The resistance model connects the top of the surface atmospheric layer with concentration χa to the surface, as seen in Fig. . The total resistance Rt=Ra+Rb+Rc is decomposed into a series of different resistances: The aerodynamic resistance Ra which characterizes transport through the surface layer to the quasi-laminar sublayer, the quasi-laminar resistance Rb that models the transport through the quasi-laminar sublayer, and finally the surface resistance Rc that models the air-surface exchange. The surface resistance is often decomposed into different parallel pathways that characterize different modes of atmospheric-surface transport. For this general model, we consider Nc different surface types, where each surface pathway i is connected to a surface concentration χs(i) associated with a surface resistance Rc(i).

This resistance model can be represented in terms of total equivalent quantities, as represented on the left side of Fig. , where the total surface resistance Rc and the equivalent total surface concentration χs are expressed in terms of the individual surface pathways as 1aRc-1=∑i=1Nc(Rc(i))-1,1bχs=∑i=1NcRcRc(i)χs(i). Written in terms of these total quantities, the flux Ft at the top of the surface layer can be expressed as 2Ft=vd(χs-χa), where vd=1/Rt is the deposition velocity. Details of the derivation of Eqs. () and () can be found in Appendix . When χs>χa net flux is to the atmosphere and when χs<χa net flux is to the surface. For unidirectional models where the resistance model is used only for deposition, all surface concentrations χs(i) are set to zero and emissions are supplied separately to the atmospheric chemistry model.

Lastly, it will be convenient to introduce the pathway-weighted deposition velocity vd(i) for surface pathway i defined as 3vd(i)≡RcRc(i)vd, so that we have vd=∑i=1Ncvd(i) and the total flux Ft can be expressed as 4Ft=∑i=1Ncvd(i)(χs(i)-χa). Note that vd(i)≤vd, which reflects that only a portion of the ammonia leaving from the ground may be emitted at top of the surface layer as some ammonia can be (re)absorbed through other surface pathways that act as a sink.

Under equilibrium conditions, combining Henry's law with the aqueous dissociation equilibrium between NH3 and NH4+ relates the gaseous ammonia concentration χs(i) and aqueous ammonium concentration [NH4+(aq)]. This relation (at 1 atm) can be expressed as 5χs(i)=Γs(i)ATe-B/T, where Γs≡[NH4+]/[H+] is known as the emissions potential, the gaseous ammonia concentration at equilibrium is known as the compensation point, and A = 161 500 mol K L⁻¹ and B = 10 380 K are constants . As in the previous section, the superscript (i) labels particular surface pathways.

The relation in Eq. () is commonly used to model the exchange of atmospheric ammonia with the ground, which can include exchange between soil pores and soil water, exchange with plant litter, and volatilization from fertilizers , as well as the exchange between the gaseous ammonia in the sub-stomatal cavities of plants and ammonium in the apoplastic fluid .

Our bidirectional flux model for ammonia utilizes the same dry deposition resistance model in GEM-MACH that was described in Sect. , but has non-zero values set for the source concentrations χs(i). The details of how the source concentrations χs(i) are specified are described in this section.

The circuit diagram for our bidirectional flux model is shown in Fig. . The top portion of the figure shows the standard surface resistances used in GEM-MACH, comprised of deposition pathways to plant stomata, cuticles, other plant surfaces, and the ground, associated with resistance Rst, Rcut, Roth, and Rg, respectively.

We make the simplifying assumption that only the ground source concentration χg has a non-zero value, while the other three surface pathways have their source concentrations set to zero so that they act only as sinks. As mentioned in Sect. , the bidirectional exchange with stomata can be non-negligible. However, while in non-agricultural settings the stomatal emissions potential can be as large or larger than the ground/soil emissions potential , the ground emissions potential for fertilized grounds can exceed these values by many orders of magnitude . As it is unlikely that our satellite ammonia retrievals contain enough information to differentiate between the stomatal and ground ammonia emissions, and agricultural emissions are the dominant ammonia emissions source within our model domain, we set the stomatal emissions potential to zero to reduce the number of parameters in the bidirectional model. However, this does not necessarily imply that there is no bidirectional ammonia exchange between the atmosphere and the other three pathways, only that we fold all contributions to the bidirectional exchange into the ground pathway for the sake of computational simplicity.

included a model for the bidirectional flux of ammonia from plant stomata based on extending the resistance-based deposition model to include a capacitor that can store and release ammonium. Inspired by this model, we model the ground/soil ammonium pool by a capacitor with capacitance C, as depicted in Fig. . The capacitor is placed in parallel with Np ammonium sources/sinks, where each pathway j is associated with a source concentration χp(j) and a resistance Rp(j), as illustrated at the bottom of Fig. . At this point, we do not specify the specific ammonium sources (with χp(j)>0) or sinks (with χp(j)=0), but sources can include fertilization and livestock and examples of sinks are nitrification in soil and ammonium lost due to runoff or drainage. Instead, we gather these pathways into a single equivalent source/sink with a total resistance Rp and an equivalent concentration χp, given by 6aRp-1=∑j=1Np(Rp(j))-1,6bχp=∑j=1NpRpRp(j)χp(j). We note that the expressions above for Rp and χp have the same form as Rc and χs in Eqs. () and ().

The ammonium ground surface concentration at the fluid/gas boundary Qg can be expressed by Qg=h[NH4+], where h is the fluid depth. Using Eq. (), the capacitance C=Qg/χg of the ground ammonium pool is given by 7C=hTgAeB/Tg[H+], where Tg is the ground temperature.

As the flux coming out of the capacitor can be expressed by -dQg/dt, Kirchhoff's current law implies 8dQgdt=χc-χgRg+χp-χgRp. Equation () shows that the aqueous ammonium pool evolves with time to establish an equilibrium between the canopy ammonia concentration χc and the ammonium source χp. To find the time evolution of the ground emissions potential Γg, we make the simplifying assumption that the product h[H+] is constant in time, so that dQg/dt=h[H+]dΓg/dt. After some algebra, which can be found in Appendix , Eq. () can be written in terms of the ground emissions potential Γg as 9dΓgdt=1τa(Γa-Γg)+1τp(Γp-Γg), where we have defined Γa and Γp analogously to Γs in Eq. () as 10aΓa≡χaTgAeB/Tg,10bΓp≡χp1+αTgAeB/Tg. In the equation above, α is a factor that accounts for the flow into the other surface pathways (i.e. through Rst, Rcut, and Roth) given by 11α≡RpRtRcRg1-RcRgRtRc-1. We have also identified the RC time constants τa and τp for the atmosphere and ammonium source, respectively, as 12aτa=RgRcRtC,12bτp=11+αRpC, which are the time scales for the ammonium pool to reach equilibrium with the atmosphere and ammonium sources, respectively. Equation () shows that Γg evolves in time to establish an equilibrium between the atmospheric ammonia source Γa and the aqueous ammonium source Γp, which is shown diagrammatically in Fig. .

With the prognostic equation for the ground emissions potential given by Eq. () and χg related to Γg by Eq. (), the bidirectional exchange of ammonia between the surface and atmosphere given in Eq. () can be computed. These equations reference total parameters such as χp and Γp, but do not reference parameters for individual pathways (i.e. parameters with the superscript (j)). Therefore, if we can adequately set these total parameters, then it is not necessary to determine the individual source/sink quantities of the ammonium pool. In our case, as the bidirectional flux model parameters will be determined through top-down inversions that use retrievals from satellite-borne instruments that are only sensitive to these total parameters, this yields a bidirectional flux model with minimal parameters that is tailored for use in inversion systems.

In summary, the evolution of the ground emissions potential in our bidirectional flux model is determined by the parameters Γp and τp, which sets the ammonium source level and response timescale, respectively. Although the intention is to deal with these total parameters directly, we note that Γp and τp can be expressed in terms of individual pathways as 13aΓp=TgAeB/Tg∑j=1Npχp(j)/Rp(j)∑j=1Np1/Rp(j),13bτp-1=∑j=1Np(τp(j))-1.

With the prognostic equation for the emissions potential derived from the capacitance model outlined in the previous section, we now set the parameters for the ammonium pool. From Eq. (), the model parameters that need to be set are Γp, τp, and τa.

Γp, which describes the sources of ammonium to the ground, is taken as a spatially-varying 2D field on the same horizontal grid as described in Sect. , allowing for a unique field for every month. In this work, Γp will represent agricultural sources (both fertilizer and livestock), while separate non-agricultural ammonia emissions that account for ≲1 % of total emissions will be accounted for using the unidirectional emissions model. As ammonia emissions from cattle originate primarily from excreted nitrogen in manure , we make the same assumption as in in which all ammonia emissions from livestock are assumed to originate from manure. We can then treat both the livestock and fertilizer emissions using the same emissions modeling framework. As such, model parameters that reference the ground or soil should be understood to be inclusive of manure produced by livestock.

The RC constant τp is the time scale for the ground emissions potential to equilibriate to Γp. Although this time scale can depend on temperature and soil moisture , we simply set this parameter to a constant value. In their bidirectional flux model, proposed using a value just under three days for the time constant associated with the dynamics of the ground emissions potential following fertilizer application. Following , we set τp to a value of three days.

Using the expression for capacitance in Eq. (), the time constant τa given in Eq. () can be computed by 14τa=RgRcRtdsθTgAeB/Tg[H+], where we have expressed the fluid depth h in soil pores in terms of the soil depth ds and the volumetric water content of soil θ as h=dsθ. The resistances Rg, Rc, and Rt are given by the standard deposition resistance parametrization used in GEM-MACH described in Sect. , while the soil water content θ and ground/soil temperature Tg are computed using the two-level Interaction Soil-Biosphere-Atmosphere (ISBA) surface model that has been incorporated into GEM-MACH. The soil depth ds is set to 2 cm, as done in and . Lastly, [H+] is parametrized through the ground/soil pH, which was set as a 2D field on GEM-MACH's horizontal grid.

In summary, the free parameters for the bidirectional flux model are the ammonium source term Γp that sets the level of ground ammonium and the ground pH which determines the rate at which the ammonium pool reaches equilibrium with ammonia in the atmosphere. Figure shows the values of τa as a function of pH and ground temperature (with RtRg/Rc=300 s m⁻¹ and θ=0.1 chosen to represent typical values). In this figure, we can see that high values of pH and ground temperature result in a low capacitance and short atmosphere/ground equilibrium time scales. In this example, for a pH of 8 and a ground temperature of 25 °C, τa is just under three hours, so in this case the ammonium pool will reach equilibrium with atmospheric ammonia relatively quickly. In contrast, for a pH of 6 and a ground temperature of 10 °C, τa is over 100 d and so the ground emissions potential will be relatively insensitive to sub-seasonal changes in ammonia levels.

If τa≫τp, then Γg≈Γp after a few multiples of τp in time. As Γp was chosen to be a 2D field that is constant within a month, if τa is much longer than a month, Γg will only evolve to reach equilibrium with Γp and not with the atmospheric ammonia. In this limit, the bidirectional flux model is similar to models where the Γg values are static. Accordingly, the dynamic nature of the ammonium pool is only evident when τa is (roughly) less than or equal to one month.

CrIS is a Fourier transform spectrometer that measures the infrared spectrum with a swath width of ∼ 2200 km and a ∼ 14 km spatial resolution at nadir. CrIS instruments are currently aboard the Suomi National Polar-orbiting Partnership (SNPP), NOAA-20, and NOAA-21 satellites. As the year 2016 is used for evaluation, before NOAA-20 and NOAA-21 were launched, all retrievals used in this study come from the instrument aboard the SNPP satellite. SNPP is in a sun-synchronous orbit with local overpass times at approximately 01:30 and 13:30, although only day-time retrievals are used in the inversions.

Ammonia retrievals are made using version 1.6.4 of the CrIS Fast Physical Retrieval (CFPR) algorithm that is based on the TES ammonia retrieval algorithm , which minimizes the difference between observed radiances and radiances generated by a radiative transfer model . This minimization also includes an a priori regularization term, where the a priori profile is chosen as one of three possible profiles that aim to represent ammonia profiles for background, moderate source, and high source regions. Retrievals are made on 14 pressure levels, with the averaging kernel typically peaking in the boundary layer. Version 1.6.4 of the CFPR algorithm also accounts for non-detects where the ammonia signal is below the detection limit of the instrument . Only retrievals over land with a minimum degrees of freedom of 0.1 are used in the inversions.

Surface observations of ammonia from Canada's National Air Pollution Surveillance (NAPS; https://www.canada.ca/en/environment-climate-change/services/air-pollution/monitoring-networks-data/national-air-pollution-program.html, last access: 25 May 2026) network and the US's Ammonia Monitoring Network (AMoN; http://nadp.slh.wisc.edu/amon, last access: 13 May 2026) are used for validation. AMoN, part of the US National Atmospheric Deposition Program (NADP), uses Radiello^® passive diffusion samplers to measure ammonia levels over two-week periods. The NAPS stations that measure ammonia do so using citric acid-coated denuders that measure ammonia levels over a 24 h period every three or six days. The locations of the 101 AMoN stations and 12 NAPS stations, which account for 3065 and 1080 observations in total, respectively, are displayed in Fig. .

Validation with the surface observations is done by examining the changes in the bias, standard deviation of errors (STDE), the Pearson correlation coefficient (ρ), and root mean square error (RMSE), the definition of which can be found in Appendix . When the bias, STDE, or RMSE are expressed as a percentage, the denominator is taken as the annual mean observation value.

The goal of an inversion is to find an optimal synthesis of observational and a priori (background) model information. For our application, the model state x is comprised of ammonia atmospheric concentrations c and bidirectional flux model parameters and/or other emissions parameters β, which can be written in block form as 15x=cβ. The inversion algorithm seeks a change Δβ, known as the increment, that results in a change Δc to the atmospheric ammonia concentrations that improves the agreement with the observations y. In a variational algorithm, the inversion is performed by minimizing a cost function J, which for our case is given by 16J=12ΔβTBββ-1Δβ+12(y-H(xb)-HΔc)TR-1(y-H(xb)-HΔc), where xb is the background state of the model and H and H are the nonlinear and linearized observation operators, respectively, that map the model state into observation space. The first term on the right-hand side of Eq. () measures the deviation from the background model state and is weighted by the inverse of the univariate background error covariance matrix Bββ for the parameters β. The second term measures the deviation from the observations and is weighted by the inverse of the observation error covariance R.

In the inversion, uncertainties in the atmospheric ammonia concentrations are attributed entirely to uncertainties in the model parameters β, so that Δβ and Δc are related to each other by 17Δβ=BβcBcc-1Δc, where Bcc is the univariate background error covariance matrix for the atmospheric ammonia concentrations and Bβc is the error cross-covariance between β and c.

Retrievals are compared to the GEM-MACH model output (available at 15 min increments) closest to the retrieval time. As the uncertainties on individual ammonia retrievals are relatively large , inversions were performed using one month's worth of retrievals to ensure an adequate amount of observational information was present in each inversion. As such, the parameters β produced by each inversion are monthly-mean parameter values.

The previous section described the inversion procedure in a general framework. In this section, we specify our choice of β, as well as the background and ensemble fields used in the inversions.

In Sect. , the parameters Γp and pHg were set as 2D horizontal fields and will be the fields optimized in our inversions, so that β={Γp,pHg}. The values for Γp and pHg can then be used in Eqs. () and () to determine the ground emissions potential Γg.

The background (a priori) values for Γp were set to the values that yield the inventory-derived monthly mean emissions described in Sect. . The background error covariance used for Γp was nearly identical to that described in , in which background error standard deviations were set to 50 % of the background values (with a minimum standard deviation corresponding to emissions of 0.26 kg ha⁻¹ month⁻¹ to ensure all locations have a non-negligible deviation) and homogeneous and isotropic correlations with a half-width at half-maximum of 40 km.

The background values for pHg were chosen so that the resulting values for τa were much longer than a month. This makes our a priori equivalent to assuming that the emissions potentials are time-independent. This choice was made in part due to time-independent emissions potentials being used in previous studies of ammonia bidirectional fluxes in GEM-MACH as well as with other models . As a pH value of 5 results in τa values much longer than a month, a uniform background value of 5 was set for pHg (we also note that the mode of the distribution of soil pH values from the World Soil Information Service (WoSIS) is near 5, see Fig. S1). The background error covariance for pHg was constructed in a similar manner to that for Γp, but with a standard deviation of 3. Additionally, a minimum value of 1.5 and maximum value of 8.5 was imposed on the pHg distribution to limit the number of pH values far outside of the range of pH values found in the WoSIS database (see Fig. S1). While this distribution results in more pH values below 5 as compared to the WoSIS database, as all pH values below 5 yield very large values for τa, these values should only be interpreted as indicating a static emissions potential (and not necessarily to be interpreted as physical values).

The background error covariance was constructed using an ensemble of 80 members. For each ensemble member, perturbed value of Γp and pHg were drawn from their respective distributions and GEM-MACH was then run with these perturbed values to yield the atmospheric ammonia concentrations for that ensemble member. The ensemble GEM-MACH output was saved only at the top of the hour, in contrast to the unperturbed GEM-MACH runs that had output saved at every 15 min time step, to reduce the storage requirements of the ensemble.

The inversion increments for Γp and pHg are shown in panels (a) and (b) of Fig. , respectively. The increments for Γp are positive in most locations and times, with some exceptions. For March to June, many areas on the East coast and midwestern US have background values for Γp on the order of 103 to 104 increase to ∼2×104–3×104 in the inversions (see Fig. S2 for plots of the background and analysis of Γp). The largest increments for Γp occur in April, where increases of up to 2×104–4×104 are seen in a number of locations (Iowa, Pennsylvania, North Carolina, Southern Ontario), which represent increases of more than 100 % as compared to background values. For the handful of locations where the inversions significantly decrease Γp, such as North Carolina in July, Canadian prairies in September and October, and Northern Iowa/Southern Minnesota in November, the inversions decrease Γp by up to 2×104, representing decreases of up to 60 % compared to background values.

The largest increments for pHg occur in the central US and Canada for April to June, where the increment reaches values of 3.5, resulting in total pHg values of 8.5. Although the increments for pHg are generally more concentrated in the middle of North America as compared to the increments for Γp, large pHg increments do occur along the East coast of North America and in California's Central Valley in the spring and summer. The pHg increments are also generally more spread out spatially and smoother as compared to the Γp increments.

With the inversions described in the previous sections, GEM-MACH was run with the revised parameters for the bidirectional flux model, as well as with the unidirectional flux models for comparison. For the unidirectional flux run, emissions are set so that their monthly means are equal to those in the bidirectional flux run (instead of the inventory values to have the benefit of the inversions), but its intramonthly emissions time profiles are set using the unidirectional scheme as described in Sect. .

The monthly mean values for the ammonia ground emissions potential Γg and the time constant τa from GEM-MACH using the inversion-derived bidirectional flux parameters are shown in Fig. . The values displayed in this figure are the weighted means across the 15 land-use categories in the model. Regions with significant agriculture and consequently ammonia emissions, such as California's Central Valley, the Canadian prairies, the midwestern US, and North Carolina, can have ground emissions potential values ranging from 104 to 5×104. Ground emission potential values vary greatly with ground condition, vegetation type, and (if applied) fertilizer type and time since application. Reported ground emission potential values for fertilized ground range between 103 and 106 , placing the higher end of the inversion values for Γg roughly at the midpoint of this range (in terms of the order of magnitude).

We expect the atmospheric ammonia concentrations to have a non-negligible influence on the emissions potential when τa≲τp. From Fig. b, we can see that the inversions place τa at or below τp=3 d for large swaths of the central US for April to June, as well as some smaller regions within the model domain. In these regions, the emissions potential will equilibrate with the atmospheric ammonia concentrations within hours to days, while the regions with τa≫3 d will have emissions potentials that are approximately static within each month and will be relatively insensitive to the atmospheric ammonia levels.

The changes in the monthly mean ammonia emissions when GEM-MACH is run with the bidirectional flux model with Γp and pHg set from the inversions as compared to using their background values are displayed in Fig. . Changes in the emissions are most significant from March to October. Emissions increases exceeding 5 kg ha⁻¹ can be seen in the Central Valley, the American midwest, and northern Texas, which represent increases of over 80 % as compared to the case without inversions (see Fig. S3 in the Supplement for plots of the monthly mean emissions with and without the inversions). Although overall the inversions increase emissions, significant decreases can be seen in some regions, such as a decrease of 2.6 kg ha⁻¹ in North Dakota in September (∼60 % decrease) and a decrease of 4.5 kg ha⁻¹ (∼96 % decrease) in southern California in April.

Many of the regions where the inversions give large changes to Γp results in significant changes in emissions, but there are a few regions where this is not the case, most notably for the significant increase to Γp in Alberta and Saskatchewan in January (seen in Fig. a) that do not produce sizable changes in the ammonia emissions (shown in Fig. ). This is primarily due to the strong temperature dependence of χs, as seen in Eq. (), so that the same value of Γg may result in significantly fewer emissions in the winter as compared to the warmer seasons.

Figure shows the total emissions over the continental United States and Canada from the inversions along with inventory emissions for comparison. In the winter months, when ammonia emissions are low, the emissions from the inversions are close to the inventory emissions. For April to September, the total emissions from the inversions increase between 9 % and 61 % in Canada and between 9 % to 41 % in the continental United States, with the largest total increases occurring in April for the continental United States and in May for Canada, which is generally the start of the crop growing season when fertilizer is applied to fields.

We now turn our attention to comparing the resulting atmospheric ammonia concentrations from the unidirectional and bidirectional flux models. For the remainder of this section, all results presented for both the unidirectional and bidirectional flux models use inversion-derived parameters (as opposed to inventory-derived parameters).

Figure shows the root mean square differences in ammonia surface concentrations between the unidirectional and bidirectional flux model runs. Significant differences between the flux schemes are seen in the midwestern US, the Central Valley, the Eastern seaboard, and Canadian Prairies during the spring, summer, and fall. A number of regions have large root mean square difference on the order of 10 ppbv. For comparison, the monthly mean ammonia surface concentrations for the unidirectional model generally vary between 0.5 and 30 ppbv (see Fig. S4). The root mean square differences between flux models exceeds 50 % in many regions and can range from 100 % to 150 % in the Central Valley during the spring (in comparison to the monthly mean unidirectional values).

The previous section examined the differences in surface concentrations between the unidirectional and bidirectional flux models using parameters set via inversions. We now compare these model results with the ammonia surface observations described in Sect. .

Figure shows the differences in RMSE values for surface ammonia observations from AMoN and NAPS stations for the unidirectional and bidirectional flux model runs, taken over all months in 2016. Red markers indicate that the RMSE for the bidirectional run is smaller than for the unidirectional run and blue markers indicate larger RMSE values for the bidirectional run. Statistically significant/insignificant differences at a confidence level of 1σ≈68.2 % are denoted by circular/square markers in this figure. Out of the 113 stations in the combined AMoN and NAPS dataset, 28 stations show a statistically significant improvement in the RMSE for the bidirectional run, most of which are on the East coast or American midwest. Four stations show statistically significant increases in RMSE from using the bidirectional flux model. The median station RMSE for the unidirectional run of 38 % decreases to 32 % for the bidirectional run. The statistically significant changes in RMSE range from decreases of 82 % to increases of 31 %.

The changes in bias, error standard deviation, correlation, and root mean square error between the unidirectional and bidirectional runs for the AMoN and NAPS ammonia surface observations are illustrated in Fig. . Red upward-pointing triangles indicate an improvement in the statistic (a reduction for bias, STDE, and RMSE, an increase for ρ) for the bidirectional flux model as compared to the unidirectional flux model. Blue downward-pointing triangles indicate a degradation in the statistic (an increase for bias, STDE, and RMSE, a decrease for ρ) for the bidirectional flux model run as compared to the unidirectional flux model. Filled triangles indicate that the change in the statistic is statistically significant at the 1σ confidence level. Statistics are compiled for each month individually as well as for the whole year. These statistics can be found in full in Tables S1 and S2 of the Supplement.

For AMoN, the top panel of Fig. shows that the bias decreases for March to July and for September and October for the bidirectional model run as compared to unidirectional model. In these months, the bias for the unidirectional model ranges from 13 % to 50 %, but is between 14 % and 26 % for the bidirectional model (see Table S1 for all values and uncertainties). A statistically significant degradation of the bias is seen for January, February, August, and November. The bidirectional model reduces the bias taken over the whole year by 1.4 %. The unidirectional model has error standard deviations that range from 53 % to 193 %, which are reduced in the bidirectional run for March to November (second row from the top in Fig. ), with the reductions ranging from 5 % to 20 %, and is reduced by 11 % over the year. While the bidirectional run degrades the error standard deviations in January, February and December, these changes are not statistically significant (as indicated by the unfilled triangles). Some improvements to the correlation ρ can be seen for AMoN (third row from the top in Fig. ), increasing from 0.62 for the unidirectional model to 0.64 for bidirectional model over the year. The RMSE for the bidirectional model is smaller than for unidirectional model for March to November (although the improvement is not statistically significant for November), and over the whole year the bidirectional model reduces the RMSE value by 11 % (90 % reduced from 101 %) as compared to the unidirectional model.

The bidirectional model improves nearly every statistic for AMoN over the spring, summer, and fall months, as well as over the whole year, but degrades the statistics somewhat in the winter months. This difference in performance during the winter as compared to the rest of the year may be due to the quality of the model's surface temperature (as described in Sect. ), which for the continental US in 2016 performs worse in the winter than for the rest of the year.

In contrast to AMoN, few differences in the statistics for the NAPS observations between the unidirectional and bidirectional models are statistically significant, as seen in the bottom panel of Fig. . The STDE, ρ, and RMSE values for the bidirectional model in June are better than for the unidirectional model, while opposite is true for ρ in March and May, but otherwise the statistics for the different flux models are not significantly different from one another. As seen in Fig. , most of the stations that show improvement from using the bidirectional flux model are concentrated in the mid-to-eastern US, where there are stations for AMoN but not NAPS. Overall, the bidirectional flux model improves the agreement of the GEM-MACH model with the ammonia surface observations in the spring, summer, and fall, while degrading the model performance somewhat in the winter.

We conclude with a brief examination of the relative sensitivity of the inversions to the inversion model parameters. While there is a noticeable reduction in the posterior uncertainties for both Γp and pHg, the reduction in uncertainties for Γp was much larger than that for pHg in many regions (see Fig. S5). Furthermore, the number of degrees of freedom for the inversions associated with Γp are 2 to 8 times larger than that associated with pHg. Overall, the inversions are better able to constrain Γp than pHg.

While the ammonia retrievals from CrIS do constrain Γp more than pHg, including pHg as an inversion parameter can nonetheless have significant effects on the atmospheric ammonia concentration. Inversions were repeated with β={Γp} as the sole inversion field, with the values of pHg set to values from the WoSIS pH database. The root mean square differences in the ammonia surface concentrations between inversions with β={Γp} compared to inversions with β={Γp,pHg} (shown in Fig. S6) can be as large as 10 ppbv in some regions, such as the midwestern US and California's Central Valley, which is comparable to the differences between the unidirectional and bidirectional models (as seen in Fig. ), although with not as large of a horizontal extent. Including pHg as an inversion parameter also decreases the biases with AMoN observations in May, June, September, and October, but increases the bias in July and August (see Fig. S7).