Analysis and Improvement of Tipping Calibration for Ground-Based Microwave Radiometers

1260
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 38, NO. 3, MAY 2000
Analysis and Improvement of Tipping Calibration for
Ground-Based Microwave Radiometers
Yong Han and Ed R. Westwater, Senior Member, IEEE
Abstract—The tipping-curve calibration method has been an
A limiting factor in evaluating LBLRTM is the accuracy
important calibration technique for ground-based microwave ra-
of the humidity profiles used as input to the model. Research
diometers that measure atmospheric emission at low optical depth.
reported by Liljegren and Lesht [6] showed that there were
The method calibrates a radiometer system using data taken by
substantial differences between relative-humidity sensors on
the radiometer at two or more viewing angles in the atmosphere.
In this method, the relationship between atmospheric opacity and
radiosondes that were obtained from different calibration lots
viewing angle is used as a constraint for deriving the system cali-
(i.e., batches of radiosondes that were calibrated at different
bration response. Because this method couples the system with ra-
times). Thus, a limiting factor in evaluating LBLRTM is the
diative transfer theory and atmospheric conditions, evaluations of
accuracy of the humidity profiles used as input to the model.
its performance have been difficult. In this paper, first a data-sim-
Recognizing the need to overcome this limitation, two water
ulation approach is taken to isolate and analyze those influential
factors in the calibration process and effective techniques are de-
vapor intensive operating periods (WVIOP) were conducted at
veloped to reduce calibration uncertainties. Then, these techniques
the ARM cloud and radiation testbed (CART) site in 1996 and
are applied to experimental data.
1997. This paper focuses on the 1997 observations obtained at
The influential factors include radiometer antenna beam width,
or near the ARM Southern Great Plains (SGP) central facility
radiometer pointing error, mean radiating temperature error, and
(CF) near Lamont, OK [5]. A primary goal of these experiments
horizontal inhomogeneity in the atmosphere, as well as some other
factors of minor importance. It is demonstrated that calibration
was the comparison of the absolute accuracy of both remote
uncertainties from these error sources can be large and unaccept-
and in situ humidity sensors.
able. Fortunately, it was found that by using the techniques re-
Ground-based microwave radiometers (MWR) have been
ported here, the calibration uncertainties can be largely reduced
widely used to measure atmospheric water vapor and cloud
or avoided. With the suggested corrections, the tipping calibration
liquid water. Frequencies on the 22.235 GHz water vapor
method can provide absolute accuracy of about or better than 0.5
K.
absorption band and in the 31 GHz absorption window region
are commonly used in the systems. These frequency channels
Index Terms—Calibration, ground-based, microwave radiome-
differentiate in their response to water vapor and cloud liquid
ters.
water and provide brightness temperature measurements from
which precipitable water vapor (PWV) and integrated cloud
I. INTRODUCTION
liquid water (ICL), as well as low-vertical-resolution water
vapor profiles, are derived [7]–[9]. The absolute calibration is
DUETOrecentemphasisontheapplicationofline-by-line fundamental in determining the accuracies of these retrievals,
radiative transfer models (LBLRTM) to climate models
although some other factors are important as well [10]. For a
[1] and to assimilation of satellite data in weather forecasting
dual-channel radiometer at 23.8 and 31.4 GHz, 1 calibration
[2], the accuracy of forward models to calculate absorption and
error may cause about 1 mm error in PWV.
emission spectra during clear sky conditions is of increasing im-
The importance of the PWV measurements and thus, the
portance. Measurement of water vapor profiles is fundamental
importance of the system calibration, has increased in recent
to these and to many other atmospheric and climate problems.
years as the MWR measurements are often served as references
With the increasing deployment of Fourier transform interfer-
and comparison standards for other water vapor measuring in-
ometric radiometers (FTIR) [3], [4] at observation sites around
struments, such as radiosondes, Raman water vapor lidars, and
the world, an excellent data base of well-calibrated radiance data
global positioning systems (GPS) [11]. Part of the motivation
is becoming available through the U. S. Department of Energy’s
for the WVIOP’s was the suggestion by Clough et al. [12] that
Atmospheric Radiation Measurement (ARM) program [5]. The
MWR measurements of PWV could be used to scale radiosonde
conventional way of evaluating and improving models is to mea-
observations to more realistic values. The possibility also exists
sure vertical profiles of temperature and relevant emitting con-
for using the MWR or the GPS to help calibrate Raman lidar
stituents, use these measurements as input to LBLRTM, and
measurements of mixing ratio profiles. Such a referencing
compare measured and calculated radiance.
role has been played recently at the Department of Energy’s
ARM CART CF site. It has been observed at the site that the
spectra measured by an FTIR are closer to the results from
Manuscript received January 20, 1999; revised July 20, 1999. This work was
supported by the U. S. Department of Energy, Environmental Sciences Division,
the LBLRTM that uses radiosonde water vapor profiles scaled
Atmospheric Radiation Measurement Program, Washington, DC.
by the PWV from the MWR than those from the LBLRTM
The authors are with the Cooperative Institute for Research in Environmental
without using the scaling [12].
Sciences (CIRES), University of Colorado/NOAA, Environmental Technology
Laboratory, Boulder, CO 80303-3228 USA (e-mail: yhan@etl.noaa.gov).
From September 15 to October 5, 1997, WVIOP’97 was
Publisher Item Identifier S 0196-2892(00)02481-5.
conducted at the ARM CART site. During WVIOP’97, the
U.S. Government work not protected by U.S. copyright

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HAN AND WESTWATER: ANALYSIS AND IMPROVEMENT OF TIPPING CALIBRATION
1261
NOAA’s Environmental Technology Laboratory (ETL) op-
defined as the radiation power per unit area, per unit frequency
erated two microwave radiometers at the CART site. During
interval at a specified frequency , and per unit solid angle at
the same time, NOAA’s forecast systems laboratory (FSL)
a specified direction. However, in the microwave frequency re-
operated two GPS’s, one at the SGP CF and one at NOAA’s
gion, the intensity is usually expressed as brightness tempera-
wind profiler site near Lamont, OK, about 9 km away from
ture, denoted as
. In microwave radiometry, there are two
the CF. NASA’s Goddard Space Flight Center (NASA/GDFC),
popular definitions of the brightness temperature. Because of
Greenbelt, MD, operated a Raman water vapor lidar, and the
the differences between these two difinitions, some confusion
ARM operational Raman lidar was operated as well. At the CF,
and mistakes may arise when measurements or models are com-
ARM has also routinely operated an MWR for several years
pared. Because it is important to be aware of the issue, we in-
[7]. The primary goal of these deployments was to quantify the
clude a brief discussion in the following. A detailed discussion
absolute accuracies of the MWR’s and GPS’s in PWV and to
is given in Janssen [16] and Stogryn [17].
compare these measurements with in situ measurements made
The so-called Rayleigh–Jeans equivalent brightness temper-
every 3 h by ARM’s balloon borne sounding systems (BBSS).
ature is defined as
Preliminary intercomparisons among these instruments showed
that the PWV from the ARM’s MWR was consistently higher
(1)
than that from all other instruments. With other instruments
where , , and
are the speed of light, Boltzmann’s constant,
being in agreement with one another, a logical step was to
and frequency, respectively. Note that (1) is not the traditional
examine the calibration process performed on both ARM and
Rayleigh–Jeans approximation.
is a scaled intensity in
ETL MWR’s during the experiment. This led to the investiga-
temperature units. The second definition is the so-called ther-
tion of the calibration method that we present here.
modynamic brightness temperature, defined as
Microwave radiometers that are used in space are usually cali-
brated using known calibration reference targets [13]. However,
(2)
it is desirable to have a target temperature close to the brightness
where
is the inverse of the Planck function
at a
temperatures that an MWR measures during regular observa-
tions. For the upward-looking radiometer channels considered
temperature
. The Planck function
is the radiance
here, the observed brightness temperature can be as low as 10
emitted at
from a blackbody at temperature
and is given by
K. Two calibration methods are often applied for these channels:
using a liquid nitrogen LN -cooled blackbody target or a tipping
(3)
calibration method that uses a clear atmosphere as a calibration
reference. If applied with care and caution, the LN method can
be useful. However, it is not practical to be applied and auto-
By way of analogy,
is an equivalent temperature at
mated in long-term, routine operations. With the tipping method
which a blackbody emits the amount of radiation at the intensity
[14], [15], the radiometer takes measurements at two or more el-
. The conversion from
to
is
evation angles in a horizontally-stratified atmosphere. The cal-
ibration is accomplished by adjusting a single numerical cali-
(4)
bration parameter that is required by the system software until
the outputs of the system comply with a known physical rela-
Unlike
,
is not linearly related to
. Expanding
tionship. In addition, as discussed in Section IV-G, strict quality
the Planck function
in terms of
, one obtains
control methods must also be applied to the data before the cal-
ibration parameter is changed. With suitable scanning, the cal-
(5)
ibration procedure can be automated. The ETL’s and ARM’s
radiometer systems were all calibrated using the tipping cali-
bration method during the experiment.
where
is the Planck constant. The first term is the traditional
The tipping method couples a radiometer equation that is spe-
Rayleigh–Jeans approximation, and the second may be consid-
cific to the radiometer in use and the theory of the radiative
ered at a first-order correction to the Rayleigh–Jeans approxi-
transfer in the atmosphere. Calibration uncertainties may arise
mation and depends not on
, but only on frequency. In
from both the system and the application of the theory. To in-
the region
GHz, the first two terms are usually suffi-
vestigate and reduce the uncertainties, we first take a simulation
cient for applications, except when
, is very low (e.g.,
approach to reveal problems and develop techniques to solve
for the cosmic background at 2.75 K). Thus, under normal con-
them. We then apply these techniques to the data taken during
ditions, for the amount of radiation received by ground-based
WVIOP’97. The simulations are based on radiosonde data col-
radiometers, the difference between
and
is well
lected around the area of the CART site. Although the data are
represented by the second term in (5),
, which is propor-
site specific, results of the analysis are general, and the tech-
tional to frequency. At 20.6 GHz, this term has a value of 0.49 K
niques developed can be applied to all tipping calibrations.
and 31.4 GHz, a value of 0.75 K. Even when
is as low
as the cosmic background at 2.75 K, the sum of the remaining
terms is not significant for the frequencies considered here. For
II. RADIATIVE TRANSFER AND RADIOMETER EQUATIONS
example, for
K, the third term is equal to 0.030
The fundamental quantity a radiometer measures is radiation
K at 20.6 GHz and 0.069 K at 31.4 GHz. However, it may be
power, which is related to the specific intensity (or radiance)
,
significant at higher frequencies.

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IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 38, NO. 3, MAY 2000
Following Janssen [16], the definition of
may be ap-
where
is the optical path, and
is the opacity
plied to scale the uplooking radiative transfer equation
at an elevation angle . The opacity is then calculated as [for
convenience, we will drop the symbol
and express the opacity
(6)
as
]
into an equation with temperature units as
(12)
(7)
Note that in (12), the sky brightness temperature is defined as
where
is the cosmic background temperature and is a
that of the Rayleigh–Jeans equivalent
using (4) because
function of fequency, and
is defined as
it is defined as a thermodynamic brightness temperature. If we
express each of the quantities in (12) in terms of the temper-
(8)
atures given by (4) and expand the Planck functions, we may
calculate the opacity as
Note that
is not equal to the physical air temperature
.
The radiometer equation relates the system output voltage to
input radiation power. Usually, an MWR system views one or
(13)
more internal or external blackbody targets for initial calibra-
tions. Taking a simple example, if a linear system has two cali-
after neglecting the third and higher terms, where
bration reference targets at temperatures
and
, the intensity
(averaged over an antenna beam) measured by the radiometer
observing the sky, may be given by the following radiometer
equation
(9)
(14)
where
,
, and
are detected voltages from the sources of
the sky, target 1, and target 2, respectively. The radiometer equa-
In (13),
may be substituted with the measurements
tion may be scaled in the same way as that for obtaining the ra-
given by (10) directly. However, since (13) is an approximation
diative transfer (7). Thus, the measurements given by the scaled
of the exact (12), we should caution on the treatment of the
radiometer equation are consistent with the radiative transfer
quantity representing the cosmic background or sky bright-
equation (7). Note that in the scaled equation, the quantities
ness temperature with very low value. In some applications,
(
or
) are not equal to
or
.
neglecting the third term may not be valid.
It is common, however, that physical temperatures are di-
When mapping the measurements to opacity, it is important
rectly used in the radiometer equations. This is done by lin-
that the parameters in the mapping function be consistent with
earizing
in terms of
(i.e., using the first two terms in
the radiometer equation. For example, if the cosmic background
the expansion similar to (5)). If we substitute each of the radia-
is given in Rayleigh–Jeans equivalent brightness temperature,
tion intensities,
,
, and
and the sky temperature is given in thermodynamic brightness
, where
and
into
temperature, an inconsistency occurs that will fold a bias of 0.75
(9), we have
K at 31.4 GHz into the calculation of opacity.
(10)
The tipping calibration method is usually applied to deter-
mine the values of some constants in the radiometer equation
We see that this radiometer equation is consistent with the
that models an MWR system’s input–output relationship. We
model that computes the thermodynamic brightness tempera-
assume the MWR systems are linear and characterized by two
ture
but is not consistent with the radiative transfer
constants such as a system gain and an offset [18]. If accurate
(7) that calculates
. As discussed earlier, under normal
calibration reference targets are available at two temperatures
conditions, (10) is sufficient for applications. However, in
that span the range of observable brightness temperatures, then
situations such as zenith radiometric observations from air-
the two constants can be determined. In many cases, the systems
craft, the atmospheric radiation power may be too low to be
have partial calibration information, with one of the constants
linearized in terms of the thermodynamic temperature and thus,
approximately known, and leave the other to be determined in
the radiometer (10) may not be valid.
the calibration processes. For example, the ETL radiometers
In many applications of ground-based radiometry, including
have two internal calibration references, but the transmission
tipping calibrations, there is a need to map the atmospheric radi-
by a window and a segment of a waveguide need to be deter-
ation power into optical depth or opacity. This can be achieved
mined. In the following (and thereafter), we use the ETL’s and
after defining a mean radiating temperature based on the radia-
ARM’s systems as examples to illustrate the tipping calibration
tive transfer equation (7)
analysis. The results are general for other types of radiometers.
Each of the two ETL systems contains two independent mi-
crowave radiometers: one operates at 20.6 or 23.87 GHz and the
(11)
other at 31.65 GHz. Other system parameters are listed in Table

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1263
TABLE I
SELECTED RADIOMETER PARAMETERS: FREQUENCY (GHz), BANDWIDTH (GHz), 
–FULL WIDTH AT HALF-MAXIMUM POWER OF THE
ANTENNA PATTERN (DEGREES)
I. Each radiometer has two internal blackbody loads, one at tem-
which the atmospheric antenna temperature
may be calcu-
perature
, near 300 K, and the other at
, about 100 K higher
lated using a radiative transfer and an antenna model. We will in-
than
. The radiometer equation of the system is given as [14]
vestigate the four frequency channels at 20.6, 22.235, 23.8, and
31.4 GHz, respectively. Although the frequency channel 22.235
(15)
GHz is not used in the ETL and ARM’s systems, it is included
where
is the antenna temperature [16], [17] being mea-
here for reference.
sured,
is the temperature of the radiometer waveguide, and
,
, and
are voltages of a square law detector, corre-
III. TIPPING CALIBRATION METHOD
sponding to the radiation sources of the two internal loads and
We first define the atmospheric airmass as the ratio of the
the sky, respectively. All these voltages and temperatures on the
opacity at the direction
and the opacity at the zenith (
)
right side of the equation are measured. The unknown param-
eter
is the parameter determined through the calibration and,
(18)
as discussed in [14] and [18],
describes transmission losses
due to a portion of waveguide and a microwave window in the
In (18), for convenience, we have dropped the frequency depen-
system.
dence in notation. The tipping calibration method uses measure-
The ARM’s system operates at 23.8 and 31.4 GHz. Other
ments of opacity, derived from measurements of
, as a func-
main system parameters are also listed in Table I. The system in-
tion of airmass to derive a calibration factor. We will discuss
cludes a noise diode injection device and an external blackbody
later how
is derived from measurements of
. Equation
reference target at an ambient temperature
. The radiometer
(18) can be used to derive the calibration factor if the relation-
equation of the system is given as [7]
ship between airmass
and the observation angle
is known.
In a plane-stratified atmosphere and if we ignore the bending
(16)
of radiation rays caused by the gradient of refractive index, we
have
where
is the voltage when the radiometer views the sky,
is the voltage when viewing the reference target with the noise
(19)
diode on,
is the voltage when viewing the reference with the
noise diode off, and
is the noise injection temperature deter-
Sometimes, (19) is taken as the defining equation for airmass,
mined through the calibration. The radiation loss and emission
but (18) is more general and includes (19) as a limiting case.
of a microwave window in front of the radiometer antenna were
Since (18) involves two opacities, the calibration procedure re-
measured in the laboratory and are assumed to be known and,
quires observations of at least two different angles
and
.
because they are multiplicative with
, are simply incorpo-
The measurements,
and
at the two angles
rated into it.
are then mapped into
and
in opacity space by using (12)
The radiometer (15) and (16) may be simplified for the sake
or (13). Note that we have explicitly expressed the measure-
of convenience to simulate the tipping data. We may write the
ments as a function of the calibration factor . If we normalize
parameters
in (15) as
and
in (16) as
, where
and
by their corresponding airmasses as
and
and
are correct calibration factors, and
represents the
, the normalized opacities
and
should theoret-
correctness of the estimations of
or
, with
repre-
ically have the same value, and any difference between them is
senting a perfect calibration. With this consideration, (15) or
due to an incorrect calibration factor. We may adjust
until an
(16) may be rewritten in the form as
agreement between
and
is reached.
(17)
The calibration factor may also be derived equally well in
where
is an estimate of the true antenna temperature
the brightness temperature space. We may map the normalized
in the pointing direction,
for the ETL’s systems, and
opacities
and
back to zenith brightness temperatures, re-
for ARM’s system. We see that the measured antenna
ferred to as normalized brightness temperature
and
.
temperature is equal to true antenna temperature
when a cal-
They too should be equal to one another. Any difference be-
ibration is performed without error (
). We will refer to the
tween them can be adjusted with the calibration factor . The
factor
in (17) as the calibration factor. Equation (17) provides
concept of the normalized brightness temperature will be used
a convenient way to simulate the radiometric measurements in
in later analysis.

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IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 38, NO. 3, MAY 2000
To reduce measurement uncertainties, tipping observations
are often taken at more than two angles. Let
be the set of normalized opacities obtained from a set of obser-
vations. We may apply the least-square technique to obtain
by
minimizing the sum of square differences between any pair of
opacities
and
, (
;
)
(20)
where
(21)
Minimization with respect to
yields the equation
(22)
where
Fig. 1.
Function given by (22), whose zero is the calibration factor.
and
line’s offset is not equal to zero, the line’s slope is used as an ap-
(23)
proximation to the zenith opacity, which is then mapped into the
zenith brightness temperature. Next, the zenith brightness tem-
Solving (22), we obtain the calibration factor . Finding the zero
perature is used to obtain a new value of
from the radiometer
of
is a well-posed mathematic problem as the curve of
equation. The process may be repeated until the offset of the line
, shown in Fig. 1, suggests.
is near zero. The flaw of the method is that some of the infor-
We can look at the calibration procedure in another
mation contained in the tipping data is lost in the process. For
way.
Tipping
data
is
a set
of
airmass-opacity
pairs
example, if the tipping observations are taken at angles with air-
. The basic idea is that if we
mass 1, 2, and 3, the data at airmass 2 will never be used in the
consider
as a function of ,
, then
calibration process.
(i.e.,
extrapolated to airmass 0 is 0). Thus, these points of
the pairs lie on a straight line that must pass the origin on the
IV. CALIBRATION ERRORS AND METHODS TO REDUCE THEM
airmass-opacity plane, and the slope of the line is the zenith
opacity. If the fitted line does not pass the origin, it implies
Calibration errors may be caused by sources from the ra-
an incorrect calibration factor. We may adjust the calibration
diometer system and the violations of the assumptions in the
factor
until the line passes the origin. The mathematical
theory on which the calibration is based. The former include the
derivation is similar to the previous one. The difference is that
effect of radiometer antenna pattern, radiometer pointing error
the previous derivation minimizes the differences among the
and system random noise. The latter include the uncertainty in
normalized opacities, while the line fitting method minimizes
the mean radiating temperature and the uncertainties in the fun-
the differences between the opacities and a linear line that
damental relationship (19) between the airmass and the observa-
passes the origin. If only two angles are involved, the two
tion angles, which can be affected by nonstratified atmospheric
methods are identical.
conditions and the earth’s curvature.
As pointed out by an anonymous reviewer, the metric of (20)
We simulated these error sources and developed and tested
could be replaced by a similar one involving the root-mean-
effective techniques to reduce them. The simulations were per-
square (rms) of the antenna temperature residuals. This metric
formed by using a radiative transfer model [22] and the simpli-
would result in giving an rms residual that could be compared
fied radiometer (17) for a clear-sky atmosphere, which is repre-
with the noise level of the radiometer. However, the present
sented by radiosonde pressure, temperature, and humidity pro-
method could also give a similar rms residual after a mapping
files. A statistical ensemble of radiosonde data, referred to as
back from opacity space to antenna temperature space. Since the
with a size of 16 380 soundings, was collected from five stations
same basic information enters into both metrics, we don’t think
around the area of Oklahoma City, OK, from 1966 to 1992.
that the results would differ substantially.
To evaluate a calibration, we first need to define the calibra-
There has been a less rigorous method that appears similar to
tion error. Although the quantity
(the difference
the second method. Since it has been often used [15], [19]–[21],
between the calibration factor
and its correct value
),
it is worth a brief discussion here. The method fits a line to a set
can be used as a measure to the error, it may be more intuitive to
of tipping data with an approximate calibration factor . If the
express the error in terms of the brightness temperature. How-

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TABLE II
COLUMNS A: rms DIFFERENCES BETWEEN TWO MODEL CALCULATIONS OF T , ONE WITH REFRACTIVE INDEX CONSIDERED AND THE OTHER WITHOUT;
COLUMNS B: rms DIFFERENCES BETWEEN TWO MODEL CALCULATIONS, ONE WITH EARTH CURVATURE CONSIDERED AND THE OTHER WITHOUT. THE
DIFFERENCES (K) ARE LISTED AS A FUNCTION OF FREQUENCY (GHz) (ROW 1) AND AIRMASS (COLUMN 1)
ever, for a given error
, the error in corresponding antenna
Thus, the effect of the refractive index profile on system cali-
temperature depends on the magnitude of the temperature itself.
bration is negligible. Table II is computed from the statistical
For simplicity, we evaluate the error at a reference temperature
ensemble
.
. That is, for a calibration with an error
, the calibration
Earth curvature has a relatively large effect, which causes air-
error is defined using (17) as
mass at an angle
to be smaller than that of an atmosphere with
a flat surface at the same angle. The differences between the
(24)
two brightness temperatures with and without the earth curva-
ture are about one or two tenths of a degree at airmass 3 and
We use the climatological mean of the brightness temperature
three- to five-tenths of a degree at airmass 4 (see the B Columns
as the reference temperature. The values of the reference tem-
in Table II). Although the effect is still small when compared to
peratures are 27.2, 40.0, 35.0, and 18.5 K for the four selected
those from other sources, the airmass can be conveniently cor-
channels at 20.6, 22.235, 23.8, and 31.4 GHz, respectively.
rected to a large degree. In a spherically stratified atmosphere
Theoretically, any set of angles with two or more being dis-
and neglecting the gradient of refractive index, the atmospheric
tinct can be included in the tipping observations. However, be-
opacity
is given by
cause of the various uncertainties, at least two of the angles (or
their corresponding airmasses) should not be too close. In ad-
(25)
dition, low elevation angles or large airmasses should also be
avoided due to the finite beam width. Our data simulations in-
where
is the radiometer elevation angle,
(
km) is
clude five elevation angles at 90 , 41.8 , 30 , 19.5 , and 14.5 ,
the earth’s radius, and
is the absorption coefficient. Since the
corresponding to airmasses 1, 1.5, 2, 3, and 4. To reveal the
absorption coefficient
decreases with height
almost expo-
angular dependence of the tipping calibration, we perform cal-
nentially with a scale height of 2–3 km [23], the value of the
ibrations using tipping data at two elevation angles, with one
ratio
is a small quantity in the range where the absorp-
fixed at zenith and the other varying among the remaining an-
tion has a contribution to the integrand. Thus, we may expand
gles. Thus, from a set of tipping data taken at the five angles,
the denominator in (25) with respect to
and derive an ap-
we may have four values of the calibration factor for a single
proximation as
radiometer channel.
(26)
A. Effect of Earth Curvature and Atmospheric Refractive Index
where
is the zenith opacity. If we define an effective
As we discussed earlier, (19) is for a plane horizontally strat-
height
such that
ified atmosphere with a refractive index profile independent of
height. For the earth’s atmosphere, however, the earth curvature
and the vertical gradient of the refractive index cause the amount
then the integral in the above equation may be written as a mul-
of airmass at an elevation angle to differ from that given by (19)
tiple of the zenith opacity
and
. We note that if the ab-
[22]. Under normal conditions, the vertical gradient of the re-
sorption decays exponentially, then the effective height is equal
fractive index “bends” a radiation ray downward and thus re-
to the absorption scale height. Thus, we have
sults in a larger airmass than if there were no such gradient. We
computed
using a computer code [22] that uses ray tracing
(27)
methods based on Snell’s Law for a spherical atmosphere and
ran the code twice for each profile in our statistical ensemble:
According to its definition (18), the airmass in a spherical at-
once with the refractive index
calculated from a radiosonde
mosphere is
profile and the second time with
set to unity. As shown in the
(28)
A Columns in Table II, the rms differences between the two with
and without considerations of the gradient are very small at all
where
is airmass in a plane stratified atmosphere, given by
of the selected airmasses, about a few hundredths of a degree.
(19). As an example, we derived the average effective height

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1266
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 38, NO. 3, MAY 2000
TABLE III
H–EFFECTIVE HEIGHT (km) AND ITS STANDARD DEVIATION (km)
and its standard deviation from a subset of the statistical en-
semble
, which is about a half of the size of
. The subset
is referred to as set
. The effective height is computed using
(27), where the opacity is calculated from the model in which
both the earth curvature and refractive index are included. Thus,
the obtained effective height
contains the effect of refractive
index profile, and its value is slightly less than one that does
not consider the effect of refractive index, due to the reasons
discussed earlier. Table III lists the effective height for the four
frequency channels.
The use of the airmass-angle relationship (19) without a cor-
rection causes calibration errors, resulting in measurements with
values lower than their true values as shown in A Columns in
Table IV. The error increases with the increase of airmass. After
Fig. 2.
Differences between antenna temperature and the brightness
the airmass corrections using (28), the errors are reduced about
temperature at the beam center direction as a function of antenna beam width.
a factor of 5 (see Table IV). Results of Table IV are obtained
The antenna gain pattern is modeled for a corrugated feed horn. The three
from a test data set, referred to as
, a subset half of the size of
curves correspond to the differences when the antenna views a clear sky with
PWV = 3:2 cm at airmass 2, 3, or 4, respectively.
, independent of
.
B. Effect of Antenna Beam Width
the amount of PWV. At large airmass (or low elevation angle),
The antenna temperature
of a radiometer at a
the magnitude may reach a peak, as the figure shows.
specified frequency and direction
is a weighted average of
Due to the finite antenna beam width, the airmass-angle re-
incoming brightness temperature
over all directions
lationship is no longer correctly given by (28), which is appli-
[18]
cable for a radiometer with an infinitely small beam width. Cal-
ibrations using (28) for a radiometer with a finite antenna beam
width are accompanied by errors, as shown in the A Columns in
Table V(a) for a 4 beam width and Table V(b) for a 5.7 beam
width. The errors from a 5.7 beam width are about twice as
(29)
large as those for a 4 beam width. Another significance in the
table is the increase of the errors with the increase of airmass.
For example, for the same beam width, the calibration errors at
In (29),
is the power pattern of the radiome-
airmass 3 is about 50% more than those at airmass 2. For a 23.8
ters antenna. Under normal atmospheric conditions and at the
GHz radiometer with a 5.7 antenna beam width, the calibra-
weakly absorbing frequencies considered here, due to the non-
tion error is about
K when airmass 3 is used along with
linear increase of the brightness temperature when lowering the
airmass 1. The significance of these errors calls for a correction
elevation angle,
is larger than that of the brightness tem-
for the airmass given by (28) or, equivalently, an adjustment of
perature
at a cone-like antenna beam center direction. Fig.
the measured antenna temperature to that at the antenna beam
2 shows their difference
as a function of the antenna
center. We adopt the latter to adjust
to
. After the adjust-
beam width (3-dB beam width of an antenna with a corrugated
ment, (28) can then be applied. The amount of adjustment
feed horn) for a 23.8 GHz radiometer under a typical sky condi-
is derived (see Appendix) by assuming a Gaussian beam and is
tion with PWV
cm at three elevation angles. The antenna
given as
power pattern in the calculation is for a corrugated feed horn
[24]. We see that the difference increases with the increase of
the beam width. For a 5.7 antenna beam width, the difference
is 0.47, 1.05, or 1.72 K at the angle with an airmass of 2, 3,
(30)
or 4, respectively. The difference also depends on the amount
of atmospheric water vapor. Fig. 3 shows the difference (filled
where
is in radians and is the full width at half-maximum
symbols) as a function of PWV for about 4000 selected sam-
power of the power pattern. Note that the
in (30) is the slant
ples in the ensemble
. The concave-down features shown in
path opacity at an elevation angle . The observed antenna tem-
the figure reflect that the magnitude of the nonlinear variation of
peratures should be corrected by the amount given by (30) be-
the brightness temperature across the antenna beam varies with
fore being used in the calibrations:
. Note that

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HAN AND WESTWATER: ANALYSIS AND IMPROVEMENT OF TIPPING CALIBRATION
1267
TABLE IV
Rms CALIBRATION ERRORS BEFORE (A) AND AFTER (B) AIRMASS CORRECTIONS, IN WHICH THE SCALE HEIGHTS IN TABLE III ARE USED.
THE ERRORS (K) ARE LISTED AS A FUNCTION OF FREQUENCY (GHz) (ROW 1) AND AIRMASS PAIR (COLUMN 1) AT WHICH THE
TIPPING-CURVE DATA ARE SIMULATED AND USED IN THE CALIBRATIONS
Fig. 3.
Differences between antenna temperature and the brightness temperature at the beam center direction as a function of PWV. The filled symbols are those
without beam effect corrections. The open symbols are those with the corrections, in which the antenna temperature is adjusted by an amount given by (30). The
airmasses at which the differences are calculated are indicated in the figure. Data used in the simulations are explained in the text.
the amount of brightness temperature adjustment is itself a func-
some others discussed later, tipping observations should avoid
tion of the brightness temperature, whose correct value is un-
low elevation angles. Our experiences suggest that angles with
known before the calibration. In practice, we may derive
by
airmasses larger than 3 should be avoided, especially for the 6
an iteration process. First, the calibration is carried out without
antenna used by ARM.
the adjustment. Then, the calibration is repeated but with an ad-
justment. Two steps are usually sufficient for the calibration.
C. Errors Caused by Uncertainties in Radiometer Pointing
The open symbols in Fig. 3 represent the differences between
Angle
the adjusted calculated antenna temperature
and the
In a Cartesian
coordinate system where a radiometer
brightness temperature at the beam center
. The differences
is located at the origin, we define zenith as the
direction and
are reduced by one order of magnitude over those without such
let the
plane contain the antenna beam center. The pointing
adjustment (filled symbols).
angle or elevation angle
is then defined as the angle between
The corrections of antenna beam effect using (30) signifi-
the positive
axis and the beam center direction. For a ra-
cantly reduces the calibration errors as shown in B columns in
diometer that is able to point any direction in the
plane,
Table V. However, the effect of antenna beam width is a com-
we refer to a pair of symmetric angles as a pointing angle
and
plicated issue. In reality, the beam’s side lobes may pick up
its reflection in the
plane, the angle
. We may also
radiation at low elevation angles from sources that are unpre-
refer the one side or two sides as the one or two of the half spaces
dictable. For example, radiation from the ground that enters the
in the
plane separated by the
axis (zenith vector). Ra-
antenna sidelobes is a contaminating factor. For this reason and
diometer systems often use reflectors to direct radiation to the

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1268
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 38, NO. 3, MAY 2000
TABLE V
rms CALIBRATION ERRORS BEFORE (A) AND AFTER (B) ANTENNA BEAM WIDTH CORRECTIONS. (a) FOR A 4 BEAM WIDTH ANTENNA AND (b) FOR A 5.7 BEAM
WIDTH ANTENNA. THE ERRORS (K) ARE LISTED AS A FUNCTION OF FREQUENCY (GHz) (ROW 1) AND AIRMASS PAIR (COLUMN 1), AT WHICH
THE TIPPING CURVE DATA ARE SIMULATED AND USED IN CALIBRATIONS
TABLE VI
Rms CALIBRATION ERRORS CAUSED BY A 1 SHIFT OF THE ELEVATION ANGLES AS A FUNCTION OF THE FREQUENCIES (GHz) (ROW 1) AND AIRMASS PAIR
(COLUMN 1). (a) CALIBRATIONS IN WHICH TIPPING DATA ARE TAKEN AT ONE SIDE ONLY AND (b) CALIBRATIONS IN WHICH DATA FROM BOTH SIDES ARE USED
antenna. The slant path measurements are accomplished by ro-
by performing tipping observations at symmetric angles. If the
tating the reflectors. The reflector rotating angles, the alignment
measurements at an angle consistently differ from those at its
between the antenna and the reflector, and the positioning of the
symmetric angle, it usually implies the existence of the pointing
total system all affect the pointing (elevation) angles.
error (except in the situations when there is a persistent hori-
To see the impact of the pointing error on the calibration, we
zontal inhomogeneity in the atmosphere). However, if one were
use data in
to simulate the calibrations in which all the obser-
to perform tipping calibrations on a moving platform, the diag-
vation angles in the plane of scanning are off by 1 . As shown
nosis of angular errors would be much more complex.
in A Columns in Table VI, the pointing errors could have se-
Fortunately, the effect of the pointing error can often be re-
rious impact on the performance of the tipping calibration if
duced significantly by using tipping data taken on both sides
only one-side tipping calibration is used. Also shown in the
under the condition that the differences among those angles are
table is that the same pointing errors cause larger calibration
known precisely. This is due to the effect that the uncertainties
uncertainties when data with larger airmasses (lower elevation
in the measurements on one side due to the pointing error are
angles) are used than those with small airmasses (higher eleva-
partially canceled out by the uncertainties of those on the other
tion angles). The pointing angle errors can often be identified
side. As shown in the B Columns in Table VI, the calibration

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HAN AND WESTWATER: ANALYSIS AND IMPROVEMENT OF TIPPING CALIBRATION
1269
Fig. 4.
Calibration errors at 23.8 GHz channel as a function of errors of the
mean radiating temperatures. The calibration errors also show dependence on
the observation airmasses (or angles), at which data are taken and used in the
calibrations. The atmospheric state for this example has a zenith brightness
temperature of 59.2 K.
Fig. 5.
Nocturnal temperature profile showing temperature inversion (solid
errors are reduced to a negligible level after we use tipping data
line) and a daytime profile with normal lapse rate (dashed line).
from both sides. This strongly suggests that tipping data should
be taken in pairs on the symmetric elevation angles.
ture inversion, while the dashed curve is a daytime profile with
normal lapse rate. Using the regression coefficients in Table
D. Effect of Mean Radiating Temperature
VII(b) and the surface temperatures of the two profiles in Fig. 5,
The mean radiating temperature
plays a role in mapping
the prediction yields a
for the daytime profile differing by
the brightness temperature
to the opacity
. Traditionally,
about 1.6 K from the true value and that for the nocturnal pro-
is treated as a constant and is determined climatologically.
file differing by about 7.9 K. The prediction can be improved
For zenith observations, the uncertainties
are usually not a
significantly by using boundary temperature profiles from a 60
crucial factor [10], because the brightness temperatures are usu-
GHz radiometric temperature profiler [25], which accurately re-
ally small, resulting in small uncertainties in opacity, as seen
covers boundary layer surface temperature inversions or a radio
from the mapping function (12) or (13). But in tipping observa-
acoustic sounding system [26].
tions, the brightness temperature can be large at a low elevation
To see the effects of the
uncertainties, we applied
angle. Fig. 4 shows an example of how the
uncertainties
separately from Table VII to the calibrations using data from
(which are generated by adding the same error to both
at the
. The results are summarized in the A columns for the uses
two airmasses) affect the calibrations. As listed in Table VII(a),
of climatological mean and B Columns for the uses of surface
the climatological variations of
are about 9 K, estimated
temperature in Table VIII, which strongly suggests that the use
using data in
. Thus, we see from Fig. 4 that the uncertain-
of surface temperature can significantly improve the calibration
ties in
may cause significant calibration errors when large
accuracy.
airmasses are used. The
uncertainties can be reduced by
E. Errors Caused by System Random Noise
dividing the
climatology into seasons, a method that has
been practiced at ETL for many years.
The system random noise also affects calibration accuracy.
Another method that reduces the uncertainties significantly
To estimate its influence on system absolute accuracy, we per-
is the predicting
from the surface air temperature
, using
formed tipping calibrations using simulated tipping data from
regression analysis. Surface-based temperature measurements
with a 0.1 K Gaussian white noise. The results are shown in
along with calculated
using radiosonde measurements are
Table IX. The calibration uncertainties are about 0.1 to 0.4 .
used to derive linear regression coefficients relating surface tem-
Also shown in the table is that the uses of larger airmass differ-
perature to
. Table VII(b) lists the linear regression coeffi-
ences suffer less than the uses of smaller airmass differences.
cients for prediction of
and the standard error of estima-
The impact of the system noise can always be reduced by time
tion (SEE) after the regression. To simulate the effect of errors
averaging of a time series of calibration factors.
in the measurement of
, a 0.5 K Gaussian random noise
is added to the radiosonde value, and
is used as a pre-
F. Errors Caused by Uncertainty in the Offset of the
dictor. As shown in the table, the
uncertainties are cut in
Radiometer Equation
half by using the
measurements. However, the
predic-
In the analysis so far, we have assumed that the offset of the
tion from
is often poor when strong temperature inversions
radiometer equation is known precisely. In reality, however, un-
occur. In Fig. 5, the solid curve represents a nocturnal tempera-
certainty may exist. To estimate the impact of the uncertainty on

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