Let LνB be the radiance at the sensor for a scene with climatological background levels B of the target trace gas and a corresponding HRI of zero. Dividing the atmosphere in n appropriately spaced layers z, we denote by Bz the corresponding partial columns (B=∑zBz). We can then calculate LνB from the following sequence of equations: 7L1νB=t1ν(L0ν-B1ν)+B1ν8L2νB=t2ν(L1ν-B2ν)+B2ν9⋮LνB=tnνL(n-1)ν-Bnν+Bnν. Forward substitution of the above relations yields LνB as a function of the surface term L0ν. Here tzν are the layer transmittances that account for the absorption of all atmospheric species. The z dependence of this parameter is related to vertical variations in the atmospheric constituents and the pressure and temperature dependence of the line intensities. Bzν=Bν(Tz) corresponds to the Planck's blackbody function for an averaged layer temperature Tz.

For this scene, we now introduce an additional trace amount X-B and write the corresponding observed radiance as Lν. In each layer, the transmittance will decrease by a factor tzνXz-Bz=e-τzν≈1-τzν, with τzν the small optical depth caused by the excess Xz-Bz. With this, the sequence of equations becomes 10L1ν=t1ν1-τ1νL0ν-B1ν+B1ν11L2ν=t2ν1-τ2νL1ν-B2ν+B2ν12⋮Lν=tnν1-τnνL(n-1)ν-Bnν+Bnν. For optical thicknesses of the target trace gas τzν well below one, the second (τiντjν) and higher order terms in optical depth can be neglected. Combining both sets of equations, one can verify that 13L1ν=L1νB-τ1νL0ν-B1νt1ν14L2ν=L2νB-τ1νL0ν-B1νt1νt2ν-τ2νL1ν-B2νt2ν15⋮Lν=LνB-∑z=1nτzνL(z-1)νB-Bzν∏j=zntjν. Each of the terms in the sum expresses the effect of the absorption in one layer due to the excess Xz-Bz and the attenuation by the layers above. Note that the absorptions are proportional to both the optical depth and the local thermal contrast L(z-1)νB-Bzν, which are two parameters that drive the measurement sensitivity in the infrared . The optical thickness in turn is proportional to the partial column of the target species τzν∝(Xz-Bz) and thus 16Lν=LνB-∑zczν(Xz-Bz). The constants czν depend on the state of the atmosphere at level z and above, but are independent of the excess trace gas amount X-B.

The HRI is by definition a linear combination of spectral channels ν, from Eq. (), 17HRI=∑νwνLν+C, with C and wν numerical constants. Using Eq. () and the fact that the HRI is zero on the background (HRI=∑νwνLνB+C=0), we find 18HRI=∑νwνLνB-∑ν,zwνczν(Xz-Bz)+C19=-∑ν,zwνczν(Xz-Bz). This equation implies that for optically thin absorbers, the HRI can be written as a weighted sum of the partial column enhancements. This linearity and additivity of the NH3 HRI as a function of partial columns is illustrated in Fig.  by means of simulations with a radiative transfer code. The blue line illustrates the nearly linear increase in HRI for increasing NH3 columns in the first atmospheric layer of the model. Starting from about 7×1016 molec. cm-2, slow departure from linearity is observed. The green line shows the same for the second atmospheric layer. The overall larger HRI in this second layer results from higher thermal contrast (TC) higher up in the atmosphere. Finally, the red line represents the HRI when NH3 is increased simultaneously in both layers. The dash-dotted line is the sum of the blue and green curve and for low columns is almost indiscernible from the simulated HRI, illustrating additivity in the optically thin limit.

To make the link with Eq. (), Eq. () can be rewritten as 20HRI=∑zSFz(Xz-Bz)21=∑zSFzXz-BzX-B(X-B)22=SF(X-B), with SFz local scaling factors and 23SF=∑zSFzXz-BzX-B, total scaling factors that depend on the scene conditions (e.g. surface temperature, atmospheric temperature and pressure profiles, vertical profile of the target species). We note that whereas SF depends on the normalized vertical profile shape Xz-BzX-B, both SFz and SF are independent of the total column for a fixed profile shape.

Finally, introducing 24HRIz=SFz(Xz-Bz), we see from Eq. () that the HRI can be decomposed in different partial HRIz: 25HRI=∑zSFz(Xz-Bz)=∑zHRIz. These HRIz quantify how much each layer contributes to the total HRI and they are, again in the optically thin limit, independent from each other.

Equation () motivates the NN retrieval formula, 26X^a=HRISFa+B, with SFa an estimated scaling factor which depends on the best estimates of all the dependencies of SF, including the vertical profile of the target species. As we have just shown, in the optically thin limit SFa is independent of the total column amount. This is no longer the case for strong absorptions when non-linear effects become increasingly important. In the ANNI retrieval framework this is taken care of by including the HRI as an input parameter in the calculation of the SF. In what follows, we will first derive the AVK formalism in the optically thin limit, assuming both linearity and additivity (and therefore SF that are independent of the (partial) column). In Sect.  we will then show how to correct for small errors that arise when these conditions are not met.

The quantities X^z-Bz express the local enhancement of the trace gas column at an altitude z, and it is their relative proportions that are constrained in the retrieval by the use of an a priori profile shape. We write 27az=X^za-BzX^a-B for the normalized a priori vertical partial column profile, or partial column profile shape, with X^za the partial columns corresponding to the retrieved X^a and ∑zaz=1. In the previous section we demonstrated that in the optically thin limit, the scaling factor SFa equals the weighted sum of the different local scaling factors defined by the profile shape: 28SFa=∑zSFzXza-BzXa-B=∑zSFzaz. Defining the averaging kernel as 29Aza=SFzSFa=SFz∑zSFzaz, we can express X^a as a function of Xz by eliminating the HRI from Eqs. () and (): 30X^a=HRISFa+B31=∑zSFz(Xz-Bz)SFa+B32=∑zAza(Xz-Bz)+B. As can be seen from this equation, the averaging kernel Aza fully characterizes the measurement, and can be used to mathematically map the true profile Xz to the measured total column X^a.

Two example AVKs for an NH3 retrieval are shown in the middle panel of Fig. , for a retrieval over land and over ocean, with a priori profiles shown in the left panel. Both AVKs logically increase with altitude, as the temperature difference between the surface and a given atmospheric layer, and therefore the scaling factor, increases with altitude. Apart from a multiplicative constant (1/SFa), the AVKs are independent from the a priori profiles, which explains why both land and ocean AVKs increase similarly with altitude. The multiplicative constant determines the altitude for which the AVK is one. This altitude can be interpreted as an equivalent effective NH3 altitude, where all NH3 can be thought to be located. For the ocean and land retrieval, this altitude is located respectively around 1.6 and 0.7 km, consistent with the a priori profiles shown in the left panel.

It is instructive to compare the total column averaging kernel, as defined above, with the one arising in optimal estimation retrievals : 33X^a=I-AXa+AX. Here, we use matrix and vector notation, with X and X^ respectively the true and retrieved partial columns and A the AVK matrix. I is the identity matrix. Both Eqs. () and () allow simulating the retrieval process for any hypothetical X (e.g. from a model, or an independent measurement). However, this is largely where their similarity ends, as there are important differences when it comes to interpreting these two types of AVK.

The first difference is the role of the a priori. For the total column retrieval, the a priori fixes only the vertical profile shape, while for the optimal estimation retrieval, the a priori affects both the vertical profile shape and the retrieved value at each altitude separately. Equation () expresses that the retrieved profile is a weighted sum of the a priori and the true profile, with the weights provided by the AVK. When the information content of the measurement is low or the retrieval is too heavily constrained, the AVK will tend toward zero and the solution will remain close to the a priori. Conversely, when the information content is high or the retrieval loosely constrained, the AVK will approach the unit matrix. The optimal estimation AVK is therefore a measure of how much information is extracted from the measurement, with its trace commonly denoted “degrees of freedom for signal”. By contrast, the total column averaging kernels introduced above are not a measure of how much information is extracted from the measurement, as they accompany an unconstrained retrieval. A perfect measurement would correspond to an all-ones vector Az=1 for all z. However, inherent to (infrared) sounding, sensitivity varies as a function of thermal contrast, and thus altitude, so this ideal can never be met.

The second important difference relates to the fact that a vertical profile is retrieved in the optimal estimation type retrievals. The rows of A express the vertical resolution of the retrieval, where the ideal is a narrow function that peaks at its corresponding altitude. As no vertical information is extracted in the total column retrievals, this interpretation also does not apply. Az is a (normalized) vector that shows which layers of the atmosphere offer in principle the greatest sensitivity. It naturally peaks high up in the atmosphere, irrespective of the location of the trace gas. However, combining the total column AVK with the a priori profile does allow extracting a vector which characterizes the vertical sensitivity, as we now show. Starting from Eqs. () and () we have 34HRI=SFaX^a-B35=∑zSFzX^a-Baz36=∑zSFzX^za-Bz37=∑zHRI^za, where we defined HRI^za=SFzX^za-Bz similarly to Eq. () where we defined HRIz=SFz(Xz-Bz). A retrieval that assumes a given a priori vertical profile az therefore implicitly assumes that the HRI (the trace gas signal) can be decomposed into partial HRI^za corresponding to spectral change at each altitude. Note that while HRI=∑zHRI^za=∑zHRIz, the same does not necessarily hold for each individual level HRI^za≠HRIz, as the assumed profile can differ from the actual profile. With this we can define the normalized assumed HRI profile or, equivalently, the probable vertical partitioning Vza of the signal: 38Vza=HRI^zaHRI,=SFzX^za-BzSFa(X^a-B)=Azaaz. From the first equality, it follows that ∑zVza=1 (and also follows the last term and Eq. ). The three profiles az, Aza and Vza are illustrated in Fig.  for a typical NH3 retrieval. As can be seen, the maximum of Vza is shifted upwards compared with az, due to the more favourable thermal contrast higher up in the atmosphere.

There are two alternative ways in which averaging kernels can be exploited to remove the impact of the vertical profile assumption of the retrieval .

In the ANNI retrieval formalism, the total scaling factor SFa is calculated directly by the NN, not via intermediate SFz and application of Eq. (). However, these SFz, which are required to calculate the AVKs, can be estimated by exploiting the flexibility of the NN. For NH3, the NN is trained for a wide variety of Gaussian profiles, with peak altitudes ranging from 0 to 20 km and σ from 100 m to 3 km. The SFz can be estimated from the network using the input parameters zpeak=z and σ=100 m for the Gaussian profile. For this calculation, an HRI input parameter is also required and the choice was made to use the observed HRI. The corresponding column that can be calculated from this satisfies 50X^|z=HRISF|z+B, where X^|z is the retrieved total column assuming all the trace gas enhancement is situated in the narrow Gaussian band around altitude z. SF|z is the corresponding total scaling factor, which is used to approximate the local scaling factor SFz. With this the AVK can be constructed as 51Aza=SFzSFa≈SF|zSFa=(X^a-B)HRI(X^|z-B)HRI=X^a-BX^|z-B.

The formulas provided above are exact in the linear limit, but for large columns, SFa and SF|z have an increasingly high dependence on the value of the HRI. The NN takes into account this dependence so that Eqs. () and () are always good approximations of the true X, provided that either the assumed a priori profile is correct or that the tracer is confined to a narrow layer. However, there is no guarantee that 52SFa=?∑zSFzaz, or thus that ∑zAzaaz equals one. A consistent AVK can be obtained, however, as 53Aza=1NX^a-BX^|z-B, with N=∑zAzaaz a normalization factor. This factor also guarantees that applying the averaging kernels on the a priori vertical profile returns the retrieved column: 54∑zAzaX^za-Bz+B=∑zAzaazX^a-B+B=X^a. The normalization factors are shown in Fig.  for 1 day and 1 year of global measurements. The histograms shown in the insets indicate average values around 0.98–0.99 (±0.03–0.04), illustrating the consistency of the approach and the fact that non-linear effects are modest. The areas where the normalization factors are furthest from 1 are affected by low clouds (e.g. off the west coast of South Africa) or sea ice.

The necessity of using normalization factors follows from the fact that in the non-linear regime, the SFz are not uniquely defined, depending on the concentration in the layer z (non-linearity) and the other layers (non-additivity). However, as we have shown, in the neighbourhood of the solution, a fully consistent AVK can be obtained after renormalization. It is this AVK that is recommended when applying method 1 in model comparisons. However, in case method 2 is employed and when the modelled profile concerns a narrow layer at high altitude (e.g. a pyro-convective fire plume) or significantly deviates from the a priori, it can be better not to renormalize. In particular, when we have a narrow modelled profile layer at an altitude z′, with mz=δzz′, the second method, without renormalization, yields the expected 55X^m=X^a-B∑zAzaδzz′+B56=X^a-BAz′a+B57=X^|z′-B+B58=X^|z′. The output files of the ANNI retrieval contain X^a, X^|z, Bz and N. With this, Aza can be calculated if required from Eq. () or renormalized via Eq. ().

The generalized error covariance matrix S plays a key role in the calculation of the HRI. As a symmetric matrix, S has real eigenvalues λi and can be decomposed as 59S=∑i=1nλisisiT, with all si orthogonal to each other. It follows that its inverse can be written as 60S-1=∑i=1n1λisisiT. This expression leads to an intuitive interpretation of the HRI : it can be seen as a weighted projection of the spectrum onto the Jacobian, with the directions that usually exhibit the most variability carrying the lowest weight.

The distribution of the eigenvalues of the covariance matrix used for the NH3 HRI (with a spectral range covering 812–1126 cm-1) is shown in Fig. . Three domains can be distinguished: (i) the 30 highest values corresponding to the principal components, (ii) around 1200 values corresponding mainly to instrumental noise and (iii) around 20 very small eigenvalues. These smallest eigenvalues are of the order of the numerical precision at which the covariance matrix is calculated and, in essence, correspond to directions not occurring in IASI spectra. While random instrumental noise would be expected to occur in all directions, apodization and L1 post-processing remove some. Such directions carry the most weight in S-1, but as they are not found in real spectra they do not contribute much to the total HRI (as can easily be verified numerically).

However, small changes to the instrument calibration or post-processing can alter the contribution of these directions in the IASI spectra, and because they carry such a large weight in the HRI, they can affect its value considerably. This explains why the HRI in the past has been found to be very sensitive to such changes . It also explains the occurrence of (small) biases between the different instruments. The solution is fortunately simple and is obtained by disregarding the terms corresponding to the very small eigenvalues in Eq. (). As we will show later, using such a pseudoinverse does not eliminate the effects of L1C changes completely but reduces their magnitude considerably.

After the pseudoinverse was implemented, an unexpected change was observed in the value of the HRIs on spectra from the period on which the covariance matrix was built. It turns out that while the scalar product of observed IASI spectra with the eigenvectors corresponding to the lowest eigenvalues LTsi/λi is near zero, this is not the case for spectra generated with the forward model. This is due to small discrepancies between spectra generated by the forward and actual spectra that are magnified by the 1/λi factors. Hence, synthetic HRIs calculated on the training set of the neural network have in the past been overestimated, resulting in low biases in the retrieved columns. As we will show in Sect. , the magnitude of this bias was around 18 % in ANNI-v3. For the retrieval of other trace gases presented in , especially those operating on a smaller spectral range, the bias has been evaluated to be much smaller.

As the mean spectrum and covariance matrix that are used for the HRI are calculated from spectra measured within 1 reference year (2013), long-term changes in atmospheric composition that affect the spectral region of interest can have unwanted effects on the HRI. This was first noted in , where a spurious trend was seen in the HRI NH3 data over remote regions. It was attributed to the increase in global carbon dioxide (CO2) concentrations, because of the presence of a weak CO2 absorption band in the 920–990 cm-1 spectral region where NH3 has its strongest absorption. A linear correction on the HRI of the order of 0.03 yr-1 was introduced to compensate for this effect. However, because of seasonal variations, and possible temperature dependence of the interference an HRI which is less sensitive to CO2 changes is preferable. One option is to build the covariance matrix from spectra spanning the entire period of IASI measurements.

An alternative approach is to account directly for the effects of CO2 in the calculation of the HRI. The HRI formula is related to generalized least squares estimation and can be expanded to include multiple variables that are simultaneously estimated . In our case, the Jacobian vector becomes a two-column matrix, one column corresponding to NH3 and the other to CO2. The HRI formula Eq. () remains formally identical (with only the first component of the two-element HRI vector of interest). The effect of this change on the long-term trend of the HRI is detailed in Sect. .

The ERA5 model output replaces satisfactorily the IASI L2 for all input parameters, except for the surface temperature and cloud cover. These are spatially and temporally too variable for model output to be representative for an IASI footprint at a given time. All previous reanalysed ANNI-NH3 products still relied on different versions of the IASI L2 cloud product. Recently, developed an NN-based cloud flag. Trained with data from the latest version (v6.5) of the official L2 cloud product, it inherits all its advantages as a proven and well-validated product. The NN utilizes carefully selected IASI channels as input (excluding channels affected by long-lived tracers CO2, N2O, CH4, CFC-11 and CFC-12) and was shown to be temporally consistent, and coherent across the three IASI instruments. The network presented in was trained to distinguish completely clear scenes (0 % cloud cover) from the rest. For the NH3 processing, two additional networks were trained to distinguish scenes with a cloud cover below 10 % and 25 % respectively. With this, three cloud flags are available and these have now been integrated in v4 of the reanalysed ANNI-NH3 product. The results presented in the rest of this paper utilize the flag corresponding to the 10 % threshold.

The stability of the HRI was evaluated over 10 remote regions where only background columns of NH3 are expected. Their average monthly HRI is shown in Fig.  for the three IASI instruments separately. The top panel shows the average as obtained with the HRI setup as described above, i.e. with pseudoinverse and with a CO2 Jacobian. As with previous versions of the product, a spurious linear trend is observed, but thanks to the introduction of the CO2 Jacobian, its magnitude is reduced to about 0.01 yr-1, compared with 0.03 yr-1 previously. A slightly steeper decrease is observed for Metop B. We correct for these trends by adding a time-dependent offset as in . The result after correction is shown in the second panel of Fig. .

A detailed analysis was made of this time series to detect offsets between the different instruments and shifts that coincide with known changes in the IASI L1C data. The largest of these shifts is the offset of 0.11 seen between IASI-C and the two other instruments. Small offsets in the HRI time series of IASI-A were found in 2010, 2015 and 2017, and in the HRI of IASI-B in 2015. For each of these, offset corrections were calculated in the range of 0.01–0.03. Thanks to the pseudoinverse, their magnitude is drastically reduced. Previously, offsets as large as 0.6 were observed. The resulting corrected time series is shown in the third panel. This time series is temporally stable and shows an excellent consistency between the three instruments, but it exhibits a weak seasonal cycle, likely due to the combined effect of seasonal changes in the concentrations of H2O and volatile organic compounds that absorb in the same spectral range as NH3. To remove this seasonality an offset depending on latitude and month of the year was calculated from 2012 to 2014 IASI-A data and applied on all data. The HRI after correction is shown in the bottom panel of Fig. . Thanks to the improved setup of the HRI, and the new cloud product, the magnitude and therefore also uncertainty of all these corrections is lower than in the previous product, which results in much improved temporal consistency.

As outlined before, v4 has an improved temporal consistency compared with v3. In this section, we provide a short assessment of the new NH3 spatial distributions and how they compare with previous versions. As an illustration of the new product, a seasonal average over 2007–2022 is presented in Fig. . The distributions follow closely the ones of previous versions . Comparisons with version 3 are provided in Figs.  and . The main differences are: (1) overall larger columns, especially in areas with high columns. As explained in Sect. , this is due to the more robust way of calculating the HRI, which makes it less sensitive to small errors in the forward model. Comparing individual observations, the new version is about 10 % (for low HRI) to 20 % (for high HRI) larger. (2) A notable improvement over high latitudes thanks to the improved bias correction (see Sect. ). Averaged columns were clearly overestimated in ANNI v3 for such observations, especially over Greenland and Antarctica but also Canada and Russia. (3) Slightly higher concentrations over deserts, partly due to the overall increase in v4 and partly due to the larger weight of deserts in the construction of mean background spectra and associated covariance matrix (see Sect. ). In v3, negative HRIs were consistently observed over certain deserts, resulting in negative average columns or low biases. This problem is not entirely gone (e.g. the average HRI over parts of the Arabian desert is still negative), but much improved, resulting in more consistent pronounced transport patterns over, for example, the Sahara Desert. (4) Overall larger concentrations over oceans. The improved bias correction (last step of the HRI correction presented in Sect. ) enables removal of practically all negative values on a long-term average. Average columns over remote ocean are now around 7×1014 molec. cm-2 almost everywhere. An adverse effect of the correction might be overestimated columns over the Red Sea, Persian Gulf and Mediterranean Sea.

The most obvious remaining artefact in the v4 distribution concerns the continuity of the land–sea transitions. While they are reasonable for some regions of outflow (Gulf of Mexico, Mediterranean Sea), off the west coast of Africa, over the Arabian Sea, Gulf of Bengal or Yellow Sea, the transition is too abrupt to be realistic. The origin of this problem is that different NH3 profiles are used for land and ocean. With the introduction of the AVKs, this does not constitute a problem in model comparison or assimilation. However, for stand-alone use of the product, it would be desirable in the future to improve the parametrization of the a priori vertical profile shape.

Given the low bias in ANNI NH3 v3, it is important to exclude the presence of other biases related to the HRI calculation. Here, we present the results of an independent intercomparison that was conducted between the ANNI v4 retrieval output and that of an optimal estimation approach which relies on spectral fitting.

For the optimal estimation retrieval, the Atmosphit forward and inverse model was used , which is the same tool whose forward model is used for the construction of the ANNI training database. The optimal estimation was set up as follows: the retrieval range was set to 900–975 cm-1. Total columns of NH3 were retrieved with a fixed vertical profile, using the same parametrization as in ANNI NRT. The NH3 variance was set to 1000 %, corresponding to an almost unconstrained retrieval. Together with NH3, H2O was retrieved in 10 partial columns, with the a priori coming from the IASI L2. Total columns of CO2, O3 and CFC-12 were retrieved as well as the surface temperature. Spectral emissivity was taken from . Before presenting the results, it should be emphasized that despite the similarities in both retrieval approaches (same input parameters, vertical profiles, forward model), no perfect agreement is expected because of (1) use of a narrower spectral range in the optimal estimation retrieval, (2) different propagation of instrumental noise to the retrieval result, (3) limitations of the fitting model (e.g. with respect to fitting water vapour or surface emissivity) and (4) errors related to the imperfect training of the neural network.

For the comparison, 2 days were selected, one over Europe and one over North America, with relatively high NH3 columns. The results are shown in Fig. . Intercepts, mean and median differences are all of the order of 1015 molec. cm-2 or below. Regression slopes, calculated using iteratively reweighted least squares to remove the impact of outliers, are 0.97 and 1.05. While the scatter around the 1–1 lines is not negligible (with standard deviation of the differences around 3–4 ×1015 molec. cm-2), these numbers demonstrate the overall consistency of both retrieval approaches and do not indicate a significant bias. Note that the ANNI retrieval approach has numerous advantages over optimal estimation, as discussed in .

The last detailed global validation of the IASI NH3 product was based on a comparison of ground-based FTIR measurements of NH3 with the LUT-based NH3 product, where a low bias around 35 % was found . Since then, two independent validation studies have been conducted. One study compared IASI ANNI v3 with in situ measurements in Colorado, USA, and found regression slopes ranging from 0.78 to 1.1 and intercepts of the order of 1–2×1015 molec. cm-2. A second study compared IASI NH3 columns with columns obtained from FTIR measurements in Hefei, China. Here, mean differences around 3.5×1015 molec. cm-2 (IASI being lower) were found and regression slopes close to one. Given the results of the comparison with the optimal estimation method, we do not expect any significant bias in v4 for columns above 1×1016 molec. cm-2 in comparisons that correct for the vertical profile assumption of the retrieval. A comprehensive validation of the v4 product is foreseen within the framework of ESA's CCI+ precursors for the aerosol and ozone ECV project, which should confirm this, as well as assess the performance of the algorithm on low columns. Apart from validation of the columns in an absolute or relative sense, comparison with FTIR columns will also allow evaluating regional NH3 trends derived from IASI data. Such an evaluation could also be made with bottom-up inventories or with data derived from in situ measured concentrations. However, in that case, there is the additional difficulty that long-term trends of other inorganic pollutants (NO2, SO2) affect NH3 columns differently than emissions or local concentrations (e.g. ), necessitating the intervention of a (chemistry transport) model.

In previous ANNI versions, an estimated uncertainty σX^ was calculated for each individual measurement X^ via 61σX^2=∑i∂X^∂pi2σpi2, with σpi the uncertainties of the different input parameters pi. This equation assumes uncorrelated uncertainties, but as this cannot always be justified, in ANNI v4, we switch to the more general 62σX^2=JTSpJ, with Sp the error covariance matrix of the input parameters (with covariances Sp,ij=σpipj) and J the Jacobian of the retrieval, with components ∂X^∂pi.

In the ANNI retrieval framework, the input parameters include the skin temperature, the surface pressure, the HRI, the surface emissivity, the zenith angle, the width and the peak of the Gaussian vertical NH3 profile, the temperature profile (15 levels) and the water vapour profile (7 levels). After some preliminary analysis, it was concluded that only the correlations between the uncertainties in the temperature profile cannot be neglected. We therefore employ a block diagonal covariance matrix, block diagonal for the elements pertaining to the temperature profile, and diagonal for all other input parameters. As for uncertainty on the vertical profile, this source of uncertainty is removed when applying averaging kernels. For this reason, uncertainties are reported with and without the vertical profile uncertainty, to be used according to whether AVKs are applied.

In total, we report four types of uncertainty for each observation: random or systematic, and with or without the vertical profile uncertainty included. Reporting random and systematic uncertainties separately is a generally recommended practice . Random uncertainties describe errors specific to a single measurement, and assuming a normal distribution, these average out over many repeated measurements. Systematic uncertainties are those that exhibit correlations in time or space, and are thus associated with more than one measurement. This type of error can lead to biases in the measurement dataset. In ANNI v4, both random σrX^ and systematic σsX^ uncertainties are calculated using Eq. () and estimates of the random and systematic uncertainties and covariances of the input parameters.

Random and systematic uncertainties can be combined and averaged in different ways according to the needs of the user. In particular, for a given measurement X^, a total uncertainty estimate can be obtained as 63σX^2=σrX^2+σsX^2. An average measurement uncertainty can be associated with an average X‾ of a series of n measurements X^i as 64σX‾2=σrX‾2+σsX‾265=∑i=1nσrX^in2+∑i=1nσsX^in2. For the special case where all random and systematic uncertainties are the same, we obtain 66σX‾2=1nσrX‾2+σsX‾2, which tends to the expected σX‾=σsX‾ for large n.

As most input parameters come without an uncertainty budget, let alone covariances, we made best-effort estimates of the co(variance) based on the limited information that is available. For now, the same (co)variances were used for the near-real time as for the reanalysed NH3 product. It is also important to note that the systematic uncertainties of the input parameters vary according to the time and space scales that are considered . Temperature profiles, for example, may be more biased monthly than annually. Here, we estimate systematic uncertainties with a typical L3 gridded data product in mind, i.e. for spatial scales of the order of 1∘ latitude and longitude or less, and for time periods of the order of 1 month or less.

The (co)variances, summarized in Table , were determined as follows: HRI

By definition, the random uncertainty on the HRI equals one. We estimate a systematic uncertainty of 0.1 due to potential and residual interferences (e.g. surface emissivity, other species). To account for potential biases in the spectroscopy and generalized error covariance matrix, we add to this an additional 10 % on the calculated HRI value.

It is useful, remembering the general form X^a=HRI/SFa+B of the retrieval, to rewrite the propagation of uncertainty in terms of the uncertainty of the nominator and denominator (see also ). Neglecting the small dependence of the SF on the HRI, we obtain 67σX^2=∂X^∂HRI2σHRI2+∂X^∂SF2σSF268=σHRI2SF2+σSF2SF2(X-B)2. Taking into account both random and systematic uncertainties, we see from Table  that the uncertainty on the HRI has an absolute (constant) and a relative (proportional to the value of the HRI) component, so that 69σX^2=σabs,HRI2SF2+σrel,HRI2SF2+σSF2SF2(X-B)270=12+0.12SF2︸σabs2+0.12+σSF2SF2(X-B)2︸σrel2.