Radiometric sensitivity of SMOS
1. Purpose
We need some basic informations about the MIRAS sensitivity in order to be pertinent in contributing to choices for SMOS.
This note is mainly based on informations communicated by J.M. Goutoule (MMS). The purpose, then, is to describe how the radiometric sensitivity DT is estimated, and to give a few figures.
2. DT formulation
2.1) General formula. For an elementary MIRAS measurement (i.e. a single radiometric temperature in a single pixel), the radiometric DT sensitivity is given by :
DT = [(T_{sys} + T_{geo}) / sqrt(2)] x [A_{tot} / A_{thin} ] x [1 / sqrt (B x t )] (1)
This formula naturally separates into 3 factors, linked respectively to the input temperatures, the collecting area and the time/frequency integration..
2.2 Input temperature factor
T_{sys} is the system noise temperature ; for Lband, the performance claimed for receivers is about 60 K. However, this will probably be increased by the need for a commutating device (for calibration purposes), with a resulting 90 K estimation for T_{sys} . For Cband, by the same token, a 170 K figure would be probably conservative.
T_{geo} is the signal radiometric temperature. This quantity is of course (fortunately !) variable ; its range is however not the same over the ocean (# 60 – 160 K) and over the land surfaces (# 160 – 260 K). These figures may be taken for both L and Cband
The 1/sqrt(2) factor is due to the fact that two independant signals (for real and imaginary parts) are available.
2.3 Collecting area factor
While introducing the ratio [A_{tot} / A_{thin} ] is intuitive, the demonstration is not straigthforward. This demonstration, however, has been made !
A_{tot} is the total area of the circular antenna just containing the Yshaped MIRAS antenna. If r is the armlength :
A_{tot} = pi x r^{2}
A_{thin} is the actual collecting area. There are three arms, each consisting of adjacent elementary, circular receiving elements with diameter e x l, where l is the wavelength (for Lband, F=1.43 GHz, l =21 cm). The spacing factor e between adjacent elements has values near to unity (between 0.8 and 1.2 ; a typical value adopted here is 0.89). Then the number of elements along an arm is :
N = r / (e x l) ; this has to be an integer !
A_{thin} = {3 x N – 2} x pi x [e x l / 2]^{2} (2)
And : A_{tot} / A_{thin} # 4/3 x r / (e x l) assuming N large.
2.4 Time – frequency integration factor
B is the receiving bandwidth. In the Lband case, B cannot be broader the the 27 MHz band protected for radiometry around 1.43 GHz ; it is estimated safer to include a safety margin and bring the actual bandwidth down to 19 MHz
For Cband, there are no protected ranges on the spectrum, only regions allowed for shared use. In any case, a practical limitation to 28 MHz is set by the sampling frequency (56 MHz) of the correlators.
t is the integration duration. Actually, correlation products are summed ,in turn, for each polarisation ; in a next step, it is possible to postintegrate these products on board in order to reduce the uncertainty.
The limit to postintegration comes from the fact the the satellite moves (with a speed V_{sat} of about 7.3 km/s) ; hence, time integration tends to elongate the footprint along the track, thus deteriorating the space resolution.
The preliminary design for SMOS includes a post integration time on board of 1.5 s for each polarisation. This comes from the wish to achieve compatibility between the elementary integration time of the "correlation box" (which turns out to be set to 0.3 s through hardware) ands the datation process on board (which works best by entire second numbers)
3. Numerical applications and discussion
3.1 Numerical application outlook : see tables I & II for L & C bands.
Table I : L Band
F=1.43 MHz ; averaging time = 0.5x3 s ; T_{sys}= 90 K ; Bandwidth= 19 MHz ; e = 0.89
C 
D 
H 
I 
J 
K 
L 
M 
N 
O 
Q 
R 
S 

SEA or LAND 
arm length (m) 
Tgeo min (K) 
Tgeo max (K) 
Ttot min (K) 
Ttot max (K) 
A tot (m^{2}) 
N el / arm 
A thin (m^{2}) 
ant Fr 
int fact *10^{3} 
DT min (K) 
DT max (K) 

1 
SEA 
3,0 
60 
160 
106 
177 
28 
16,1 
1,3 
22,4 
0,19 
0,44 
0,74 
2 
LAND 
3,0 
180 
260 
191 
247 
28 
16,1 
1,3 
22,4 
0,19 
0,80 
1,04 
3 
SEA 
4,5 
60 
160 
106 
177 
64 
24,1 
1,9 
33,0 
0,19 
0,66 
1,09 
4 
LAND 
4,5 
180 
260 
191 
247 
64 
24,1 
1,9 
33,0 
0,19 
1,18 
1,53 
5 
SEA 
6,0 
60 
160 
106 
177 
113 
32,1 
2,6 
43,8 
0,19 
0,87 
1,45 
6 
LAND 
6,0 
180 
260 
191 
247 
113 
32,1 
2,6 
43,8 
0,19 
1,56 
2,03 
Table 2 : C Band
F=5 MHz ; averaging time = 0.5x3 s ; T_{sys}= 170 K ; Bandwidth= 25 MHz ; e = 0.89
C 
D 
H 
I 
J 
K 
L 
M 
N 
O 
Q 
R 
S 

SEA or LAND 
arm length (m) 
Tgeo min (K) 
Tgeo max (K) 
Ttot min (K) 
Ttot max (K) 
A tot (m^{2}) 
N el / arm 
A thin (m^{2}) 
ant Fr 
int fact *10^{3} 
DT min (K) 
DT max (K) 

1 
SEA 
0,86 
60 
160 
163 
233 
2 
16,1 
0,1 
22,4 
0,16 
0,59 
0,85 
2 
LAND 
0,86 
180 
260 
247 
304 
2 
16,1 
0,1 
22,4 
0,16 
0,90 
1,11 
3 
SEA 
1,29 
60 
160 
163 
233 
5 
24,1 
0,2 
33,0 
0,16 
0,88 
1,26 
4 
LAND 
1,29 
180 
260 
247 
304 
5 
24,1 
0,2 
33,0 
0,16 
1,34 
1,64 
5 
SEA 
1,72 
60 
160 
163 
233 
9 
32,1 
0,2 
43,8 
0,16 
1,16 
1,67 
6 
LAND 
1,72 
180 
260 
247 
304 
9 
32,1 
0,2 
43,8 
0,16 
1,77 
2,17 
We have selected three posible arm lengths (the nominal proposal option is around 4.5 meters). In each case, the table indicates DT values for the lower and upper limits of likely geophysical radiance temperature values.
3.2 Receptor noise : the intermediate case.
Both on land and sea, while the contribution of signal power to the receiving noise is most of the time larger than the receiving system contribution, the latter can definitely not be neglected. Therefore, when trying to simulate the noise, it will be necessary to combine both a uniform component and a component proportional to the signal power, making thus full use of formula (1). Similarly, biases in the measured temperatures may originate either ifrom an erroneous estimation of the system noise, or from a long term variation of the receiver gain..
3.3 Integration time effects
Since data are post integrated over a 3 seconds period, the pixels will be spreaded along the track, and there will be some loss in space resolution along this direction, at least for cases where this instantaneous space resolution is the best that can be achieved (the lower limit, for the nominal configuration, is very near to 30 km).
In every case however, a 3 seconds integration time corresponds to some "oversampling" (note this means that successive "views" corrspond to pixel that overlap ; on the other hand, thes data are statistically independant), by a factor of the order 2 on the average. Therefore, provided a clever way is found to take advantage of this oversampling without further degrading the space resolution, then it may be said that a margin still exists to reduce the radiometric resolution by a factor of about sqrt(2).
3.4 Radiometric sensitivity versus antenna size
For an elementary measurement, the radiometry sensivity varies like the armlength r, due to the collecting area factor. Assuming the integration time on board exactly matches the space resolution (i;e. no oversampling in space), then DT actually varies like r^{3/2} , because the space resolution improves as the armlength and the time available for averaging along the track diminishes accordingly, hence a supplementary r^{1/2} factor.
However, when the space resolution improves, for MIRAS this means that there are more instantaneous independant views of the same pixel available ; accounting for this in the retrieving process compensates approximately a r^{1/2} factor.
Therefore, one is back to a proportionality of DT to r.
Consider now the case where it is anyway planned to average over many adjacent pixels, in order to improve the overall radiometric resolution. Then the number of pixels available for averaging, over a fixed area, varies approximately as r^{2} ; in terms of reducing DT, this compensates the remaining r factor.
In summary, the situation is as follows : over the land or coastal areras, where pixel averaging is not recommended, the radiometric resolution worsens as the space resolution improves due to an increase of the armlength. Over the open ocean, for averaged areas, the dependance of the resulting radiometric sensitivity on the armlength essentially disappears.
3.5 Other factors to take into account
On the tables above, the figures likely to be the most relevant have been framed with a double line. Here they are again :
SEA or LAND 
arm length(m) 
DT min (K) 
DT max (K) 
SEA 
4,5 
0,66 
1,09 
LAND 
4,5 
1,18 
1,53 
There are three reasons why these figures are optimistic.
The only thing we know is that the absolute accuracy is warranted to be better than 3 K over the mission lifetime. Therefore we can only hope that over 200 seconds this will be much smaller, indeed substantially smaller than the radiometric sensitivity. This topic, however, deserves a thorough discussion of the factors contributing to the receiver's short term stability (power, temperature).
4. In conclusion
There is simply no way to give a single figure for what we are really concerned with, i.e. the standard deviation of measured radiance temperatures. However, all the indications given above suggest that selecting 1 K over the ocean and 2 K over land surfaces is a sensible approximation for average, typical values.