AMTAtmospheric Measurement TechniquesAMTAtmos. Meas. Tech.1867-8548Copernicus GmbHGöttingen, Germany10.5194/amt-8-2521-2015A novel approach for absolute radar calibration: formulation and theoretical validationMerkerC.claire.merker@uni-hamburg.dePetersG.ClemensM.LengfeldK.AmentF.Meteorological Institute of the University of Hamburg, Hamburg, GermanyMax Planck Institute for Meteorology, Hamburg, GermanyMETEK Meteorologische Messtechnik GmbH, Elmshorn, GermanyC. Merker (claire.merker@uni-hamburg.de)22June2015862521253020January201505February201504June201506June2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://amt.copernicus.org/articles/8/2521/2015/amt-8-2521-2015.htmlThe full text article is available as a PDF file from https://amt.copernicus.org/articles/8/2521/2015/amt-8-2521-2015.pdf
The theoretical framework of a novel approach for absolute radar calibration
is presented and its potential analysed by means of synthetic data to lay out
a solid basis for future practical application. The method presents the
advantage of an absolute calibration with respect to the directly measured
reflectivity, without needing a previously calibrated reference device. It
requires a setup comprising three radars: two devices oriented towards each
other, measuring reflectivity along the same horizontal beam and operating
within a strongly attenuated frequency range (e.g. K or X band), and one
vertical reflectivity and drop size distribution (DSD) profiler below this
connecting line, which is to be calibrated. The absolute determination of
the calibration factor is based on attenuation estimates.
Using synthetic, smooth and geometrically idealised data, calibration is found
to perform best using homogeneous precipitation events with rain rates high
enough to ensure a distinct attenuation signal
(reflectivity above ca. 30 dBZ). Furthermore, the choice of the interval width (in
measuring range gates) around the vertically pointing radar, needed for
attenuation estimation, is found to have an impact on the calibration results.
Further analysis is done by means of synthetic data with realistic,
inhomogeneous precipitation fields taken from
measurements. A calibration factor is calculated for each considered
case using the presented method. Based on the distribution of the
calculated calibration factors, the most probable value is
determined by estimating the mode of a fitted shifted logarithmic
normal distribution function. After filtering the data set with
respect to rain rate and inhomogeneity and choosing an appropriate
length of the considered attenuation path, the estimated uncertainty
of the calibration factor is of the order of 1 to 11 %,
depending on the chosen interval width. Considering stability and
accuracy of the method, an interval of eight range gates on both sides
of the vertically pointing radar is most appropriate for
calibration in the presented setup.
Introduction
In many domains, for example in weather prediction, nowcasting, or
hydrology, accurate rainfall monitoring is an ongoing
issue. Accurate rain rate estimates can be obtained by using rain
gauges, which are continuously measuring at one point and have
achievable measurement uncertainties of about
5 mm h-1, down to 5 % for precipitation above
100 mm h-1. Even a dense
network of these devices only provides point measurements which are
not able to describe the high temporal and spatial variability of
rainfall events e.g.. Radar networks, such as those from national weather
services or smaller ones operated by research institutions, can
provide spatially and temporally highly resolved, area-covering
rainfall data e.g. and are
already used to improve drainage control and flash-flood warning
systems e.g.. However, a disadvantage of using weather radar data is
the lower accuracy of the retrieved rain rate in comparison to
measurements from rain gauges, since rain rates are not measured
directly but derived from reflectivity measurements. Beside errors
induced by Z–R relations, attenuation, noise, ground clutter, blocking
or interferences, one major limiting factor for precise rain rate
determination is radar calibration
e.g.. Although recent weather radars making use of
polarimetric variables for precipitation estimates are more
accurate and not affected by calibration, polarimetric methods
are not applicable in all conditions, and a demand for absolute
calibration remains.
For relative radar calibration it is common practice to compare
reflectivity measurements from radars that simultaneously
monitor the same rainfall event
e.g.. This adjustment only works
for radars within a network and do not allow for an absolute
comparison of data with other networks or
instruments. Furthermore, the retrieved rain rate cannot be used
quantitatively. A frequently used approach is a calibration with
respect to rain rate using point measurements from rain gauges
e.g. or disdrometers
e.g. at ground
level. This implies the disadvantages of point-to-area
comparison and differences in measuring height
e.g.. Additionally, the obtained calibration is highly
dependent on the chosen Z–R relation
e.g.. In order to avoid this
source of error, calibration with respect to radar reflectivity
is preferable.
Vertically pointing micro rain radars (MRRs), using a frequency
modulated continuous wave (FM-CW) measuring principle and
operating at K band (24.1 GHz, λ= 12.4 mm) , allow a comparison of data
at same height levels, but compared measuring volumes are still not
necessarily equal, and micro rain radars also lack an absolute
calibration e.g..
A novel method for absolute calibration using a setup of three radars,
performed without previously calibrated reference device, and
calibrating with respect to reflectivity is presented here. It
requests a radar network setup and takes advantage of the attenuation,
which is generally seen as perturbing effect on measurements. The aim
of this paper is the theoretical formulation and the proof of concept
validation of the presented method, which has not been investigated before.
Schematic network setup including horizontally oriented radars R1
and R2 and vertically oriented radar R3 (top panel) and schematic reflectivity
measurements for R1 and R2 along the connecting line (bottom panel).
The analysis of this absolute calibration method focuses on the
application on MRRs. Nevertheless, considered instrumental setup and
theoretical framework of the method presented in
Sect. are applicable to any strongly attenuated radar
type. A proof of concept validation is realised in
Sect. , and validity and potential of the
method are analysed further by means of synthetic data presenting
realistic measurement structures in
Sect. . From this study, criteria of
appropriated rainfall events for calibration are worked out for
subsequent utilisation.
Theoretical framework
The network setup required in order to apply the absolute calibration
method presented here is depicted schematically in
Fig. . Two horizontally oriented radars (R1 and
R2) measure along the same connecting line from opposite directions
at a certain height. A third, drop size distribution (DSD) profiling
device (R3) is positioned below the measuring path in order to
provide measurements at one point of the connecting line. For the
sake of simplicity, the focus of the study presented in the following
is on the calibration of R3. However, it is straightforward to
calibrate R1 and R2 once R3 is calibrated.
Considering that the two horizontally oriented radars R1 and R2
(Fig. ) operate at a strongly attenuated
frequency (e.g. K band) and
measure on the same path from opposite directions, the measured
reflectivity provided at a point s on the connecting line,
Z1(s) and Z2(s), can be expressed as follows e.g.:
Z1(s)=C1⋅Z(s)⋅exp-2∫s0sk(s′)ds′,Z2(s)=C2⋅Z(s)⋅exp-2∫ssmaxk(s′)ds′.
The positions of R1 and R2 are denoted s0 and smax,
respectively. Similarly, the measured reflectivity provided by the
vertically pointing radar R3 at a height z is
Z3(z)=C3⋅Z(z)⋅exp-2∫0zk(z′)dz′.
The measured reflectivity differs from the intrinsic reflectivity
Z(s) by the multiplicative calibration factors C1,
C2 and C3 comprising device characteristics
and by the two-way attenuation, with k the specific attenuation. The
latter is given by
k(s)=∫DminDmaxN(D,s)σe(D)dD,
where N(D, s) is the drop size distribution (droplet number per unit
volume and per unit size interval) dependent on the drop diameter D
and σe(D) the extinction cross section.
The beam of the vertically pointing Doppler radar R3 crosses the
connecting line at s3, which will be called the reference point in the
following. At this point, not only the reflectivity but also the
discrete DSD N3(Dj, s3, h) can be derived from the measured Doppler
spectra according to the method of , using an
analytical relation between drop terminal velocity and drop size
in the absence of vertical winds. Because N
is proportional to Z, the relation between measured and
intrinsic DSD is analogous to the expression for the reflectivity:
N3Dj,s3,h=C3⋅NDj,s3,h⋅exp-2∫0hk(z′)dz′,
where h is the height of the reference point above R3 and
C3 the same as in Eq. (). Now we
determine the measured attenuation using Eq. ()
with discrete size classes Dj:
k3s3=∑DminDmaxN3Dj,s3,hσeDjΔDj.
Here, ΔDj is the width of the size classes. Replacing the
measured DSD N3(Dj, s3, h) by Eq. () yields
k3s3=∑DminDmaxC3NDj,s3exp-2∫0hk(z′)dz′σeDjΔDj=C3⋅exp-2∫0hk(z′)dz′⋅∑DminDmaxNDj,s3σeDjΔDj=C3⋅exp-2∫0hk(z′)dz′⋅ks3.
In the following, assuming homogeneous conditions in the environment
of the reference point allows for resolving Eq. () for C3:
C3=k3s3exp-2ks3h⋅ks3.
This implies the necessary condition of a constant specific
attenuation in the vertical
section between R3 and the height of the measuring
path (about 40 to 80 m, depending on the network setup).
While k3(s3) is known from Eq. (), k(s3) is
derived by comparing reflectivity measurements from R1 and R2
along a selected section of the measuring path. Again, constant
specific attenuation along this particular
section is required. Considering the high spatial variability of
rainfall on small scales e.g., one important challenge of the
method becomes obvious here. The section
bounds are located at s3-Δs and s3+Δs on both sides
of R3. From Eq. (), the ratio between measured
reflectivity at s3-Δs and measured reflectivity at
s3+Δs for each radar R1 and R2 separately
yields
Zs3-ΔsZs3+Δs=Z1s3-ΔsZ1s3+Δs⋅exp-4ks3Δs
and
Zs3-ΔsZs3+Δs=Z2s3-ΔsZ2s3+Δs⋅exp4ks3Δs.
Notice that the calibration factors C1 and C2 cancel out at this
point, ensuring the absolute determination of the specific attenuation
k(s3) needed for absolute calibration.
Equalising Eqs. () and () and
rearranging terms gives an expression in which only the specific
attenuation k remains as a function of known values:
Z1s3-Δs⋅Z2s3+ΔsZ1s3+Δs⋅Z2s3-Δs=exp8ks3Δs.
By assuming a constant attenuation factor k(s3) along the
considered section, the latter can then be expressed by
ks3=lnZ1s3-Δs⋅Z2s3+ΔsZ1s3+Δs⋅Z2s3-Δs⋅18Δs.
Having determined the absolute specific attenuation at s3,
comparison with the specific attenuation k3(s3) obtained from DSD
measurements of R3 (Eq. ) allows for absolute
calibration of R3. Combining Eqs. () and ()
yields the equation for the absolute
calibration factor C3. The total expression for C3 now only
depends on measured values.
This calibration approach is valid
provided that the DSD measured by R3 is representative for the DSD
along the measuring path between s3-Δs and s3+Δs.
Also, it is worth mentioning that errors in DSD
measurements from R3 propagate directly to the obtained
calibration factor through the specific attenuation k3(s3). The main source of errors
for DSD when measuring under real conditions with MRRs is
vertical wind. Considering a study from , specific
attenuation can be overestimated by a factor of 2 for
1 m s-1 vertical wind, which is substantial. However,
an analysis of vertical wind data with 10 s resolution at a height of
50 m (Wettermast Hamburg site, Germany) yields a
standard deviation of only 0.49 m s-1 for rainfall
events. This strongly reduces the possible error to a
factor of about 1.4. Since convective precipitation events (inducing
strong inhomogeneity) are not suited for calibration,
the typical vertical wind variance in considered cases, and thus
the measuring error, is even lower. Furthermore, when taking into
account the measuring volume of the MRR, which also reduces
fluctuation, the error in specific attenuation should be very
small.
In the following, the inverse of C3 is considered, since
C3-1 is the factor with which data is corrected after calibration.
Proof of concept
In order verify the theory of the presented method, it is evaluated
using synthetic data obtained from a forward model generating
reflectivity out of a given rain rate. These data represent perfectly
calibrated devices for validation purpose, which means correction
factors C1-1, C2-1 and C3-1 are implicitly set to 1.0. The
measuring path described in Fig. is divided into 31
range gates of Δr= 200 m width each. Simulated
data for R1 and R2 are discretised accordingly. After the
discretisation, 1 ≤i≤ 31 describes the range gates along the
measuring section, starting at R1. We also define the interval
between s3-Δs and s3+Δs to be (2 n+ 1) Δr,
where n is the number of considered range gates on both sides of the
range gate comprising R3.
A rain rate pattern R(i) is freely defined according to the
requirements of the simulated case. The rain rate is assumed to be
constant within one range gate. Out of this, the DSD N(Dj, i) is
calculated for each range gate using the Marshall–Palmer standard
distribution . Drop diameter classes Dj from
0.15 to 6.5 mm with a class width of 0.05 mm are used
here. Then, the theoretical, intrinsic reflectivity Z(i),
depending on DSD and backscattering cross section, is
calculated. Finally, attenuated reflectivities
Z1(i), Z2(i) and Z3(h) – describing measurements from
R1, R2 and R3, respectively – are calculated in analogy with
Eq. (). The specific attenuation k3(i) required to
simulate the attenuated reflectivity is derived from
Eq. (). The extinction cross section is
calculated using Mie theory according to ,
considering droplet flattening.
Idealised synthetic data simulated along the measuring
path. Homogeneous precipitation pattern with a rainfall intensity of
15 mm h-1 (top panel) and corresponding reflectivity fields simulated using
forward operator (bottom panel). Intrinsic reflectivity is shown in green and
simulated, attenuated reflectivity in red for R1 and blue for R2. The
vertical black line marks the position of R3.
This study aims at analysing how the calibration accuracy depends on
rainfall intensity and structure, and on the width of the interval
chosen for determination of specific attenuation k(s3) around
R3, defined by n. Synthetic data are simulated for two idealised
rainfall patterns (homogeneous rain intensity along
(Fig. ) or featuring a maximum in the middle of
(Fig. ) the measuring path) with 15 different
rainfall intensities in each case according to the method described
above. For these rainfall patterns calibration is performed using
12 different widths (2 n+ 1) Δr, with n varying from 1 to 12 range
gates on both sides of R3. Hence, the sensitivity study comprises
180 different combinations of rain rates R(i) and interval widths
(2 n+ 1) Δr. In order to take into account measurement
uncertainties, which are inherent to data, calibration is run in
a Monte Carlo simulation with 100 repetitions for each combination,
assigning random errors to the data for each of the 100 repetitions.
For this the simulated radar reflectivity fields are
overlaid with a Gaussian distributed noise with a standard deviation
of 2 dBZ for each range gate. For both rainfall patterns,
mean and standard deviation of the correction factor C3-1 are
calculated from the Monte Carlo simulation results.
In a first analysis, a homogeneous rainfall pattern is considered with
rainfall intensities varying between 1 and 15 mm h-1. Mean
correction factors C3-1 (Fig. a) are close to the
expected value of 1.0 (perfectly calibrated radar), between 0.995
and 1.005 for most of the tested combinations. Higher deviations can be
found at short interval widths with 1 ≤n≤ 3 range gates and
for rain rates below 5 mm h-1 (corresponding to
approximately 30 dBZ) where values range between 0.95 and 1.25.
The stability of these results, quantified by the standard
deviation of the obtained factors (Fig. b), increases
towards higher rain rates and larger (2 n+ 1) Δr as the standard
deviation decreases. Starting at high values above 2.00 (3.00 for
n= 1 and lowest rain rate), which indicate a spread of more than
twice the expected value of the calibration factor, the standard
deviation reaches values below 0.1 for R≥ 5 mm h-1
and n≥ 4. The standard deviation decreases with increasing rain
rate and range gate number n, reaching down to 0.016. Hence,
calibration of R3 is found to be best for high rain rates and large
n, both having the effect of producing a clear and detectable
attenuation signal along the considered path section required by the
calibration method. Note that, when considering real conditions,
the assumption of a homogeneous DSD within the section of
interest becomes less applicable the larger the section is;
i.e. the larger n is chosen.
In analogy with Fig. , rain rate and simulated
reflectivity fields for a precipitation pattern with a maximum intensity above
R3 (maximum rain rate is 13.3 mm h-1).
Mean (a) and standard deviation (b) of the
correction factor C3-1 for homogeneous precipitation patterns,
calculated from Monte Carlo simulations including 100 repetitions with random
measuring error. The Monte Carlo simulations are performed for combinations of
15 different intensities of the homogeneous precipitation field and 12 considered
interval widths (2 n+ 1) Δr for attenuation determination
above R3.
Mean (a) and standard deviation (b) of the correction
factor C3-1 for Gaussian-shaped rain fields, calculated from Monte Carlo
simulations including 100 repetitions with random measuring error. The Monte
Carlo simulations are performed for combinations of 15 different rainfall
intensities and 12 considered interval widths (2 n+ 1) Δr for attenuation determination
above R3.
Distributions of the correction factor C3-1 calculated using
synthetic data with realistic precipitation patterns and fitted logarithmic
normal distribution functions for six values of n. The total amount of
considered time steps is indicated by N. The mode, 25th and 75th
percentile of the fitted distributions are shown in green, blue and red,
respectively.
The study of the second analysed precipitation pattern, characterised
by a maximum value in rain rate above the location of R3,
investigates possible impacts of precipitation maxima or minima on the calibration
results. This precipitation field (Fig. ) is
created by using the shape of a normal distribution in order to define
a rain rate R(i) in each range gate from a maximum rain rate. The
maximum rain rate (value above R3) is varied between 4.0 and
13.3 mm h-1, creating 15 fields with different rainfall
intensities. Figure a and b depict the results for
averaged correction factor C3-1 and corresponding standard
deviation (note that the colour scales of Figs. a
and a are different). The calibration results differ from the one discussed above
for homogeneous rain fields. Here, the mean calibration factor is well
determined only for high rainfall intensities and small intervals
(2 n+ 1) Δr where it lies in the interval between 0.95
and 1.05. For high rain rates and increasing interval width, the
correction factor shows a large negative bias with values down to 0.313,
which represents an error of almost 70 % in the
calibration. The standard deviation is less dependent on rainfall
intensity. Stable results (standard deviations below 0.1) are achieved
for n> 6. A slight tendency toward lower standard deviations at
higher rain rates is still visible. This obtained bias in the
correction factor C3-1 can be explained by the shape of the
precipitation field. Maximum rain rate above R3 induces an observed
(attenuation-corrected) DSD, and consequently attenuation, at the
position s3 that is not representative for the whole interval
(2 n+ 1) Δr. Attenuation corrected specific attenuation
k3(s3)⋅exp(2k(s3)h) calculated from DSD measurements of
R3 (Eq. ) is then higher than specific attenuation
k(s3) calculated from measurements of R1 and R2 along the
considered interval (Eq. ), which assumes
a constant, averaged specific attenuation. Consequently, correction
factor C3-1 is erroneously found to be smaller than 1.0
(Eq. ). The opposite effect occurs in the case of
a minimum in the rainfall intensity above R3.
Test on synthetic data with realistic precipitation patterns
Since the presented method is found to be valid when using idealised,
smooth precipitation patterns, a further study is realised based on
data with realistic rainfall patterns. For this purpose, measured MRR
data from a network installed at the Meteorological Observatory
Lindenberg (MOL) operated by the German Meteorological Service (DWD)
are used in order to create synthetic data with a realistic texture as
given by measurements. The network fits the conditions introduced in
Sect. , and data were recorded between the beginning of May
and end of June 2013. Within this period 15 rainfall events,
comprising 4220 10 s time steps in total, are chosen for
testing the calibration method. For slanted devices R1 and R2,
reflectivity is calculated directly by integration of the power
spectrum, as done before by e.g. and .
The values given by the standard MRR software are derived from the
measured DSD and are only valid for vertically pointing MRRs.
In order to obtain realistic precipitation fields, reflectivity
measurements from both horizontally oriented MRRs are used to
generate synthetic, intrinsic reflectivity fields along the
path. These synthetic, intrinsic reflectivity fields are created
by comparing and combining measurements from R1 and R2
such that the highest reflectivity value of both is selected in
each range gate. Using measurements from just one MRR would
yield synthetic, intrinsic reflectivity fields showing a
systematic decrease in reflectivity toward one side of the
measuring path, as an artefact of attenuation present in real
measurements. From the obtained reflectivity fields, rain rate
and synthetic, attenuated reflectivity for all three radars are
simulated according to the procedure described in
Sect. . All devices are still considered
to be perfectly calibrated for this analysis, yielding
a correction factor C3-1 of 1.0.
Since two characteristics of reflectivity fields have been detected to
be disadvantageous for calibration in Sect.
(high heterogeneity and low rain rates along the measuring section),
the simulated reflectivity fields are filtered using two parameters in
order to remove unsuited cases. The prerequisite for a good
calibration is an attenuation effect strong enough to be detected
reliably. Therefore, the rainfall intensity along the measuring path
has to be high enough to achieve the required signal
extinction. Averaged reflectivity along the measuring path is
calculated and a threshold is set to 30 dBZ. Furthermore,
strong inhomogeneities, evidence of high noise or disturbances in
the measurements, can falsify calibration. Therefore these data are
filtered out. This is done by using the texture of the reflectivity in dBZ (TDBZ) according to :
TDBZ=∑i(dBZ(i)-dBZ(i-1))2⋅1I.
TDBZ is an indicator for fluctuation of reflectivity along the path
calculated by summing up the squared differences in reflectivity
between adjacent range gates i. Here, I represents the total
number of range gates considered for the TDBZ calculation, which has
to be chosen appropriately. A TDBZ threshold over the considered I= 11
range gates in the middle of the measuring path is set to
1.4 dBZ2. This threshold is defined by studying the quality of calibration
results running a Monte Carlo simulation as presented in Sect.
for the 4220 chosen time steps. Ninety percent of the cases having at most 10 % error in the
determination of the correction factor and standard deviation
below 1.0 have to show TDBZ lower than the threshold. After filtering with averaged rain rate and TDBZ,
3246 suited time steps remain for calibration and are used in the
following. After generation and selection of synthetic reflectivity
fields, calibration of R3 is performed in order to analyse the
behaviour of the calibration method when applied to data showing
realistic patterns. The calibration is performed once for each time
step, with an added random, Gaussian-shaped fluctuation with
a standard deviation of 2 dBZ which simulates measurement
uncertainties. Eleven different intervals (2 n+ 1) Δr are
considered, with n varying from 2 to 12 (n= 1 was found to provide
unsatisfactory results in Sect. ).
Mode, 25th and 75th percentiles and interquartile range calculated
from the logarithmic normal distribution functions fitted to the results for
C3-1 for 2 ≤n≤ 12 (rounded values). Calibration was
performed using synthetic data with realistic precipitation structures.
Calibration using the selected time steps and intervals (2 n+ 1) Δr
does not achieve precise results in all cases. Resulting correction
factors are spread over a wide range of values. In order to define
a method for the determination of the wanted correction factor,
distributions of obtained C3-1 are studied
(Fig. ). For each considered number of range gates
n, the calculated correction factors are considered among 20 classes
with a width of 0.2. Time steps providing negative calibration factors
are removed, resulting in a different number N of remaining
calibration results for the 11 different interval widths. Those
negative results have no physical meaning, as corrected reflectivity
fields would then also appear to be negative
(Eq. ). According to Eq. () negative
correction factors appear if k(s3)≤ 0. Since this study is based
on synthetic data, as described above, non-physical, negative specific
attenuation values are an artefact of the added random noise creating
strong inhomogeneities and missed by the TDBZ filter. These cases
should not be considered. Distributions of the correction factor have
their maximum within the class including 1.0 (except for n= 10),
which is the expected correction factor for a perfectly calibrated
radar, and are positively skewed. As discrete distributions only allow
for the estimation of a median within the given resolution, a shifted
logarithmic normal distribution function is fitted to the discrete
distribution in order to provide more precise results. The most
probable value of the wanted correction factor is given by the mode of
the distribution function, and thus the most frequently calculated
value. In order to describe the width of the distribution, which
describes the accuracy of calibration, the interquartile range
(describing the distance between the 25th and 75th percentile) is
also considered (Table 1). Obtained distribution
functions have the lowest interquartile range for 4 ≤n≤ 8,
indicating the most stable results for these settings. The widest
spread of the calculated correction factor is found using n= 2. The
error in the estimation of the correction factor, knowing the true
value is 1.0, reaches 11 % for n= 2 and is lowest for
n≥ 8, where C3-1 is determined by the mode with an error of only
1 %. It is exactly 1.0 for n= 9. These results
suggest the possibility to achieve satisfactory calibration results
when applying the presented calibration method to carefully chosen
data and settings. Since a considered interval (2 n+ 1) Δr with
4 ≤n≤ 8 yields the most stable results, and the most accurate results
are achieved with n≥ 8, n= 8 is possibly the most appropriate
setting for calibration in the network considered here.
Summary and conclusions
A novel method for absolute radar calibration with respect to
reflectivity and without reference device is analysed in this
study. The method is first
tested using synthetic data with idealised, smooth precipitation
patterns to prove the validity of the concept. Homogeneous and
Gaussian-shaped precipitation patterns are analysed and calibration
performed considering simulated data with a measuring uncertainty of
2 dBZ. Furthermore, different rainfall intensities and
interval widths (2 n+ 1) Δr for determination of the attenuation
k(s3) are taken into account. Calibration using homogeneous
precipitation patterns yields precise results. Mean correction factor
C3-1 for R3 takes values between 0.995 and 1.005, except from
calibration at rain rates below 5 mm h-1 and with n≤ 3.
This represents an error of 0.5 % in the determination of
the true calibration factor, which is 1.0 for a perfectly calibrated
radar as simulated in this analysis. The standard deviation,
expressing the stability of the procedure, stays below 10 %
for R≥ 5 mm h-1 and n≥ 4. It decreases towards
higher rain rates and larger interval widths, reaching down to under
2 %. Inhomogeneous precipitation patterns, featuring
higher (lower) rainfall intensity than average above R3,
reveal one weakness of the method. Due to less likely high (low)
rain rates above R3, its measurements are erroneously
corrected toward lower (higher) average reflectivity values,
inducing a negative (positive) bias in the calculated correction
factor. This bias is stronger the higher the rain rate and the
larger the interval along which the attenuation is
determined. Since a minimum in rainfall intensity above R3 is
likely to occur as often as a maximum, inducing a positive
bias, this effect will be cancelled out when averaging over numerous rainfall events.
The theoretical validity of the presented absolute calibration method
has been proved for adequate precipitation patterns. Ideal cases are
preferably homogeneous, intense rainfall along the measuring path,
leading to attenuation strong enough to be determined reliably.
Since these first promising results are obtained focusing on idealised
synthetic data, a further study is done with synthetic data featuring
realistic structures. Those structures are taken from reflectivity
measurements from a real deployment of MRRs following the concept
introduced in Sect. . Considered time steps are
filtered using averaged reflectivity along the measuring path and
texture of the reflectivity field TDBZ in order to guarantee high
attenuation and smooth measurement structures. Calibration over a
3246 sample of filtered time steps of reflectivity measurements leads to
distributions of calculated correction factors which can be described
using shifted logarithmic normal distribution functions. These
functions are fitted to the obtained, results and the mode is
calculated in order to describe the sought-after correction factor for
each considered n. Here, the calibration results vary between 0.89
and 1.01, i.e. up to 11 % error. The best results are achieved using n= 8.
Future analysis applying the method to selected network data,
including comparison with reference devices and established
calibration methods, have to be performed in order to prove its
applicability in practice. Some problems will arise from using
real, measured data, and their impact on calibration accuracy
must then be evaluated. Regarding the network setup, accurate
alignment of R1 and R2 and the height of the measuring
path should be considered in order to ensure consistent
measurements and minimise beam blockage from the ground. Since
measuring volumes are different, the vertical variability of DSD
can induce errors when comparing measurements in strongly
inhomogeneous cases. Furthermore, errors in the DSD measurements
from R3, caused mainly by vertical winds, should be
analysed. In order to obtain precise calibration, it will be
important to improve the criteria to select suitable
data. Beside testing the presented parameters (rain intensity
and TDBZ for rain homogeneity), removing convective events (with
strong turbulence) and time steps with vertically strongly
varying DSD (making use of measurements from R3) is probably
necessary. The sensitivity of calibration with respect to the
integration time, finding the optimum between minimising noise
and still resolving rainfall variability, also needs further study.
Nevertheless, the fundamental analysis of the presented novel
method for absolute radar calibration in a network proves its
theoretical validity. The method could offer great opportunities
for absolute calibration of radar networks operating in strongly
attenuated frequency ranges (e.g. K and X band),
providing accurate and comparable data for application.
Acknowledgements
Thanks are due to Ulrich Görsdorf and the team of the Meteorological
Observatory Lindenberg (MOL) of the German Meteorological Service (DWD) for
supporting the project and enabling the setup of a micro rain radar network
at MOL as well as to Hans Münster for installing and maintaining the
installed devices.
This work is part of the Precipitation and Attenuation Estimates
from a High-Resolution Weather Radar Network (PATTERN) project and
is funded by the Deutsche Forschungsgemeinschaft (grant AM308/3-1).
Edited by: G. Vulpiani
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