Significant uncertainty exists in the magnitude and variability of ammonia (NH3) emissions. NH3 emissions are needed as input for air quality modeling of aerosols and deposition of nitrogen compounds. Approximately 85% of NH3 emissions are estimated to come from agricultural nonpoint sources, which are suspected to have a strong seasonal pattern. Because no seasonal information is available in current NH3 emission inventories for air quality modeling, the emissions are often distributed evenly over the year by default. Doing so can adversely affect air quality model-predicted concentrations of nitrogen-containing compounds, as shown here. We apply a Kalman filter inverse modeling technique to deduce monthly 1990 NH3 emissions for the eastern United States. The U.S. Environmental Protection Agency (USEPA) Community Multiscale Air Quality (CMAQ) model and ammonium (NH4+) wet concentration data from the National Atmospheric Deposition Program network are used. The results illustrate the strong seasonal differences in NH3 emissions that were anticipated, where NH3 emissions are more than 75% lower during the colder seasons fall and winter as compared to peak emissions during summer. The results also suggest that the current USEPA 1990 National Emission Inventory for NH3 is too high by at least 20%. This is supported by a recent USEPA study of emission factors that proposes lower emission factors for cattle and swine, which are two of the largest sources of NH3 emissions in the inventory.
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 Air quality models rely on emission inputs, including ammonia (NH3), to predict concentrations of fine particulate matter (PM2.5) as a criteria pollutant under the National Ambient Air Quality Standard (NAAQS). The NAAQS includes both a 24-hour and an annual standard for PM2.5; therefore modeling studies are needed to capture both episodic as well as annual levels of PM2.5. Sulfate and nitrate aerosols are two major components of PM2.5. NH3 gas neutralizes sulfuric acid to form ammonium sulfate aerosols, and if excess NH3 is present after reaction with sulfuric acid, NH3 then reacts with nitric acid to form ammonium nitrate [Dentener and Crutzen, 1994]. Given this, the representation of NH3 emissions can have a large effect on air quality model predictions of aerosol nitrate and ammonium concentrations. Since it is necessary to consider an annual as well as a 24-hour standard for PM2.5, model simulations must cover different times of the year, and NH3 emissions under different seasonal conditions must be represented.
 Current NH3 emission inventories are limited to total annual estimates with no intra-annual variability [U.S. Environmental Protection Agency (USEPA), 2000; Bouwman et al., 1997] (the former is available at http://www.epa.gov/ttn/chief/trends/trends98/); however, we qualitatively know that NH3 emissions should vary seasonally because of the nature of the major sources. The USEPA National Air Pollutants Emissions Trends report [USEPA, 2000] states that NH3 emissions predominantly come from agricultural sources, primarily from livestock animal waste (Table 1). Livestock sources should vary during the year since volatilization of NH3 from the animal waste is a function of temperature. Observations have confirmed large seasonal variations in NH3 emissions at hog waste lagoons and swine finishing buildings [Aneja et al., 2000; Heber et al., 2000]. Seasonality in NH3 emissions is also expected since field application of manure and synthetic fertilizers occurs during specific seasons [Asman et al., 1998] (available at http://www.gsf.de/eurotrac/nl20.htm). While it is qualitatively accepted that NH3 emissions vary substantially during a given year, limited information exists to quantitatively deduce the seasonal variability of these emissions for a regional-scale domain. In addition to the intra-annual uncertainty in NH3 emissions, it should also be noted that many of the emission factors used to develop U.S. emission inventories are based on European studies. This adds to the uncertainty in the U.S. emission inventories because of differences in the agricultural practices between Europe and the United States.
Table 1. National Emission Inventory (NEI) NH3 emission estimates for 1990a
Other (chemical and allied product manufacturing, waste disposal, other industrial sources, etc.)
Total (all sources)
 The purpose of this study is to use an inverse problem approach to quantify seasonal variations in NH3 emission estimates on a regional scale for the eastern United States. Observed and modeled wet concentrations of NH4+ are used in an optimization formula to estimate emissions. Inverse modeling techniques have been used in a variety of applications to estimate quantities that are not directly available. They have been used to estimate global-scale emissions of CFC-11 [Hartley and Prinn, 1993], CO [Bergamaschi et al., 2000], CO2 [Bousquet et al., 1999], and CH4 [Houweling et al., 1999; Zhang et al., 1999]. Inverse methods have also been applied to regional- and urban-scale domains to estimate air quality emissions for isoprene [Chang et al., 1996], CO [Chang et al., 1997], NOx and VOCs [Mendoza-Dominguez and Russell, 2001a], as well as aerosol precursors SO2 and NH3 [Mendoza-Dominguez and Russell, 2001b]. As traditionally done in air quality modeling, these regional- and urban-scale studies focus on episodic simulations during summer conditions to estimate hourly, daily, or episodic scaling factors for the emissions.
 The lack of a seasonal or monthly NH3 emission inventory is one of the larger known uncertainties in air quality modeling of the annual PM2.5 standard. This is the first study that has used inverse modeling to quantitatively estimate the monthly and seasonal variability in NH3 emissions. Preliminary results for two months from this study were presented by Gilliland et al. ; however, those results were limited to only January and June 1990 and relied on an older version of the National Emissions Inventory (NEI) that has since been updated. In this paper, NH3 emission scaling factors will be presented for each season of 1990 along with air quality model results based on these estimates to provide the most comprehensive estimate of seasonal NH3 emissions to date.
 The Fifth Generation Penn State/NCAR Mesoscale Model (MM5) is used to generate the meteorology data fields for CMAQ [Grell et al., 1994; Dudhia et al., 1998]. In addition, the CMAQ cloud module accounts for the presence of ice and graupel. Emissions fields for all chemical species were based on the NEI inventory version 3.01a [USEPA, 2000], Mobile 5b [USEPA, 1994] for mobile emissions, and BEIS2 [Pierce et al., 1998] for biogenic emissions. The annual average gridded NH3 emission field is presented in Figure 1 for the CMAQ domain. Since NEI NH3 emissions data are total annual values (Table 1), the a priori NH3 emission fields used in these simulations have no seasonality, and an inverse modeling approach (as described in the next section) will be used to estimate NH3 emission adjustments for the specific month.
 Precipitation-weighted NH4+ wet concentration, or wet [NH4+], will be used as the primary chemical indicator for NH3 emissions in the inverse modeling approach. Wet [NH4+] data include NH3 that entered precipitation through direct absorption into water and NH4+ aerosols that have been readily scavenged. NH3 gas reacts with sulfuric or nitric acid to form these NH4+ aerosols in the atmosphere. Therefore the wet [NH4+] essentially represents the full air concentration budget of NH3 and NH4+ (i.e., NHx). Precipitation-weighted wet [NH4+] is calculated by dividing the accumulated wet deposition of NH4+ by the total precipitation, and using the appropriate conversion factors to result in units of mg/L. The National Atmospheric Deposition Program National Trends Network (NADP) [National Atmospheric Deposition Program (NADP), 2001] (available at http://nadp.sws.uiuc.edu) has collected precipitation chemistry data for a dense network of monitors for more than twenty years. Wet concentration of NH4+ was chosen over wet deposition because the wet concentration of an efficiently scavenged species like NH3 is less sensitive to rainfall amount than wet deposition [Lindberg, 1982; Stein et al., 1993; Styer and Stein, 1992]. Predicted precipitation uncertainty is a concern because, as a process, it is difficult for mesoscale models to predict clouds and the correct amount of precipitation in the right place at the right time. Using wet concentration, rather than deposition, reduces our sensitivity to this important factor of model uncertainty.
 Ambient concentrations of [NH3] + aerosol [NH4+] ([NHx]) could also be used as a chemical indicator for inverse modeling of NH3, but there are no continuous networks collecting both NH3 and aerosol NH4+. Observational data for NHx are available from January through May 1990 from the Eulerian Model Evaluation Field Study (EMEFS) [McNaughton and Vet, 1996], and NHx will be used with the inverse method for these months to independently estimate NH3 emissions adjustments as a check against the results using wet [NH4+]. [NH3] is not used in the inverse modeling application to avoid introducing model partitioning errors between [NH3] and aerosol [NH4+]. Ambient aerosol [NH4+] data were found to not be suitable for inverse modeling of NH3 emissions. Aerosol [NH4+] responses to NH3 emission changes are associated with changes in the partitioning between NO3− and HNO3 due to the semi-volatility of ammonium nitrate, but the bulk of NH4+ aerosols in the eastern United States are from ammonium sulfate which is left relatively unchanged with NH3 emission changes. Thus the effect of NH3 emission changes on aerosol [NH4+] was shown to be dampened compared to effects on [NHx] or wet [NH4+].
 Observed wet [NH4+] data from the NADP network have a weekly sampling frequency, and the NH4+ wet deposition and precipitation data are aggregated up to monthly or 4-week values for both CMAQ and NADP to calculate precipitation-weighted mean monthly wet [NH4+]. The 4-week periods that are used for monthly values coincide with the beginning and end of the NADP collection time periods. The specific dates that represent each month are listed in Table 2.
Table 2. Dates Used to Represent Each Month
NADP Four-Week Collection Time Periods
9 January to 6 February 1990
6 March to 3 April 1990
3 April to 1 May 1990
1–29 May 1990
5 June to 3 July 1990
3–31 July 1990
31 July to 28 August 1990
2–30 October 1990
 Originally, we had planned to perform the inverse modeling tests for one month per season; however, the study was extended to include all months during the spring and summer seasons to capture the peak monthly emission period. Ideally, we would like to extend this study to include all months of 1990, but the time and resource requirements for running CMAQ are large and limit our ability to expand the study. These monthly simulations require about 2 weeks of runtime on a CRAY T3E system, and a minimum of 3 simulations are needed for each month, not including the resources required to develop the meteorological and emission fields.
 In comparing NADP weekly versus daily sampling protocols, Butler and Likens  and Lamb and Comrie  have shown that wet [NH4+] weekly observations from the NADP network exhibit a low bias because some ammonium is removed from the sample while it remains in the field during the week because of microbial activity. For this reason, we apply a 15% upward adjustment to the NADP wet [NH4+] observations based on an average bias estimate from comparisons of collocated daily versus weekly sampled data at 4 sites within the NADP network [Butler and Likens, 1998]. The variability in the bias suggests that the bias can range by about ±5% intra-annually [Gilliland et al., 2002].
 When reviewing observational data from the NADP monitors, sites were removed if 1 or more of the weekly samples were missing for the 4-week period. Also, data from monitors located within 9 grid cells (i.e., 324 km) of the western boundary of the model domain are not considered so that boundary conditions do not unduly influence the inverse modeling estimates of NH3 emission adjustments.
 In addition to comparing NADP wet [NH4+] and EMEFS [NHx] data with CMAQ results in the inverse modeling analysis, observations of aerosol [NH4+], [NO3−], and [SO42−] from the Clean Air Status and Trends Network (CASTNet) [Clarke et al., 1997] will also be compared to model results before and after adjustments to the NH3 emissions.
2.2. Inverse Modeling Approach
 The inverse modeling approach chosen for this study is based on a Kalman filter approach where the filter is applied independently at each time increment. This approach is similar to the method discussed by Gilliland and Abbitt  and Chang et al. [1996, 1997] except that it is not applied iteratively as in those studies because of the linear response of wet [NH4+] to NH3 emission changes. Without iterations, the approach has some similarities to that of Jacob et al. . To calculate emission adjustments, the following equation is used:
t,posterior represents the integral of the a posteriori emissions estimate for time t within the model domain. To estimate monthly NH3 emissions, the time increment t is monthly in this study. t,prior represents the a priori integral emission estimate within the model domain. The Gain Gt in equation (1) determines the sensitivity of the emission adjustment to differences between observed and modeled concentrations, χtobs and χtmod, at each monitor n. The optimal Gain Gt is given by:
Pt is the Jacobian of the change in concentration with respect to emissions, (Pt = ∂f/∂Et), and f( ) is the function that calculates (χtmod − χt−1mod). The function f( ) represents the chemical transport model. The approach for quantifying Pt is similar to Haas-Laursen et al. , Chang et al. [1996, 1997], and Gilliland and Abbitt  where two parallel simulations are performed for the time increment t where the only difference is a 10% change in emissions. The Jacobian then reduces to Pt = ∂f/∂Et = ∂χt/∂Et because the lifetime of NH3 is short relative to the month-long simulation. The variance of error Ct for t,prior is based on a 40% uncertainty estimate from Asman et al. . Following Chang et al. [1996, 1997], the standard approach first used to quantify the Nt variance of error in concentration is based on the observations' uncertainty, Nt = [σobs, σobs]. The analytical uncertainty in the wet [NH4+] observations (σobs) is estimated to be 4% of the concentration value following Electric Power Research Institute (EPRI) . Because the uncertainty in the a priori emissions is much larger than Nt in this approach, the a posteriori emission estimates that are generated should primarily be a result of a least squares fitting to the observations without much influence from the a priori. As a test of these results in section 3.3, a posteriori NH3 emission adjustments are also calculated with model uncertainty included in Nt so that the a priori information should play more of a role.
 As in previous studies using a regional- or urban-scale air quality model [Chang et al., 1996, 1997; Gilliland and Abbitt, 2001; Mendoza-Dominguez and Russell, 2001a, 2001b], the entire domain is treated as one source region, so that t,prior and t,posterior are scalar. By treating the entire domain as one source region, the entire emission field for all sources will be adjusted by a single factor, leaving the spatial distribution of emissions (as illustrated in Figure 1) within the domain unchanged. This was taken as the first approach because we anticipated less uncertainty in the spatial distribution of emissions than the total magnitude, based on how the emissions are calculated. For animal husbandry and fertilizer emissions, the two largest sources of NH3 emissions, a source-specific emission factor is applied to the county-level number of animals and amount of fertilizer purchased. While this county-level information should be fairly accurate, the lack of temperature dependence in the current emission factors does introduce a spatial uncertainty into the emission estimates that is not addressed in the current approach but is planned as a refinement to this study.
 When presenting the approach, it should be noted that equation (1) assumes that the modeled and observed concentrations at the previous time step are equal. This assumption can pose a risk if initial condition concentrations are substantially different from observed concentrations. For relatively long-lived chemical species (e.g., CO), emission adjustments can compensate for discrepancies in initial conditions, and this can subsequently affect related chemical fields and cause feedbacks [Gilliland and Abbitt, 2001]. However, initial condition discrepancies are inconsequential in this application because we are using monthly time increments t and the lifetime of NH3 in the troposphere is on the order of a few days.
 Pseudodata tests or twin experiments [Hartley and Prinn, 1993; Haas-Laursen et al., 1996; Gilliland and Abbitt, 2001] were performed to test the approach described above. Model-generated data are used as reference observations or pseudodata in equation (1). Another simulation is then performed with a perturbed emission field. If the inverse modeling technique is applied correctly, adjusted emissions should closely approximate the original emissions that were used to produce the reference observations. Pseudodata tests were successful, thus confirming that the methodology was applied correctly and that the application was suitable for the technique. Therefore we could proceed with real observational data.
3. Results and Discussion
3.1. A Posteriori NH3 Budget
 The a priori and a posteriori NH3 emission estimates are summarized in Table 3. For reference, the Bouwman et al.  NH3 inventory for the North American continent is approximately 4.37 × 106 tons/year as compared to the a priori NEI NH3 emission estimates of approximately 4 × 106 tons/year for the contiguous United States. Where EMEFS data are available, a posteriori emission estimates are presented based on [NHx] along with results based on wet [NH4+]. The estimated adjustments for [NHx] and wet [NH4+] are all within 8% of each other, suggesting that wet [NH4+] is behaving fairly well as an indicator for the inverse modeling application. The results in Table 3 illustrate the strong seasonal variation in the a posteriori NH3 emission estimates, where the emissions are lowered substantially in January and October and slightly adjusted during the summer months of June and July. A less expected item of note is that the a posteriori emission estimates suggest that the annual total NEI is too high for 1990.
Table 3. Adjustments to NH3 Emissions Based on Inverse Modeling Results
U.S. NH3 Emission Inventory, 1000 t/yr
A Posteriori Scaling Factors
A Priori (Based on NEI)
3.2. Comparison of Model Results Against Precipitation Chemistry and Ambient Data
 Scatterplots of CMAQ wet [NH4+] based on a priori and a posteriori NH3 emission estimates will first be presented for one month per season. Results for winter and fall are available for January and October. For spring, March is presented since it has the largest NH3 emission adjustment and therefore the strongest response. For summer, the month of June is presented since June has the highest a posteriori NH3 emissions estimates. Responses of [NHx] and aerosol [NO3=] for January will also be shown in scatterplots. Monthly median results will then be presented for wet [NH4+], [NHx] (where available) and aerosol [NO3−], [SO4=], and [NH4+] to illustrate how well the modeled and observed seasonal variations compare when using annually constant versus inverse-adjusted NH3 emissions. For clarification, no emissions other than NH3 have been adjusted in these simulation results.
Figure 2 presents the [NH4+] wet concentration results from CMAQ against NADP data for January 1990. The 68% reduction in NH3 emissions in the a posteriori case reduces the Root Mean Square Error (RMSE) to less than half of its original value. The correlation remains very similar before and after NH3 emission adjustments, which is expected because no changes are made to the spatial distribution of the emissions. March (Figure 3) is similar to January, where the model using a priori emissions tends to substantially overpredict. The 58% a posteriori decrease in NH3 emissions brings the results more closely in line with the observations. Again, the RMSE has improved considerably after the a posteriori decrease in emissions. Since the adjustment to NH3 emissions was only a minimal 6% increase in NH3 emissions during June, Figure 4 does not show any sizable changes after the emission adjustment. Similar to results for January, October results show overpredictions of wet [NH4+] based on a priori annually constant NH3 emissions (Figure 5) that are improved after the 75% a posteriori decrease in NH3 emissions. The RMSE has also decreased to half the original value. While these results suggest that the modeled concentrations agree more closely with observations if using the a posteriori NH3 emissions, a tendency to underpredict is apparent after the emission adjustment, and will be discussed further in section 3.3.
 The response of [NHx] to the NH3 emission adjustments is shown for January in Figure 6, where dramatic improvements in overpredictions and RMSE are evident. A substantial response in the aerosol nitrate [NO3−] is also evident with the decrease from the a priori to a posteriori NH3 emissions for January (Figure 7). In January, large overpredictions in [NO3−] are improved substantially after the decrease in a posteriori NH3 emissions, and the RMSE improves by an order of magnitude. The NH3 emission adjustments not only decreases the amount of particulate nitrate but also the total nitrate, which includes [NO3−] and [HNO3], because the shift from ammonium nitrate to HNO3 has a secondary effect: HNO3 has a much larger dry deposition velocity, so more HNO3 is removed from the atmosphere via dry deposition. Similar responses of [NHx] and [NO3−] to the NH3 emission reductions during other winter and fall months are also illustrated in the next series of figures.
 A concise picture of how well the emission adjustments capture the seasonal variability in the observed data is shown in Figures 8–12. These figures present median results from all monitors along with the 25% and 75% quartiles of NADP data and the corresponding CMAQ grid cells for each simulated month. Directly relevant to the inverse calculations themselves, Figure 8 shows the a priori and a posteriori CMAQ model results for wet [NH4+] against the NADP wet [NH4+]. When using the annually constant a priori NH3 emissions, CMAQ shows large ranges of overpredictions during January, March, April, and October, and the seasonal variability evident in the observational data is poorly captured (Figure 8a). The a posteriori NH3 emissions greatly improve the seasonal pattern in wet [NH4+] (Figure 8b). If the mean monthly perturbations around that annual mean are compared between CMAQ and the observational data, the annually constant a priori NH3 emission estimates results in a correlation of R = 0.12, and the a posteriori NH3 emission estimates increase the correlation to R = 0.98. Clearly, the adjusted NH3 emissions better capture the seasonal variability in wet [NH4+] observed in the NADP network. However, an obvious shortcoming of the results in Figure 8b is that the a posteriori model results consistently underpredict the observed values, an issue already raised in the scatterplots previously presented. The determined cause for this underprediction will be discussed in section 3.3.
Figure 9 compares the [NHx] CMAQ results based on a priori and a posteriori NH3 emission estimates against EMEFS [NHx]. Overpredictions are substantially corrected by the emissions adjustments for January, April, and May 1990. However, a posteriori [NHx] results for March 1990 are noticeably lower than observed. When [NHx] was used in equation (1), the March NH3 emission adjustment was 8% less than the wet [NH4+] results.
 The effects of the NH3 emissions adjustments on aerosol [NH4+] in CMAQ are presented in Figure 10. Substantial improvements are evident in the winter, spring, and fall months. During summer months in the eastern United States, most of the aerosol [NH4+] is in the form of ammonium sulfate, and sulfate aerosols are not affected to a noticeable extent by the modifications in NH3 emissions (Figure 11).
 Improvements in the a posteriori aerosol [NO3−] results are dramatic during the winter, spring, and fall periods (Figure 12b). These improvements in [NO3−] are related to the improvements seen in aerosol [NH4+] predictions during winter, spring, and fall in Figure 10b (as compared to Figure 10a). The dramatic improvements to [NO3−] result from less NH3 gas being available to form ammonium nitrate, so more remains as HNO3 and is removed via dry deposition. When performing annual or seasonal model simulations to evaluate model predicted concentrations of nitrate aerosols, the need for seasonal NH3 emissions is obvious from these results.
3.3. Biases and Uncertainties in a Posteriori NH3 Emission Estimates
 As previously mentioned, the CMAQ results with inverse-adjusted NH3 emissions impose an underprediction in wet [NH4+] during most months (Figure 8). Since the intention with the inverse modeling methodology is to optimize the modeled wet [NH4+] against observed values, the underprediction of the directly compared indicator is troubling. Based on tests shown here, we have determined that the underprediction is caused by the uniform relative error used when quantifying the Noise Nt, where the same 4% uncertainty value is used at all monitors. We test this by calculating the emission adjustments using a uniform absolute error for Nt, based on the average wet [NH4+] value from all monitors multiplied by the same 4% analytical error. Because of resource constraints, no additional simulations were made here, but we compare the adjustment estimates themselves. As presented in Figure 13, the adjustments based on a uniform absolute error are clearly higher in the spring, summer, and fall, with June having the largest difference of a 24% increase versus the previous 6% increase.
 With a uniform relative error, model discrepancies have more influence on the emission adjustment when the observed concentrations are low. This is because the low concentrations result in smaller error values (and larger Gain Gt values). If the modeled concentrations had a similar spatial pattern as the observed concentrations, the uniform relative error approach would not create a noticeable bias in the results because there would not be sporadic locations with large model overpredictions and low observed concentrations. This is why the problem was not detected in the pseudodata tests, where the spatial pattern of the emissions was not perturbed. However, in reality, the model does not consistently overpredict or underpredict at all monitors. Figure 14 shows the most extreme example, June 1990. Gt is much smaller in areas where the observed concentrations are high, but the model substantially underpredicts wet [NH4+] in these areas. On the other hand, there are five NADP monitors (AL99, MN18, TN11, MI99, and PA42) where the model overpredicts wet [NH4+] and the Gt weighting is quite high.
 Analytical precision tests in the NADP 2000 Quality Assurance Report show that the uncertainty is indeed higher for [NH4+] at low concentrations [NADP, 2002] (available at http://nadp.sws.uiuc.edu/lib/qa/qa2000.pdf), which suggests that a uniform relative error is not a correct representation of the uncertainty for these data. These test results show that the representation of error throughout the range of concentrations should be carefully considered where dense networks of observed data are used within much coarser source regions.
 Going back to a point made in the previous section, the emission adjustments estimated here by the inverse modeling approach suggest that the current NEI's annual NH3 emissions are too high. Seasonally weighted a posteriori NH3 emissions based on wet [NH4+] and the initial inverse modeling approach (Table 3) suggest that the NEI annual NH3 emissions should be 47% lower to optimize modeled wet [NH4+] against observations. The low bias that has been detected in the a posteriori model results in Figure 8 suggests that this decrease may be too large. We have also shown that a uniform relative error does not correctly represent the uncertainty at lower concentrations and puts too much weight on model performance at lower concentrations. Based on the results using a uniform absolute error for the observed data, the NEI annual NH3 emissions should be approximately 37% lower to optimize modeled wet [NH4+] against observations. This may be a more reasonable estimate, but it is still based on an air quality model that has its own uncertainties. Two areas of model uncertainty are considered here, precipitation and the dry deposition of NH3.
 To test the sensitivity of the emission adjustment factors to model uncertainty, we calculate the standard error of the estimate relative to the total model precipitation,
where Pdailyobs is the daily precipitation value from the NADP monitors and Pdailymod is the same from MM5. At monitors with many days of no precipitation, extremely large relative error values resulted if the error was calculated relative to the mean precipitation. For the purpose of a sensitivity test, using the total accumulated model precipitation for the month Ptotalmod was chosen as a practical solution around this problem. The precipitation uncertainty was calculated separately for each monitor, and the average across all monitors was about 15% and ranged between about 5% and 100%. The precipitation uncertainty must then be translated into an uncertainty in the wet [NH4+]. To do so, we use a uniform absolute error approach by multiplying the precipitation uncertainty at each monitor against an average wet [NH4+] for the domain. This way, the model uncertainty parameter and Nt values were weighted primarily by the model's ability to predict precipitation, and the same relative uncertainty is not assumed throughout the concentration range. The resulting NH3 emission adjustment factors for most months are approximately 5 to 10% higher than results based only on a uniform absolute error for the observed concentrations (Figure 13). The results from this test including model uncertainty suggest approximately a 30% decrease in the annual total NEI NH3 emissions based on a seasonally weighted average of the emission adjustment factors.
 Another area of model uncertainty that could affect the CMAQ predicted wet [NH4+] is NH3 dry deposition. A review of published field data [Y. Wu, personal communication, 2002; Poor et al., 2001; Pryor et al., 2001; Hansen et al., 1998] indicates that the NH3 dry deposition velocity, vd, is too low in the standard dry deposition model used in CMAQ [Wesely, 1989]. To test the sensitivity of the inverse results to this discrepancy, we performed CMAQ simulations for January and June 1990 by doubling the NH3 vd. This change resulted in about a 10% higher adjustment in the computed emission adjustment factors. Further research is needed in the area of NH3 dry deposition modeling before a more conclusive impact could be assessed for these results; however, at a minimum, we believe that our a posteriori adjustments to NH3 emissions may be compensating for an underprediction of NH3 dry deposition removal in the model.
 If modeled precipitation uncertainty and potential biases in the NH3 dry deposition velocity are considered, the NH3 emission adjustments suggest that as an upper bound, the annual NEI inventory is potentially about 20% too high. This is supported by the USEPA  report that proposed modifications to the NH3 emission factors currently used in the NEI. The proposed modification that has the largest impact is the emission factor for nondairy cattle. Cattle are the largest source of NH3 in the inventory, and the NEI currently uses the same emission factor for both dairy and nondairy cattle. The report suggests that the emission factor for nondairy cattle should be decreased by approximately 50%. This change, along with a proposed decrease of 24% in the NH3 emission factor for swine, would result in a total 23% decrease to the NEI NH3 emission inventory. This 23% decrease provides some grounding for the decrease in emissions that is suggested based on these inverse modeling tests.
 Monthly NH3 emissions are estimated using an inverse modeling approach that compares wet [NH4+] data from NADP network against USEPA CMAQ model results. The motivation is that the EPA NEI inventory for NH3 emissions currently provides NH3 emission estimates only on an annual timescale, but we suspect seasonal variability in these emissions. The a posteriori NH3 estimates presented here suggest a strong seasonal variation in the NH3 emissions where emissions are approximately 70% higher in June as compared to fall and winter months. Similarly, NH3 flux observations from two swine finishing buildings were about 75% higher during summer as compared to early spring conditions [Heber et al., 2000], and fluxes observed from a hog waste lagoon were approximately 80–90% higher during summer as compared to winter and fall conditions [Aneja et al., 2000]. The only other inverse modeling study that has considered NH3 emissions focused on May and July 1995 episodes [Mendoza-Dominguez and Russell, 2001b], but a different emission inventory was used for that study so that a direct comparison with their results cannot be made here.
 The results presented include independent comparisons of modeled data against observations for related chemical species including [NHx], aerosol [NH4+] and [NO3−], as well as the wet [NH4+] used in the inverse applications. These comparisons suggest that the a posteriori NH3 emission adjustments provide considerable improvements in the CMAQ model results over using the annually constant a priori NH3 emissions, particularly during the fall and winter periods. As no official seasonal distribution is currently available for the NEI NH3 inventory, a default approach has been to distribute the emissions evenly throughout the year. The results presented here elucidate the large biases that can be introduced into the air quality modeling results if constant NH3 emissions are applied. Annual simulations of PM2.5 should carefully consider the temporal allocation of NH3 emissions as annually constant emissions are inadequate for winter, early spring, or fall conditions.
 In addition, the resulting NH3 emission adjustment presented here suggests that the annual 1990 NEI NH3 emissions could be too high. This is supported by the USEPA  report that proposed modifications to the NEI NH3 emission factors for nondairy cattle and swine, which would result in a total 23% decrease to the NEI NH3 emission inventory. Considering the results of this study along with the USEPA  report, we believe that it is plausible to consider as a bounding estimate that the1990 NEI NH3 inventory could be at least 20% too high on an annual basis.
 While quantitative results are presented, emphasis should be placed on the qualitative conclusions from this study regarding the seasonal variations in the emissions and the over-estimate of NH3 emissions in the NEI. The uncertainties within the air quality model can influence the results, and not all model uncertainties are or can be included when estimating a posteriori NH3 emissions. Furthermore, these results are specific to the domain and time period considered. Other aspects that will be considered as extensions of this study include larger (e.g., national) domains, individual source regions, interannual variability, and additional testing of model uncertainties. Field measurements linking environmental conditions to different livestock categories should also provide improvements to a national emission inventory and hopefully help to develop emission models that relate emission fluxes to climatic conditions. In the meantime, this study provides the most rigorous seasonal NH3 emission estimates to date for urban and regional air quality modeling.
 We would like to thank Lucille Bender from DynCorp, Inc., for her extensive help with the CMAQ model simulations. Our appreciation also goes to Ruen Tang and George Pouliot of DynCorp, Inc., for their assistance with emissions calculations and Renee Jaramillo of ASI, Inc., for assistance with the NADP data. This work was supported by the USEPA Office of Research and Development's National Exposure Research Laboratory. This paper has been subjected to U.S. Environmental Protection Agency peer review and approved for publication. Mention of trade names or commercial products does not constitute endorsement or recommendation for use.