2.1.1

Modulation efficiency

We first assess the impact of an integration of the interferogram in the opd domain. The integration can be understood as a convolution with a boxcar.

Transition to the spectral domain gives a multiplication of the original spectrum with the Fourier transform of the boxcar, which is a sinc function.

The integration in the OPD domain scales the raw spectral signal. The scaling depends on the wavenumber, and also on the integration window length. The scaling does not directly cause a radiometric error, as the sinc factor cancels when the same integration window length is chosen in the calibration interferogram.

The term M = sinc (π · Δx · σ) is called modulation efficiency. We understand the naming, when looking at a monochromatic interferogram, which is a cosine function. The amplitude of the cosine reflects the usable signal strength. The integration over the window effectively reduces the amplitude of the cosine.

2.1.2

Integration over constant time windows

We now look what happen for a real system where the integration is done in the temporal domain. For perfectly constant OPD speed, the integration is equally done over constant OPD windows. For a varying OPD speed, the OPD windows scale with the OPD speed:

As the window in Eq (3) is not constant, it cannot replace the convolution kernel in Eq. (1). To do so, we must explicitly write the integral form:

As Eq. (4) is not a convolution in the mathematical sense, the derivation of the spectrum is not as easy, as in Eq. (2). We ignore this fact, and develop the spectrum for the special case of a monochromatic input scene with a harmonically perturbed OPD speed.

We write the harmonically varying OPD speed as:

We further give the (normalised) interferogram of a monochromatic source (wavenumber σ₀) as:

The resulting interferogram, after integration is shown in black in Fig. 1, compared to the same interferogram integrated over constant OPD-windows (the averaged one) in magenta. In order to compute the interferogram (4), we convolve over the ordinary, unperturbed averaging window, factoring out the OPD-dependence into a scaling function K(x). After some transformation and a first order Taylor development we get a good estimation of K.

Fig. 1

The figure compares the interferogram resulting from integration over constant time windows (magenta) with that after integration over constant OPD-windows (black). The interferogram sampling is on an equi-spatial grid in both cases. The parameters of the spectrum, the sampling, and the exaggerated perturbation have been chosen to be well distinguishable in the plot.

We can now do the Fourier transform of IFG(x).

With the setting of x=v₀·t in the sine term of the OPD variation, we analytically can give the FT of K(x). The induced error is of second order, and it is easily shown that the ignored ghosts have much smaller amplitude than the two considered ones. With the given approximations, the ghosts due to a harmonic OPD speed variation become classical amplitude ghosts.

We realise from Eq. (9) that the ghosts’ amplitude depends on the amplitude of the OPD variation a, but also on b, the sensitivity of the modulation efficiency wrt. speed variation. This briefly brings us to the correction idea; we simply make b close to zero. How this can be done will be described in detail later in this paper.

Fig. 1 compares the monochromatic interferograms, resulting from integration over constant opd-windows (magenta curve) and from integration over constant time-windows (black curve) for harmonic OPD speed variations. Fig. 2 shows the corresponding spectra with the ghosts left and right of the original line.

Fig. 2

The figure shows the Fourier transformation of the interferograms in Fig. 1. The spectra are normalized such that the magnitude of the original line in the centre, i.e. at 1260cm^-1, is one.

The monochromatic interferogram could be easily corrected by dividing the interferogram value by the modulation efficiency. All we need to know is the OPD speed at the interferogram sampling positions. This information is needed for the resampling anyway.

The continuous spectrum can be decomposed into monochromatic lines:

The difficulty of the correction for the continuous spectrum is that the modulation efficiency depends on the wavenumber. We can calculate the scaling of the interferogram at every sampling point by summing all wavenumbers over the monochromatic interferogram and the (monochromatic) modulation efficiency at that point. To set up a correction, we can use an iterative process, where the measured spectrum is used to get the first interferogram scaling. The first corrected spectrum is transformed from the measured interferogram divided by the first interferogram scaling. This spectrum can be used again to get the second interferogram scaling, and so on. This correction works, but is very costly in terms of processing.

2.2.1

The correction method

The method consists in performing a convolution of the equi-time sampled interferogram with a special 3-tap filter. This convolution is applied before the resampling on the equal OPD grid. The filter is of the type:

Because the filter is symmetrical, it does not cause phase changes, i.e. it does not affect the positions of the samples. For the special choice of q=1 the sum of the elements is unity, thus the filter is neural at zero frequency.

We get the function g(v) by Fourier transformation of the filter. The Fourier transformation can be more easily seen, for the filter separated into its two basic components:

The first filter term is symmetrical and with single frequency (only one tap on each side of the centre tap), and is then represented in the Fourier domain as a cosine. The period of this cosine is the inverse of the sampling period. The second filter term only multiplies the input signal by a constant.

The first filter term must be chosen so as to compensate the slope of the modulation efficiency, i.e. so that the slope of M·g is zero. This defines the value of q·k. The second filter term has no impact on the ghosts and can be chosen for convenience. Either choice of it must be identically chosen in the calibration spectrum, so that the scaling with a constant cancels during radiometric calibration. In case the interferogram amplitude does not matter, the simple choices of q=1 or q(1 – k) = 1 are rather convenient. Another sensible setting is q(1 – k) = 1 – qk · cos(2πv₀Tσ₀). The latter results in 〈g〉 = 1, therefore the energy of the interferogram is not changed by the convolution. Therefore this setting was chosen in all presented figures. Fig. 3 shows the modulation efficiency (blue) together with the Fourier transform of the correction filter (magenta) for a speed variation of 10%. The black curve in the same figure shows that the filter works and the slope of the product of the functions is zero for the mean OPD speed, and the central wavenumber. Fig. 4 shows the equi-time sampled interferogram (·) and interferograms (o) resampled on an equi-spatial grid with and without preceeding filter. The filter is switched off by setting k=0. Comparing the resampled interferograms in this figure with the interferograms in Fig. 1 illustrates how the filter works. Without the filter, the resampled interferogram is similar to an interferogram sampled on the correct positions but with constant integration time windows, while the resampling with preceeding filter leads to an interferogram similar to that sampled at the correct OPD positions and integrated over constant OPD windows.

Fig. 3

The modulation efficiency M, the correction function g, and their product are shown for one wavenumber σ_o=1264cm^-1 and an OPD speed in the range 0 to 1cm/s. The circles indicate the velocity range ± 10% around the nominal OPD speed , for which the correction filter is optimized.

Fig. 4

The solid line is the measured, equi-temporal sampled ifg, with the dots giving the position of the sampling. The circles give the resampled ifg, with and without (k=0) preceeding filter.

In the monochromatic case, the selection of q·k is straightforward. The situation changes for a broad band spectrum. Generally, a first guess can be derived for the central wavenumber, already keeping the variation of M·g small over the range [min(v)·σ_min, max(v)·σ_max]. Dependent on a given set of requirements, the parameter can be further fine-tuned, e.g. to minimize the radiometric error for an applicable reference scene.