ORIGIN and REDUCTION of THE RECONSTRUCTION BIAS. Eric ANTERRIEU

Transcription

1 6th SMOS WORKSHOP May 15-17, 2006 Technical University of Denmark - Lyngby, DK ORIGIN and REDUCTION of THE RECONSTRUCTION BIAS Eric ANTERRIEU Observatoire Midi-Pyrénées Laboratoire d Astrophysique de Toulouse 14 avenue Edouard Belin TOULOUSE -FRANCE

2 Reconstruction procedure band-limited imaging device Taking into account the limited resolution of SMOS, a regularized approach finds the temperature map T r which has its Fourier transform confined to the experimental frequency coverage H. This band-limited solution realizes the minimum of the constrained optimization problem: min V - GT 2 T E ( I - PH ) T = 0 which unique solution is: T r = U * ZA + V with A + = (A * A) -1 A * and A = GU * Z.

3 Reconstruction procedure apodized map In order to filter out the Gibbs effects due to the sharp frequency cut-off associated to the limited H, the solution is damped by an appropriate window W: τ r = U * WUT ^ r This map has to be compared to: τ = U * WUT ^ which is the temperature map to be reconstructed, i.e. the map reconstructed with an ideal instrument (identical antenna patterns, no fringe washing, no modeling errors, no radiometric noise and same window W).

4 Reconstruction procedure the two components of the reconstruction error The corresponding error map τ r = τ r - τ is characterized with its mean (or bias) τ r its standard deviation σ τr which are computed in the extended alias-free field of view (up to some margins).

5 Origin of the reconstruction error in the absence of any errors/noise The reconstruction bias is one component of the systematic reconstruction error which has been observed even in the absence of modeling errors and/or radiometric noise. It is an important issue because it cannot be reduced by space/time averaging. It has been attributed to the high frequencies components of the observed scene which contribute to the complex visibilities while the instrument provides only the capability to retrieve a band-limited brightness temperature map.

6 Origin of the reconstruction error remote sensing vs. radio astronomy With regard to this band-limited imaging property, the situation of SMOS is very similar to that of any interferometer in radio astronomy (such as the VLA, for example). The truncation of H is responsible for the standard deviation component of the systematic error because of the presence of the well-known Gibbs oscillations in the reconstructed maps. Specific features may be responsible for the systematic bias

7 Origin of the reconstruction error remote sensing vs. radio astronomy Due to the short spacing between the antennas, the synthesized field of view of MIRAS is much larger than with any interferometric array in radio astronomy. This short spacing and the geometry of the array are also responsible for Earth and sky aliases in the synthesized field of view. Since the antenna patterns are also wider than in radio astronomy, non-negligible effects of individual antenna patterns should be taken into account.

8 Origin of the reconstruction error remote sensing vs. radio astronomy In radio astronomy, typically observed fields have a background level close to zero which is, in addition, attenuated by narrowness of antenna patterns. For the SMOS mission the average temperature of the Earth is not close to zero and the antennas are not so directive. Furthermore, the boresight of the instrument is tilted, in such a way that part of the transition between Earth and cold sky is seen by every antenna. This transition is therefore very critical from the point of view of the high frequencies because it is not damped. This does not preclude, of course, the effects of sharp details in the observed Earth scene itself.

9 Origin of the reconstruction error remote sensing vs. radio astronomy As a consequence, two origins, or two components, of the reconstruction bias have been investigated: an instrument dependent component, which has revealed the key role played by the voltage pattern of the reference radiometer in the reconstruction process; a scene dependent component, which has shown the influence of the high frequencies of the observed scene when using band-limited instruments such as MIRAS onboard SMOS.

10 Origin of the reconstruction error an instrument dependent component FWHM = 65 τ r location of the antenna on the arm

11 Origin of the reconstruction error an instrument dependent component FWHM = 70 τ r location of the antenna on the arm

12 Origin of the reconstruction error an instrument dependent component τ r FWHM of the voltage antenna pattern

13 Origin of the reconstruction error an instrument dependent component The reference antenna plays a key role regarding the reconstruction bias because its measurement is directly linked to the average temperature of the scene. Surprisingly enough, for the bias to be minimal, it is not necessary that all the voltage patterns be the same. From the linear algebra point of view, the condition for minimizing the bias is that the voltage pattern of the reference antenna should be as close as possible to the average pattern of the other antennas of the interferometer (whatever the spread between these patterns).

14 Origin of the reconstruction error an instrument dependent component The reference antenna plays a key role regarding the reconstruction bias because its measurement is directly linked to the average temperature of the scene. Surprisingly enough, for the bias to be minimal, it is not necessary that all the voltage patterns be the same. From a practical point of view, for a set of antennas manufactured under some industrial specifications, the best one should be assigned the role of the reference antenna.

15 Origin of the reconstruction error a scene dependent component The brightness temperature distribution of any observed scene can be decomposed into 3 orthogonal components: T = T DC + T LF + T HF with: TDC = T average value T LF = P H T low spatial frequencies T HF = ( I-P H )T high spatial frequencies T T DC T LF T HF

16 Origin of the reconstruction error a scene dependent component According to linearity of FOURIER operator U, we have: UT = UT DC + UT LF + UT HF i.e. with: T ^ = T ^ DC + T ^ LF + T ^ HF T ² ^ = T ^ DC ² + T ^ LF ² + T ^ HF ² T ^ T ^ DC T ^ LF T ^ HF

17 Origin of the reconstruction error a scene dependent component According to linearity of modeling operator G, we have: GT = GT DC + GT LF + GT HF i.e. but: V = V DC + V LF + V HF V ² V DC ² + V LF ² + V HF ² V V DC V LF V HF

18 Origin of the reconstruction error a scene dependent component NIR = 64 T o τ r τ r τ r ^T HF / ^AFOV T HF

19 Origin of the reconstruction error a scene dependent component NIR = 64.2 τ r ^T HF / AFOV ^ T HF

20 Origin of the reconstruction error a scene dependent component NIR = 64.4 τ r ^T HF / AFOV ^ T HF

21 Origin of the reconstruction error a scene dependent component NIR = 64.6 τ r ^T HF / AFOV ^ T HF

22 Origin of the reconstruction error a scene dependent component NIR = 65 τ r ^T HF / AFOV ^ T HF

23 Origin of the reconstruction error a scene dependent component This scene dependent component of the bias cannot be reduced from the hardware, except by enlarging the experimental frequency coverage H with additional antennas since an imaging interferometer sees all the frequency components of a scene under observation but only those lying in H can be retrieved (with more or less success). Only a numerical approach can reduce this bias by removing, as far as possible, the contribution of the high frequency components of the observed scene from the visibility measurements prior to inversion.

24 Reduction of the reconstruction error the basic idea (sub and add) The driving idea is to subtract, from the complex visibilities V, ~ ~ the contribution V of a brightness temperature distribution T, which is as close as possible to the observed scene T.

25 Reduction of the reconstruction error the basic idea (sub and add) Instead of solving the inverse problem V = GT the differential system δv = GδT ~ ~ ~ is solved for δt with δv = V - V and V = GT. The same regularized inversion is used and the temperature ~ map T is added to the solution δt r thus obtained, so that the solution now writes T r = U * ZA + ~ ~ (V - GT) + T This is not a new idea!

26 Reduction of the reconstruction error the basic idea (sub and add) ~ The artificial scene T should be synthesized without any additional measurements. It should contain as far as possible the high frequency components responsible for the scene dependent component of the bias so that, ideally, the regularized inversion procedure is accountable for retrieving only the frequency components inside the experimental frequency coverage H from complex visibilities measured in the same coverage.

27 Reduction of the reconstruction error a simple approach ~ ~ the T E /T S correction ~ The brightness temperature distribution T is made of a ~ constant temperature T E over the Earth and a constant ~ temperature T S over the sky, so that the effect of the transition between the observed scene and the sky should be reduced. ~ ~ Since T E and T S are not known, they are obtained through an optimization process which minimizes the quadratic error ~ between the complex visibilities V and V.

28 Reduction of the reconstruction error a simple approach ~ ~ the T E /T S correction τ r σ τ r τ r σ τ r ~ V V ~ T E

29 Reduction of the reconstruction error a very simple approach the V(0) correction ~ ~ The experimental value V(0) can be substituted to T E and T S can be set to a value taken from sky maps so that there is no optimization to perform.

30 Reduction of the reconstruction error an improved approach ~ ~ ~ the T O /T L /T S correction The previous approach can be improved by introducing a priori information on the subtracted brightness temperature ~ distribution T like coastlines or known variations with the incidence angle. The former information is introduced with the ~ ~ ~ aid of an ocean/land mask so that T is now equal to T O or T ~ L over the Earth and to T S over the sky. ~ ~ ~ Here again, T O, T L and T S are obtained through an optimization process which minimizes the quadratic error ~ between the complex visibilities V and V.

31 Reduction of the reconstruction error τ r, V V an improved approach ~ ~ ~ the T O /T L /T S correction ~ ~ σ τ r, V V ~ T L ~ T L ~ T O ~ T O

32 Reduction of the reconstruction error Africa test scene T X T Y

33 Reduction of the reconstruction error Caspian test scene T X T Y

34 Reduction of the reconstruction error Europe test scene T X T Y

35 Reduction of the reconstruction error Pacific test scene T X T Y

36 Reduction of the reconstruction error Sahara test scene T X T Y

37 Reduction of the reconstruction error reconstruction error before any correction Africa T X T Y τ r best / worst 0.48 / 2.83 K 0.25 / 3.61 K σ τ r best / worst 2.10 / 2.41 K 2.05 / 2.28 K Caspian T X T Y 0.45 / 4.17 K 0.47 / 5.66 K 3.01 / 3.49 K 3.01 / 3.26 K Europe T X T Y 0.47 / 3.85 K 0.05 / 4.34 K 2.55 / 2.95 K 2.89 / 3.08 K Pacific T X T Y 0.22 / 1.73 K 0.04 / 1.67 K 1.33 / 1.48 K 1.20 / 1.27 K Sahara T X T Y 0.50 / 4.44 K 0.15 / 4.96 K 2.85 / 3.31 K 3.24 / 3.45 K

38 Reduction of the reconstruction error reconstruction error after V(0) correction Africa T X T Y τ r best / worst 0.04 / 0.68 K 0.07 / 1.16 K σ τ r best / worst 0.72 / 0.76 K 1.03 / 1.21 K V(0) / K / K T S 3.7 K 3.7 K Caspian T X T Y 0.07 / 0.46 K 0.08 / 0.84 K 0.65 / 0.69 K 0.97 / 1.12 K / K / K 3.7 K 3.7 K Europe T X T Y 0.03 / 0.31 K 0.01 / 0.39 K 0.37 / 0.47 K 0.36 / 0.56 K / K / K 3.7 K 3.7 K Pacific T X T Y 0.01 / 0.26 K 0.02 / 0.60 K 0.40 / 0.43 K 0.44 / 0.53 K 98.4 / K / K 3.7 K 3.7 K Sahara T X T Y 0.06 / 0.40 K 0.09 / 0.52 K 0.51 / 0.61 K 0.54 / 0.77 K / K / K 3.7 K 3.7 K

39 Reduction of the reconstruction error ~ ~ reconstruction error after T E /T S correction Africa T X T Y τ r best / worst 0.05 / 0.57 K 0.08 / 1.02 K σ τ r best / worst 0.70 / 0.73 K 1.01 / 1.07 K ~ ~ T E T S / K 3.7 K / K 3.7 K Caspian T X T Y 0.03 / 1.04 K 0.01 / 1.39 K 0.84 / 0.96 K 0.87 / 0.91 K / K / K 3.7 K 3.7 K Europe T X T Y / 0.07 K / 0.17 K 0.31 / 0.32 K 0.32 / 0.33 K / K / K 3.7 K 3.7 K Pacific T X T Y 0.06 / 0.46 K 0.02 / 0.58 K 0.53 / 0.55 K 0.48 / 0.51 K 83.4 / 83.7 K / K 3.7 K 3.7 K Sahara T X T Y 0.04 / 0.18 K 0.01 / 0.29 K 0.46 / 0.47 K 0.51 / 0.52 K / K / K 3.7 K 3.7 K

40 Reduction of the reconstruction error ~ ~ ~ reconstruction error after T O /T L /T S correction Africa T X T Y τ r best / worst 0.01 / 0.44 K 0.03 / 0.22 K σ τ r best / worst 0.43 / 0.45 K 0.29 / 0.30 K ~ ~ ~ T O T L T S 94.9 / 95.0 K / K 3.7 K / K / K 3.7 K Caspian T X T Y 0.02 / 0.37 K 0.01 / 0.35 K 0.34 / 0.36 K 0.33 / 0.35 K 69.0 / 69.2 K / K / K / K 3.7 K 3.7 K Sahara T X T Y 0.01 / 0.15 K 0.01 / 0.23 K 0.46 / 0.47 K 0.44 / 0.45 K 0.02 / 0.03 K / K / K / K 3.7 K 3.7 K

41 Reduction of the reconstruction error visualization of the reconstruction error before any correction τ r τ r = 2.00 K σ τ r = 2.30 K τ r

42 Reduction of the reconstruction error visualization of the reconstruction error after V(0) correction τ r τ r = 0.41 K σ τ r = 0.75 K τ r

43 Reduction of the reconstruction error visualization of the reconstruction error ~ ~ after T E /T S correction τ r τ r = 0.16 K σ τ r = 0.71 K τ r

44 Reduction of the reconstruction error visualization of the reconstruction error ~ ~ ~ after T O /T L /T S correction τ r τ r = 0.02 K σ τ r = 0.43 K τ r

45 Conclusion The systematic reconstruction error has been characterized by its average (or bias) and standard deviation across the alias free field of view. The origin of the bias has been studied and two major generating mechanisms have been investigated. The first one is related to the voltage pattern of the reference radiometer used for measuring the visibility function for the zero spacing. The second one depends on the high frequencies content of the observed scene which contribute to the complex visibilities though they are outside the experimental frequency coverage of the imaging device.

46 Conclusion An efficient approach for reducing both the bias and the standard deviation has been presented. The driving idea is to subtract, from the complex visibilities, the contribution of an artificial brightness temperature distribution which is as close as possible to that of the observed scene. Two approaches have been presented for synthesizing artificial scenes that have been used for reducing significantly the Earth/sky transition without requiring any additional measurements and without any significant additional numerical cost for the reconstruction process.

47 Conclusion The first approach is a 2-parameters linear optimization based on a flat target model for the Earth while the second is based on a 3-parameters model which accounts for the difference between land and ocean so that the effects of the corresponding transition are also reduced. The latter approach might be improved with two additional parameters in order to take into account variations of the brightness temperature with the incidence angle. Other strategies, including iterative ones where the subtracted temperature distribution is updated after each reconstruction process, can also be implemented.