displayed on the upper horizontal axis. For the cloud species, the
normalization parameter N0 is not fixed by any additional constraint, and
is therefore determined at each Dn 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. b. 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 ΔβpartRayleigh=6πLIm(Kw)(f-f0)/(ρwc)=0.2 dB km-1 for f=174.8 GHz, where ρw is the
density of liquid water, Kw=(mw2-1)/(mw2+2), mw is the complex
refractive index of water, and c is the speed of light. For larger values
of Dn, 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.
FMCW radar basics and instrument details
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. . The
transmitted signal is frequency-modulated with a linear chirp waveform of
bandwidth ΔFchirp and duration Tchirp. After
scattering off of a target at a distance r from the radar, the received
chirp is delayed in time by an amount 2r/c, leading to a fixed frequency
offset of δf=2ΔFchirpr/(cTchirp) 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 fIF(r)=5 MHz ± δf(r), where the positive(negative) sign applies for
decreasing(increasing) frequency chirps.
Basic FMCW radar schematic. See
Sect. for
discussion.
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
. The architecture is similar to that presented in an
earlier work 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
(), 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 ΔFchirp=60 MHz and duration Tchirp=1 ms, resulting in a range
resolution of Δr=c/2ΔFchirp=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 .
Hardware and radar signal acquisition parameters used in this work.
The noise figure reported is for a complex radar signal detected using a
double-sideband front-end mixer.
Parameter
Value
Unit
Transmit frequency
167–174.8
GHz
Transmit power
140
mW
Noise figure
<8
dB
Primary aperture diameter
6
cm
Antenna gain
39
dB
Far-field beam width (FWHM)
1.9
degrees
Side lobe level
-23
dB
Chirp bandwidth
60
MHz
Chirp time
1
ms
Range resolution
2.5
m
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 δf(r)/r=400 kHz km-1.
Power measurement uncertainty
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 Pn 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 (Pn) and radar echo (Pe) power,
one can show that the statistical uncertainty of the detected power,
Pd=Pe+Pn, is given by (see
Appendix )
σd=1NpPe2+2PePn+Pn21/2,
where Np 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 Pd. To see why this is, consider
Eq. () with the left-hand side replaced by
Pd(r2,f)/Pd(r1,f), which is equivalent to
interpreting the detected power as the true echo power, set Z(r2)=Z(r1)
for simplicity, and consider the limit Pe≪Pn (i.e.,
Pd→Pn). In this case we would find that
exp(-2β(r1,r2,f)R)→1 regardless of the actual value of
Pe, 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 SNR≡Pe/Pn, this would lead to a systematic underestimate of
the true humidity. Therefore, after subtracting the separate noise power
measurement from Pd we obtain a measurement of Pe
with total uncertainty σe=(σd2+σn2)1/2,
where σn=Pn/Np is the noise power
measurement uncertainty (see Eq. with
Pe=0). The relative uncertainty in the measured echo power is
therefore
σePe=1Np1+2SNR+2SNR21/2.
As will be discussed in Sect. , the range
dimension is purposefully oversampled in our measurements, allowing us to
decrease the statistical power uncertainty at a given range by averaging
Nb adjacent range bins. The resulting relative power uncertainty
is given by
σePe=ξ(Nb)NpNb1+2SNR+2SNR21/2,
where ξ(Nb)≥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
ξ(Nb)=1+Nb-1Nb891/2.
Inversion algorithm for profile retrieval
Under the simplifying assumptions introduced in Sect. , 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 . To invert the radar
spectra, we consider a set of measured echo powers Pe(ri,fj)
for ranges {r1,r2,…,rm} and transmission frequencies
{f1,f2,…,fNf}, where ri+1-ri=Δr is the radar
range resolution. We note that in most circumstances we employ a retrieval
step size R that is larger than Δr, 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=ri+S-ri for some integer S, we form the frequency-dependent measured
quantity
γi(fj)=-12Rlnri+Sri2Pe(ri+S,fj)Pe(ri,fj)
for each starting range ri. From Eq. (), we see that
we can extract the average humidity between ri and ri+S by performing
a least squares fit of the function
γ^(f)=ρ‾κv(f)+B
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
{ρ‾1,ρ‾2,…,ρ‾m-S} have
a corresponding range axis
{r‾1,r‾2,…,r‾m-S}, where
r‾i=(ri+ri+S)/2, and have associated uncertainties
determined from the fitting procedure.
Using standard error propagation, the estimated uncertainty in the measured
quantity γi(fj) defined in Eq. () is
σγi(fj)=12RσePeri+S,fj2+σePeri,fj21/2.
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, f1 and f2. In this case,
we can combine Eqs. (), (), and
() to obtain the humidity directly,
ρ‾(r‾i)=κv(f2)-κv(f1)-1γi(f2)-γi(f1),
with the associated relative uncertainty
σρ‾ρ‾r‾i=12Δτ∑j=1,2σePeri+S,fj2+σePeri,fj21/2,
where Δτ=κv(f2)-κv(f1)ρ‾(r‾i)R is
the differential optical depth for f1 and f2 between range bins ri
and ri+S. Equation () reveals that there
are three linked quantities determining the sensitivity of the system:
(1) the magnitude of the DAR signal quantified by Δτ, (2) the
statistical uncertainty of the power measurements given by the quadrature sum
of relative errors in Eq. (), 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 γi(fj) is that Eq. () relies on a
Taylor expansion in the relative error σe/Pe, and
therefore is only valid for measurements with SNR above some
critical value that depends on the number of measurements Np.
Because there is no closed-form expression for the probability distribution
function (PDF) of γi(fj), we resort to a Monte Carlo analysis, which
is described in Appendix , to generate the relevant PDFs for the parameters used in this work numerically. From this
analysis, we find that for Np=2000 pulses and Nb=11 averaged bins, the Taylor expansion method is accurate for measurements
with SNR>-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 γi(fj) with
the model γ^(f), while a two-frequency approach would mask
inconsistencies between measurements and model, or systematic hardware
effects, since the two free parameters ρ‾ 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.
DAR measurement spectra. (a) The bidirectional frequency
chirp technique provides for accurate, real-time characterization of the
background noise floor within the radar's IF band, with no loss of
measurement duty cycle. Here the detected power spectrum for fj=167 GHz
is shown. The IF frequency to range conversion factor is 400 kHz km-1.
(b) Echo power spectra normalized to their value at 100 m for the
12 transmission frequencies. The large variability in the signals near
1.4 km indicates the system reaching the noise floor. (c) Echo
power spectra after averaging Nb=11 adjacent bins, and filtered
for points with SNR>-10 dB. (d) Measurement relative
error (blue circles) for all traces in (c) compared with the
statistical model (Eq. , dashed black
line).
Boundary-layer measurements and analysis
Radar characteristics, spectra, and filtering
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. . 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. a, 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 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 Np=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. .
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. a), and subsequently
combining the mirrored spectra, we obtain our estimate of the echo power
spectra. In Fig. b, we plot the echo power spectra
scaled by r2 for the 12 transmission frequencies before bin averaging,
which reveals the range dependence of the quantity Z(r)exp(-2τ(r,f)).
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 ri2Pe(ri,fj) within a swath of size
Nb=11, we filter the spectra based on the Monte Carlo analysis
in Appendix , keeping only those points with
SNR>-10 dB, and are left with the smoothed profiles shown in
Fig. c. Figure d shows the
relative error in the binned (Nb=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. () (black dashed line), indicating that our
statistical model based on speckle noise, which underlies the Monte Carlo
simulations implemented in this work, is accurate.
Water vapor profile retrieval for DAR spectra from
Fig. . (a) Three examples of least-squares
fits of the millimeter-wave propagation model to DAR measurements. Artificial
offsets are imposed in order to plot all three on the same graph.
(b) The retrieved profile exhibits roughly constant absolute
humidity error until SNR≈10 dB (1 km). See
Sects. and for
retrieval details. The green line shows the saturated water vapor density
range dependence using a near-surface temperature of 11 ∘C and lapse
rate of 6 ∘C km-1. The shaded regions correspond to deviations
of ±2 ∘C.
Water vapor profile retrieval
Using the averaged, filtered spectra in Fig. c, we
proceed towards retrieving the water vapor density profile using the
procedure outlined in Sect. . For the profiles
presented in this section, we utilize a retrieval step size of R=200 m.
Beginning with an initial range of r1=100 m, we form the 12 quantities
γi(fj) for each starting ri in the set {r1,r2,…,rm-S}, and perform a least-squares fit of the function
γ^(f) to the data at each range point. Note that the retrieved
water vapor density ρ‾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. and
), the retrieved ρ‾ values are quite
insensitive to the assumed thermodynamic profiles (see
Appendix ).
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. a 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 ri and ri+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 χred2≈1 for these fits. The retrieved water vapor density profile is shown
in Fig. b, where the range r‾i assigned
to each fitted value ρ‾i is the midpoint of ri and
ri+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 ri and ri+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 σρ‾=2.9 g m-3. The value of σρ‾ in the
high-SNR regime (i.e., the first 1 km) remains roughly constant, even
though ρ‾ 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
κv(174.8 GHz) - κv(167 GHz), the retrieval step size
R, and the power measurement uncertainty (see
Eq. ).
Retrievals for different cloud and precipitation conditions.
(a) Averaged DAR power spectra (Nb=11) for light rain
near the surface, with a cloud extending from 1 to 2 km.
(b) Retrieved humidity profiles for two independent data sets
corresponding to the same scene from (a). (c) Averaged DAR
power spectra (Nb=11) for heavy precipitation near the surface
with strong particulate extinction. (d) Independent retrievals from
two data sets for the scene in (c).
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 Np=4000 pulses at each frequency with a total
measurement time of 50 s, and parse the data into two groups of
Np=2000 pulses both spanning the full 50 s. The results are
shown in Fig. , where we also present measurements of
different cloud and precipitation scenarios than that presented in
Figs. and . In
Fig. , 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. , the relative
error in the retrieved humidity σρ‾/ρ‾
(see Eq. ) 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 ρ‾.
Alternatively, one can look at the absolute error and rearrange
Eq. () to find that, for a given pair of
frequencies and power measurements at two ranges, the product of
σρ‾ 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.