About the MIRAS  SMOS field of view
1. Issues
It is clear for everybody interested in 2D interferometric radiometry that it is a very versatile technique, with a lot of potential. Still, this technique is complex, and there are things we are not clear enough about.
Specifically, the basic presentation of MIRAS is along the lines that it allows to "emulate" the operating way of a classical scanning radiometer, with a pseudoconical scanning pattern. Then, the main comparative advantage is the improved ground resolution (given the constraints of putting in space and deploying a large antenna), and indeed this is a major advantage. The balancing drawback is that the radiometric sensitivity is worse than for a real antenna, since the actual receiving area is maller for a given size. This can be accepted over the ground, less easily over the ocean.
More recently, emphasis has been given to the ability of MIRAS to simultaneously provide independent signals radiated by pixels seen over a range of elevation angles (i.e., ultimately, incidence angles). Indeed this opens two possibilities :
Most of the information on which the following analysis
is based (specifically : sections 2 & 3) was provided by Eric Anterrieu
(OBS_MIP  CERFACS)
2. Reference frames
In order to describe how the MIRAS field of view is bounded,
it is necessary to introduce two reference frames. The geometry of the
problem is depicted by Graph 1.
Graph 1 : geometrics of the measurement
On this graph, S is the satellite, H the subsatellite point ; SH is the satellite altitude. The geographic reference frame is Hxyz, with Hz along the SH line towards the center of earth C. Hxy is the horizontal plane, with Hy in the orbit plane and Hx perpendicular to it.
The target point on the earth surface is T ; elevation e is the (ST,SH) angle. The incidence angle i is the sum of e with the (CT, CH) angle.
In order to explicit the field of view (FOV), we next
introduce the antenna axis Sx_{z}_{,
which belongs to the orbit plane Syz, and is tilted with respect to the
vertical by the tilt angle t}
(in a typical configuration, the value of t
is 2530 °) ; O is the intersection between the antenna axis Sx_{z}_{
and earth surface.}
_{The Yshaped antenna is located in a plane perpendicular to the antenna axis, with one of its legs in the orbit plane.}
_{We have to complete the antenna reference frame with a plane parallel to the antenna plane. Here we have chosen a plane which includes the Hx axis. This plane is intersected by the antenna axis at point A. Then, the antenna reference frame is [A}x_{x}_{, A}x_{y}_{, A}x_{z}_{].}
_{The Earthsatelliteline of view ST intersects the [A}x_{x}_{,
A}x_{y}_{]
plane (the }x plane)
at point U. The (Sx_{x}_{,
SU) angle is the elevation angle q
}in
the antenna rerefence frame ; the (Ax_{x},
AU) angle is the azimut f
in the antenna r.f.
3. Field of view in the x plane
Graph2 a, b FOV in the x plane
The effective boundaries of the field of view are expressed in the x plane. In this plane, a view is referenced by the directing cosines x_{x , }x_{y }:
x_{x}_{ }=sin(q) cos(f) ; x_{y} = sin(q) sin(f)
The right half of the x plane is shown on graph 2a (the problem is symmetric with respect to the x_{y} axis). The location of the subsatellite is shown by a square on the x_{y} axis.
D_{hex} = 1/sqrt(3)/e
where e is the ratio : distance between adjacent elementary antennas / wavelength. Its value is near to unity.
The dotted curve on graph 2a is (the right half) of) the limit of the earthly horizon. In the x plane, this is an ellipse, defined by the equation :
[x_{x} / sin (e_{M}_{)]2 + [ (}x_{y} + cos(e_{M}_{)sin(}t) ) / (sin(e_{M}_{)cos(}t) )]^{2} = 1
where e_{M}_{ is the grazing elevation angle : sin(}e_{M}_{) = R / (R + SH)}
_{B The FOV is next bounded by "earth replicas" or aliased earth images. In the }x plane, these replicas are defined by ellipses, which are translated from the earth horizon ellipse by quantities simply related to the hexagon size.
(Whenever the tilt angle is such that ellipses cross over the Earth horizon, replica's boundaries take more complicated shapes, as they become partly circular).
4. Alias free
field of view in the x y coordinates
The FOV boundaries
transposed into the geographic reference frame, and the resulting FOV,
are shown respectively on graphs 3a & 3b. The deformation with
respect to the x
plane is self explanatory.
Graph 3 a, b : FOV in the x,y frame
Same as in graph 2, continuous lines for the reconstruction field limits ; hyphenated lines for the Earth's replicas.
5. Space resolution at ground level
The radiometer angular resolution de is approximately given by :
de # 0.8 l / r,
Where l is the wavelength, and r is the length of each antenna arm.
In what follows, we have taken r = 4.5 m ; hence, de = 0.0373 (i.e. 2.14 °).
It is understood that this formula yields an angular width that can be assimilated to the "3 dB angular width" in the case of the main lobe of a real antenna. This ought to be confirmed ; the 0.8 value of the coefficient seems rather high when compared to the classical results for uniform aperture illuminations.
The ground resolution Dx then is :
Dx = ST de / [ cos(i) cos(q) ]
where ST is the satellite to target distance, i is the incidence angle, q the elevation angle with respect to antenna axis. The origins of the cos factors are easy to unterstand : increasing the elevation angle decreases the antenna area normal to the line of sight ; increasing the incidence angle results in a larger earth area for a given area normal to the line of sight.
Graph 4a shows
the variation of the ground resolution throughout the FOV ; the contours
for equal Dx
values are actually cercles or near cercles centered on a point intermediate
between the subsatellite point.and the Earthantenna axis intersection
Graph 4 : (a) ground resolution contours (b) : independant views along the y axis
6. Independant views along the satellite motion
It is worthwhile knowing the number of actual independant views that could be combined in order to provide views of the same ground area with various incidence angles as the satellite moves on its orbit.
This is given by graph 4b. On this graph, the FOV is restricted by requiring the ground resolution to be better than 120 km.
It is clearly seen that, due to the specific shapes and orientations of the the lines bounding the FOV, the latter consists of 2 regions :
Note also that, in
any case, obtaining independant view of the same pixel for various
incidence angle values is not strictly possible, since the pixel size
will vary with i.
This will deserve attention, since this variation covers a factor exceeding
two (the best space resolution, at subsatellite point, is 29 km in the
present case).
7. Independant views over the whole FOV
One should not be mistaken by graph 4b : all the crosses shown do not correspond to independant views, since there is redondancy along the x axis (the x grid spacing has been chosen here equal to 10 km for illustration purposes).
Still, is it interesting, having in mind the cases where the target is assumed to be essentially homogeneous (i.e. the ocean) to obtain an estimate of the total number N_{i} of independant views over the whole instantaneous FOV.
In the present case, the value found for N_{i} is : N_{i} = 356.
A comparison then becomes possible with the performance of a "full" antenna (i;e. an interferometric antenna, the area of which would be completely filled with elementary antennas !) , in terms of sensitivity. For an interferometric instrument such as MIRAS, the radiometric sensitivity is deteriorated by a factor F_{A} = [A_{full} / A_{real} ], where A_{ full} is the area of the full antenna including the Yshaped array, the actual area of which is A_{real}.
The approximate expression for the area factor is : F_{A} # 4/3 / (e x l) x r
For numerical values chosen here (e = 0.83, r=4.5 m), F_{A} # 34
In order to compensate fully this factor, it would be necessary to obtain simultaneously (in order to compound them) F_{A}^{2} # 1000 independant measurements. This is the figure to be compared to N_{i}. Therefore, while obviously the sensitivity performance remains below what would be provided by a real antenna, it is seen than when making use of the full FOV capability, one goes quite a long way in this direction.
Actually, the comparison
turns out still more favorably, because useful independant views are restricted,
for a full as well as a thinned antenna, to the Earth horizon. Taking this
into account, the figure for available views is brought down to # 720 rather
than 1000.
Table I : sensitivity trends











Reference :
tilt = 25°, arm length = 4.5 m, flight altitude = 757 km, element spacing = 0.83 


1 view à 10 views à D i > 10°à 
230 150 170 
330 270 270 
450 390 360 
510 480 450 
Tilt angle from 25° to 30° 







Flight altitude from 757 to 650 km 







Arm length from 4.5 to 3.8 m 







Element spacing from 0.83 to 0.88 







* for ground resolutions better than 120 km. Grid resolution is 10 km
** Taken as the minimum distance to track of the lateral FOV boundaries
Details given for the reference case illustrate that the "swath" (and consequently the revisit time) is significantly dependant upon the requirements : at least one view, at least N independant views (here N=10), at least a gance range for incidence angles (here 10°). For other configurations, only the values for 1 view is indicated.
Obviously, in the context of this note, decreasing the arm length offers only disadvantages : particularly, the swath for various km sizes narrows significantly, while total N_{i} diminishes.
Increasing the tilt angle does not change things much : the swath width decreases slightly for useful ground resolutions.
Decreasing the flight altitude narrows the overall swaths. As shown by the table, ground resolution is improved for low values, but deteriorates for high values.This results from combined effects of the tardet to satellite distance and incidence angle.
Increasing the element
spacing has a major (negative) impact on the FOV ; howver, this is
only true for the largest ground resolutions.
9. Conclusion
Discussing the MIRASSMOS FOV is an important element in assessing the potential of this versatile instrument. The results presented above help to build a scheme in order to optimize the final configuration ; in carrying out this optimisation however, other elements, such as the radiometric resolution and several technical issues, has to be considered.