Canada has major sources of atmospheric methane (

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Atmospheric methane (

Canada has both natural and anthropogenic

Canada's anthropogenic

Environment and Climate Change Canada (ECCC) has been expanding the GHG monitoring program over the past decades across Canada. These observations have been used in regional inversion studies to estimate

This section provides a brief description of the atmospheric

The ECCC atmospheric measurement sites used in this study.

Measurement sites. North, West, East, and South indicate the four subregions defined in Sect.

This study utilized the records of continuous atmospheric

The ECCC continuous measurement of atmospheric

Atmospheric

Time series of atmospheric

After the initial quality control, all the atmospheric

In this study, we used the Bayesian inversion approach to estimate the regional

The Bayesian inversion optimizes the scaling factors of posterior fluxes by minimizing the mismatch between modelled and observed mixing ratios with constraints and given uncertainties using the cost function (

The posterior error covariance matrix,

We optimize the total

In an LPDM, air-following particles travel backward from the measurement location at a given initiation time (corresponding to the time of observation) and provide the relationship between surface fluxes and atmospheric mixing. This relationship is called footprint, source–receptor relationship, or flux sensitivity. To estimate the transport model errors in the flux estimate, three different models were employed in this study, combining two different LPDMs – FLEXible PARTicle dispersion model (FLEXPART) (Stohl et al., 2005) and Stochastic Time-Inverted Lagrangian Transport Model (STILT) (Lin et al., 2003; Lin and Gerbig, 2005) – and three different meteorological datasets. These three model setups are here named FLEXPART_EI, FLEXPART_JRA55, and WRF-STILT. FLEXPART_EI is FLEXPART v8.2 driven by the European Centre for Medium-Range Weather Forecasts (ECMWF) ERA-Interim (Dee et al., 2011; Uppala et al., 2005), FLEXPART_JRA55 is FLEXPART v8.0 driven by the Japanese 55-year Reanalysis (JRA-55) from the Japanese Meteorological Agency (JMA) (Kobayashi et al., 2015), and WRF-STILT is STILT driven by the Weather Research and Forecasting (WRF) model (e.g., Hu et al., 2019). The WRF-STILT footprints used in this study were provided by the NOAA CarbonTracker-Lagrange project (CT-L;

LPDMs simulate surface contributions for a certain period prior to the measurements at sites by air-following particles. In this study, at the endpoints of the particles after 5

Prior fluxes used in this study. Eight prior flux scenarios were made as combinations of two anthropogenic fluxes (EDGAR and ECAQ) and four wetland fluxes (WetCHARTs, GCPwet, CLASSICdiag, and CLASSICprog). For other natural fluxes, the same prior datasets were used in all the scenarios.

There are many factors governing the transport-model-simulated concentrations at the measurement sites used in this study, such as the spatial distribution of emissions and meteorological conditions, including winds and atmospheric stability. Since we are focusing on the synoptic variability in our observations, these are the results of the regional emissions (typically within the synoptic spatial scale of

We considered eight scenarios of prior emissions, combining four different wetland fluxes and two anthropogenic emission inventories (Table

The first wetland ensemble model, WetCHARTs, derives wetland

Wetland and anthropogenic prior

The two sets of anthropogenic

Subregion masks for inversion and the ECCC atmospheric measurement sites. Canada is divided into

The

The regional domain of interest for this study is Canada. We set up two subregion masks for Canada, mainly based upon climate zone with a provincial–territorial division and industrial activities also considered (see Fig. S2 for the Canadian provinces and territories). Outside Canada was treated as one outer region. The first mask consists of four subregions: North, West, East, and South (defined in Fig.

The sensitivity of the flux estimation to the number of subregions was examined; increasing the spatial resolution to six subregions revealed some model instability problems. With the limited observational constraints during this study period, subregions with weak prior fluxes or lack of measurement sites demonstrated larger uncertainties in estimated fluxes, with frequent unrealistic negative fluxes. Similar relations in resolving power of inversions were reported in Ishizawa et al. (2019) and Chan et al. (2020). Therefore, this study focuses on the inversion results of the four subregions and the two larger subregions.

In our model testing, the statistical Bayesian inversion model worked well if the basic model assumptions were satisfied. The important assumptions are (1) no transport errors and (2) a large dataset for robust statistics. We found that the main reason for the negative posterior fluxes in our model is transport errors (the inversion model yields the best statistical fit of the observations without accounting for transport biases). For atmospheric transport with random errors (unbiased), the model still works well if there are sufficient constraining data (“observations”) to allow the statistical model to robustly estimate the scaling factors. Imposing positive flux constraints (usually for negative solutions resulting from a scarcity of constraining data; e.g., Michalak and Kitanidis, 2003) does not appear to address the problem of transport biases. Positive flux constraints or imposing non-negativity constraints on the scaling factors could violate the statistical assumptions in our linear Bayesian inverse model, namely linearity and normality.

There are inversion studies doing grid-scale inversions (using non-zero off-diagonal covariance constraints) to address the aggregation error issue (e.g., Gourdji et al., 2012; Hu et al., 2019; Thompson et al., 2017). These grid-scale inversions are limited by the lack of observations and systematic transport errors (Gourdji et al., 2012). Their discussions about them are typically on the aggregated fluxes in larger regions and temporally averaged estimated features. This is consistent with our inversion model sensitivity analysis; we found that inversion flux errors from the transport model errors appear larger than aggregation errors in our case. For example, in the worst-case scenario, the difference of the inversion results calculated by different transport models could be greater than 100 %.

In this study, we tried to reduce the effects of transport model biases and insufficient observations for statistical Bayesian inversion analysis. We used multiple transport models to lessen the effects of individual transport model biases and to provide flux uncertainty estimates associated with the choice of transport model. This inversion model employed a limited number of subregions to allow the abundant observations to provide sufficient constraints to obtain flux estimates that appear robust and positive without the added model complications like positivity constraints and non-zero off-diagonal covariance constraints.

Flux estimations by inversion models can be complex: the flux estimate uncertainties depend on the tracer bio-geochemical characteristics; quantity and quality of the observations; and model formulation, setup, and assumptions. Having a wide range of models (including grid point inversion, non-negative constrained inversion, and multi-transport with abundant observational constraint inversion used here) could be helpful in understanding the strengths and weaknesses of inversion modelling.

Diagram of inversion experiment settings in this study. Eight prior-emission scenarios out of four wetland fluxes and two anthropogenic fluxes are applied to three different transport models. These flux–transport combinations yield the 24 experiments as listed in Table S1. The experiments with a 4-subregion mask and 12 observation sites are conducted as the reference inversion (Inv_4R12S). The experiments with a 2-subregion mask and 12 observation sites (Inv_2R12S) or 2 sites (Inv_2R2S) are performed as additional inversions. The mask maps are defined in Fig.

Figure

The posterior

Here, considering the possible winter wetland

Next, these annual fluxes could be expanded in terms of the fraction,

If

Monthly posterior

The monthly posterior

The subregion West (on the south side of the subregion North; see Fig.

Figure

Trend of estimated yearly

In Sect.

To investigate the presence of trends over 2007–2017, the mean annual posterior

Comparing all the inversion results for long-term trends for the 11 years (Fig.

Our result of no significant long-term trend for the national total

A model–data comparison of atmospheric mixing ratios at the measurement sites is commonly employed to evaluate the posterior fluxes. Figure S7 shows the model–data comparison by measuring the mean biases and correlation coefficients between the simulated and observed mixing ratios at each site in all 24 experiments for the reference inversion Inv_4R12S. The results of the simulated prior mixing ratios are overall dependent on the prior fluxes and transport models. The simulated posterior mixing ratios show an improvement in matching with the observations at most of the sites, except ESP. Also, DWS exhibits a notable transport model dependency. The FLEXPART_JRA55 cases show larger biases than the other transport model cases. This might be related to the resolution of the driving meteorological data of FLEXPART_JRA55 (1.25°

In this study, we explored another type of data for posterior flux evaluation. There is flux information in the observed mixing ratio difference (or gradient) between sites. Fan et al. (1998) noted that the downwind and upwind mixing ratios' difference for a given region should reflect the source/sink strength within the region. For example, for

For the case of the mixing ratio difference (

For reference, the mixing ratio differences from two other global inverse models (CT-

For the posterior north–south differences

Overall, the mixing ratio differences over these larger spatial scales are useful as evaluation or verification data for the model results. Our regional inverse model shows better agreement with observed mixing ratio differences than the global inverse models examined. Some of the issues could be due to the differences between global and regional cost functions, the mixture of fluxes in each basis region, and the number of observations for the global inversions. More work remains to understand the differences among the models tested here.

As presented in the previous sections, Sect.

Seasonal cycles of

Dependency of mean posterior monthly

Figure

In contrast to the reduced posterior summer fluxes, the posterior fluxes are higher during the cold winter season than the prior fluxes in both East and West. The presence of higher fluxes in East with little anthropogenic fluxes suggests the wetland emission in the winter is higher than the ecosystem model results used as priors. Also, our winter flux results are not consistent with the previous regional inversion results (e.g., Miller et al., 2014; Thompson et al., 2017), which do not show any large winter fraction (

Flux–temperature relationship in the summer season. Fluxes are the anomalies of estimated monthly fluxes from the 6-year (2012–2017) monthly mean fluxes. Temperature is the regional monthly anomaly from the same 6-year (2012–2017) monthly mean temperature. The summer season is defined as July and August for North and July to September for the remaining subregions. Closed green circles are the ensemble mean flux anomalies, error bars denote the standard deviation (SD), and open green circles are the anomalies of 24 individual experiments. The yellow circles are from the estimated fluxes, which are excluded because the nearby forest fires apparently affected the flux estimates.

As presented in Sect.

Thus, we examined the statistical correlation between flux anomalies and temperature anomalies by subregion for the period of 2012 to 2017. For this, surface air temperature anomalies from NCEP reanalysis (Kalnay et al., 1996) are aggregated to the respective subregions. The correlations between the monthly flux anomalies and monthly surface air temperature anomalies for the summer months are shown in Fig.

In South, no robust correlation is found between the posterior fluxes and climate on seasonal and monthly scales (Fig.

The correlation of inter-annual variability in posterior

For comparison, the same analysis was done for the different prior fluxes with inter-annual variations (WetCHARTs, CLASSICdiag, CLASSICprog) used in this study, and the results are shown in Fig. S12. Only CLASSICdiag in East shows a positive temperature dependence (

We also examined the correlation between flux anomalies with the precipitation anomalies with no lag to a 2-month lag, but no significant correlation was found (

Figure

Estimated mean total

Some spatial differences can be seen in this study compared to the priors and global inverse models. At the subregional level, the mean prior flux distribution shows larger emissions in East than West (the spatial distributions of the prior scenarios are shown in Fig. S13). Similarly, global inversions (CT-

As a first attempt of total emission breakdown into natural and anthropogenic sources with the scheme presented in Sect.

Next, we explored an alternative approach assuming that the natural

As seen in Sect.

Mean spatial distribution of prior total flux (top), posterior total flux (middle), and the difference between posterior and prior (bottom). Posterior emissions partitioned into natural and anthropogenic sources by subregion, along with the respective prior-emission means and ranges (min–max), are shown on the left and right sides.

The resultant mean natural and anthropogenic

One assumption in the flux partition analysis is that the posterior anthropogenic fluxes for the subregions North and East are the same as their priors in Eqs. (

Results for cold-season natural

The partition of total

Seasonal cycles of normalized diurnal amplitude and standard deviation (SD) of observed atmospheric

The monthly mean diurnal amplitude at each measurement site was obtained as follows. Firstly, we calculated a normalized diurnal cycle, defining the mean afternoon mixing ratio over the local times between 14:00 and 16:00 LT as a reference. Secondly, the individual normalized diurnal cycles were averaged by month over the measurement periods (Fig. S14). Then, we obtained the monthly mean diurnal amplitude as a maximum mixing ratio difference from the respective reference afternoon mixing ratio during 24

For the baseline (coastal) sites ALT, ESP, and WSA with negligible

For the remaining arctic and boreal forest sites (INU, BCK, CHL, CHM, CPS, LLB, ETL, FSD), the strong summer diurnal cycles of mixing ratios are clearly exhibited with amplitudes of

In this study, we estimated the

Sensitivity experiments comparing different subregion masks (up to six subregions) were done to examine the variability in the posterior flux estimates. Results indicate that, with the set of 12 observation sites, the inverse model yields more stable and physical results (with no unphysical negative fluxes) for the 4-subregion mask setting (the reference inversion). The earlier period (2007–2011) with fewer measurement sites (without BCK or INU) has more variability in the flux estimates and unphysical negative posterior fluxes, primarily in western Canada.

The reference inversion experiment ensemble mean estimate of total

The reference inversion results for 2012–2017 were analyzed for other physical characteristics including the temporal, spatial, and statistical properties as well as a possible relationship to climatological forcing. Compared to other inversion studies, some notable results in our flux estimates include a quantifiable amount of winter wetland

The measurement network across the nation is essential to improve our ability not only to quantify how Canada's natural

The ECCC observations are available from the World Data Centre for Greenhouse Gases (WDCGG) at

The supplement related to this article is available online at:

MI and DC designed the research. MI conducted all inversions and data analysis, wrote the initial draft, and edited it together with DC. DW led the ECCC GHG measurement program. MI, DC, and EC provided and processed footprint data for the inversion. JRM and VKA provided the CLASSIC wetland

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

We are grateful to the NOAA CarbonTracker-Lagrange (CT-L) program for providing the WRF-STILT footprint data of Canadian sites for our inversion study. We acknowledge the Global Carbon Project

This paper was edited by Robert McLaren and reviewed by two anonymous referees.