Methane mixing ratios measured at a tall tower are compared to model predictions to estimate surface emissions of CH4 in Central California for October–December 2007 using an inverse technique. Predicted CH4 mixing ratios are calculated based on spatially resolved a priori CH4 emissions and simulated atmospheric trajectories. The atmospheric trajectories, along with surface footprints, are computed using the Weather Research and Forecast (WRF) coupled to the Stochastic Time-Inverted Lagrangian Transport (STILT) model. An uncertainty analysis is performed to provide quantitative uncertainties in estimated CH4 emissions. Three inverse model estimates of CH4 emissions are reported. First, linear regressions of modeled and measured CH4 mixing ratios obtain slopes of 0.73 ± 0.11 and 1.09 ± 0.14 using California-specific and Edgar 3.2 emission maps, respectively, suggesting that actual CH4 emissions were about 37 ± 21% higher than California-specific inventory estimates. Second, a Bayesian “source” analysis suggests that livestock emissions are 63 ± 22% higher than the a priori estimates. Third, a Bayesian “region” analysis is carried out for CH4 emissions from 13 subregions, which shows that inventory CH4 emissions from the Central Valley are underestimated and uncertainties in CH4 emissions are reduced for subregions near the tower site, yielding best estimates of flux from those regions consistent with “source” analysis results. The uncertainty reductions for regions near the tower indicate that a regional network of measurements will be necessary to provide accurate estimates of surface CH4 emissions for multiple regions.
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 Changes in atmospheric methane play an essential role in Earth's climate. CH4 is now associated with a direct radiative forcing of ∼0.48 Wm−2 [IPCC, 2007] and an indirect radiative forcing of ∼0.13 Wm−2 [Lelieveld et al., 1998], which accounts for about of the non-CO2 radiative forcing (0.98 W m−2 in 2004 [Hofman et al., 2006]) and about of the total radiative forcing (2.64 W m−2 from IPCC ) from all greenhouse gases (GHGs). It has been argued that reducing anthropogenic emissions of methane may be an important component of an initial strategy for avoiding the most severe impacts of global warming [Hansen et al., 1998; Hansen, 2004; Shindell et al., 2005]. In particular, reduction of anthropogenic methane emissions may be possible (e.g., improving CH4 recovery from landfills and waste treatment, reducing industrial emissions, and improving agricultural practices) [Harriss, 1994]. In view of methane's role in the climate system, increased attention has been brought recently to assessing CH4 sources [Gimson and Uliasz, 2003; Miller and Tans, 2003; Houweling et al., 2006; Kort et al., 2008].
 In California, total GHG emissions in 2004 were approximately 480 MMT CO2 equivalent, with CH4 contributing approximately 6% [CARB, 2007]. Now that California has passed Assembly Bill 32, which requires that greenhouse gases emissions be reduced to 1990 levels by 2020, careful accounting of current CH4 emissions and of their future reductions is essential. Unfortunately, current inventory and model-based estimates of CH4 emissions are uncertain because many of the factors controlling emissions are poorly quantified. Atmospheric measurements and inverse modeling may provide an independent method to quantify local to regional CH4 emissions from California.
 In this study, we employ an approach originally developed to estimate regional CO2 emissions [Gerbig et al., 2003a, 2003b] that combines calculation of surface footprints [Lin et al., 2004] with procedures to estimate transport model uncertainty [Lin and Gerbig, 2005] using the Stochastic Time-Inverted Lagrangian Transport (STILT) model. Of particular relevance to our work, Kort et al.  recently used observations of CH4 and N2O from an airborne platform in combination with STILT to infer CH4 and N2O emissions from the continental interior of North America in May–June 2003. Our study also uses STILT, but applies it to a smaller model domain at finer spatial and temporal resolutions, taking advantage of unique computational benefits offered by the Lagrangian approach.
 To address the problem of estimating CH4 emissions from different sources in Central California, we conducted coordinated CH4 measurements and modeling as part of the California Greenhouse Gases Emission Measurement (CALGEM) project. Section 2 describes the methods for the measurements of CH4 mixing ratios, profiler-based estimates of wind fields and boundary layer heights, spatially resolved a priori CH4 emission maps, meteorological transport fields and resulting surface footprints, an analysis of measurement and model errors, and the Bayesian inverse model used to estimate CH4 emissions. Section 3 describes the results of the measurements, bias corrections and error estimates, and the best estimates of CH4 surface emissions implied by the measurements. Section 4 discusses the estimates of CH4 emissions in the context of current inventories, examines the spatial region in which the tower measurements effectively constrain CH4 emissions, and concludes with initial recommendations for additional measurement sites to constrain other important emission regions in California.
2. Data and Methods
2.1. CH4 Measurements
 The CH4 measurements were made at 91 and 483 m on a tall tower near Walnut Grove, CA (WGC, 121.49°W, 38.27°N, 0 m above sea level), beginning in September 2007. The measurements were made using a sampling and analysis system combining pumps, air driers, and three gas analyzers. Briefly, air samples are drawn continuously from the different heights on the tower, are partially dried by a condensing system that lowers water vapor to a 5°C dew point, are sequentially applied on a 5 minute interval to a temperature stabilized membrane drier (Purmapure Inc.) which dries air to a −33°C dew point, and then are supplied to the gas analyzers. The first 4.5 minutes of each measurement interval are used to allow equilibration of the gas concentrations and instrument response, while the last 30 seconds is used as the measurement interval. In particular, CH4 is measured using a cavity ring-down spectrometer (Picarro EnviroSense 3000i) with an accuracy and precision of approximately 0.3 ppb in the 30 second averaging interval. To quantify and correct instrument drifts, the offset is measured and corrected every hour using a reference gas, while the gain (and linearity) is checked and corrected every 6 hours using 4 NOAA gas primary standards. In addition, flask samples were collected twice daily (1000 and 2200 hr GMT) from a separate sample line at the 91 m level and analyzed at NOAA-ESRL. To provide additional quality assurance, the in situ CH4 measurements were compared with the flask measurements. This redundancy allows the detection of small (∼ppb) sampling errors. In general, the difference between in situ and flask analyses was consistent with the precision of the in situ instrument. During some periods, particularly during late night and early morning, variability in CH4 mixing ratios was larger. For these periods, the difference between flask the in situ CH4 measurements was generally consistent with the standard deviation of the in situ CH4 measurements averaged over a 30 minute window centered on the flask sample.
Figure 1 shows 3-hour averages of measured CH4 mixing ratios at 91 m (black) and 483 m (red) in October 2007. Diurnal cycles due to changing boundary layer height are apparent in the data. The daily maximum CH4 mixing ratio measured at 91 m often occurs when the minimum is obtained at 483 m. This would be expected to occur in cases when the boundary layer lies between 91 and 483 m, trapping surface emissions within a shallow layer that is measured by 91 m sample height, while the 483 m sample height observes comparatively decoupled background air. In the following work, we will use the daily minimum CH4 measurements at 483 m to provide a check on the CH4 background analysis. Moreover, we limited the inverse model study to only include measurements collected during well-mixed periods. Henceforth, we define the well-mixed periods by using the criteria that the difference of measurements at 91 m and 483 m are less than 100 ppb, as shown by the black points in Figure 1. This criteria will also be evaluated in the following analysis.
2.2. Wind Profiler Measurements
 To quantify uncertainties in modeled atmospheric transport, hourly boundary layer heights and vertical profiles of winds were obtained from a radar wind profiler (RWP) operated by the Sacramento Metropolitan Air Quality Management District. The profiler is located (38.30°N, 121.42°W) within 8 km of the tower used for the CH4 measurements. Given the level terrain of the Sacramento delta region, we expect that errors in modeled winds and PBL heights for the region surrounding the tower can be accurately assessed by comparing the wind profiler measurements with corresponding meteorological simulations for profiler (winds) and tower (PBL) locations. The RWP acquires data in two different settings, high-resolution and low-resolution mode with vertical resolutions of 60 m and 105 m respectively. Boundary layer heights were estimated from subhourly RWP vertical velocity and returned signal strength (signal-to-noise ratio) data using objective algorithms and qualitative analysis following techniques found in the works of Wyngaard and LeMone , Bianco and Wilczak , and Bianco et al. . In the used configuration, the RWP can detect boundary layer heights from about 150 m to 4000 m with an accuracy of ±200 m [Dye et al., 1995].
2.3. The a Priori CH4 Emissions
 We used two methods to estimate CH4 emissions. As a base case, we used the North American maps of total anthropogenic CH4 from the EDGAR 3.2 model with 1 × 1 degree spatial resolution [Olivier et al., 2005]. To provide finer spatial resolution inside California, we also estimated California CH4 emissions separately for six sources sectors: landfills (LF), livestock (LS), natural gas production and use (NG), petroleum refining (PL), crop agriculture (CP), and wetlands (WL). CH4 emissions from landfills were estimated by the California Air Resources board (L. Hunsaker, private communication, 2008) using IPCC methods [IPCC, 2006], which is driven by landfill specific waste application statistics from the CA Waste Management Board [e.g., Carr, 2004] and site-specific estimates of CH4 recovery. CH4 from livestock was estimated using United States Department of Agriculture (USDA) county level animal stocking densities [Census, 2002] and animal specific emission factors for dairy and beef cattle separately [Franco, 2002]. CH4 from natural gas production and use and from petroleum refining activities were estimated as the difference of total minus reactive hydrocarbon (typically between 0.2 and 0.4 of the total) emissions estimated from the California Air Resources Board (CARB) emission criteria pollutant emission inventory for those source sectors (http://www.arb.ca.gov/app/emsinv/fcemssumcat2006.php). CH4 emissions from crop agriculture were assumed to follow emissions from the DNDC model for an average climate year with high irrigation as described by Salas et al. . CH4 emissions from wetlands were assumed to follow the National Aeronautics and Space Administration Carnegie-Ames-Stanford Approach (NASA-CASA) estimates from Potter et al. . Although some of these sources are expected to vary on a seasonal basis, we calculated mean emissions and did not attempt to resolve temporal variations over the relatively short period of this three months study. Maps of the a priori CH4 emissions are shown in Figures 2a–2f for these six California-specific source sectors. For comparison, Figure 2g shows total EDGAR 3.2 emissions for the Western United States, while Figure 2h shows the sum of the California-specific CH4 emissions. Last, Figure 2i shows a set of California subregions that roughly correspond to air basins that are nearby or distant from the measurement locations and will be used in following analysis. Table 1 summarizes the CH4 emissions from different California-specific sources in the 13 subregions. CH4 emissions are scaled to equivalent CO2 forcing using a global warming potential of 25 (gCO2eq gCH4−1 [IPCC, 2007]). The total of the California-specific emissions is similar to total CH4 emissions (∼31 MMT CO2eq) reported by the California Air Resource Board [CARB, 2007], but is approximately half the total emissions from California pixels in the Edgar 3.2 inventory. Inspection of the Edgar 3.2 emissions shows that the largest sources are from natural gas (22.5 MMT CO2eq) and landfills (19.3 MMT CO2eq), suggesting very different assumptions about emissions from these sources. To assign an uncertainty to the a priori emissions, we follow previous work on uncertainty analysis [USEPA, 2004; Farrell et al., 2005] and assign a 30% uncertainty to each of emissions sources. We consider the uncertainties in US total CH4 emissions only a rough estimate to the uncertainties for subregions of California (and over the time period of this study) because the 30% estimate was derived for more aggregated emissions over annual cycles and the entire continental US.
Table 1. A Priori CH4 Emissions (MMT CO2eq) From Six Different Sources and 13 California Regions in Figure 2i
CH4 (MMT CO2eq)
2.4. WRF-STILT Model
 As mentioned in the Introduction, the work presented in this paper employs the STILT model, run in the time-reversed (receptor-oriented) mode, as the atmospheric transport model. STILT is a Lagrangian Particle Dispersion Model (LPDM) that has been specifically developed and applied to regional simulations and inverse flux estimates for CO2, other greenhouse gases, and CO. Its detailed description is provided elsewhere [Lin et al., 2003, 2004; Gerbig et al., 2003a; Matross et al., 2006; Kort et al., 2008; Miller et al., 2008] and, consequently, only the most pertinent features will be summarized here. As in all LPDMs, transport in STILT includes both advective and turbulent components, with turbulence being responsible for the dispersion of particles. In this application, given input meteorological data, the STILT model transports ensembles of 100 particles (air parcels) backward in time 5 days for a receptor point (WGC site here). We calculate the response of the target gas concentration at the receptor point to surface sources (“footprint”), in units of ppb/(nmol m−2 s−1). The footprint, which represents the adjoint of the transport field, is calculated by counting the number of particles in a surface-influenced region (defined as of the estimated PBL height in the STILT model, for example, see Gerbig et al. [2003a] and Kort et al. ) and the time spent in the region (for details, see Lin et al. ). When multiplied by the a priori field of surface flux, the footprint gives the associated contribution to the mixing ratio measured at the receptor, hence the footprints can be used to estimate parameters of the source functions and their respective uncertainties.
 We calculate the footprints relating surface fluxes to measured CH4 mixing ratios using the meteorological output from a customized version of the Weather Research and Forecasting model [Skamarock et al., 2005] to drive STILT. This combined model will henceforth be referred to as WRF-STILT. Specifically, the WRF model version 2.2 has been modified to output time-averaged (hourly in this study) values of the mass-coupled velocities, which significantly improve mass conservation in STILT (compared with the instantaneous advective velocities), as well as convective mass fluxes that are used directly in the STILT calculations. The main physical options are set as following: (1) Radiation: RRTM scheme [Mlawer et al., 1997] for the longwave and Goddard scheme [Chou and Suarez, 1994] for the shortwave; (2) Planetary Boundary Layer: Yonsei University (YSU) scheme coupled with the NOAH land surface model and the MM5 similarity theory based surface layer scheme. (3) Microphysics: Purdue Lin scheme [Lin et al., 1983; Chen and Sun, 2002] (4) Convection: Grell-Devenyi ensemble mass flux scheme [Grell and Devenyi, 2002]. The initial and boundary meteorology conditions for WRF are provided by the North American Regional Reanalysis (NARR [Mesinger et al., 2006]). A one-way nesting WRF running with 3 nest levels is used for the meteorology simulations around the WGC tower location, which is shown in Figure 3 (Domain 1: −149.16° < lon < −102.21°, 17.82° < lat < 50.53° on a 40 km grid; Domain 2: −123.53° < lon < −120.66°, 36.76° < lat < 38.94° on a 8 km grid; Domain 3: −121.71° < lon < −121.23°, 38.09° < lat < 38.45° on a 1.6 km grid). The vertical resolution is taken from the input meteorology from NARR with 30 layers. Each day was simulated separately using 30-hour run (including 6 hours from the previous day for spin-up) with hourly output. Growth in transport model errors were minimized by nudging the forecast fields to the gridded NARR analysis fields every 3 hours.
2.5. WRF-STILT Transport Errors
 As a first approximation to evaluate the transport errors in the WRF-STILT predictions of surface influence footprints, we compared the modeled estimates of WRF winds and WRF-STILT boundary layer heights (Zi) with corresponding profiler measurements of wind velocity and Zi. Errors in modeled winds are estimated by comparing WRF predictions with profiler measurements of the u and v wind components at a height of 137 m, close to the height of the air sampling. Using data from the October to December 2007, the root mean square (RMS) errors in horizontal winds at 137 m are 2.21 (σu) and 2.86 m s−1 (σv) for the u and v directions respectively. Some of this difference can be attributed to the fact that profiler winds are measured at a single site while the WRF winds are the averages over a grid of 1.6 × 1.6 km. We note that the RMS error decreased by approximately a factor of 2 between 137 m and 1000 m above the ground, though the decrease was nonlinear with most of the decrease occurring between 137 and about 500 m. Henceforth, we assume errors in u and v are constant with height and randomly distributed with an RMS magnitude of 3.6 m s−1, which is obtained as .
 Measured and predicted daytime boundary layer heights in October through December 2007 are shown in Figure 4. Profiler data were selected to match the time of the WRF predictions to within 1 hour. In addition, the WRF-STILT simulations impose a lower limit value of 215 m on Zi, while the radar profiler has a minimum detection height of 120 m. To avoid biasing the comparison and make sure CH4 well mixed from surface till heights above 483 m, we included WRF-STILT predictions of Zi greater than 215 m in the analysis. The resulting best fit geometric linear regression of WRF-STILT on radar profiler PBL heights yields a slope of 1.25 ± 0.10 and intercept of −138 ± 70 m. Based on this result, we obtain a scale factor of 1/1.25 which is then applied to Zi when calculating footprints using STILT. This result is similar to that found in the work of Lin et al. , where STILT predictions of Zi were about 1.09 higher than Zi measurements at a site in Wisconsin. After scaling STILT Zi by a factor of 1/1.25, the RMS residual error between scaled WRF-STILT and profiler Zi is reduced by a factor of 1.5 to ∼200 m, roughly consistent with the estimated error in the profiler measurements. In the following work we calculate particle trajectories and resulting footprints using the scaled parameterization of PBL height. It is possible that an additional error in the effective wind field may be introduced by the Zi scaling for particles near the top of the boundary layer if there is significant wind shear at that altitude but expect that this is small compared to the first order errors already identified for winds and PBL heights.
2.6. Footprints and Predicted CH4 Signals
 Particle trajectories were calculated using STILT driven by the WRF winds. One hundred particles are released every 3 hours (from UTC hour 00) at the WGC tower and transported backward in time 5 days to insure a majority of the particles reach positions representative of the marine boundary layer. Footprints are then calculated from the particle trajectories as in the work of Lin et al. . The time-averaged footprint is shown in Figure 5 for the period between October and December in 2007. The high footprint values within approximately the Central California area near the tower site indicate that CH4 signals measured at 91 m and 483 m at WGC will be strongly influenced by the California emissions.
 Predicted local CH4 signals Cl(Xr, tr) (index “l” denote local and “r” denote receptor) from land surface emissions are calculated using the product of the footprint maps and the a priori emission maps, as
where Xr and tr are receptor (WGC tower) location and time, f(Xr, tr∣X, tm) is the footprint and F(xi, yj) is the surface emission map at location (xi, yj) and time tm. The total CH4 mixing ratio at the receptor can be expressed as
where CBG(Xr, tr) is the upstream CH4 background mixing ratios.
2.7. Inversion Technique
 The posterior CH4 emissions were estimated by optimizing scaling factors for the a priori CH4 emissions to provide a best fit between measured and predicted CH4 mixing ratios. This was done in two ways: (1) as a standard least square optimization of an overall scaling factor for all land surface emissions and (2) in a Bayesian approach that scales each source type or subregion separately and incorporates individual estimates for the uncertainties in different a priori emissions.
 Combining equations (1) and (2), the difference between measured and predicted background CH4 relates to the surface emission flux as
where is footprints, F is surface CH4 emission, C and CBG is CH4 mixing ratios from tower measurements and background calculations, respectively. Assuming mixing ratio measurements from local sources as y = C − CBG. Following Gerbig et al. [2003a], we introduce a model parameter or a state vector of scaling factors, λ, for the surface flux, F(λ). The inversion adjusts the model parameters λ such that the modeled changes in CH4 concentrations are optimally consistent (in standard least square sense) with the observed values. In the study of surface CH4 emissions from different sources (“source analysis” hereafter), λ represents the scaling factor for different sources; in the study of surface CH4 emissions from different regions (“region analysis” hereafter), λ represents the scaling factor for different areas. For both the “source analysis” and “region analysis” study, F(λ) is linearly dependent on λ:
where is the a priori emissions for different sources or regions in this study.
where = , ɛ is measurement error covariance matrix λprior and are the a priori and a posteriori vectors, and prior and λ are the a priori and a posteriori error matrices for λ. Corresponding to our initial estimate of 30% uncertainty in the CH4 emission maps, the initial value of prior is 0.09. Note that the measurements and a priori emission error matrices are diagonal, equivalent to assuming that the prior errors are uncorrelated. The measured and predicted CH4 signals are computed and compared on a 3 hour interval.
2.8. Error Covariance Matrix
 The equivalent “measurement” error covariance matrix Sɛ is formed as the sum of different components
Here as in the work of Lin et al. , the contribution of instrumentation error in the CH4 measurements is assumed to be random, uncorrelated, and negligible in magnitude relative to the other sources of error, and hence not considered further in the inverse model estimates. We consider each of the terms in equation (6) below.
 The particle number error (Spart) is due to the finite number of released particles at the receptor location. It can be estimated by comparing the simulated signals from the STILT running with release of 1000 particles and those from the STILT running with release of 100 particles. Using the WRF simulated meteorology in October 2007 and the total a priori emission map, we found that the standard error is about 3 ppb, indicating ∼5% particle number error. This value is less than ∼13% particle number error for CO2 indicated by Gerbig et al. [2003a]. Considering the ∼5% error determined by us here and ∼13% error determined by Gerbig et al. for signals in the mixed-layer, Spart for 100 particles is assumed as 10% in this study. For all of the following error analyses, we used 1000 particles in order to minimize the effect of particle number error.
 The “aggregation error” (Saggr) arises from aggregating heterogeneous fluxes within a grid cell into a single average flux [Kaminski et al., 2001]. Gerbig et al. [2003b] demonstrated that a rough estimate of the aggregation error can be derived from the observed “representation error,” which is derived from the difference between a point observation and a value averaged over a specific grid size [Gerbig et al., 2003a]. Without multiple observation stations over a specific grid, we try to estimate the aggregation error based on the a priori CH4 emissions. Although we do not have high-resolution emission maps for all of the CH4 sources, we estimate aggregation error using landfill emissions, which are fully resolved. Here the aggregation error is estimated by comparing the unaggregated landfill signal from to the landfill signal estimated after averaging emissions over each county (the maximum aggregation used for the other sources). The average aggregation error, estimated as the RMS difference between the unaggregated and aggregated signals, is 11% of the mean landfill signal.
 The transport error (Trans = TransWND + TransPBL) denotes the errors in modeling transport, which can be caused by the uncertainties in wind speeds and directions, and the uncertainties in PBL heights. Following Lin and Gerbig , the transport error due to winds TransWND is calculated as the RMS difference between signals predicted from simulations with and without input of an additional stochastic component of wind error σV (3.6 m/s; section 2.5) in STILT. The resulting RMS signal is equivalent to 8% of the average predicted CH4 signal. This estimate of uncertainty assumes that the wind error at the radar profiler location can be used to represent the wind error within the modeling domain. While we have not evaluated the wind errors for other locations, we note that the 3.6 m/s wind error used here is comparable to the mean wind error of 3.08 m/s, determined from radiosonde observations over the coterminous U.S. between 0 and 3 km in altitude [Lin and Gerbig, 2005].
 Uncertainty due to errors in modeled PBL heights TransPBL is estimated by propagating the residual error Zi into the predicted CH4 signals. Here we use the estimate of residual error in Zi determined from the comparison between predicted WRF-STILT PBL height and PBL height measured with the wind profiler. The sensitivity of CH4 signal to Zi is expressed as a first order perturbation in C as
where γ is estimated by calculating STILT footprints and then variations in C for small perturbations in Zi. The error due to error in Zi can then be estimated as
where ΔZi is the residual error in WRF-STILT Zi, and 〈C〉 is the mean total CH4 signal. Note that this error is calculated for well-mixed conditions. Using equations (7) and (8), the estimated transport error due to PBL uncertainties is 22% of the mean signal.
 The background error (bkgd) is due to the uncertainty in estimating the background contribution to the CH4 measurements at WGC 91 m. For this study, we estimate the upstream background CH4 mixing ratio using the final latitude of each particle as a lookup into the latitudinally averaged marine boundary layer (MBL) CH4 for October–December 2007 (NOAA Globalview CH4). Only time points (>95% of the total) for which more than 80% of the particles reached longitudes 1.5 degrees from the coast were included in the study. We expect that the NOAA MBL average will be a reasonable approximation for the CH4 background because it is heavily weighted to the Pacific and the typical CH4 gradients between Pacific and Atlantic are less than 10 ppb. We evaluated the error in CH4 background using the daily minimum CH4 mixing ratio measured at 483 m. The reason that the daily minimum CH4 mixing ratio at 483 m often reflects that of background air is because the 483 m sample height decouples from the surface at night (when 91 m < Zi < 483 m) as indicated in Figure 1. A comparison of the CH4 mixing ratios determined from the NOAA MBL average and WGC 483 m minimum estimates is shown as a function of time in Figure 6. Figure 6b shows that there is no systematic bias, although the minimum CH4 mixing ratio at 483 m is occasionally enhanced relative to the NOAA MBL average, likely due to local CH4 contributions. We estimate the error due to CH4 background as the RMS difference in Figure 6b, which is 15% of the mean background-subtracted measurements at 91 m.
 The eddy flux error (Seddy) specifies the fluctuations in CH4 mixing ratios due to contributions from turbulent eddies. Gerbig et al. [2003a] observed it is ∼0.2 ppm for CO2. For CH4 studied here, a value of 1% is assumed. The error due to omitting ocean emissions (ocean) is assumed to be negligible. To evaluate this assumption, we calculated the expected CH4 signal from the Coal Point field near Santa Barbara, the largest known coastal natural gas field near California [Mau et al., 2007], and found the signals to be less than 1 ppb.
 In order to combine the above errors from different sources, we need to know their correlations, which are unfortunately unknown. Assuming the errors from different sources are independent, the above errors are combined in quadrature to yield a total expected model prediction mismatch error of 32%.
3.1. CH4 Mixing Ratios
 Predicted CH4 signals and background-subtracted measurements at 91 m are shown in Figure 7. As described in sections 2.1 and 2.8, data are selected to only include times with well-mixed conditions and when background CH4 can be reliably, which are shown as black points in Figure 7. Diurnal cycles due to changing boundary layer height and synoptic variations due to frontal passages are apparent in the data. The data gap in early–mid December resulted from a leak in the sampling system that was diagnosed and repaired. The measured and predicted CH4 mixing ratios show similar temporal variations, indicating partial success of the model. However, the predicted signals do not always capture the large CH4 measurements, indicating some combination of errors in the a priori emission model (e.g., spatial pattern or limited resolution) and atmospheric transport (e.g., wind fields, boundary layer height).
3.2. Inferred Surface Emissions
 We compare the tower measurements and WRF-STILT simulations at WGC site during winter (October–December) 2007. Three analyses are reported here: (1) a linear analysis for total CH4 emissions; (2) a “source analysis” for the six CH4 source sectors; and (3) a “region analysis” for thirteen regions in CA. For the linear analysis, we employ a Chi-square linear regression analysis by assuming equal relative errors of 32% in both variables. For the “source analysis” and “region analysis,” the Bayesian analysis from equations (7) and (8) is applied. Note that the “region analysis” used the same a priori spatial distributions of CH4 emissions as the “source analysis,” and same total effective measurement errors of 32% are used in the following analyses.
3.2.1. Linear Regression Analysis
 Results of the regression analyses using California specific emission applied to the October through December 2007 period are shown in Figures 8a and 8b. Without Zi scaling (Figure 8a), the best fit slope between predicted and measured CH4 mixing ratios is 0.46 ± 0.07. After applying the Zi scaling to WRF-STILT (Figure 8b), the slope between predicted and measured CH4 is 0.73 ± 0.11. The change in slope between Figure 8a and Figure 8b demonstrates that scaling the PBL heights affects the predicted CH4 signals, and any residual uncertainty in PBL height should be considered as a source of uncertainty in the Bayesian analyses that follow. After the Zi scaling, the slope obtained in Figure 8b suggests that the actual emissions are higher than inventory estimates by a factor of 1.37 ± 0.21. We note that the normalized Chi-square value for Figure 8b is 1.17, suggesting that the estimated errors do not completely explain the residual variance in the differences between the predictions and measurements. CH4 signals based on Edgar 3.2 emissions are also simulated and compared with the tower measurements in Figure 8c, yielding a slope of 1.09 ± 0.14. This slope is roughly consistent (p > 0.1 in a t test) with the slope (0.92 ± 0.03) obtained by Kort et al.  in their comparison of measured and predicted CH4 signals using Edgar 3.2. However, the slopes obtained with the California specific (Figure 8b) and Edgar (Figure 8c) emissions are significantly different (p < 0.01), as might be expected given the large difference in the a priori emissions shown in Table 1. For the central California region, the total emission estimated by Edgar 3.2 is about 75% more than that estimated from California specific sources, which is roughly consistent with the difference (∼50%) of fitting slopes between Figure 8b and Figure 8c.
 To evaluate the effect of the well-mixed data selection criteria, we also examined the slopes obtained with a more stringent requirement that the difference between CH4 mixing ratio measured at 91 m and 483 m is less than 50 ppb. This subset of data are shown as triangles in Figure 8. Using the selection criteria of 50 ppb in Figure 8 obtains a slope of 0.86 ± 0.17, which is quite consistent with that obtained using the selection criteria of 100 ppb. The following analyses include data based on the 100 ppb selection criteria.
3.2.2. Bayesian Analysis
 The Bayesian “source” inverse analysis was carried out for the six source sectors for October through December 2007. As shown in Figure 9a, the a posteriori scaling factors for the crop agriculture (CP), landfill (LF), wetland (WL), petroleum (PL), and natural gas (NG) are not significantly different from unity (at 95% confidence). The scaling factor for livestock is 1.63 ± 0.22, suggesting the emissions from livestock are significantly (95% confidence) larger than the a priori inventory estimates. The Bayesian “region” inverse analysis of emissions from the 13 California regions is shown in Figure 9b. The a posteriori uncertainties are noticeably reduced relative to the a priori uncertainties only for regions 6, 7, and 8, which have a strong influence on the CH4 measurements either because the land surrounds the tower site (regions 6 and 8) or has a teleconnection through the prevailing wind (region 7). The a posteriori scaling factor for region 6 is 1.08 ± 0.06, indicating that the posterior emissions agree well with the a priori inventory estimates. Posterior scaling factors for region 7 and 8 are 1.55 ± 0.17 and 1.37 ± 0.15 respectively, indicating that the a posteriori emissions are greater than the a priori estimates for these two regions.
 After applying the scaling factors obtained from Bayesian analyses, the posterior predicted CH4 mixing ratios are compared with measurements in Figure 10. Figure 10a shows the comparison for results from the “source analysis” with measurements. Compared to Figure 8b (before inverse optimization), the fitting slope is closer to unity, and the normalized Chi-square value is slightly reduced from 1.17 to 1.11. This suggests that the inverse optimization has slightly improved the agreement between the measured and predicted CH4 signals but that on order 10% of the variance remains unexplained. It is possible that the apparent underestimation of the errors may be due to positive correlation between the error sources that we assumed independent. Similar results are obtained for the region analysis, as shown in Figure 10b. In both cases, the slopes after optimization are still slightly less than unity, likely because of the weight on the a priori scaling factors. We note that relaxing the a priori uncertainties on the scaling factors from 30% to 50%, allows the optimization to adjust the posterior scaling factors further from their a priori values.
4. Discussion and Conclusions
 Here we discuss the impact of error in PBL height on uncertainty in estimated CH4 emissions, the implications of our results on estimated CH4 emissions from Central California, and conclude with recommendations for additional measurement sites that would help quantify CH4 emissions from more regions in California.
 First, the results of this work highlight the need for careful estimation and minimization of errors in the transport model. Our work is really only a first step in this regard because we have only evaluated wind and PBL height errors for one site, albeit at the location where the CH4 measurements were made. The comparison between the radar profiler measurements and WRF-STILT predictions of PBL height show a systematic overestimation in the WRF-STILT predictions, while the sensitivity test shows that predicted CH4 emission estimates are sensitive to PBL height. The error in WRF-STILT predictions of PBL height may be a result of imperfect land surface parameterization in WRF that does not account for a suppression of PBL height in the Central Valley. Possible causes for overestimation of PBL height include the Pacific low over California's interior and low ratios of sensible to latent heat (Bowen ratios) driven by agricultural irrigation as shown in recent model studies of California [Kueppers et al., 2007; Lobel and Bonfils, 2008]. Because of the limited amount of PBL height data, the present work should be considered a first step toward a more comprehensive analysis employing profiler data from additional profiler sites and over longer periods. We expect that this effort will substantially improve the fidelity of the WRF-STILT PBL predictions and hence accuracy of GHG emission inversions.
 Second, the linear regression estimates suggest that October–December CH4 emissions from Central California are estimated to be 37 ± 21% higher than the annually averaged California specific a priori inventories. Examining the source sector results, the increase in overall emissions is largely due to the 63 ± 22 (1σ)% increase in estimated emissions from livestock. State-wide a priori livestock emission are 9.7 MMT CO2eq (see Table 1), which includes 5.6 MMT CO2eq from dairies and 4.1 MMT CO2eq from other cattle. Scaling the a priori CH4 emissions from dairies suggests that actual dairy emissions are 9.1 ± 1.3 MMT CO2eq. This result is nominally consistent with or slightly less than the results of a recent study by Salas et al. , which estimated total CH4 emissions from dairies in CA to be approximately 9.8 MMT CO2eq. We note that the source sector and regional analyses are consistent with each other in that CH4 emission from region 8, which is dominated by livestock, shows a large and statistically significant increase relative to the a priori inventory. Some other sources also showed smaller but not significant differences from inventory estimates. For example, inferred CH4 emissions from crop agriculture are smaller than the annually averaged inventory, consistent to the expectation of higher CH4 emissions from the north-central Valley during the summer due to flooded rice agriculture [Salas et al., 2006]. Finally, the “region” analysis shows that emissions from regions 6, 7 and 8 are constrained by the measurements. This is because they either surround the tower (i.e., regions 6 and 8) or have a strong influence on air reaching the tower through prevailing winds from the Bay Area to the Sacramento Valley (i.e., region 7). This observation provides an insight into the spatial domain that can be effectively investigated with the tower measurements and suggests that a network of towers would be required to accurately constrain the multiple regions and air basins in California. In principle, measurements from multiple towers would also be combined in a larger inverse analysis to provide more stringent constraints on emissions from regions that influence several towers. We consider a model-based design of a dedicated tower network to be a natural extension of the work described here.
 We thank Dave Field, Dave Bush, Edward Wahl, and particularly Jon Kofler for assistance with installation and maintenance of the instrumentation at WGC, Edward Dlugokencky for advice and assistance in verifying the Picarro instrument performance at NOAA, John Lin and Steve Wofsy for generously sharing the STILT code and providing expert advice, Chris Potter and William Salas for sharing their models of CH4 emission for use as a priori estimates, Larry Hunsaker and Webster Tassat for providing the CARB estimates of landfill CH4 emissions, Ken Massarie for providing the global CH4 background data, and Susan James for assistance in running WRF on the LBNL-ASD computer cluster. We gratefully acknowledge NOAA Air Resources Laboratory (ARL) for the use of the HPSPLIT model underlying STILT and NCEP for the provision of the NARR meteorology. We also thank Jean Bogner, Nancy Brown, Eric Crosson, Guido Franco, Ling Jin, Ying-Kuang Hsu, Eileen McCauley, Tony VanCuren, James Wilczack, and three anonymous reviewers for valuable comments. This study was supported by the California Energy Commission (CEC) Public Interest Environmental Research Program and the Director, Office of Science, Office of Basic Energy Sciences of the U.S. Department of Energy under contract DE-AC02-05CH11231. The findings, views, and opinions presented in this paper do not necessarily represent the views and opinions of the California Energy Commission or the State of California.