## Abstract

In this paper we propose a new approach to operate two-dimensional sensitivity studies on the observations of MIPAS, an experiment on board the ENVISAT satellite. The proposed analysis system is intended to evaluate the amount and the spatial distribution of the information that is carried by MIPAS observations with respect to the target atmospheric parameters. The new approach enables the definition and assessment of the target-dependent atmospheric sampling of the measurements. The amount of information is evaluated by merging MIPAS measurements, relative to different limb-scans, in a two-dimensional analysis that models the sensitivity of the spectral signals combined with the geometrical redundancy introduced by different observation geometries. The spatial distribution of the information that is obtained with our analysis highlights the advantage of using a two-dimensional retrieval system. Furthermore, within the two-dimensional context, this analysis provides crucial indications for the definition of the optimal retrieval grid and, therefore, for the best exploitation of existing measurements. The proposed analysis is also suited for the design of optimized observation strategies. The sensitivity analysis, applied in this paper to MIPAS observations, can be extended to other orbiting limb sounders that, like MIPAS, continuously measure the atmospheric emission along the orbit track.

©2009 Optical Society of America

## 1. Introduction

Limb-sounding measurements are widely used to detect and quantify the presence of atmospheric constituents. Space-borne instruments are suited for this observational strategy especially when they measure the atmospheric emission from a polar orbit that allows full geographical coverage; in these experiments the line of sight of the spectrometer is often oriented along the orbit track. A significant number of this kind of measurements has been realized in the past (e.g. MLS [1], MIPAS [2,3], TES [4]) while others are considered by space agencies for the coming years (e.g. PREMIER [5]). In a limb-sounding space-borne experiment, the observation strategy is defined by acting on a set of observation parameters that, however, can be chosen within some constraints imposed by the peculiar characteristics of the experiment itself. A major objective pursued with the observation strategy concerns the atmospheric sampling. To date, the concept of “atmospheric sampling” has not a rigorous definition; it is based on the requirement that the observations should provide a coverage as wide as possible of the sounded atmosphere. In order to meet this requirement, only geometrical aspects are usually considered. Nowadays the choice of the observation strategy should be operated also considering the retrieval stage; indeed the atmospheric sampling should be as much as possible consistent with the capability of the retrieval system to extract information from the observations. A rigorous definition of “atmospheric sampling” should be based on the amount of information carried by the observations and its lay-out in the geometrical domain spanned by the observations themselves. The advent of two-dimensional (2D) retrieval algorithms [6,7,8,9], has extended the retrieval domain to the along track dimension making these algorithms capable of exploiting the 2D geometrical distribution of the information carried by the observations. However, the concept of “amount of information” is not, itself, univocal so that its definition should be the starting point for a study devoted to assessing the atmospheric sampling of limb-sounding observations. These considerations raise the issue of defining the “amount of information” and, possibly, its lay-out in the 2D domain spanned by the observations.

In this paper we provide a definition and the assessment of the amount of information and, therefore, the effective atmospheric sampling in the case of the MIPAS-ENVISAT observations. When the MIPAS experiment was designed, 2D algorithms were not yet available neither for the retrieval of atmospheric targets nor for analyses of the type introduced in this study. The choice of the vertical sampling-grid was mainly driven by the amplitude of the instrument finite Field Of View (FOV) of about 3 km while, in the horizontal domain, the sampling grid was determined by the frequency of repetition of the limb scans which (due to the motion of the orbiting platform) is connected with the time spent measuring a single limb scan. An analysis based on the amount of information of these observations is then expected to help in the identification of suitable choices that (a-posteriori) make the retrieval grid consistent with the spatial distribution of the information. Furthermore, the possibility to assess the 2D distribution of information for new observation modes, is expected to help in the design of further observation strategies that concentrate the information in regions of the atmosphere that are of interest for specific studies.

In Sect. 2 of this paper we recall the main features of MIPAS that are relevant for the application of the proposed analysis system to this experiment. In Sect. 3 we recall the rationale for the two-dimensional discretization of the atmosphere and describe the procedure that leads to determine, in each atmospheric parcel, the gathering of information that derives from the ray-tracing of different MIPAS observations. Section 4 is dedicated to define the sensitivity functions in the horizontal domain and to show an example of their behavior in a meaningful case study. In Section 5 we extend the definition of sensitivity function to the two-dimensional domain and we define the “information load” (IL) of an atmospheric parcel that is the basic element for the analysis that we propose; meaningful maps of IL are also reported in this section for MIPAS observations. In Sect. 6 we discuss possible applications of the IL analysis. Finally, in Sect. 7 we draw conclusions and focus on the main findings of the study.

## 2. The MIPAS experiment

#### 2.1. The instrument

MIPAS (Michelson Interferometer for Passive Atmospheric Sounding) has been developed by the European Space Agency (ESA) for the study of atmospheric composition. MIPAS operates onboard the ENVISAT satellite placed on a nearly polar orbit since March 1^{st}, 2002. MIPAS measures the emission of the atmosphere, in the spectral interval from 680 cm^{-1} to 2410 cm^{-1}, with the limb-scanning observation technique. In its original configuration the maximum optical path difference of the interferometer was 20 cm that corresponds to an unapodized spectral resolution of 0.025 cm^{-1} Fourier Transform (FT) and 0.035 cm^{-1} Full Width at Half Maximum. The “nominal” observation mode with this spectral resolution consisted of consecutive backward-looking limb-scans with the line of sight approximately lying in the orbit plane. Each limb scan consisted of 17 observation geometries (sweeps) with tangent altitudes ranging from 6 to 68 km with steps of 3 or 5 km. These observation parameters, combined with the ENVISAT orbit period of 101 min, imply about 72 limb-scans per orbit with a separation of about 510 km between consecutive limb-scans. The nominal mode was operated during most of MIPAS measuring time from July 2002 up to March 2004. However, “special” observation modes were also defined for the study of specific events or atmospheric phenomena; depending on the scientific objective, the special modes may differ from the nominal mode in the adopted spectral resolution, the altitude coverage, the vertical sampling steps, and the azimuth direction of the line of sight.

Due to the deterioration of the interferometric slides, starting from January 2005 all MIPAS observation modes have been re-defined for a “new” configuration in which the instrument is operated at 41% of its maximum spectral resolution. The nominal mode of the new configuration consists of 27 observation geometries whose tangent altitudes range from 6 to 70 km with increasing steps of 1.5, 2, 3, and 4 km. The shorter time required to measure an observation geometry in the new configuration is not fully compensated by the increased number of observation geometries in a limb-scan; this leads to about 96 limb-scans per orbit with a separation of about 410 km between consecutive limb-scans. Different observation modes have been defined also for the new MIPAS configuration; they were operated for a significant number of orbits during a testing period characterized by a conservative observational strategy that led to discontinuous measurements. Since January 2008 MIPAS is 100% operational in the new configuration mainly operated with the nominal observation mode.

Throughout this paper we will report examples relative to the nominal observation mode of the original MIPAS configuration.

#### 2.2. Data analysis

MIPAS spectra are analyzed by the ESA ground processor that determines, at the tangent points of each limb scan, the values of pressure, temperature and Volume Mixing Ratio (VMR) of six key atmospheric species (H_{2}O, O_{3}, HNO_{3}, CH_{4}, N_{2}O and NO_{2}). The ground processor uses a retrieval algorithm [10,11], based on the Global-Fit approach [12]. In this algorithm the portion of atmosphere sounded by the line of sight of the instrument is assumed horizontally homogeneous and observations of a full limb scan are simultaneously processed in order to determine the vertical distribution of the analyzed target. With this strategy a one-to-one correspondence exists between the measured limb scans and the retrieved profiles: the latter are naturally associated with the average geographical coordinates of the tangent points of the corresponding limb-scan.

A 2D “Geo-fit” algorithm [7] was developed for the analysis of MIPAS measurements and implemented in the operational code named GMTR [13]; the study reported in this paper refers to this kind of retrieval analysis. The Geo-fit approach is based on the simultaneous inversion of observations selected (see below) from all the limb scans measured along a whole orbit; this strategy makes it possible to model the horizontal variability of the atmosphere. In the Geo-fit the 2D retrieval grid is fully independent from the measurement grid (that is the grid identified by the tangent points of the measurements); by exploiting this feature atmospheric profiles can be retrieved with horizontal separations that are different from those of the measured limb scans. This property made it possible to carry out trade-off studies aimed at the identification of optimal values of the horizontal resolution for a given observation scenario [14,15]. Nevertheless, adopting the assumption that the information is mostly concentrated on the tangent point of the observations, also for the 2D algorithm the default choice is a horizontal retrieval grid defined by the average geographical coordinates of the tangent points of the analyzed limb-scans.

Irrespective of the specific retrieval algorithm, MIPAS observations are analyzed using a non-linear least squares fit based on the Gauss-Newton method [10,13]. Since the background of most of the reasoning in this paper is the performance of the retrieval analyses, we recall that the iterative solution expression of the Gauss-Newton method is:

where:

**x**is the vector containing the corrections to the state vector assumed in the previous iteration,**y**is the measurement vector,**f**is the forward model prediction,**S**is the variance-covariance matrix (VCM) associated to vector_{y}**y**,**K**is the matrix (usually denoted as Jacobian matrix) containing the derivatives of each observation with respect to the retrieved parameters,*i*denotes the iteration index,- T denotes the transpose of the matrix.

At the last iteration the errors associated with the solution of the inversion procedure are characterized by the VCM of **x** given by:

Matrix **V _{x}** maps the experimental random errors (represented by

**S**) onto the uncertainty of the values of the retrieved parameters; in particular, the square root of the diagonal element

_{y}*h*of

**V**provides the Estimated Standard Deviation (ESD or σ) of the corresponding parameter:

_{x}In operational retrievals the analysis is carried out on a limited number of observations; actually, it is possible to select narrow (less than 3 cm^{-1} wide) spectral intervals containing the best information on the target quantities [16]. The use of narrow spectral intervals, called microwindows (MWs), allows to limit the number of analyzed spectral elements and to avoid the analysis of spectral regions which are characterized by uncertain spectroscopic data, interference by non-target species, or are influenced by unmodeled effects [10].

An exhaustive description of the MIPAS experiment can be found in Ref. 2.

## 3. Multiplicity of the atmospheric parcels

#### 3.1. Discretization of the atmosphere and ray tracing

In a 2D approach the discretization of the atmosphere must be operated on both the vertical and the horizontal extension of the atmosphere. In the vertical domain the discretization is operated using altitude levels as it is done in a 1D approach. For operational applications the spherical approximation cannot be adopted for the Earth’s shape which, instead, must be represented by an ellipsoid; accordingly also the altitude levels have the elliptic shape. The definition of concentricity does not apply to ellipses; however, in the Earth’s dimensions scale, the objective of building two evenly spaced levels can still be attained, with an approximation of a few centimeters, using ellipses that differ in both semi-axes by the desired altitude separation [13]. For what concerns the horizontal discretization, it can be built using segments perpendicular to the Earth’s geoid and extended up to the boundary of the atmosphere. The process of building a horizontal discretization starts from a set of evenly-spaced radii that originate at the Earth’s center. The radii identify a set of points on the surface of the Earth’s ellipsoid; from these points, segments perpendicular to the Earth’s surface are originated and extended up to the atmospheric boundary. This kind of 2D discretization leads to a web-like picture in which the intersections between levels and radii identify points denoted as “nodes”. Furthermore, consecutive levels and radii define plane figures that are denoted as “cloves”. A sketch of this kind of two-dimensional discretization of the atmosphere is reported in Fig. 1 where a sample clove is highlighted in green.

The ray tracing operated on a 2D discretization of the atmosphere starts from both the coordinates that define the position of the satellite and the elevation angle corresponding to a Line Of Sight (LOS) of the considered limb-view. These two geometrical elements define a segment (representing the considered ray) that extends from the satellite to the boundary of the atmosphere. From this point on, while crossing the atmosphere, the propagating ray will intersect levels and radii of the 2D discretization. At each intersection, the direction of propagation of the ray is modified by effect of refraction (modeled by the Snell’s law applied to the boundary between the two cloves involved in that intersection). The resulting shape of a LOS is therefore a piecewise linear made by segments each representing the optical path within a clove. Each limb-view will require the ray tracing of some LOSs whose number depends on the application (see below). In Fig. 1 we show, on a highly distorted scale, the ray tracing of the two LOSs that delimit the FOV of a limb-view in the case of 2D discretization of the atmosphere. For the purpose of forward model calculations the atmospheric fields are defined by assigning to each node of the 2D discretization values of pressure, temperature and VMR of the atmospheric constituents; these quantities are interpolated within each clove starting from the values on the four nodes that define it.

A detailed description of the discretization and ray tracing implemented in the 2D Geo-fit retrieval system for the MIPAS data analysis can be found in Ref. 13.

#### 3.2. Multiplicity analysis

The sketch of a limb-view is reported in Fig. 1; the cloves crossed by the corresponding FOV are emphasized with a light shadowing in the figure. Figure 1 highlights that the observation corresponding to this limb-view will be affected by the state of the shadowed cloves. As we have seen in Sect. 2, MIPAS measures consecutive backward-looking limb-scans with the line of sight lying in the orbit plane. It is then legitimate to imagine that, in a real case, cloves can be crossed by the FOV of different observation geometries possibly belonging to different limb-scans.

As a case study we consider ENVISAT orbit 2081 of July 24, 2002 when MIPAS was operated in the original configuration with the nominal observation mode. The ray tracing was carried out, for all the observation geometries of the 72 limb-scans measured in this orbit. The 2D discretization of the atmosphere was operated with layers 1 km thick, up to 80 km, and radii equispaced by 0.25 latitudinal degrees (about 28 km at Earth’s surface). Figure 2 shows (with different colors) the ray tracing of the LOSs that delimit the FOV of three out of the eight limb-views that cross a sample clove (highlighted in Fig. 2) of the 2D discretization. Each FOV line is marked in Fig. 2 by two order numbers that identify the limb-scan and the sweep the represented FOV belongs to.

Since the particular clove considered in Fig. 2 is crossed by a total of eight different observation geometries, we can define a “multiplicity” (*M*) value that, for this clove is 8. With this kind of analysis we can assign a multiplicity value to each clove of the 2D atmospheric discretization; a color scale assigned to the *M* values permits to draw the map, reported in Fig. 3, that shows the distribution of *M* along the full orbit. In Fig. 3, as well as in all the figures of this type that follow in this paper, the vertical dimension of the atmosphere is expanded by a factor of 10 (for the sake of clarity) with respect to the extension of the Earth’s radius. In this type of figures is reported the Orbital Coordinate (OC) that is the angular coordinate ϑ of the polar reference frame having origin on the Earth’s center and ϑ = 0 at the North pole.

Figure 4 shows a blow-up of Fig. 3 meant to provide a better detail of the color structures generated by the distribution of *M*. The most prominent features in Figs. 3 and 4 are patterns that indicate uneven distribution of *M* characterized by peak values (up to 19) at high altitudes. The origin of these patterns can be explained looking at Fig. 5 that provides a sketch for the LOSs of four observation geometries within a limb-scan. The red dots in Fig. 5 mark the position of tangent points. Although with a highly distorted scale, Fig. 5 properly represents the fact that all the LOSs of a limb-scan converge toward a common air mass at high altitudes where, as a consequence, the highest values of *M* occur.

The results of the multiplicity analysis could suggest that MIPAS observations provide diversified amounts of information with respect to different regions of the atmosphere; being the amount of information proportional to the *M* value. If this argument is correct we should expect favorable retrieval conditions at the altitudes where high multiplicities occur. However, looking at Fig. 5 we can notice that the highest multiplicities occur in the part of the LOSs that are the farthest from the observation point. Therefore we expect that the radiative transfer process operates a meaningful masking effect on the signals that come from the highest-multiplicity regions. As a consequence, we can forecast a reduced sensitivity of the observations, with respect to the behavior of the target parameters, in the parcels of atmosphere characterized by high *M* values.

## 4. Horizontal sensitivity functions

The qualitative considerations reported at the end of the previous section (about the sensitivity of the observations) can be supported in a quantitative manner by calculating the horizontal sensitivity functions (SF) of the observations.

In the vertical domain the SF has been defined [17] as:

where:

*S* is the signal that reaches the spectrometer at frequency *ν*,

*q _{z}* is the value, at altitude

*z*, of the atmospheric parameter

*q*that is spectroscopically active at frequency

*ν*,

*x* denotes the set of observation parameters.

For any observation defined by a set of parameters (*x*,*ν*) the function *ϕ* measures the sensitivity of the observation to a variation of the atmospheric parameter *q* at different altitudes. For this reason *ϕ* provides, as a function of altitude, a measure of the amount of information carried by that particular observation with respect to the parameter of interest. Examples of vertical SFs can be found in the literature (see e.g. Ref. [17]).

In a similar way we can define the SF in the horizontal domain as:

where *q _{ϑ}* is the value of atmospheric parameter

*q*at orbital coordinate

*ϑ*(see Sect. 3.2). By analogy with the vertical domain, the values of Φ provide, as a function of the OC, a measure of the amount of information carried by that particular observation with respect to the parameter of interest. The process of computing Φ at a given value of

*ϑ*(see e.g. the observation geometry in Fig. 1) starts with the simulation of the spectral intensities by means of a forward model. A perturbation is then introduced to the atmospheric field of the target quantity by giving an increment on all the nodes that lie along the radius identified by ϑ. The intensities calculated for the unperturbed atmosphere are then subtracted to those corresponding to the perturbed atmosphere to build the incremental ratio that approximates the derivative of Eq. (5). The whole SF is calculated by repeating the above process for all radii that are crossed by the FOV of the analyzed observation geometry.

In order to provide an example of horizontal SFs we consider a MW used by the MIPAS ground processor for the retrieval of CH_{4} VMR (we will use this target as case study also in the next sections). Figure 6 shows this MW as measured at about 24 km tangent altitude; the CH_{4} spectral feature at about 1228.8 cm^{-1} has been used to calculate the Φ functions for four spectral points (identified with numbers in Fig. 6) in all the sweeps of a selected limb-scan. In order to ease the interpretation of the resulting functions we have assumed a homogeneous field with 1 ppm constant VMR for CH_{4}.

Examples of the horizontal SFs calculated for this CH_{4} test case are provided in Fig. 7. Each row of Fig. 7 corresponds to a spectral point; the left-hand panel shows SFs relative to higher tangent altitudes while lower tangent altitudes are represented in the right-hand panel with a different scale expansion. The lowest tangent altitude of each left-hand panel is also reported in the corresponding right-hand panel. The units in the abscissa axes of Fig. 7 are latitude degrees with respect to an origin placed at the latitude where the first LOS enters the atmosphere. The position of tangent points is marked in Fig. 7 with a dot; according to the sketch reported in Fig. 5, in Fig. 7 the tangent points move closer to the origin while tangent altitudes decrease. It can be seen in Fig. 7 that the SFs are dominated by a bell-shaped profile at high tangent altitudes where the radiative transfer is, in a good approximation, a linear process. Actually, in these conditions, absorption is negligible with respect to emission and emission, as well as its derivative, increases while penetrating the atmosphere along the LOS (see e.g. Fig. 1). This is because the concentration of the active target increases with pressure at descending altitudes. At lowest tangent altitudes most of the radiation still originates from near the tangent point. An increase of the VMR between this point and the observer generates a local increase of emission that, however, is surpassed by the increase of absorption the extra VMR operates on the radiation emitted near the tangent point; this mechanism explains the negative lobes in Fig. 7. On the other hand the radiation emitted from the far side of the LOS is absorbed to some extent along its path and an increase of the VMR in these locations does not affect the radiation emitted from near the tangent point; this explains the weak intensity of the SFs beyond the tangent point.

The behavior of the horizontal SFs observed in Fig. 7 confirms the conjecture made in Sect. 3.2 about the reduced sensitivity to signals that come from the atmospheric regions characterized by high multiplicities.

## 5. Two-dimensional sensitivity functions and information load

#### 5.1. Two-dimensional sensitivity functions

The SF definitions provided by Eqs. (4) and (5) are relative to the distribution of an atmospheric parameters along just one dimension (vertical and horizontal respectively). We notice that the rows of the Jacobian matrix **K** in Eq. (1) can be regarded as a discrete form of the SFs. Moving to the two-dimensional approach, the SF must be defined for the distribution of the parameter on the 2D domain. The 2D sensitivity function requires the calculation of derivatives with respect to the value that the atmospheric parameter assumes in the infinitesimal elements of a surface; it can be defined as:

where *h* denotes the element of the considered surface. For numerical calculations *h* can be identified with the clove of the 2D discretization and *q _{h}* with the average value that the atmospheric parameter

*q*assumes within clove

*h*.

Equation (6) provides a measure of the amount of information carried by the observation with respect to atmospheric parameter *q* in clove *h*. Each 2D sensitivity function represents the behavior of a single spectral point within a single observation geometry; we will not show examples of 2D sensitivity functions because the corresponding maps would provide a limited picture of the amount of information. However, as we will see in the next sub-section, the 2D sensitivity functions represent the starting element for the definition of the overall amount of information carried by the observations in each clove of the 2D atmospheric discretization.

#### 5.2. Information load

The analysis of MIPAS observations is operated on a set of MWs (see Sect. 2), it follows that all the considered spectral points contribute to accumulate information about the state of each clove of the 2D discretization. Furthermore, we have seen in Sect. 3.2 that a multiplicity value can be assigned to each clove. The outcome of these considerations is that we can assign to clove *h* an overall “information load” (IL) that results from the gathering of information from both the spectral elements of the analyzed MWs and the observation geometries that contribute to its multiplicity. We call Ω this new quantifier defined as:

where:

Ω(*q*,*h*) = overall information load of clove *h* with respect to atmospheric parameter *q*, *S _{ijk}* = spectral signal of observation geometry

*i*at frequency

*j*of the analyzed MW

*k*,

*l*= number of observation geometries that define the multiplicity of clove

*h*.

*m* = number of analyzed MWs in observation geometry *i*,

*n* = number of spectral points in MW *j*,

The fraction within round brackets in Eq. (7) is an element of the 2D sensitivity function defined in Eq. (6). In Eq. (7) we have adopted the rule of “summation in quadrature” because the column vector containing the set of elements within the triple summation is the Jacobian matrix (see Eq. (1)) corresponding to the retrieval of target parameter *q* in clove *h*. Therefore the term in parenthesis of Eq. (2) turns into:

In Eq. (8) we assume uncorrelated observations all characterized by constant uncertainty and we neglect the multiplicative constant equal to the reciprocal of the uncertainty. In a retrieval analysis the uncertainty on the value of the target quantity *q* in clove *h* is then given by 1/Ω (see Eq. 3).

For a given atmospheric target we can calculate a value of Ω for each clove of the 2D atmospheric discretization; by defining a color scale for these values we can draw a map of the 2D distribution of Ω with respect to the considered target.

The process of computing a map of Ω starts with the simulation of all the spectra of the considered orbit by means of a forward model. The cloves of the 2D discretization are then “switched on”, one at a time, by giving an increment to the target quantity (see e.g. the clove considered in Fig. 2). The observation geometries that define the multiplicity of the considered clove are then calculated again and the spectra corresponding to the unperturbed atmosphere are subtracted to those corresponding to the perturbed atmosphere to build the incremental ratios that approximate the derivatives of Eq. (7). At this point the terms are all available for the calculation of the triple summation and of the Ω value as defined in Eq. (7).

Equation (7) implies that the values of Ω are mainly determined by: the observation geometries, the atmospheric field assumed for the analyzed target, the set of observations (MWs) selected for the analysis (*M* is unambiguously determined by the observation geometries). It is then expected that different targets will display different distributions of Ω because only the first of these three elements does not depend on the properties of the target. This expectation is confirmed by the set of targets and observation modes analyzed so far.

As an example, Fig. 8 shows the distribution of Ω with respect to the VMR of CH_{4} along the full orbit 2081. For the calculations leading to Fig. 8:

It can be noticed in Fig. 8 that the summer hemisphere (upper part of the figure) shows higher values of Ω with respect to the winter hemisphere; this is correlated to both the CH_{4} distribution and the higher temperatures that occur in the summer hemisphere. Details of the structures generated by the distribution of Ω can be appreciated in Fig. 9 that shows a blow-up of Fig. 8 in a region around the equator.

## 6. Discussion

The maps reported in Figs. 8 and 9 represent the effective sampling that MIPAS observations operate on the atmosphere when measuring CH_{4} in the set of MWs used by ESA for the operational analysis; this kind of maps provide an objective definition of the “atmospheric sampling”. As pointed out in Sect. 5.2, the layout of the IL, and therefore the atmospheric sampling, is determined by not only the observation parameters but also depends on other quantities that make it target dependent.

The information provided by the IL analysis can be exploited for several applications; we will discuss three of them in the next sub-sections.

#### 6.1. choice of the retrieval grid

The 2D retrieval grid of a Geo-fit is fully independent from the measurement grid (see Sect. 2). This feature can be used to obtain detailed modeling of the horizontal structures of the atmosphere [15], however it poses the problem of choosing a strategy for the definition of the retrieval grid. For what concern the horizontal domain, the best available criterion (to date) is to pose the retrieval grid at the position of the mean tangent point of each limb-scan where, it is assumed, the information carried by the measurements peaks. The blue triangles in Fig. 9 mark the position of the tangent points of the limb-scans while the vertical lines indicate the OC where the altitude profiles would occur if retrieved at the position of the mean tangent point of each limb-scan. It can be seen in Fig. 9 that, with the aforementioned layout, the profiles would be retrieved at positions that miss the maximum information load. The consequence is that the retrieved profiles will be assigned to positions where they are possibly different from the real profiles, but they can still produce a good fit of observations that sample the atmosphere elsewhere. The assessment of this error is beyond the purpose of this paper.

The results of the IL analysis carried out for the CH_{4} case-study suggest two strategies that could be adopted to define a retrieval grid that better copes with the IL distribution of the observations:

- Figure 9 shows that the tangent points of the limb-scans identify lines of high IL; therefore it suggests that the altitude profiles could be retrieved along these lines. The IL analysis carried out on different atmospheric targets (see the example of temperature below) shows that the matching between tangent points and high IL is not always verified. On the other hand the option of altitude profiles identified by lines of high IL is not suitable for practical applications because it would produce curved (or slant) profiles that are of difficult interpretation.
- The IL patterns in Fig. 9 indicate positions where vertical profiles could be retrieved with a matching of the IL that is better than that of the positions of the mean tangent points. On the basis of these indications a suitable position of vertical profiles can be identified also considering the altitudes that are of maximum interest for the analyzed target (e.g. the position of the lowest tangent points is a good choice if the interest is focused at low altitudes).

We remark that temperature is often used as an input for VMR retrievals; it is therefore advisable to adopt common retrieval grids for temperature and VMR targets.

#### 6.2. choice of the observation strategy

A straightforward consequence of the discussion in Sect. 6.1 is that the optimal scenario for the choice of the retrieval grid is provided by a uniform distribution of the IL that does not show significant patterns along the OC. In this case the choice of the retrieval grid would be driven only by the trade-off between the precision of the retrieval products and their horizontal resolution [14,15]. The target that, in this respect, provides the best performance is “temperature” whose IL distribution is shown, for a segment of orbit 2081, in Fig. 10 with the same graphical notations as in Fig. 9. The MWs that generate Fig. 10 are those used by the MIPAS ground processor for the retrieval of temperature analyzing CO_{2} spectral features. It can be seen in Fig. 10 that, in the case of temperature, the IL distribution is more uniform with respect to the case of CH_{4}; for temperature the tangent points no longer identify clear preferential lines characterized by the highest IL values.

The strategy of obtaining a uniform distribution of the IL can be pursued by acting on the parameters that define the observation modes and by seeking for the optimal performance of the MWs to be analyzed. In the case of MIPAS also the spectral resolution enters in the list of parameters that define the observation modes because, in a FT spectrometer, the frequency of repetition of the limb-scans depends on the time spent by the moving mirror to reach its maximum displacement. Within the exercise of obtaining the optimal retrieval scenario the performance of different observation modes and different MWs can be tested in terms of the IL distribution that they generate.

The discussion developed so far refers to MIPAS so that it is bounded to the observational degrees of freedom of this experiment. However the IL analysis can also be exploited to evaluate the performance of other experiments (operational or in their design stage) that sound the atmosphere along the orbit track using the limb-scanning observation technique.

#### 6.3. 1D vs. 2D retrievals

The IL analysis can be used to visualize the different scenarios that are encountered in a parcel of atmosphere when the retrieval is carried out using a 1D or a 2D retrieval system. Figure 11 shows the IL distribution that results when the analysis on temperature is carried out considering only one of the limb-scans that occur in the atmospheric segment represented in Fig 10; Fig. 11 represents the scenario available to a 1D retrieval operated on the considered limb-scan. Figure 12 shows the IL distribution in the same set of cloves as in Fig. 11 but calculated considering all the limb-scans as in Fig. 10.

We notice here that the IL accumulation visible in Figs. 10 and 12 for the 2D scenario increases in the case of observations operated with the nominal mode of the “new” MIPAS configuration; this is a consequence of the reduced separation between consecutive limb-scans (see Sect. 2.1). We do not show examples in this paper, however horizontal discontinuities are still evident in the IL distribution of the “new” configuration even if they are less marked than those visible in the “old” configuration.

In Fig. 11 are also reported the tangent points of the considered limb-scan and the position where the 1D retrieval analysis would assign the retrieved profile (the position of the mean tangent point). It can be seen in Fig. 11 that the imaginary line joining the points with highest IL precedes (from the observer point of view) the line identified by the tangent points; this offset suggests that the adopted MWs are affected by a saturation that makes the atmosphere opaque nearby the tangent points. Figure 11 indicates an appropriate position where to assign the retrieved profile also for the 1D retrieval analysis. Actually, in Fig. 11, the shift between the vertical line and the line of highest IL exceeds 3 latitudinal degrees (about 330 km) in the middle stratosphere. This discrepancy may appear of minor entity since in the 1D retrieval the atmosphere is assumed horizontally homogeneous within the whole air parcel, however an error of the order of 300 km becomes significant for targets, as temperature, that display marked horizontal variability; such an error is not negligible if, for example, the objective is the comparison with in-situ measurements that have very accurate geo-location. The handling of this geo-location problem requires (to date) the use of the horizontal averaging kernels [19] of the 1D retrieval. Averaging kernels are not a user friendly tool but permit to evaluate the size of the geo-location discrepancy [20], and to apply a correction that partially compensates for the mismatch with the help of a reference atmospheric model [21,22].

The IL analyses presented in this section have highlighted the problem deriving from an incorrect geo-location of the retrieved profiles. In the case of 2D retrievals, the IL analysis itself indicates a new retrieval approach that overcomes the problem. The hint is to abandon the usual definition of “retrieval parameter” as: “the value that the atmospheric quantity assumes in the element (point) of a line that describes the altitude profile”. The new definition would be: “the value that the atmospheric quantity assumes in the element (clove) of a surface that describes the 2D distribution of the atmospheric quantity”.

## 7. Conclusions

We have designed and implemented a new analysis system that permits to define and to measure the information load of MIPAS observations. In the proposed approach observations relative to a whole orbit are merged in a two-dimensional analysis that combines the sensitivity of spectral signals with the geometrical redundancy that is introduced by different limb-views that observe the same air parcel. We have found that, for a specific observation mode, the 2D distribution of the information load depends on the atmospheric quantity that is to be retrieved from the observations. The IL analysis makes it possible to assess and visualize the effective “atmospheric sampling” that is operated by the observations with respect to the considered target quantity. We have shown that the knowledge of the atmospheric sampling helps in the problem of identifying a suitable layout of the horizontal retrieval grid; this permits to minimize the error that derives from the attribution of the retrieved profiles to wrong geo-locations. The optimal scenario for a retrieval analysis is provided by a uniform distribution of the IL because this scenario makes easy the definition of the retrieval grid; in this context the IL analysis allows to test the performance of different observation modes and spectral intervals on the light of the IL fields that they generate. The IL analysis also highlights the advantage of a 2D retrieval approach in terms of the larger amount of information exploited in a parcel of atmosphere with respect to a 1D approach. Finally, the IL distribution corresponding to a single limb-scan shows that the error in the geo-location of the retrieved profile can also occur in the case of a 1D retrieval analysis.

The IL analysis presented in this paper has been designed and implemented for MIPAS observations but its use can be extended to other orbiting limb-sounders that, like MIPAS, continuously measure the atmospheric emission along the orbit track.

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