AMTAtmospheric Measurement TechniquesAMTAtmos. Meas. Tech.1867-8548Copernicus PublicationsGöttingen, Germany10.5194/amt-11-6409-2018Can turbulence within the field of view cause significant biases in radiative transfer modeling at the 183 GHz band?Turbulence and 183 GHz biasesCalbetXavierxcalbet@googlemail.comhttps://orcid.org/0000-0003-0058-8425Peinado-GalanNiobeDeSouza-MachadoSergioKursinskiEmil RobertOriaPedroWardDaleOtarolaAngelhttps://orcid.org/0000-0002-9789-2564RípodasPilarhttps://orcid.org/0000-0003-2509-9337KiviRigelhttps://orcid.org/0000-0001-8828-2759AEMET, C/Leonardo Prieto Castro, 8, Ciudad Universitaria, 28071 Madrid, Madrid, SpainPhysics Faculty, University of Valencia, Carrer del Dr. Moliner, 50, 46100 Burjassot, Valencia, SpainJCET, University of Maryland, Baltimore County, Baltimore, MD, USASpace Sciences and Engineering, Boulder, CO, USAHydrology and Atmospheric Sciences Department, University of Arizona, Tucson, AZ, USATMT International Observatory, Pasadena, CA, USAFinnish Meteorological Institute Arctic Research Centre, Tähteläntie 62, 99600 Sodankylä, FinlandXavier Calbet (xcalbet@googlemail.com)30November20181112640964174June201830July201826October201829October2018This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://amt.copernicus.org/articles/11/6409/2018/amt-11-6409-2018.htmlThe full text article is available as a PDF file from https://amt.copernicus.org/articles/11/6409/2018/amt-11-6409-2018.pdf
The hypothesis whether turbulence within the passive microwave
sounders field of view can cause significant biases in radiative transfer
modeling at the 183 GHz water vapor absorption band is tested. A novel
method to calculate the effects of turbulence in radiative transfer modeling
is presented. It is shown that the turbulent nature of water vapor in the
atmosphere can be a critical component of radiative transfer modeling in
this band. Radiative transfer simulations are performed comparing a uniform
field with a turbulent one. These comparisons show frequency dependent biases
which can be up to several kelvin in brightness temperature. These biases can
match experimentally observed biases, such as the ones reported in
. Our simulations show that those biases could be
explained as an effect of high-intensity turbulence in the upper troposphere.
These high turbulence phenomena are common in clear air turbulence, storm or
cumulus cloud situations.
Introduction
Radiative transfer models (RTMs) are key tools in the microwave and infrared
atmospheric remote sensing of the atmosphere. They are used to model the
radiances at the top of the atmosphere as measured from satellites. The main
inputs to the RTMs are atmospheric profiles of temperature, water vapor and
trace gases, as well as surface properties, such as surface temperature and
emissivity. The RTM simulate the absorption and emission of the molecular
constituents of the atmosphere in a layer-by-layer approach. Their accuracy
depends on the accuracy to which the spectral properties of the molecular
absorption/emission lines are known, as well as the quality and vertical
resolution of the vertical profiles of temperature, pressure and
concentration of absorbing gases. By applying inversion techniques to these
RTMs, such as Optimal Estimation , the physical parameters
of the atmosphere can be obtained from radiances observed from satellites.
These inversion techniques are commonly used in retrieval processes, where
atmospheric properties are directly estimated from the measurements. They are
also used in Numerical Weather Prediction (NWP) models, where measured
radiances are assimilated as corrections to forecasts to form what are best
estimates of the atmosphere, known as analyses. RTMs are, therefore, elements
that bridge the gap between measured satellite radiances and atmospheric
physical parameters such as profiles of temperature and water vapor
concentration.
Although RTMs have achieved a high degree of accuracy, there is often, in
practice, a systematic mismatch between what is observed and what is
calculated from the RTMs. Their cause is varied and can range from an
incorrect or incomplete implementation of the radiative transfer model setup,
to uncalibrated instrumental effects or deviations in their nominal
performance. These systematic mismatches are usually solved, in practice, by
determining, for a particular satellite instrument, an offset or bias between
measured and calculated radiances. This bias is then later corrected for
retrievals and NWP assimilation systems. This practical fix is far from
perfect, since deviations between systems are corrected with a simple bias or
offset, when the reality can be more complex, depending on the underlying
physical principles behind these systematic mismatches. For example, if a
satellite system is deviating from its nominal behavior, ideally, the
physical principles behind it should be sought for and corrected. If, on the
other hand, radiative transfer is not modeled properly, the missing physical
pieces should be put in place such that satellite measurements and RTM
calculated radiances are consistent.
One of such mismatches occur in the 183 GHz water vapor absorption band,
which has been noted by and . To discuss
this effect, a “Joint workshop on uncertainties at 183 GHz”
was convened to discuss biases observed between
measurements at 183 GHz and calculations using different RTMs plus either
radiosondes or short range forecasts from NWP systems. The results were
reflected in a paper published one year later . The
values of the differences between observed and calculated brightness
temperatures between the SAPHIR Megha-Tropiques instrument vs. Météo
France NWP profiles plus the RTTOV v11 RTM are reproduced here as blue dots
in Fig. . For further information and more results, with
data from other instruments and other NWP models, please refer to
.
The potential causes behind these biases can be varied. Many plausible
hypothesis could be formulated, such as poorly modeled gas absorption,
effect of clouds, etc. In this paper we will discuss whether these
significant detected biases can be caused by turbulence effects in the
atmosphere. The rationale behind this hypothesis is based on the fact that
radiative transfer is an extremely non-linear process. It is then possible
that the average of the radiances at the top of the atmosphere, obtained from
several neighboring different atmospheric columns, are not necessarily equal
to the radiance computed from the average of all atmospheric columns. Which
in turn, implies that radiances coming from a turbulent medium within the
field of view of the instrument may be different from the radiances
originating in a uniform medium.
Turbulence is a well known phenomena occurring in the atmosphere. Its spatial
properties are commonly measured from the ground using in situ or remote
sensing instruments. Turbulence in the troposphere is measured using sonic
anemometers on meteorological towers , hot wire
anemometers suspended on tethered lifting systems , wind
profiling radars , passive microwave sounding
, Doppler wind lidar (see review from
and references therein), elastic backscatter lidar
(e.g.,
), ozone differential absorption lidar ,
water vapor differential absorption lidar (e.g., ;
; ; ), and water vapor
Raman lidar (e.g., ; ;
). Individual sonde measurements have also proved to be a
suitable instrument to measure turbulence (;
).
In the stratosphere, the measure of turbulence is tightly linked to the
measure of gravity waves, since they share similar time and spatial scales.
Detection of turbulence or gravity waves is usually made with remote sensing
instruments. One of them is the satellite based microwave limb sounder (MLS;
) with which gravity waves can be detected (e.g.,
; ). Passive microwave sounders on board
of polar orbiting satellites have also been used to detect gravity waves
(e.g., ). Airglow measurements can also be used to detect
gravity waves, either using ground-based measurements (e.g.,
) or space-borne ones (e.g., ). Infrared
hyperspectral sounders such as AIRS and IASI have also proven to be useful
for the measurement of gravity waves ().
The effect of turbulence in the modeling of the propagation of light in the
atmosphere is taken into consideration in certain fields of astronomy and
communication, where different effects of scintillation are studied. Studies
of this sort are abundant in the literature (e.g., ;
). Although high variability effects in the atmosphere are
taken into account when looking for gravity waves in the stratosphere (e.g.,
), they are often neglected in operational radiative
transfer modeling for passive microwave sounders. The main reason for this
is that operational radiative transfer models need to be fast and any saving
in the already relatively heavy computation time is welcomed. Therefore,
phenomena that do not, at first sight, seem to be needed are excluded. One
such example is the effect in radiative transfer modeling of atmospheric
turbulence within the field of view of microwave or infrared instruments. In
this paper, an attempt to evaluate its significance in the radiative transfer
modeling of the microwave 183 GHz band is shown. It will be shown that this
effect can be quite important in some situations. The intensity of turbulence
can vary several orders of magnitude within the troposphere. The typical
values that will be used in this paper will be taken from , in
particular, its Fig. 2. Typical values in the low troposphere are usually in
the range between 1 and 10 cm2 s-3. Close to the tropopause,
values vary greatly spanning from 10-3 to 104 cm2 s-3. The latter huge values occur in clear air turbulence,
severe storm or cumulus cloud situations. It should be noted that in some
particular cases, turbulence intensity could be outside of these limits and
could well exceed the maximum values that are stated there.
In Sect. a quantitative estimation of turbulence is
made from measurements from dual sequential radiosondes. Section shows how to calculate the effects of turbulence in radiative
transfer modeling. In the discussion section, Sect. ,
its effect in the top of the atmosphere radiances is calculated and
comparisons are made with the observed biases . In the
conclusions section a summary of the results is made.
Turbulence from radiosondes
In this section, an estimation of the order of magnitude of atmospheric
turbulent variables is performed. Parameters such as turbulent energy
dissipation rates, turbulence length scales and turbulent atmospheric
parameter variance are determined from sonde measurements. The main objective
is to estimate the order of magnitude of these parameters, and, by the use of
an RTM, an estimate of the order of magnitude of turbulent effects in
microwave brightness temperature can be made.
The analysis of radiosonde data for its comparison with satellite
measurements has shown that temperature and water vapor have a spatial
behavior that is difficult to determine from just one radiosonde measurement
. More than one measurement is needed to resolve the small
scale variability of the atmosphere, such as launching two consecutive sondes
. Another alternative is to make a spatial
average of the measurements or to have a low spatial
variability to have consistency between radiosonde and
satellite measurements. The reason behind this is that spatial and temporal
scales for water vapor are extremely small, as shown, for example, by
. This kind of behavior is typical of turbulent systems,
such as the atmosphere.
Structure functions are tools that are often used in turbulence measurement, which are closely
related to spatial auto-correlation functions. To effectively determine and
quantify the turbulent nature of the atmosphere, the structure function of
temperature and water vapor was determined. This was done with radiosonde
data coming from the EUMETSAT MetOp campaigns which took place in 2007 and
2008 at Lindenberg and Sodankylä. In these campaigns, two consecutive
radiosondes were launched from the same site with 50 min time difference.
From now on, they will be referred here as sequential sondes. From Lindenberg
and Sodankylä 266 and 359 sonde pairs were launched, respectively, making a
total of 625 sonde pairs. The instrument payload analyzed here are the
conventional RS92 radiosondes. For more details of this campaign and its
instrumentation we refer the reader to . The data was later
processed by GRUAN , which, among other advantages,
greatly removes the humidity measurement dry biases usually present in RS92
measurements at the high troposphere .
The computed structure functions for temperature and water vapor are shown
in Fig. . In the case of the temperature field, differences
at the same pressure levels between the sequential sondes were calculated. To
this difference, an effective distance was assigned. This effective distance
is the real spatial distance between sequential sondes plus the time
difference multiplied by the wind speed measured by the radiosondes at that
level. The average of the square of this temperature difference is then
calculated for different effective distance bins. The same analysis was
performed for water vapor, but in this case, using the difference in water
vapor partial pressure, e, divided by the average of the two water vapor
partial pressures from the sequential sondes. In order to achieve a
significant sample size, the results for all radiosonde pressure levels have
been combined. The resulting total number of data pairs, coming from the 625 sequential sonde pairs, is 658 217. Results are shown in Fig. .
The typical behavior for turbulence, as measured in the laboratory, is
observed in the structure function plots (Fig. ). For very
small scales, which are not observable in these sequential sonde
measurements, it constitutes what is known as the dissipation range
(delineated by blue arrows in the figure). As the scale is enlarged, the
inertial range is seen (red arrows). This range spans from very small scales
to approximately 6 km. A decrease in the structure function is usually
observed at larger scales, where forcings are induced, which constitutes the
energy injection range (green arrows in the figure). Finally, at very large
scales, the synoptic differences are observed (magenta arrows).
Temperature and water vapor structure function.
Within the inertial range, the structure function typically exhibits a power
law behavior with a two third exponent following Kolmogorov's theory of turbulence
. The two-thirds law from Kolmogorov states that
〈(δv(l))2〉=Cε2/3l2/3,
where v is a parameter measured in the fluid, C is a universal
dimensionless constant, which needs to be determined from experimental data,
and l is the distance between the points where the parameter difference is
determined. The mean energy dissipation rate per unit mass, ε,
effectively constitutes a measure of the intensity of turbulence. The
measurements from sequential sondes, within the inertial range, follow this
law remarkably well. The measured slopes are 0.84±0.14 and 0.78±0.18 for temperature and water vapor, respectively. These values are
consistent with the theoretical one of two thirds.
To determine the constants in the above equation, the four-fifths law from
Kolmogorov's theory is needed
〈(δv∥(l))3〉=-45εl,
where v∥ is the velocity in the direction of the flow. This
quantity can be estimated from the radiosonde measurements by using the u
wind component. This relationship is shown in Fig. . A
value of ε=5.2 cm2 s-3 fits well within the inertial
range. This is typical of the lower troposphere, with values usually between
1 and 10 cm2 s-3.
Radiative transfer modeling
We can now try to answer the question whether turbulence can have a
significant effect on the RTM calculations. In this section, the radiance
originating from a single profile will be compared to the one generated from
this same profile perturbed by turbulence.
The turbulent perturbation from a single profile will be calculated by means
of averaging a Taylor expansion of the radiances over the inertial range
within the field of view of the instrument. The humidity variable used in
this Taylor expansion will be the same as the one used in the structure
function plot, Fig. . To simplify the notation, this
variable will be defined as δR≡δe/e. The full Taylor
expansion would then read,
〈δB〉≈∑i=1AllLevels∂B∂Ri〈δRi〉+∂B∂Ti〈δTi〉+12∂2B∂Ri2〈(δRi)2〉+∂2B∂Ti∂Ri〈δTiδRi〉+12∂2B∂Ti2〈(δTi)2〉,
where we have made the important assumption that the correlation between
different levels is 0, i.e.,
〈δBiδBj〉=〈δTiδBj〉=〈δTiδTj〉=0 if i≠j.
Third order longitudinal structure function for u.
As will be shown later in Sect. and Figs. and , the terms in
Eq. () involving the temperature are negligible, leaving the
simplified solution,
〈δB〉≈∑i=1AllLevelsdBdRi〈δRi〉+12d2BdRi2〈(δRi)2〉.
The first (linear) term in this equation is proportional to the Jacobian.
This result indicates that given a deviation of the mean humidity field of
0, the mean deviation in brightness temperature is directly proportional
to the second derivative of the brightness temperature with respect to
humidity.
It is now possible, by means of Eq. (), to estimate in practice
the magnitude of the brightness temperature deviations. Note that the average
in this equation is taken over the complete field of view of the instrument,
typically 10 km at nadir for SAPHIR. The value of fluctuations in
humidity, 〈(δRi)2〉, will vary within the field of view (inertial
range in Fig. ) and with turbulence intensity (ε
in Eq. ). To simplify, an approximate average is taken for the
humidity fluctuations as a function of turbulence intensity. A value of
〈(δRi)2〉=0.05 is adopted for a turbulence intensity of ε=5 cm2 s-3, both parameters approximated from the values obtained
from sonde measurements as in Figs. and .
For other turbulence intensities, humidity fluctuations are scaled following
Eq. (), expressing ϵ in units of cm-2 s-3,
〈(δRi)2〉≈0.05(ϵ/5.0)2/3.
The same arguments can also be applied to the temperature field fluctuations,
in which case, the full Taylor expansion equation should be used (Eq. ).
A single profile is chosen from a typical tropical location at Manus island
and it is extracted from ECMWF analyses. The reason to choose a tropical
profile is because high turbulence in the high troposphere seems to be more
likely to happen in these regions. The profile has 50 levels from the lowest
level at 1010 hPa to the highest one at 5×10-3 hPa.
Figure illustrates the lower levels of the profile, the
Jacobians and the second derivatives for a few close to SAPHIR frequencies.
Satellite zenith angle is fixed to 60∘. Derivatives of the brightness
temperature are calculated by finite differences using the AM 9.2 radiative
transfer model . Final results of the brightness temperature
differences are shown in Fig. .
Temperature and humidity profiles, as dew point temperature, used in
this paper. Highlighted are the regions denoted as “Hi Tropo” and “Lo
Tropo” in Fig. . Jacobians, dB/dR, and second
derivatives, d2B/dR2, of brightness temperature vs. humidity for a
few close to SAPHIR channels are also plotted. Precise frequencies are:
183.50 GHz ≈ S1 channel, 186.10 GHz ≈ S3 channel and
194.30 GHz ≈ S6 channel.
Brightness temperature deviations calculated for different levels in
the troposphere (as shown in Fig. ), various turbulence
intensities, ε in cm2 s-3, and adjusted offsets in
temperature and humidity. Blue dots are values of observed minus calculated
brightness temperatures between the SAPHIR Megha-Tropiques instrument vs.
Météo France NWP profiles plus the RTTOV v11 RTM (from
).
DiscussionSimulations
Simulations comparing radiances with turbulence vs. radiances without them
are shown in Fig. . Results show differences in brightness
temperatures when locating turbulence in different layers in the troposphere,
with various turbulent intensities and with varying perturbations in
temperature and humidity. Several interesting features can be observed as
follows.
When only turbulent perturbations in temperature are considered, 〈(δTi)2〉≠0, the results in brightness temperature difference are very
small and nearly negligible (dashed blue line). In this case, the full Taylor
expansion formula is needed (Eq. ).
With low turbulent intensities in humidity, ε=5 cm2 s-3, located in the high troposphere, between 170 and
370 hPa, and with an offset in humidity of δR=3×10-3
(magenta line) the effects are positive in frequencies near the center of the
183 GHz line and negative further away. The brightness temperature
differences are of only a few tenths of a kelvin.
With higher turbulent intensities in humidity, ε=10 cm2 s-3, located in the high troposphere (green line) results are slightly
higher than in the previous case.
With the highest observed turbulent intensities in the low troposphere,
ε=10 cm2 s-3, and locating the humidity turbulent
simulations in the low troposphere, between 480 and 1010 hPa, and
with an offset in humidity of δR=10-2 (orange line) the biases
show a different behavior, they are negative in frequencies near the center
of the 183 GHz line and positive further away. The brightness temperature
differences are of only a few tenths of a kelvin.
Locating the turbulence in the high troposphere and fitting the parameters to
follow the observed data (blue dots) the results are ε=1200 cm2 s-3, δT=-1 K and δR=0.18 (red
line). The brightness temperature differences are of the order of a few
kelvin. The simulations fit very well the observed data. Also, offsets in
temperature and humidity are necessary. This offset is positive and is quite
high, meaning that the average value of humidity in nature is actually higher
than in the spatially uniform model.
With the highest observed turbulent intensities in the high troposphere,
ε=104 cm2 s-3, and locating the turbulence in
the same high troposphere, the brightness temperature differences are up to
several tens of kelvin (black line).
Behavior of the different terms
The orders of magnitude and the behavior of the various terms in both Taylor
expansions (Eqs. and ) will be analyzed
here. The cross-correlation term in humidity and temperature in Eq. () is not directly described in Kolmogorov's theory of
turbulence, and, therefore, it is not easy to estimate. Since we are only
interested, at this stage, in estimating the order of magnitude, the gross
assumption that the cross-correlation is equal to the product of both
variations can be made,
〈δTiδRi〉∼〈δTi〉〈δRi〉.
Results are shown in Fig. . The following features can be
observed.
Using the full Taylor expansion (Eq. ) if only an
offset in temperature is introduced (orange line), the brightness temperature
is displaced with hardly any frequency dependence.
Using the full Taylor expansion (Eq. ) if only a
humidity offset is introduced (blue line), the brightness temperature is
displaced, but there are also differences of a few kelvin which depend on the
frequency.
Placing high values of turbulence in the high troposphere and no
perturbations in temperature or humidity, the effects of turbulence can be
clearly seen (green line). It generates an overall positive displacement in
brightness temperature of several kelvin and it is very strongly dependant on
the frequency. The full Taylor expansion formula has been used here (Eq. ).
To shift this curve down in order to fit the observed data (blue dots), an
offset is required. This can be achieved with a temperature or a humidity
bias (red line). In this paper, both have been utilized (δT=-1 K, δR=0.18), but any other appropriate combination of
offsets would also fit the data. An offset in humidity partially compensates
the turbulence effect, as can be seen from the differences between the red
and the green line. The full Taylor expansion formula has been used here (Eq. ).
To verify that the temperature terms from Eq. () are
negligible, the same parameters as in the red line are used to plot the
reduced Taylor expansion from Eq. () (black line). Due to their
small difference, the black and red curve are nearly indistinguishable. The
black line in this plot is identical to the red line from Fig. .
Finally, as a simple test to check whether temperature and humidity offsets
would fit the observed data by themselves, without any turbulence effect, the
purple curve is shown. This curve comes close to the observed data (blue
dots), but with unreasonable values of temperature and humidity offsets
(δT=-8 K, δR=-0.8).
Same as Fig. , but setting parameters to analyze
the behavior of the different terms in the Taylor expansions from
Eqs. () and (). The full Taylor expansion
(Eq. ) is used to calculate all the lines in this plot
except for the black line, which uses the reduced Taylor expansion from
Eq. ().
Conclusions
Effects of turbulence in radiative transfer modeling stems from the fact
that the process is highly non-linear. In other words, the average of the
radiances coming from different atmospheric columns located within an
instrument field of view can potentially be different from the radiance
obtained using the average of all the atmospheric columns. Effects of
turbulence in temperature fields seem to have a low impact in the radiative
transfer modeling (dashed blue line of Fig. ). Humidity
turbulent effects seem to significantly affect the radiance biases by as much
as several kelvin.
Turbulence simulations can match observed biases as summarized by
. Biases are positive close to the center of the
absorption band and negative at the wings. To achieve this match, turbulence
has to be of high intensity and located in the high troposphere (red line in
Fig. ). These high turbulence phenomena usually occur in
places with clear air turbulence, regular or severe storms and cumulus clouds
.
Turbulence simulations placed at the low troposphere create biases which have
an opposite behavior as to the ones originating at the high troposphere. In
other words, negative in the center of the absorption band and positive at
the wings (orange line in Fig. ).
Note that the calculations presented here have been made with just one
atmospheric profile and one satellite zenith angle since the sole purpose of
the exercise is to test whether the turbulence effect hypothesis can
plausibly explain the observed biases. Biases shown in more general studies,
such as , are usually the result of an average of many
different cases at different locations. To verify precisely that these biases
do originate from turbulence effects, many different cases at different
locations should be analyzed to finally calculate a global bias, which can
then be compared with the measured ones. Turbulence effects will also depend
strongly on each scene analyzed. Furthermore, in this paper, turbulence
properties have been measured with radiosondes at mid-latitudes (Sodankylä
and Lindenberg) and this is being extrapolated to other regions by only
changing the turbulence intensity (mean energy dissipation rate per unit
mass, ϵ). A more local measurement of turbulence would better
characterize the scenes under scrutiny.
have also tried to reconcile the biases found between
microwave satellite instruments and sonde measurements plus radiative
transfer modeling. By correcting dry biases from sonde measurements and
selecting fields of view which are unaffected by clouds and are spatially
homogeneous they achieved agreement between satellite observations and sonde
measurements. They are, in practice, selecting scenes with low turbulence
such that its effects in radiative transfer modeling are not significant.
These results are therefore consistent with the ones presented in this paper.
In summary, turbulence within the field of view of microwave instruments
seems to have a significant effect in the modeling of radiative transfer,
which, if ignored can give rise to significant biases up to several kelvin in
the 183 GHz band. These biases are frequency dependant. To confirm this
hypothesis, more precise and further modeling and its corresponding
comparisons with measurements at different frequencies in the microwave and
the infrared, in this latter case, under clear sky conditions, would be
needed. Coincident turbulence measurements of the atmosphere might also be
necessary to solve the problem.
Another, additional conclusion that can be envisaged from these results is
that when comparing atmospheric profiles with sondes for precise calibration
or validation either a turbulent term should be added in the uncertainty
budget or dual sequential radiosondes should be used. This will be the
subject of a future paper.
GRUAN radiosonde data has been used in this study. Please visit https://www.gruan.org/data/data-products/gdp/rs92-gdp-2/
() for more details and data access.
The authors declare that they have no conflict of
interest.
Acknowledgements
We wish to thank Scott Paine for useful discussions regarding his RTM:
AM 9.2
.
Edited by: Tanvir Islam
Reviewed by: two anonymous referees
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