Table  lists the ground-based FTIR stations included in this study, while Fig.  shows their geographical distribution. These stations take regular solar absorption measurements under clear-sky conditions, using either the high-resolution spectrometers Bruker 120 M, 125 M, 120 HR, and/or 125 HR, which can achieve a spectral resolution of 0.0035 cm-1 or better, or the Bomem DA8, which can achieve a spectral resolution of 0.004 cm-1. The only lower spectral resolution spectrometer (0.06 cm-1) used in this study is the Bruker Vertex at Mexico City. This instrument is not accepted by the NDACC FTIR standards at present; therefore Mexico City is the only site in this study that will not be part of NDACC.

The formaldehyde spectral signatures used in ground-based infrared measurements lie in the 3.6 µm region and belong to the ν1 and ν5 bands. This implies that, for HCHO, a CaF2 or KBr beamsplitter and a nitrogen-cooled InSb detector are used together with an optical filter which usually covers the 2400–3310 cm-1 region (called an NDSC-3 filter; see e.g. ). At St Petersburg a broader filter is used (1700–3400 cm-1). The spectral resolution can be reduced in order to increase the signal-to-noise ratio (SNR). In practice, the spectra used in the present study have a resolution between 0.0035 and 0.009 cm-1, except for Mexico city (0.075 cm-1).

HBr or N2O cell measurements are regularly taken to verify the alignment of the instruments. The instrument line shape (ILS) can be obtained by analysing these cell measurements using the LINEFIT programme . This ILS can impact the shape of gas absorption lines, and its determination by LINEFIT can be used as an input parameter in the forward model of the retrieval codes (Sect. ).

We refer to and/or for more details on the FTIR retrieval principles. Total columns of atmospheric gases, but also volume mixing ratio vertical profiles, are obtained from their pressure- and temperature-dependent absorption lines. As seen in Table , two retrieval algorithms are used in the NDACC FTIR community: PROFITT9 , and SFIT2 , which has been updated to SFIT4 09.4.4. It has been demonstrated in that the profiles and total column amounts retrieved from these two different algorithms under identical conditions are in excellent agreement.

We summarize the forward-model and retrieval parameters that have been harmonized in Table . The forward model uses pressure and temperature profiles from NCEP (National Centers for Environmental Prediction) for each site, except that the temporal resolution can vary depending on the retrieval team from daily means, from 6-hourly ones to even hourly interpolated ones.

The dominant source of systematic uncertainty being the spectroscopic parameters, it is crucial that all stations use the same spectroscopic database. We use the compilation from Geoffrey Toon (JPL), the atm16 line list, which is available at http://mark4sun.jpl.nasa.gov/toon/line list/linelist.html (last access: 5 September 2018). In this atm16 line list, the HCHO and N2O lines correspond to the HITRAN 2012 database . This HITRAN 2012 database includes the latest improved HCHO parameters (broadening coefficients, ), which complement the release in HITRAN 2008 of new HCHO line intensities from the same group . The spectroscopic parameters for the lines of water vapour and its isotopologues in atm16 are from Toth 2003

http://mark4sun.jpl.nasa.gov/data/spec/H2O/RAToth_H2O.tar (last access: 5 September 2018)

The vertical resolution and sensitivity of the retrieved HCHO products can be characterized by the averaging kernel matrix A : A=KTSϵ-1K+R-1KTSϵ-1K, where K is the weighting function matrix that links the measurement vector y to the state vector x: y=Kx+ϵ, with ϵ representing the measurement error. In our retrievals, we assume Sϵ to be diagonal, with the diagonal elements being the inverse square of the SNR. R is the regularization matrix, which, in this work, has been chosen as the Tikhonov L1 matrix (see Sect. ).

We give the trace of this averaging kernel matrix A for the elements corresponding to the HCHO profiles, called the DOFS, in Table  for each station. The DOFS range from 1.0 to 1.6, meaning that we can not provide more than one piece of information on the vertical profile. This is the reason that only total columns of HCHO are discussed in this paper and not vertical profiles. In Fig.  we show (upper panels) the averaging kernels (AKs, rows of A) for four different stations, with DOFS ranging from 1 (only scaling) to 1.4. Similar averaging kernels are obtained for the other stations with similar DOFS (not shown). We can observe that, in each case, the AKs peak at about the same altitude (8 km) with full width at half maximum of about 16–18 km, showing that we have limited vertical resolution, and that we are mainly sensitive to the whole troposphere, and to a lesser extent to the lowermost stratosphere. The total column averaging kernel (TotAK), associated with the FTIR-retrieved total columns, is plotted as well. The associated a priori profiles are also shown in Fig.  (lower panels) for completeness, together with the mean and standard deviation of the retrieved profiles. As expected by the low DOFS, the shape of the retrieved profiles is very similar to the shape of the a priori profiles.

The uncertainty budget is calculated following the formalism of and can be divided into three different sources: the measurement noise uncertainty (purely random), the forward-model parameter uncertainties (random and systematic), and the smoothing error expressing the uncertainty due to the limited vertical resolution of the retrieval (random and systematic). At each station, the random uncertainty (square root of sum of squares of the measurement noise error and of all the random forward-model errors) and the systematic uncertainty (square root sum of the squares of all systematic errors) are calculated for each single measurement. Except for a few cases (NCAR stations and Wollongong), for which a typical smoothing error is given, and St Petersburg, for which the mean value for 2013 is given, the smoothing uncertainty is also calculated for each individual measurement. In Table  we give the mean of the random and systematic uncertainties, the smoothing uncertainties (both random and systematic parts), and the total random or systematic uncertainties (square root sum of the squares of the random or systematic error and the smoothing random or systematic error), obtained using the FTIR complete time series at each station.

The random uncertainty given in Table  is dominated at all sites by the measurement noise with an error covariance matrix Sn calculated as follows: Sn=GySϵGyT, where Sϵ is assumed to be diagonal, with the square of the inverse of the SNR as diagonal elements, and Gy denotes the contribution matrix A=GyK. In this calculation of the measurement noise error, the SNR must be the real one from the noise in the spectra and not a regularization one as can be chosen in the retrieval process (as in Eq. ; see also Sect. ). For the HCHO spectra used in this study, this SNR can vary between 100 for the worst cases and 3000, with a mean of about 700–1000 for the Bruker 120/5 HR instruments and 500 for the Bomem DA8.

The forward-model parameters error covariance matrices Sf are calculated according to the following: Sf=(GyKb)SbGyKbT, where Sb is the covariance matrix of b, the vector of forward-model parameters. For each individual forward-model parameter, the Kb sensitivity matrix is mostly calculated by using analytic derivatives, while the covariance matrix Sb is an estimate of the uncertainty in the model parameter itself.

Effort has been made in this study to harmonize the uncertainty budget at all sites. This is done by calculating the errors from the same forward-model parameters (solar zenith angle, temperature, spectroscopic line parameters, baseline, etc.) across the network and by choosing the same Sb matrix for relevant parameters (i.e. when they are not site or instrument dependent, e.g. for the spectroscopic line parameters). However, some differences remain between the SFIT4 and PROFFIT codes that result in small differences that still occur between the two groups of users, despite the use of harmonized parameters. For the SFIT4 users, the random uncertainty given in Table  is dominated by the measurement noise (Eq. ). We see from Table  that the random error is between 1.0 and 3.6×1014 molec cm-2 for the SFIT4 stations equipped with the high-resolution Bruker spectrometers 120/5 HR or M (the higher values coming from the 120/125 M instruments at Paramaribo and PortoVelho), while it can reach 5.1×1014 molec cm-2 with the Bomem DA8 in Toronto. For the PROFFIT users, the random uncertainty is calculated to be a little bit larger (from 3.5 to 5.3×1014 molec cm-2) for the sites with high-resolution spectrometers, and 11.1×1014 molec cm-2 with the low-resolution spectrometer Bruker Vertex 80 at Mexico City. The main difference between SFIT4 and PROFFIT is the additional error calculated at the PROFFIT stations due to the channelling of the spectra. However, in Table  we also give the mean differences between two subsequent FTIR measurements taken within 30 min (Diff30) as an upper limit of the total random uncertainty: this difference can be larger than the error budget if HCHO has faster variability than 30 min, but with enough statistics, the mean differences should not be lower than the total random errors. We see that this empirical upper estimation of total random uncertainty has a median value (2.8×1014 molec cm-2) very close to the median total random uncertainty obtained by error propagation theory (2.9×1014 molec cm-2), which gives confidence in the overall FTIR error estimation. At all the PROFFIT sites, except the highly polluted one (Mexico city), the total random uncertainty is larger than the Diff30, which could be an indicator that the uncertainty calculated in PROFFIT is slightly too conservative, probably due to this additional channelling error that would be estimated to be too large. For SFIT4 users, the Diff30 values are usually close, within 0.5×1014 molec cm-2 of the calculated total random uncertainty, with the exceptions of Ny-Ålesund and Lauder, where the small calculated errors of 1.9 and 1.6×1014 molec cm-2 might be a little bit optimistic, and with the exceptions of Toronto, Wollongong, and Paramaribo, where differences of 7 to 13×1014 molec cm-2 are observed between the Diff30 values and the total random errors.

After the measurement noise error (and the channelling for PROFFIT users), the largest contributions to the random uncertainty due to the forward-model parameters come from the temperature, the interfering species, and the off-set baseline. For temperature, the Sb matrix has been estimated using the differences between an ensemble of NCEP and sonde temperature profiles at Reunion Island, leading to 2 to 4 K in the troposphere and 3 to 6 K in the stratosphere. This matrix is currently used by all SFIT4 users, while for the PROFITT users, these chosen values are smaller (1 K in the troposphere, 2 K up to the middle–upper stratosphere, and 5 K for the highest levels). For each interfering species, the associated Sb matrix is the covariance matrix obtained with the WACCM v4 climatology. At some stations, the ILS also has a high contribution to the random error budget.

If one uses the FTIR HCHO measurements to validate a model or a satellite with a fine vertical resolution, considering the random and systematic uncertainties (without smoothing) in Table  (4th and 7th columns) is sufficient to make correct comparisons, because the smoothing error due to the low vertical resolution of the FTIR measurements vanishes if one takes into account the FTIR averaging kernels and a priori profile in the comparisons . However, if one wants to have a better knowledge of the real precision of the FTIR data themselves, this smoothing uncertainty can be estimated for the random part using the smoothing error covariance Ssrand : Ssrand=(I-A)Svar(I-A)T, where Svar should represent the natural variability of the target molecule. For HCHO, this Svar variability matrix is unfortunately not well known due to the poor number of vertically resolved measurements. In Table , the smoothing errors have been calculated taking the covariance matrices obtained using the WACCM v4 profiles at each station as an approximation of the Svar matrices. However, models usually underestimate the variability, and we expect that the smoothing errors provided here may be underestimated, especially in locations where HCHO is expected to have stronger vertical gradient variability than in the model. As an example, in the study by , the Svar was taken from aircraft measurements PEM-Tropics-B, and led to a smoothing error estimation of 14 % at Saint-Denis, while the present estimation based on the WACCM model gives about only 2 % for this station. However, the Svar matrix constructed from PEM-Tropics-B showed from 33 % to 70 % of HCHO variability which seems too high compared to what is observed at Reunion Island from the FTIR measurements (about 20 %). This illustrates that, ideally, the Svar matrix should be re-evaluated at the sites whenever better model data or correlative measurements become available. Since the FTIR data sets always include their associated averaging kernel matrices, this re-evaluation can be done a posteriori by future users using Eq. ().

The smoothing systematic uncertainty, reflecting the bias that would occur on the retrieved profile if the a priori xa is biased compared to the real expected profile <x>, is calculated following : Sssyst=(I-A)(xa-<x>)(xa-<x>)T(I-A)T. The xa-<x> is obviously not known (otherwise, <x> would be chosen as the correct a priori in the retrievals). Therefore, we have chosen to use xa-<x>=-50 %, -20 %, -10 %, +10 %, +8 % and +5 % for the ground–4, 4–8, 8–13, 13–25, 25–40 and 40–120 km layers, respectively. The values have to vary with altitude to induce a different a priori profile shape: if 50 % is used at all altitudes, the a priori profile is then different from <x> by a simple scaling factor, and the systematic smoothing error is close to zero. Using the above values, we obtain smoothing systematic errors from 1 to 9 % (median value of 3.4 %), which is small compared to the other systematic error sources (Table ). These values assume that the model WACCM profile shapes are not too far from reality, which should be the case: due to the known short lifetime of HCHO and its production at or near the surface, we expect that the mean profile peaks at the ground. This is, as for the random smoothing part, only an estimate of the smoothing systematic error. As discussed in , one would even prefer to not give these smoothing errors at all. We prefer to give them to provide the reader with at least an idea of the impact of the smoothing in the precision and accuracy of our FTIR HCHO measurements. However, when making model or instrument comparisons, the appropriate use of the averaging kernel and a priori profile information, following , allows the user to implicitly take the smoothing uncertainty into account. This means that, for satellite or model comparison, if the methodology of is used, there cannot be some different systematic biases at different stations due to different xa-<x>.

The dominating systematic uncertainty sources are the spectroscopic parameters: the line intensities and the pressure broadening coefficients of the absorption lines present in our MWs. For the HCHO spectroscopic parameters, the line list in atm16 follows HITRAN 2012 , which used the work of , and we use 10 % for the three parameters (line intensity, air-, and self-broadening coefficients). The larger error source is then the HCHO line intensity parameter and to a lesser extent the HCHO air-broadening coefficient. In addition, the uncertainties in HCHO columns due to the interfering species spectroscopic parameters are calculated. The dominant ones were found to be due to the pressure broadening coefficients of CH4, HDO, and N2O, with an order of magnitude of about 20 % of the error due to the HCHO line intensity.

The other systematic error sources due to forward-model parameters are lower or within a few percent (ILS, temperature), except for the PROFFIT channelling source (from 7 % to 17 %), which also has a systematic component. We see from Table  that the total systematic uncertainty is between 12 % and 15 % at the SFIT4 stations. For the PROFFIT stations, it lies between 12 % and 27 %.

In Fig.  we show the individual HCHO total columns obtained at each station for a single year only (2016, except for Saint-Denis: 2011), in order to better see the day-to-day variability. The complete time series at each station are shown in the Supplement (Fig. S1). The error bars in Figs.  and S1 are the total random uncertainties; i.e. we do not include the systematic errors in order to better visualize the precision of the FTIR measurements compared to the observed day-to-day variability. The FTIR network samples a wide range of concentrations. Indeed, we can first distinguish the “clean” sites (shown with the same vertical axis with maximum 15×1015 molec cm-2) such as the Arctic stations (Eureka, Ny-Ålesund, Thule, Kiruna, Sodankyla), the marine stations (Izaña, Mauna Loa, Maïdo, Saint-Denis, and Lauder, the former three also being high-altitude stations), and the high-mountain sites (Zugspitze and Altzomoni). These clean sites can have HCHO concentrations at the limit of detection (few 1013 molec cm-2) with mean values of 10–25×1014 molec cm-2 (Table ), except for Saint-Denis, which reaches a mean of 39×1014 molec cm-2.

Second, we show the intermediate concentration sites (with the same vertical axis with maximum 30×1015 molec cm-2) such as the tropical coastal site Paramaribo and the midlatitude polluted sites in or close to cities and/or vegetation (Peterhof close to St Petersburg, Bremen, Paris, Boulder). These intermediate sites have mean HCHO total columns of 58–73×1014 molec cm-2. The sites with the highest levels of HCHO (vertical axis with maximum 70×1015 molec cm-2) are Toronto and Mexico City, where large anthropogenic emissions are indeed expected (means of 95 and 221×1014 molec cm-2), and places which are also affected by large biogenic emissions such as Wollongong (mean of 79×1014 molec cm-2) and the new site of Porto Velho, located at the edge of the Amazon rainforest (mean of 190×1014 molec cm-2).

As explained in the introduction, to reconcile the different results obtained using satellites observing at different times (e.g. SCIAMACHY and GOME-2 measuring in the morning and OMI in the afternoon), it is crucial to have ground-based observations of the HCHO diurnal cycles . The diurnal cycle is also important for model validation, since emissions, chemistry and other processes depend on the time of the day. Our FTIR data set is now able to provide the diurnal cycles at 21 different locations. To separate the effect of the strong seasonal cycle (shown in the next section), we give the diurnal cycle in four different seasons in Fig.  for a selection of the sites, while the other ones are provided in the Supplement (Fig. S2). As seen from Figs.  and S2, the diurnal cycles are often site and season dependent. While there is no clear diurnal cycle at the Arctic sites and at some of the midlatitude cities during winter (St Petersburg, Bremen, Toronto), we usually see an increase from the morning, which is often more pronounced in June–July–August (and December–January–February in the Southern Hemisphere) at most of the stations (in the cities but also at marine sites). A maximum is often found around midday (St Petersburg, Mexico City, Izaña, Saint-Denis, and Wollongong) or much later in the afternoon (16:00–18:00 local time, LT), as in Bremen, Paris, Toronto, Lauder, and Altzomoni. Only in a few cases is a minimum found at midday (St Petersburg in SON, Zugspitze in MAM–SON, Sodankyla in MAM). The marine sites at high altitudes (free of local pollution) have a small minimum at about 08:00 LT (Izaña, Maïdo). This diversity in the FTIR diurnal cycles is also observed with MAX-DOAS at other sites : a very weak diurnal cycle at OHP (southern France) in winter and spring; a minimum around midday at Beijing and Xianghe in spring and autumn, and a constant increase in summer (as observed with FTIR for Bremen, Toronto, and Paris). The diurnal cycles observed at the Jungfraujoch station from FTIR measurements show, for all months of the year, a midday maximum, which is very different from the ones observed at our closest station Zugspitze. The IMAGES model shows diurnal cycles in phase agreement with our FTIR measurements at Zugspitze, except for the summer, for which the model diurnal cycle is very weak (not shown). However, two sites very close together can indeed observe different diurnal cycles (as seen for Saint-Denis and Maïdo). More investigation is needed to understand the different diurnal cycles at these two mountain sites.

We see from Fig.  that the FTIR measurements at Porto Velho do not show a clear pattern, in particular if one is interested in the 09:30 and 13:30 LT differences between the overpasses of two different satellites . From the 1-year data available at present at this new site, it seems that the diurnal cycle cannot help reconciling the differences observed over Rondônia between GOME-2 and OMI . In contrast, the diurnal cycles observed over cities confirm that the observation of a positive bias between OMI (13:30 LT) and GOME-2 (09:30 LT) over urban areas can be indeed explained, at least partly, by the diurnal cycle.

The length of the HCHO time series allows trends to be derived for some stations. We have calculated the trends at each station using the monthly mean time series Ym(t) with a simple model, including a fit of the seasonal cycles:

Ym(t)=A0+A1⋅cos⁡(2πt/12)+A2⋅sin⁡(2πt/12)+A3⋅cos⁡(4πt/12)+A4⋅sin⁡(4πt/12)+A5⋅t, with A5 as the annual trend.

It turned out that, due to the very high variability of HCHO, the uncertainties in the trends are often too large to obtain significant values. A more sophisticated multi-regression model might be able to reduce the uncertainties, but this is beyond the scope of this paper. However, for a few stations, significant trends are found. They are mainly negative: at St Petersburg (-3.9±3.3 % decade-1), Mexico City (-9.6±5.1 % decade-1), Wollongong (-18.8±10.8 % decade-1), and close to significance at Zugspitze (-7.7±7.7 % decade-1). Only the marine sites Izaña and Saint-Denis show positive significant trends (+17.3±15.2 and +15.8±5.2 % decade-1). Note that at Maïdo, the trend is not significant. A careful combination of the measurements at both Reunion Island sites (Saint-Denis and Maïdo) could be carried out in the future.

For the longest time series, we observe a very good agreement with previous studies in general. The negative trends observed over the European stations St Petersburg and Zugspitze are in agreement with the negative trends observed by OMI (2004–2014) over St Petersburg and Germany . At the Jungfraujoch station, a negative trend (-6.1±2.6 % decade-1) was also observed for the 1996–2015 period . Note that the calculation of the uncertainties in the trends in our study takes into account the autocorrelation in the residuals, following , which increases the uncertainties. For Zugspitze, the uncertainty without correcting for this autocorrelation, as in or , would be 4.9 % (instead of 7.7 %), showing then a more significant trend, in agreement with these studies. The non-significant trends observed at the northern European station (Kiruna), and the midlatitude American stations (Toronto, Boulder) are in agreement with . In the Southern Hemisphere, the negative trend observed at Wollongong was also found in , as well as a positive trend at Madagascar, which is near Reunion Island, in agreement with the high positive trend observed at Saint-Denis. At Lauder, OMI also shows a non-significant trend .

The IMAGESv2 global model calculates the distribution of 170 chemical compounds gases with a time step of 6 h at 2∘×2.5∘ resolution, with 40 hybrid (σ pressure) levels in the verticals between the surface and the lower stratosphere (44 hPa level). The model calculates daily averaged concentrations of chemical compounds. The effect of diurnal variations is accounted for through correction factors on the photolysis and kinetic rates obtained from a full diurnal cycle simulation using a time step of 20 min. The same model simulation also stores the diurnal shapes of formaldehyde columns required for the comparison with FTIR data on files. Meteorological fields (winds, temperature, humidity, 3-dimensional cloud cover, solid and liquid cloud water content, large-scale and convective precipitation rates, visible downward radiation, convective updraught fluxes, boundary layer diffusivities, snow depth, sea ice fraction, surface roughness lengths, surface sensible heat flux, friction velocity, etc.) are obtained from ERA-Interim analyses of the European Centre for Medium-range Weather Forecasts (ECMWF).

Anthropogenic emissions of NOx, CO, SO2, and NMVOC are provided by the Hemispheric Transport of Air Pollution data set version 2 (HTAPv2) , with the NMVOC speciation provided by the emission inventory of the Atmospheric Chemistry and Climate Model Intercomparison Project (ACCMIP) . Emissions from open vegetation fires are taken from the last version of the Global Fire Emissions Database, GFED4s, which includes the contribution of small fires . The GFED data are available at daily frequency at 0.25∘×0.25∘ from 1997 to the present (http://www.globalfiredata.org, last access: 5 September 2018). The vertical distribution of these emissions follows . Isoprene and monoterpene emissions are obtained from the MEGAN-MOHYCAN model for all years of the study period at a resolution of 0.5∘×0.5∘ (http://tropo.aeronomie.be/models/isoprene.htm, last access: 5 September 2018). Methanol biogenic emissions are obtained from the inverse modelling study of . Besides the dependence on temperature, visible radiation, leaf area, and leaf age, the model accounts for the inhibition of isoprene emissions under drought conditions through a dimensionless soil moisture activity factor (γSM). However, the parameterization of γSM is very uncertain, as discussed in , and we assume γSM=1 in this study. The average global annual emissions are 419 Tg yr-1 isoprene, 100 Tg yr-1 methanol, and 103 Tg yr-1 monoterpenes.

The chemical degradation mechanism of pyrogenic NMVOCs is described in . The oxidation mechanism for isoprene is also based on , with a few updates. It accounts for the revised kinetics of isoprene peroxy radicals according to the Leuven Isoprene Mechanism version 1 (LIM1) and is further modified to account for laboratory findings . The formaldehyde yield in isoprene oxidation by OH is close to 2.4 mol mol-1 in high NOx (1 ppbv NO2, after 2 months of simulation) and 1.9 mol mol-1 at low NOx (0.1 ppbv NO2). The chemical mechanism for monoterpenes is simplified, with product yields of formaldehyde, acetone, methylglyoxal and glyoxal based on box model calculations using the α- and β-pinene oxidation mechanism from the Master Chemical Mechanism (MCM) . The overall formaldehyde yield is 4.2 HCHO per monoterpene oxidized, reducing to 2.3 after subtracting the contributions of acetone and methylglyoxal oxidation. This yield is further reduced by 45 % to account for wet deposition of intermediate and secondary organic aerosol formation. This fraction of 45 % is higher but of the same order as the estimated overall impact of deposition on the average HCHO yield from isoprene oxidation (28 %), based on IMAGES model calculations. The higher fraction for monoterpenes is intended to account for the impact of the more complex chemistry and larger number of oxygenated intermediates involved in their oxidation compared to isoprene. The large deposited fraction is uncertain but appears justified by the larger number and lower volatility of intermediates involved in formaldehyde formation from monoterpene oxidation.

The calculation of the model columns at the FTIR stations accounts for its location in the horizontal (nearest model pixel) for the FTIR a priori profiles and averaging kernels as prescribed in , as well as for the station altitude above sea level. The model column is calculated from the calculated formaldehyde profile, between the altitude of the station and the uppermost model level (approximately 20 km), and from the a priori FTIR profile, above that level. When the model surface lies higher than the station, the model column is increased by a partial column assuming a constant mixing ratio between the two altitudes, taken equal to the value at the lowermost model level. The monthly averaged formaldehyde columns are calculated by accounting for the temporal sampling of the observations at each site and month. Also, the local time of each observation is taken into account by rescaling the daily averaged concentration using the formaldehyde diurnal shape factors calculated by the model with a time step of 20 min.

We compare the monthly means of FTIR HCHO total columns at each station with the IMAGES columns calculated for the 2003–2016 period. The time series of both products are shown in Fig. . Since the random uncertainty of the FTIR monthly means is divided by the square root of the number of measurements within each month, the dominant contribution to the FTIR error bars in Fig.  is the systematic uncertainty (estimated at 11–26 %. The smoothing uncertainty is not included in model comparisons using ). Except for very few cases (Mexico City and Paramaribo), the model is in overall good agreement with the FTIR measurements in terms of absolute levels (Fig. ) and seasonal cycle (Fig. ).

For each station the correlation, the bias and the standard deviation (SD) of the statistical comparisons between the monthly means, mean(IMAGES(smoothed)-FTIR)/mean(FTIR), are summarized in Table . The median correlation between FTIR and IMAGES for the 21 stations is very high (0.81), with weaker values at the Mexican stations (0.4/0.5) and at Mauna Loa (0.10). The median standard deviation for all comparisons is 25 % (ranging from 11 % to 41 %). This agreement is good considering the FTIR variability (i.e. the SD) of HCHO monthly means (median of 35 %). The standard deviation of the comparisons can be explained partly by the lower variability of the model monthly means (31 %) compared to FTIR, as seen in Fig. . In addition, the variability of the model data within a month is also much smaller (median of about 11 %; this SD within a month is shown as magenta error bars in Fig. ) than the FTIR one (median of about 28 %).

The median of IMAGES and FTIR differences is small (-15 %) and within the FTIR systematic uncertainty estimated at 11 %–26 %. However, the biases range from -64 % to +51 %, which requires an investigation of their possible causes. The main source of systematic uncertainty is the spectroscopic parameters, which have been harmonized in this work, with each station using the same line parameters database, and the same spectral MWs. Therefore, it is expected that all FTIR stations should provide consistent HCHO total columns within 5 %–17 % (systematic errors due to other sources than spectroscopic ones). To check this, we divide the FTIR stations according to their concentrations levels and latitudes, and use the model for comparisons.