In this paper, we consider occultations of celestial bodies through the atmospheric limb from low Earth orbit satellites and we show how the usual change of tangent altitude associated with atmospheric refraction is inseparably connected to a variation of the observed apparent intensity, for extended and pointlike sources. We demonstrate, in the regime of weak refraction angles, that atmospheric optical dilution and image deformation are strictly concomitant. The approach leads to the integration of a simple differential equation related to the observed transmittance in the absence of other absorbing molecules along the optical path. The algorithm does not rely on the absolute knowledge of the radiometer pointing angle that is related to the accurate knowledge of the satellite attitude. We successfully applied the proposed method to the measurements performed by two past occultation experiments: GOMOS for stellar and ORA for solar occultations. The developed algorithm (named ARID) will be applied to the imaging of solar occultations in a forthcoming pico-satellite mission.

In the terrestrial and planetary atmospheres, electromagnetic waves generally do not propagate along straight lines due to refractivity gradients caused by the vertical variation of the molecular concentration. In solar, stellar, planetary and GPS radio occultations, an orbiting spectroradiometer in a low Earth orbit (LEO) measures the atmospheric transmittance as a function of the tangent altitude of the line of sight. Such measurement techniques offer a precious advantage: they are self-calibrating because the observed signal is normalized with respect to the exo-atmospheric signal.

The effect of refraction of stellar light has been used in the past
for probing planetary atmospheres (see

The refractive optical dilution (i.e. light extinction caused by atmospheric refraction)
observed during Sun occultations has been
considered by

In this paper, we focus on the exploitation of the global radiative dilution
experienced by light when crossing the Earth's atmospheric layers.
We present an original method to retrieve the refraction angle profile
from the integration of a simple differential equation, defining the
ARID algorithm (Atmospheric Refraction by Inversion of Dilution).
In the recent past, our team

In the first section, we recall the elementary principles of atmospheric refraction and we simplify the geometrical problem by defining the useful phase screen approximation. In the second section, we derive several interrelated effects of atmospheric refraction and we show the concomitance of image flattening, shift of the apparent tangent altitude and dilution of the incoming irradiance for LEO satellites. We finally derive the differential equation subtending the ARID model. In the third section, we apply the developed algorithm to the exploitation of the transmittance data observed by the GOMOS fast photometer at 672 nm. In Sect. 4, we revisit the 22-year-old solar occultation data recorded by the ORA instrument, and we demonstrate the possibility of directly using the Sun transmittance to retrieve the refraction angle profile, although the problem is more complicated due to the angular extension of the solar disc. In the last section we draw conclusions on the capacity of the proposed algorithm and the associated requirements.

We deliberately do not consider electromagnetic scintillation and fast amplitude fluctuations superimposed on signals that travel through
a turbulent atmosphere and that can be measured by using fast photometers

Refraction parameters for a standard atmosphere:

Assuming that the Fermat's principle can be used to describe the
atmospheric propagation of electromagnetic radiation in the optical
domain

The phase screen approximation: the atmosphere
acts as a refraction plane where the total refraction angle

It is worth noting that the effective distance

Considering that Eq. (

In the phase screen approximation, the refraction occurs through an
equivalent infinitesimal atmosphere reduced to a vertical plane at
the considered geolocation. The turning point position is moved horizontally
by

Atmospheric refraction is usually considered in the context of ray tracing because the most apparent effect is a shift in the apparent tangent height of any ray emitted by a distant light source. Furthermore, in this section we are going to demonstrate that this change in position must correspond to an equivalent change in the measured spectral irradiance at the receiver.

In Fig.

Simplified representation of the refraction
plane geometry for the occultation of an extended light source at
distance

The angular extension of the solar disc (

Any ray emitted from the Sun at altitude

Depending on the true altitude dependence of the refractivity profile

If there was no refraction, the solar disc would span a vertical domain

We conclude that atmospheric refraction leads to two effects:

The image of the light source is displaced by a positive elevation
with respect to the unrefracted tangent altitude, leading to a non-negligible
bias for the retrieval of the vertical concentration profile of any
remotely sensed trace gas (see Table

The atmosphere acts as a diverging lens and produces a smaller and
real image of the Sun. The compression factor

As observed from the sensor side and assuming for the moment a constant
brightness of the source and the image, the radiometric signal will
be proportional to the solid angle subtending the Sun image and hence reduced
by the same factor

In some circumstances (typically when observing stars or planets),
the angular size of the light source object is below the optical resolution
and the concept of image flattening is useless. However the three
refraction effects are still there, but it is easier to understand
the dilution effect by reasoning on the refraction of a pencil of
rays emitted from the Sun's surface and passing through the position
of the refracted image at

The refractive dilution mechanism for a pointlike distant source.

We want to emphasize here that the usual measurement of refraction
by measuring the angular displacement from

Left panel: raw and smoothed signals measured by the red GOMOS photometer during a full occultation. Right upper panel: associated transmittance profile. Right lower panel: SATU angle and re-scaled transmittance.

The GOMOS stellar occultation instrument has been extensively described
elsewhere

The GOMOS measurements are sensitive to atmospheric scintillations caused by atmospheric turbulence. To demodulate the amplitude fluctuations, two fast photometers have been added to measure simultaneously the star intensities in two spectral channels centred in the blue (500 nm) and in the red (672 nm). It turned out that this correction is almost perfect when the star sets along the orbital plane but does not work properly for very oblique occultations due to the presence of residual scintillation caused by horizontal turbulent structures. Photometer data have also been used to investigate atmospheric turbulence (Kan et al., 2014) and to derive high-resolution temperature profiles (Dalaudier et al., 2006) from the time delay induced by chromatic refraction. Here we focus on the low-frequency part of the refractive effects by showing how to apply the ARID algorithm to the photometer data for the retrieval of refractive angle profiles.

During GOMOS occultations, a part of the incoming star light is routed
toward the fast photometers. They measure the stellar intensities with
a high sampling frequency (1 kHz) in limited wavelength bandwidths (50 nm). During
one spectrometer acquisition (0.5 s), the photometers record 500
stellar intensity values, from which we extract a transmittance profile

In the following example, we have used only the measurements of the
red photometers to minimize the impact of optical extinction due to Rayleigh
scattering, and ozone and nitrogen dioxide absorptions. The NO

Therefore, the transmittance

The transmittance

Refraction angle profiles retrieved using a single GOMOS occultation (star

The following expression for

The effective length

Finally, using Eqs. (

By numerically integrating this equation, we obtain

Figure

An alternative way to obtain the refraction angles from a GOMOS occultation
is to use the pointing angle of the tracking system. Indeed, at the
beginning of the measurement, the main mirror is oriented in elevation
and in azimuth toward the target star. Then, during the occultation,
this orientation is modified to keep the star image in the centres
of the detectors. In the GOMOS data, the elevation and azimuth angles
are provided as a function of time. For a vertical occultation, the
changes in elevation angle are due to the movement of the satellite
on its orbit and to the refractive effects. Thus, the refraction angle
is simply deduced by subtracting the geometric contribution from the
sum of the elevation and SATU angles. Figure

In solar occultations from a LEO satellite without any imaging system,
the observed transmittance is integrated over the angular size of
the Sun, resulting in an apparent loss of vertical resolution for
the retrieved vertical refractivity profile. However, this loss is
apparent because the sampling rate of measurements is usually large
with a strong overlap of the observed solar discs between two successive
acquisitions that causes redundance. This redundance does not lead to an ill-conditioned
inverse problem because the very large

We consider that the Sun consists of horizontal slices spanning an angular domain
described by the angle

Table of solar limb darkening parameters valid
for

The relation between these quantities is obtained by a rotation matrix
as

Geometry for angular integration across the solar disc.

The latitudinal dependence of the limb darkening parameterization

For the inversion, it is useful to work with the complementary transmittance

The ORA instrument, launched onboard the EUropean REtrievable
CArrier (EURECA) in July 1992, had the unique opportunity to observe the relaxation
of the Mount Pinatubo stratospheric aerosols, with a measurement coverage
in the latitude range [40

ORA consisted of eight radiometers of similar design with broadband
filters centred at 259, 340, 385, 435, 442, 600, 943, 1013 nm,
each containing a quartz window, an interference filter, and a simple
optics followed by a photodiode detector. From August 1992 to May
1993 the instrument measured about 7000 sunrises and sunsets from
its quasi-circular orbit at an altitude of 508 km. The near-infrared
channel at 1013 nm was mainly dedicated to stratospheric aerosol observations
as it is very weakly affected by Rayleigh scattering (see Fig.

A typical transmittance measurement in a solar
occultation observed by ORA. Notice the strong absorption by the Pinatubo
stratospheric aerosol around

The ORA data set has been unfortunately archived with a restricted
altitude range and sampling, sufficient for an accurate retrieval
of vertical profiles of ozone concentration and of stratospheric aerosol extinction. However, above 50–60 km,
a contamination by stray light was clearly caused by the very large
field of view of the instrument (

The large vertical extension of the solar disc at the geolocation
of the occultation caused a strong overlap between successive transmittance
measurements. Each acquisition combines different Sun slices (corresponding
to paraxial rays at different tangent altitudes with respect to the
Sun centre). Therefore a first step was to build a Lagrange interpolation
matrix to map the set of paraxial rays into a 1 km regular altitude
grid. The discretization allows us to re-write Eq. (

Median profile of atmospheric refraction
angle obtained from the processing of 2836 ORA solar occultations
observed between Aug 1992 and May 1993. Full thick line: median profile.
Full dashed lines: 16 % and 84 % percentiles of the retrieved profiles
distribution. Full thin lines: estimated errors due to signal digitization
(16 bits). Full circles: refraction angles obtained by exact ray tracing
for US76 standard conditions (

Eventually, the retrieved

We have processed the full ORA data set (6821 occultations) from which
we selected 2836 transmittance profiles that showed the least stray-light
contamination. The statistical results are presented in Fig.

It has to be underlined that the ARID method used in solar occultations can produce useful data over 6 orders of magnitude. The quality of the data is strongly related to the possibility of the stray-light removal, to the level of pointing stability during the acquisition of the reference radiance and during the occultation (typically 1 min) and to the possibility of cloud screening at lower altitudes.

Geometrical optics is extensively used to describe the propagation of electromagnetic waves through an atmospheric medium. Many studies have been focused on the computation and the measurement of amplitude and phase fluctuations induced by a random medium, reflecting important properties of atmospheric turbulence. However, in this paper, we concentrated on the exploitation of the average refractive bending that is relevant to a ray tracing approach (neglecting diffraction). We demonstrate how atmospheric refraction is equivalently responsible for a change of the tangent altitude in limb remote sensing geometry and for a change (mostly an attenuation) of the apparent radiance of the light source. As occultation measurements are self-calibrating, it is then possible to process the refractive dilution curve to obtain the vertical profile of refraction angles. This is the basis of the ARID method that can be implemented in a very direct way for punctual sources like stars or planets by integration of a simple differential equation. As the numerical integration proceeds downward from the exo-atmospheric domain, the method is particularly well suited for upper atmospheric measurements to the limit of the radiometric sensitivity and the pointing stability. For extended sources like the Sun, a complementary angular inversion is necessary but it leads to a well-conditioned problem due to the very high signal-to-noise ratio.

We have applied ARID to GOMOS stellar and ORA solar occultation data with
encouraging results. Our institute is presently designing a triple
CubeSat PICASSO (PICo-satellite for Atmospheric and Space Science
Observations) that will host the spectral imager VISION (VIsible Spectral
Imager for Occultation and Nightglow). This prototype instrument for
atmospheric remote sensing from pico-satellites will be able to observe
solar occultations in inertial mode thanks to its imaging capacity
of the full solar disc. It should be able to measure refraction angles
of about 0.5

This work has been partially funded by the PRODEX programme of the Belgian Scientific Policy Office (BELSPO) in support to the ORA and GOMOS experiences. PICASSO is presently funded by ESA under GSTP program “PICASSO Mission and VISION Miniaturized Hyperspectral Imager”. Edited by: C. von Savigny