The core of this instrument is an echelette cube. Mirrors on the classical design are replaced by stepped mirrors -integrated into that interference cube- thus suppressing any moving part. The steps’ directions being set over a perpendicular axis, the overlap of both stepped mirrors creates a cluster of so-called “echelettes”, each one corresponding to a different optical path difference (OPD). Hence the Fourier transform of the incoming radiance is directly imaged on a CCD array in a single acquisition.

The frequency domain of the measurements is selected by an interferential filter disposed on the incoming optical path. A rotating wheel equipped with several filters allows the successive measurement of spectra around some bands of interest, i.e. O

_{2}, CO

_{2}and CO absorption bands.

## 1

## INTRODUCTION

There is a real need for CO_{2} monitoring in order to improve our knowledge of the global carbon cycle in the climate system, and to corroborate national and continental-wide control of carbon emissions. Spaceborne instruments are specially adapted to such a task as they provide worldwide coverage. Indeed, this opens the possibility of having a finer spatial grid of measurements than any other on-ground method. Therefore high-resolution spectrometers are needed for atmospheric sounding if we want to resolve the small variations in concentration of the gas.

MOLI is a Static Fourier Transform Spectrometer which studies reflected Sunlight from the surface of the Earth. An interferential filter in the fore optics of the instrument selects a chosen narrow band spectrum containing CO_{2} absorption lines. The output signal provided by MOLI is an interferogram which has to be treated to obtain the CO_{2} spectrum. Then an inversion method [1] can be applied to this spectrum to obtain the CO_{2} concentration columns.

This article presents MOLI design, together with a progress report of the breadboard under development at CNES. Finally, on ground data processing algorithm is introduced.

## 2

## INSTRUMENT DESCRIPTION

## 2.1

### Technical specification

The instrument has been designed around the following spectral channels :

• B1: λ = 1.6 μm CO

_{2}detection (over the earth)• B2: λ = 2 μm CO

_{2}detection (over the sea)• B3: λ = 2.2 μm CO detection

However, the purpose of the breadboard is to demonstrate the feasibility of the static Fourier transform spectrometer. Thus it is intended to operate only in the B1 channel in order to simplify the design and lower realisation costs.

Specifications for every band are:

• Bandwidth of 15 cm

^{-1}, needed to observe about fifteen absorption lines.• Sampling rate: 0.05 cm

^{-1}• Resolution: 0.1 cm

^{-1}

Pixel size is 8Km and every band is acquired with 50 km grid.

## 2.2

### MOtion Less Interferometer (MOLI)

A resolving power of 50 000, with a high signal to noise ratio cannot be achieved with a grating spectrometer in a reasonable volume. Thus we have developed a new compact concept, especially suited for high resolution spectrometry in narrow bandwidth.

The technical concept is a static Fourier transform spectrometer, which takes advantage of the Fourier transform spectroscopy (high etendue) without the difficulty of a mechanism.

This spectrometer can be viewed as a multitude of interferometers in parallel, (see Figure **1**). Each channel acquires a unique path difference. All the channels are imaged at the same time on a HgCdTe CCD array. Hence the interferogram is acquired instantaneously without any moving part.

However a rotating wheel with filters is needed to acquire the three different spectral bands. It is a simple mechanism re-using existing Polder developments, which is located just in front of the detector.

At the targeted wavelengths, the ocean surface is relatively dark (reflectance<0.01). Targeting the sun glint with a tracking mirror therefore greatly increases the signal to noise ratio over the oceans.

## 2.3

### Data processing

The spectrometer will acquire the positive interferogram of the spectrum. With each interferogram performed, an obscurity level reading will be acquired. Both data sets will be downloaded on earth. Correction of the interferogram and the Fourier transform will be done on earth, to obtain a final spectrum of around 300 elements. On-ground data processing is approached in section 4.

## 2.4

### Performance budget

With the instrument regulated at 20°C, and the detector regulated at 190 K, the most important noise contributor is the background emission of the instrument itself.

The minimum reflectance of the sun glint over the ocean is 0.1. Under these conditions, the signal to noise is approximately 500 for the interferogram. It leads to a signal to noise of 100 for the spectrum and 300 for CO_{2} determination.

## 2.5

### End to end performance Simulation of the breadboard

A model has been developed which simulates the interferogram acquired by the breadboard. An atmospheric model is used to generate input data. Several sources of error related to the breadboard modify the incoming light, giving a noisy, irregularly-sampled interferogram. An example is shown in Figure **2**.

A reconstruction algorithm is then applied to obtain a regular sampled interferogram (see section 4). The FFT can then be applied to obtain the measured spectrum. In Figure **3** the spectrum resulting from the signal processing of a simulated interferogram can be seen. Several CO_{2} absorption lines are present on this spectrum. The difference between this and the reference spectrum is also plotted, showing the reconstruction error.

## 3

## THE BREADBOARD

## 3.1

### General view

The main driver of the breadboard architecture is the echelette cube. As a first validation step, the cube will not be assembled. The breadboard will operate in the air with the two sets of echelette mirrors disposed as in a classical Fourier transform spectrometer. We can see that configuration in Figure **4**, which shows a general view of the test bench.

## 3.2

### Optical layout:

The optical formula is simple. The first group of lenses forms the image of the Earth on a diaphragm. This diaphragm physically limits the field of view. After this, a second group of two lenses re-images the echelette faces of the cube on the detector array. The optical layout is presented in Figure **5**.

The optical system is achromatic for the three bands. The thickness of the filter is adjusted for each band to be in focus (a common achromatic solution is being studied). In addition, the numerical aperture is low (F/8), easing the realisation and the alignment of the optics.

## 3.3

### Echelette cube:

The echelette cube is the most critical part of the interferometer. Indeed it must remain very stable during the mission. A first mock-up of the echelette mirrors -half size- has been achieved. They are presented in Figure **6**.

The echelette mirrors have been made by molecular adherence technique. The small echelettes have a thickness of 75 μm whereas the large ones have a thickness of 1.5 mm. The tolerance of realisation is around 5μm, however, the cube will be accurately measured in the final configuration. During the manufacturing, no serious problem was encountered and reproducibility measurements show that the realisation accuracy is of 0.5 μm. Completion of the whole cube is anticipated for April 2004.

## 3.4

### Filter:

A filter with high rejection performances (about 0.1 %) is needed. The designed filter is similar to the filters used in telecom applications. The first studies indicate that it meets all the requirements. The filter is expected by middle April 2004.

## 3.5

### Detection

The focal plane used in the breadboard is based on a commercial InGaAs detector. It is a 320x256 pixel CCD array built by Indigo Systems.

There are 24x19 echelettes to be imaged. As the detector has 320x256 pixels, every echelette (i.e. every path difference) should be averaged over 13x13 pixels and thus minimise non-uniformities. However pixels at the border of two echelettes are to be excluded. Hence the echelette is averaged over 11x11 pixels, what is knownas a “superpixel”. A very first image of superpixels -coarse alignment with no calibration- is shown in Figure **7**.

## 4

## ON GROUND DATA PROCESSING

## 4.1

### Reconstruction algorithm

The main problem arising when processing the interferogram is the sampled data non-uniformity due to manufacturing constraints of the echelette cube. Indeed the FFT algorithm used to obtain the spectrum needs a uniformly sampled interferogram in input. Thus, one of the main issues is the reconstruction algorithm that provides a regularly sampled interferogram from the raw data acquired. That reconstruction algorithm is presented thereafter.

Shannon’s theorem states that a band-limited function can be reconstructed from samples if the sampling frequency is at least twice the cut-off frequency of that functio n. According to this, the MOLI concept would not be feasible because CO_{2} absorption lines impose too high a cut-off frequency. That would mean an echelette cube with too many and too small echelettes.

We are only interested, however, in a narrow band (*∆v*) atmosphere spectrum while the incoming flow outside that band is brought to zero by filtering. In this case, generalised Shannon’s theorem states that the band-limited function can be reconstructed if the sampling frequency is equal or bigger than *1/2∆v* [2]. Under these conditions, the exact interpolation formula can be written as follows:

where *v _{0}* is the first frequency of the narrow band. Therefore, from Eq. 1, the interferogram function is defined by a finite set of

*N*regular sampled points.

Then, as the interferogram function *i(x)* is defined for any *x*, we can use Eq. 1 to define a *MxN* matrix (*T’*) providing a set of *M* irregular sampled points from a set of *N* regular sampled points. The inverse of that transformation (*T ^{-1}*) gives us the reconstruction algorithm providing a set of

*N*Regular sampled points from

*M*irregular sampled points.

## 4.2

### First validation of the algorithm

The algorithm has been tested using real data acquired using IASI breadboard (IASI is a classical Fourier transform spectrometer). Incoming radiance from the atmosphere has been filtered with narrow band infrared filter.

In a first step, the algorithm has been tested in two modes: (1) Sampling and reconstructing with the same rate and (2) reconstructing with a rate two times smaller than the sampling rate. The second mode has the disadvantage of reconstructing an interferogram with half the points of the acquired one.

Mode 1 has a good performance when input data is regularly sampled. However its performance degrades rapidly if sampling is non-uniform. Indeed it is easy to notice that the reconstruction matrix (*T*) is ill conditioned; it has singular values (*s _{i}*) close to zero. Thus matrix inversion at reconstruction causes the system to be unstable.

Mode 2 overcomes the instability problem by having a system with *2N-1* equations and *N* unknowns. Now the reconstruction matrix is well conditioned, even with very irregular sampling. Figure **8** shows an example of the spectrum obtained from a reconstructed interferogram when data is acquired with an irregular sampling rate of 12.33 ± 1.54 μm. The difference between this and the reference spectrum is also plotted, showing that the reconstruction error is 0.56 % RMS. Tests have been done up to 12.33 ± 12.33 μm sampling with satisfactory results.

Nevertheless there is noise amplification due to reconstruction, even when the reconstruction matrix is well conditioned. It can be shown that a good estimator of this amplification is the mean value of the inverse of singular values of the reconstruction matrix (<*1/s _{i}*>).

There has to be a compromise between noise amplification and loss of points at reconstruction. Using the same IASI data, we have traced the noise amplification according to sampled vs. reconstructed points ratio in Figure **9**. Current value used in MOLI modelling is 456/320, although further work is required to achieve an optimal choice for that parameter.

## 4.3

### Algorithm improvement

During the validation tests of the previous algorithm, an alternative algorithm was proposed. It would allow, with the help of a light modification of the design, to reconstruct all 456 samples. The new algorithm was successfully validated by simulation. The principle of acquisition will be tested on the breadboard later on.

## 5

## CONCLUSION

MOLI instrument takes advantage of a very innovative concept of static Fourier transform spectrometer. It is a high-resolution spectrometer working over a narrow bandwidth, which is adapted to a wide range of atmospheric missions and compatible with micro-satellite platform. Volume budget for the instrument is: 600x500x400 mm including the electronic modules. The total mass is estimated at 31.5 Kg.

An optical breadboard of the instrument is under construction at CNES. It will acquire incoming radiance corresponding to the 1.6 μm CO_{2} band and generate an interferogram. This paper reports the progress in its development. First results are expected by the end of 2004. Data processing of the raw interferogram is also being tested with good performances.

MOLI instrumental concept is unique and very innovative. It is highly supported by the French national Research and Technology program and has been patented by CNES.

## 6

## 6

## REFERENCES

_{2}column by use of the differential absorption method; error analysis,” Applied Optics, 42 (18), 3595 –3609 (2003). Google Scholar