NASA-CR-202109
Reprinted from JOURNAL OF CLIMATE, Vo]. 8, No. 5, May 1995
American Meteorological Society
/
/
An Algorithm to Generate Deep-Layer Temperatures from Microwave Satellite
Observations for the Purpose of Monitoring Climate Change
MITCHELL D. GOLDBERG AND HENRY E. FLEMING*
NOAA /National Environmental Satellite. Data. attd hr[brrnation S(,twice. Satellite Research Lahoratoo,, Washington. D("
(Manuscript received 31 January 1994, in final form 6 September 1994)
ABSTRACT
An algorithm for generating deep-layer mean temperatures from satellite-observed microwave observations
is presented. Unlike traditional temperature retrieval methods, this algorithm does not require a first guess
temperature of the ambient atmosphere. By eliminating the first guess a potentially systematic source of error
has been removed. The algorithm is expected to yield long-term records that are suitable for detecting small
changes in climate.
The atmospheric contribution to the deep-layer mean temperature is given by the averaging kernel. The
algorithm computes the coefficients that will best approximate a desired averaging kernel from a linear combination
of the satellite radiometer's weighting functions. The coefficients are then applied to the measurements to yield
the deep-layer mean temperature. Three constraints were used in deriving the algorithm: 1) the sum of the
coefficients must be one, 2) the noise of the product is minimized, and 3) the shape of the approximated
averaging kernel is well behaved. Note that a trade-offbetween constraints 2 and 3 is unavoidable.
The algorithm can also be used to combine measurements from a future sensor [i.e., the 20-channel Advanced
Microwave Sounding Unit (AMSU)] to yield the same averaging kernel as that based on an earlier sensor [i.e.,
the 4-channel Microwave Sounding Unit ( MSU )]. This will allow a time series of deep-layer mean temperatures
based on MSU measurements to be continued with AMSU measurements. The AMSU is expected to replace
the MSU in 1996.
1. Introduction
For long-term monitoring of temperature change,
deep-layer mean temperatures derived directly from
satellite observations of upwelling radiance have an
advantage over traditional operational temperature re-
trievals. The advantage is that unlike operational re-
trieval algorithms (Eyre 1989; Fleming et ai. 1988;
Goldberg et al. 1988; Hayden 1988) an algorithm for
deriving deep-layer temperature directly can be made
independent of a first guess of the ambient temperature
profile. Operational retrievals are dependent on a first
guess because the satellite observations alone do not
have the vertical resolution to yield pointwise temper-
atures, which are needed for forecast models. Unfor-
tunately, the error between the first guess and the true
ambient condition is systematic and, furthermore, the
error cannot be entirely removed by the retrieval pro-
cess ( Thompson and Tripputi 1994 ). Since significant
climate change on a global scale can be on the order
of only tenths of a degree, temperature products in-
dependent of a first guess are a step in the right direc-
* Deceased.
Corresponding atahor address: Mitchell D. Goldberg, NOAA/
NESDIS, Satellite Research Laborato_, Washington, DC 20233.
tion. First guess independency provides certainty that
any observed trends in the data are not due to errors
in the first guess, which could very well have its own
interannual variation. Deep-layer mean temperatures
are appropriate for long-term monitoring of temper-
ature trends because nearly all climate models have
indicated that climate changes will occur over deep
layers and not at isolated levels (Mitchell el al. 1990).
The utilization of measurements from the Micro-
wave Sounding Unit (MSU), on board NOAA's op-
erational polar orbiting satellites, has gained much rec-
ognition during the past few years as a measure of deep-
layer mean temperature for long-term monitoring of
climate change (Spencer and Christy 1992a,b, 1993:
Spencer et al. 1990). Because radiance in this spectral
region is extremely linear with respect to temperature,
the observations can be interpreted as deep-layer mean
temperatures for the layer defined by the weighting
function. This is not true for the infrared spectral re-
gion, where temperature and radiance can be very
nonlinear. Microwave observations are usually ex-
pressed in brightness temperature, which can be ob-
tained from radiance using the inverse form of the
Planck function.
The MSU has four channels measuring outgoing ra-
diation at 50.31, 53.73, 54.96, and 57.95 GHz. Channel
1 (50.31 GHz) has a large surface component and is
generally not used for deriving temperature due to un-
994 JOURNAL OF CLIMATE VOLtJM_.8
certainty in the surface emissivity. The first MSU was
launched in 1979, and to date, its replacements have
provided nearly complete daily coverage of the earth
by scanning across the orbital track at _+_47.35 degrees
about nadir at approximately 9.47-degree increments.
The MSU's six view angles results in the projection on
the earth of i ! fields of view (FOV) for each scan line.
The weighting functions for channels 2 through 4 at
each of the six view angles are given in Fig. 1. The
highest peaking group of weighting functions is for
MSU channel 4, followed by MSU channels 3 and 2.
The higher peaking weighting functions in each channel
grouping are associated with larger off-nadir angles.
Spencer and Christy (1992a), used MSU channel 2
(53.73 GHz) brightness temperatures, adjusted to na-
dir, to monitor temperature for the layer defined by
the channel 2 weighting function on a 2.5 ° gridpoint
scale with a monthly precision of better than 0.1 °C in
the Tropics and to better than 0.2°C at high latitudes.
These estimates of precision were arrived at through
intersatellite comparisons and in comparisons with ra-
diosondes. They conclude that "the satellite precision
approaches that of individual radiosonde stations in
their ability to measure monthly temperature anom-
alies .... " In terms of monthly, zonally averaged tem-
peratures, they estimate their precision is of the order
of 0.01 °C over a 10-year period.
A deep-layer mean temperature from a single mi-
crowave observation has the equivalent vertical reso-
lution of the channel. Improved vertical resolution can
be obtained by combining different channels. The layer
D
D
100
1000
0.01
I I I I I I
......iiiiii>
0.01 0.02 0.03 004. 005 0.06 007
FJ¢i. 1. MSU weighting functions for channels 2, 3, and 4 al all
view' angles and Spencer's derived averaging kernel (dotted curve).
tO0
1000
- tli \ L J
! \L \,
m
m
0 01
I I I I
I I I I
oo; oo2 o.o3 0o4 o.o_ o.o6
Fl(;. 2. The influence of the gamma parameter
on the shape of the averaging kernel.
00)
is now defined by the averaging kernel, which is simply
derived from a linear combination of the weighting
functions ( using the same coefficients used to combine
the measurements). To remove the stratospheric com-
ponent from MSU channel 2, Spencer and Christy
(1992b) combined channel 2 measurements at different
viewing angles to create a more narrow averaging ker-
nel, shown as the dotted curve in Fig. 1, than the raw
nadir-viewing weighting function. It is interesting to
note that the raw channel 2 time series for the period
1979-90 showed a global warming trend of only
0.015°C per decade, while the combined-angle ap-
proach yielded an increased global warming trend of
0.032°C per decade. By combining different viewing
angles, Spencer was retrieving additional information
that a single channel at a common view angle was un-
able to provide. The only a priori information required
was knowledge of the weighting functions, which for
the MSU is well known and can be derived from a
standard atmosphere. Because the MSU weighting
functions are very weakly dependent on temperature
and moisture, a fixed set of coefficients can be used
globally to derive the deep-layer mean. This is not true
for infrared measurements; their weighting functions
generally have a much greater dependency on the am-
bient atmosphere.
Spencer did not use an algorithm to determine the
coefficients for his lower-troposphere deep-layer mean
temperature. He used trial and error by visual inspec-
tion of the averaging kernel to determine the appro-
priate coefficients. This technique is acceptable when
MAY I995 GOLDBERG AND FLEMING 995
considering a very few number of channels or angles.
However, as the number of different channels and view
angles increases, the determination of the coefficients
to yield a desired averaging kernel becomes a formi-
dable task. A quantitative retrieval algorithm is re-
quired to optimally solve for the coefficients. The coef-
ficients need to be optimal in the sense that the derived
averaging kernel is well behaved and that size of the
coefficients are constrained so that the noise of the
product does not become large.
The emphasis of this paper is to present an algorithm
to derive deep-layer mean temperatures from micro-
wave observations within the band 50-60 GHz. The
algorithm, derived in section 2, computes the coeffi-
cients needed to combine a set of channel weighting
functions into a desired deep-layer mean averaging
kernel. The deep-layer mean temperature is obtained
by simply applying the coefficients directly to the ob-
served brightness temperatures. Examples of averaging
kernels from the MSU are given in section 3. We will
also demonstrate that the MSU temperature time series
that Spencer pioneered can be continued with the next
generation of microwave sounders--the 20-channel
Advanced Microwave Sounding Unit (AMSU)
(Fischer 1987). The first AMSU is expected to be
launched in 1996. This will be accomplished by con-
straining the averaging kernel associated with the set
of measurements from the AMSU instrument to be
approximately equal to the averaging kernel associated
with the set of measurements from the MSU instru-
ment.
2. Algorithm
Our algorithm for computing deep-layer mean tem-
peratures and its corresponding averaging kernels is a
specialized adaptation of the Backus-Gilbert theory
discussed in Conrath (1972). The Conrath paper dis-
2.5
1,5
6
g
_t
13-
0.5
450
I
+ 400
350 ._
o
Z
300
o
250 _
200 g
a}
nc
150 m
100
5O
product noise ," _"_/"
/
/
/
i
m
/,
dr'
, A
0.001 00001 1E-05 1E-06 1E-07
Gamma Value
m
0 _ _-
1 0.1 0.;1 0
FI(;. 3. The relationship between 3' (gamma) and product noise
and the required sample to reduce the product noise to 0. I K.
I I I [ I I
msu2 msu3 rnsu4
- 0.000 0.000 0.000 --
0.000 0.000 0.000
2.310 0.000 0.000
1.069 0,000 0.000
-0.595 0.000 0.000
-1.784 0.000 0.000
starting/ending levels= 79 100 _--
lo pressures= 231.0 1000.0 --
sqcof= 10.02
act. noise= 1.05
0.1 noise sample= 110 --
gamma= O.]E-05 _
sum of coeff.= 1.00
output-input intgr, dilf= 16.08-
100
+1
-001 0 001 002 003 00+ 0.05 0.05 007
FIG. 4. Comparison of the boxcar-derived averaging kernel, based
on MSU channel 2 at view angles 3 through 6 and Spencer's derived
averaging kernel (doned curve). Also shown are the coefficients, the
starting and ending levels and pressures of the boxcar function, the
sum of the square of the coefficients, the noise of tile product, the
value of the gamma parameter, the sum of the coefficients, and the
integrated difference between the shape constraint and the derived
averaging kernel.
cusses the trade-off between instrumental noise and
the vertical resolution of the averaging kernel for a given
atmospheric level and set of measurements. The der-
ivation of our algorithm begins with the same basic
definition of the averaging kernel used by Conrath.
However, our approach differs from Conrath with re-
spect to application and constraint. Conrath's con-
straint is to derive coefficients that, when applied to
the weighting functions, attempt to reproduce the ideal
dirac delta function. In other words, he is trying to
obtain the highest-resolution averaging kernel possible,
cognizant of the effects of instrumental noise, for a
particular level in the atmosphere. This approach is
very useful for comparing the resolving power of cur-
rent and future sounders. On the other hand, our con-
straint is to yield coefficients that will reproduce a pre-
specified averaging kernel. Our averaging kernel, unlike
Conrath's, is not associated with a given level. Instead
it is "predesigned" to correspond to a desired deep-
layer mean temperature t_,derived from a linear com-
bination of n measured brightness temperatures 7",.
That is,
O. = cl T1 + • • • + c,, T,,. ( I )
where the c, are the coefficients of the linear combi-
nation.
996 JOURNAL OF CLIMATE VOLUME8
a. Algorithm constraints
To optimize the coefficients in ( 1) for a given at-
mospheric layer and a given set of channels and viewing
angles, three constraints have been imposed, which now
are explained in detail. The first constraint requires
that the sum of the coefficients is unity. Since tL of( i )
can be interpreted as a weighted average of brightness
temperatures, the weights must be normalized by con-
straining the coefficients to have sum one; that is,
ct + ''' +c.= 1. (2)
Thus, if all n of the T, in ( 1) are identical, then (2)
guarantees that tL will have that same value. Since the
7",are normalized so that a constant shift of one degree
in the temperature profile will result in a shift of one
degree in the Ti, this constraint will ensure that tt. has
the same property.
The second constraint addresses the problem that
each of the brightness temperatures used in ( I )carries
with it a measurement error. Let ¢2 be the variance of
the error associated with 7", and let a 2 be the variance
of the total error associated with tt. It is well known
that with independence of the individual errors the re-
lationship between the total error variance and the in-
dividual error variances is given by
= c + ... + (3)
Consequently, to minimize the magnitude of cr2 we
require as a second constraint that the sum of(3) be
a minimum.
E
v
o_
100
I000
!
- !
-- l
0,01 0
I I I I
msu2 msu3 msu4
0.288 -0.018 0.023
0.284 -0.024 0.027
0.271 -0.043 0.038
0.247 -0.078 0.052
0.204 -0.134 0.050
0.120 -0.224 -0.082
storting/ending levels= 72 100 --
pressures= 141.7 1000.0 --
sqcof= 0.45
oct. noise= 0,22
0.1 noise sornple= 4 --
gommo= 0.1E-05
sum of coeff.= 1.00
output-input intgr, diff= 15.75-
I I I
ool 002 003 oo_ o0_ o_os
FKI. 5. Boxcar-derived averaging kernels using
MSU 2, 3, and 4 at aH view angles,
m
0,07
100
- I
1000
001
I I I I I I
0.01 002 0.03 004 0.05 0.06 0.07
Fl(;. 6. Gaussian-derived MSU averaging kernels using
MSU 2, 3, and 4 at all view angles.
For the third constraint one must determine the
coefficients c, of( 1) in such a way that the deep-layer
mean averaging kernel agrees with the desired averaging
kernel as close as possible. The manner in which the
averaging kernel is defined is through the weighting
functions w,(x) associated with the ith channel and
which are the components of the kernel function in
the radiative transfer equation. Thus, the layer over
which tL of( 1) is defined is given by the so-called "av-
eraging kernel," given by the linear combination
a(x) = cl w, (x) + • • • + c_ w, (x). (4)
Equation (4) follows directly from ( 1 ). Note that x
can be any monotonic function of the atmospheric
pressure p. The purpose in making w,,a function of x,
instead of p directly, is that by judiciously choosing
the transformation from p to x, one can shape the
weighting function to suit specific needs. It also has
the property that the sum of vvi(x) over the range ofx
is unity. Because of the first constraint, the sum ofa(x)
over the range of x is also unity. The values of the
algorithm-derived averaging kernel represent the true
weights of the contribution of the unknown tempera-
ture profile to t_.
Note that the first two constraints were also used by
Conrath ( the first for a different reason ). It is the third
constraint and how we treat it that provides the major
relevance of this work.
b. Coc[]_cient determination
Determination of the coefficients in the linear com-
bination ( I ) of brightness temperatures, having the
MAY I995 GOLDBERG AND FLEMING 997
i'11
SUBORBITAL TRACK
t
IIT_rKM I_ "_1 I,,90KM
/'
" /,0ii
FIG. 7. The relationship between the I 1 MSU beam positions
and the ten deep-layer mean temperalures.
three properties discussed above, is now considered.
We begin by letting e and T be the vectors of coefficients
and brightness temperatures in ( 1), respectively, and
define the n-dimensional vector
u = [l, ..., l]L (5)
where the transpose superscript T is used because all
vectors are assumed to be column vectors. Then ( 1)
can be written
tl = e T T. (6)
and (2) can be written
u T c = 1. (7)
If we let
O = diag(a_, --., a_) (8)
be an n-dimensional diagonal matrix whose diagonal
elements are those indicated, then (3) can be written
a 2 = c_Dc. (9)
Furthermore, if we let W be a matrix of weighting
functions with dimensions channel (n) by level (j),
then the averaging kernel a(x) of (4) can be written
as the J-dimensional vector
a = wre. (10)
Next, a shape vector b ofj elements (i.e., the desired
averaging kernel) is defined to constrain the shape
of the resulting averaging kernel. The coefficient
vector c is determined in such a way that the shape
of a, given by (10), approximates the shape vector
as closely as possible. To do this, we minimize the
squared distance between the vectors a and b, while
at the same time satisfying the constraint (7) and
minimizing (9).
We now are ready to determine the coefficient vector
c by optimizing our solution with respect to the three
properties just discussed. This is accomplished by first
establishing a cost, or penalty, function F, which in-
corporates all three constraints. In its most general form
the cost function is
F(c) = (WTe --b)TS(WTc - b)
+ vcTDc + 2X( 1 -- uTe), ( 11)
where X and 3' are Lagrange multipliers and S
is an arbitrary symmetric, positive definite (usu-
ally diagonal) matrix of dimension J × J. Note
that the three terms on the right-hand side of ( 11 )
represent, respectively, the shape constraint of (10)
minus the shape vector b, the error variance con-
straint (9), and the coefficient normalization con-
straint (7).
To find that vector c, which minimizes F, we dif-
ferentiate F with respect to e and equate the result to
zero. This yields
2WS (WTc -- b) + 23"Dc - 2_u = 0, (12)
which implies that
c=(WSW r+TD) _(WSb+_.u), (13)