the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Estimation of CH_{4} emission based on an advanced 4D-LETKF assimilation system

### Jagat S. H. Bisht

### Prabir K. Patra

### Masayuki Takigawa

### Takashi Sekiya

### Yugo Kanaya

### Naoko Saitoh

### Kazuyuki Miyazaki

Methane (CH_{4}) is the second major greenhouse gas after carbon dioxide
(CO_{2}) which has substantially increased during recent decades in the
atmosphere, raising serious sustainability and climate change issues. Here,
we develop a data assimilation system for in situ and column-averaged
concentrations using a local ensemble transform Kalman filter (LETKF) to
estimate surface emissions of CH_{4}. The data assimilation performance is
tested and optimized based on idealized settings using observation system
simulation experiments (OSSEs), where a known surface emission distribution
(the truth) is retrieved from synthetic observations. We tested three
covariance inflation methods to avoid covariance underestimation in the
emission estimates, namely fixed multiplicative (FM), relaxation-to-prior
spread (RTPS), and adaptive multiplicative. First, we assimilate the
synthetic observations at every grid point at the surface level. In such a
case of dense observational data, the normalized root mean square error
(RMSE) in the analyses over global land regions is smaller by 10 %–15 % in
the case of RTPS covariance inflation method compared to FM. We have shown that
integrated estimated flux seasonal cycles over 15 regions using RTPS
inflation are in reasonable agreement between true and estimated flux, with
0.04 global normalized annual mean bias. We then assimilated the column-averaged CH_{4} concentration by sampling the model simulations at Greenhouse Gases Observing
Satellite (GOSAT) observation locations and time for another OSSE. Similar to the
case of dense observational data, the RTPS covariance inflation method performs
better than FM for GOSAT synthetic observation in terms of normalized RMSE
(2 %–3 %) and integrated flux estimation comparison with the true flux. The
annual mean averaged normalized RMSE (normalized mean bias) in LETKF
CH_{4} flux estimation in the case of RTPS and FM covariance inflation is
found to be 0.59 (0.18) and 0.61 (0.23), respectively. The *χ*^{2} test
performed for GOSAT synthetic observations assimilation suggests high
underestimation of background error covariance in both RTPS and FM
covariance inflation methods; however, the underestimation is much higher
(*>*100 % always) for FM compared to RTPS covariance inflation
method.

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_{4}emission based on an advanced 4D-LETKF assimilation system, Geosci. Model Dev., 16, 1823–1838, https://doi.org/10.5194/gmd-16-1823-2023, 2023.

Methane (CH_{4}) is the second major greenhouse gas, after carbon dioxide
(CO_{2}), that has anthropogenic sources. According to the contemporary
record of the global CH_{4} budget, the total of all CH_{4} sources
ranged 538–593 Tg yr^{−1} during 2008–2017
(Saunois
et al., 2020). The primary natural sources are from wetlands
(∼ 40 %). The main anthropogenic CH_{4} emissions are from
microbial emissions associated with ruminants (livestock and waste), rice
cultivation, fugitive emissions (oil and gas production and use), and
incomplete combustion of biofuels and fossil fuels. The major fraction of
atmospheric CH_{4} sinks (range: 474–532 Tg yr^{−1}) occurs in the
troposphere by oxidation via reaction with hydroxyl (OH) radicals
(Patra,
et al., 2011; Saunois et al., 2020); other loss processes include oxidation
by soil and reactions with O^{1}D and Cl. The lifetime of CH_{4} in the
atmosphere is estimated to be 9.1 ± 0.9 years
(Szopa
et al., 2021).

Regional CH_{4} emissions can be estimated from CH_{4} concentration
fields and chemistry transport models using Bayesian synthesis approaches
based on inverse modeling techniques (e.g., Enting, 2002).
In such approaches, emissions are optimized on a coarse resolution (e.g., for
a limited number of predefined regions) mostly using surface-based
observations. CH_{4} concentrations are provided by the NOAA cooperative
air sampling network sites
(Lan et al., 2022)
and other networks by the World Data Centre for Greenhouse Gases (WDCGG)
website, hosted by the Japan Meteorological Agency. In recent years,
satellite measurements have been made by the Greenhouse Gases Observing
Satellite (GOSAT) or the TROPOspheric Monitoring Instrument (TROPOMI)
(Lorente et al., 2021),
covering the globe with fine spatiotemporal scales. GOSAT has provided an
extensive global observations of column CH_{4} concentrations since 2009
(Yoshida
et al., 2013). Some of the inverse modeling studies utilize the satellite
observations for CH_{4} flux estimation
(Zhang et al., 2021; Maasakkers et al.,
2016), but this requires enormous computational resources as a result of dealing with
more flux regions and more observations.

Grid-based CH_{4} flux optimization is also performed using adjoint
technique (4-D Var data assimilation) and an ensemble Kalman filter (EnKF) but
was limited to small sets of observations
(Houweling et
al., 1999; Meirink et al., 2008; Bruhwiler et al., 2014).
Bruhwiler et al. (2014) followed the EnKF method
of Peters et al. (2005) to estimate the
CH_{4} surface fluxes that utilizes an offline atmospheric chemistry tracer model (ACTM) framework. Techniques
such as 4-D Var and EnKF are important to estimate CH_{4} fluxes since
they can assimilate a large number of observations and manage high-resolution
fluxes. In the EnKF system, a flow-dependent forecast error covariance
structure is provided by ensemble model forecasts, while it does not need an
adjoint model, which makes it a simple but powerful tool for flux estimation.
One of the limitations of the EnKF method is the dependence of the resolution of
state vector on ensemble size, which can give spurious results if the number
of ensemble members is much smaller than the rank of the error covariance
matrix (Houtekamer and Zhang, 2016).

A local ensemble transform Kalman filter (LETKF) is a type of square-root EnKF that performs analysis locally in space
without perturbing the observations
(Ott et al., 2002,
2004; Hunt et al., 2007). LETKFs are computationally efficient since the
observations are assimilated simultaneously and not serially; it is simple to
account for observation error correlation. Miyazaki et al. (2011) and Kang et al. (2012) demonstrated the
implementation of LETKF data assimilation system by coupling an ACTM for
carbon-cycle research using atmospheric CO_{2} observations. It is also
extensively applied for the emission estimation of short-lived species using
satellite data
(Skachko
et al., 2016; Miyazaki et al., 2019; Sekiya et al., 2021). In this work, we
will estimate the CH_{4} fluxes using a LETKF data assimilation system.
Assimilation windows ranging from 6 h (Kang et al., 2012)
to several months (Bruhwiler et al., 2014) have been
used, depending on the desired time resolution of the estimated emissions,
which is often limited by the observational data density. The time frame
over which the system behaves linearly and in what time frame the
observations respond to the control variables, such as atmospheric
transport, as well as observation abundance, must also be taken into
consideration. Within an assimilation window, where and when the fluxes
would be constrained by specific observations is to be ascertained by the
correlation between ensemble prior fluxes and the ensemble CH_{4}
concentration simulation from a forward model (Liu et al.,
2016).

The main objective of this work is to develop an advanced 4-D data assimilation
system based on a LETKF that simultaneously estimates atmospheric
distributions and surface fluxes of CH_{4}. Observation system
simulation experiments (OSSEs) are conducted to assess
the performance of the LETKF since it is important to test the system against
the known emissions or the truth. The OSSE LETKF setup of top-down CH_{4} flux estimation using an online ACTM is an essential step before implementation in real in situ and satellite observation.

We briefly describe the LETKF in the application of CH_{4} flux
estimation, while detailed derivation of equations and code implementation
are given elsewhere (Hunt et
al., 2007; Miyazaki et al., 2011; Miyoshi et al., 2010). The notation used
here for LETKF formulation is adopted from Kotsuki et al. (2017). In the LETKF, the background ensemble (columns of matrix
**x**^{b}) in a local region evolved from a set of perturbed
initial conditions. The background ensemble mean, ${\stackrel{\mathrm{\u203e}}{x}}^{\mathrm{b}}$,
and its perturbation, **X**^{b}, are estimated from the
ensemble forecast as

where *m* indicates the ensemble size. The background error covariance
matrix **P**^{b} in the *m*-dimensional ensemble is defined
as

The analysis ensemble mean ${\stackrel{\mathrm{\u203e}}{x}}^{\mathrm{a}}$ is derived using
background ensemble mean ${\stackrel{\mathrm{\u203e}}{x}}^{\mathrm{b}}$ and ensemble perturbations
**X**^{b} as

where *H*, *Y*, *R*, and ${\stackrel{\mathrm{\u0303}}{P}}^{\mathrm{a}}$ denote the linear observation
operator, ensemble perturbation matrix in the observation space (*Y*≡*H**x*), observation error covariance matrix, and analysis error covariance
matrix in the ensemble space, respectively. The superscripts “o”, “b”, and
“a” denote the observations, background (prior), and analysis (posterior),
respectively. *w*^{a} defines the analysis increment (or
analysis weight) in observation space and is derived using the information
about observational increment
${y}^{\mathrm{o}}-H{\stackrel{\mathrm{\u203e}}{x}}^{\mathrm{b}}$. The analysis error
covariance matrix (${\stackrel{\mathrm{\u0303}}{P}}^{\mathrm{a}}$) in the *m*-dimensional ensemble
space is spanned by ensemble perturbation (Hunt et al., 2007) and defined
as

Finally, the analysis ensemble perturbations **X**^{a} at
the central grid point are derived such as

where $\mathit{\left\{}\right(m-\mathrm{1}){\stackrel{\mathrm{\u0303}}{P}}^{\mathrm{a}}{\mathit{\}}}^{\mathrm{1}/\mathrm{2}}$ is a multiple of the symmetric square root of the local analysis error covariance matrix in ensemble space and could be computed by a singular vector decomposition method. The LETKF solves the analysis update (Eqs. 3 and 5) at every model grid point independently by assimilating local observations within the localization cutoff radius.

We have applied a gross error check as a quality control to exclude
observations that are far from the first guess; the appropriate degrees of
the gross error check are also examined. Figure 1 shows the schematic
diagram of our LETKF setup with two ensemble members for three consecutive
assimilation cycles with an 8 d assimilation window. The analysis is
obtained at the midpoint time of the assimilation window (Fig. 1). The
analyzed (updated) surface flux is used for the next data assimilation cycle
starting from the midpoint time of the previous data assimilation window.
The state vector augmentation approach is used to estimate the atmospheric
CH_{4} surface flux (Kang et al., 2012; Miyazaki
et al., 2011).

Assimilation window size and ensemble members are chosen based on computational efficiency and estimation accuracy. A larger assimilation window means fluxes are constrained by more observations; however, it requires handling of large matrix optimization which is difficult in cases of dense observation and introduces sampling errors related to transport errors. In this study, a few sensitivity experiments were performed to demonstrate the choice of assimilation window length and ensemble size when GOSAT synthetic observations are assimilated in Sect. 4.2.

## 2.1 Covariance inflation

The LETKF data assimilation needs variance inflation to mitigate the underdispersed ensemble. We tested three methods: fixed multiplicative (FM), relaxation-to-prior spread (RTPS), and adaptive multiplicative covariance inflation.

The fixed multiplicative (FM) inflation method (Anderson
and Anderson, 1999) inflates the prior ensemble by inflating the background
error covariance matrix **P**^{b} defined in Eq. (2) as

where ${\mathbf{P}}_{\mathrm{tmp}}^{\mathrm{b}}$ represents the temporary
background error covariance matrix, which is inflated by a factor *γ*.

The other inflation methods used to prevent the reduction of ensemble spread are relaxation-to-prior perturbation (RTPP) (Zhang et al., 2004) and relaxation-to-prior spread (RTPS) (Whitaker and Hamill, 2012). The RTPP method relaxes the reduction of the ensemble spread after updating the ensemble perturbations, which blends the background and analysis ensemble perturbations as

where *α*_{RTPP} denotes the relaxation parameter of
the RTPP.

The RTPS inflation method relaxes the reduction of the ensemble spread by relaxing the analysis spread to prior spread as

where *σ* and *α*_{RTPS} denote the ensemble spread
and relaxation parameter of the RTPS, respectively. The range of the
*α*_{RTPS} parameter is bounded by [0, 1]. This study focuses
mainly on the FM and RTPS covariance inflation methods.

In addition, Miyoshi (2011) applied adaptive inflation by determining the multiplicative inflation factors at every grid point at every analysis step using the observation-space statistics derived by Daley (1992) and Desroziers et al. (2005).

where the operator “*<*⋅*>*” denotes the
statistical expectation and
$d={y}^{\mathrm{o}}-H{\stackrel{\mathrm{\u203e}}{x}}^{\mathrm{b}}$
(observation minus first guess), and **R** is the error observation covariance
matrix.

The impact of using the adaptive multiplication inflation method is discussed in the GOSAT synthetic observation assimilation experiments in Sect. 4.2.

## 2.2 MIROC4-ACTM

Model for Interdisciplinary Research on Climate, version 4.0 (MIROC4)-based
ACTM (hereafter referred to as MIROC4-ACTM)
(Patra et al., 2018; Bisht et al.,
2021) is used here for CH_{4} concentration simulations. The model
simulations have been performed at a horizontal grid resolution of
approximately 2.8 × 2.8^{∘} latitude–longitude (T42 spectral
truncations) and at hybrid vertical coordinates of 67 levels (Earth's surface to
0.0128 hPa; Watanabe et al., 2008).
Bisht et al. (2021) performed multi-tracer analysis and
demonstrated the importance of very well resolved stratosphere in the
MIROC4-ACTM that illustrates better extratropical stratospheric
variabilities and simulated tropospheric dynamical fields. The
meteorological fields in MIROC4-ACTM are nudged to the JMA reanalysis
(JRA-55) data (Kobayashi et al., 2015).

## 3.1 Construction of known surface emissions (truth)

Present OSSEs intend to develop basic tuning strategies before the actual
data to be assimilated, which is useful to accelerate the operational use of
real observations. The OSSE has been discussed here by exploiting the known
“truth”. The synthetic observations to be assimilated in the OSSE are
generated from nature runs which use bottom-up surface emission (true) data
to simulate global 3-D CH_{4} concentrations. The true surface CH_{4}
emissions are prepared on the monthly scale using anthropogenic and natural
sectors, minus the surface sinks due to bacterial consumption in the soil
(Chandra et al., 2021). The anthropogenic emissions
were obtained from the Emission Database for Global Atmospheric Research,
version 4.3.2 inventory (EDGARv4.3.2) (Janssens-Maenhout
et al., 2019), which includes the emissions from the major sectors, such as fugitive sources, enteric fermentation and manure management, and solid waste and
wastewater handling. The biomass burning emissions are taken from the Global
Fire Database (GFEDv4s)
(van
der Werf et al., 2017) and Goddard Institute for Space Studies emissions
(Fung et al., 1991). The wetland and rice emissions
are taken from the process-based model of the terrestrial biogeochemical
cycle, Vegetation Integrated Simulator of Trace gases (VISIT)
(Ito, 2019), which is based on Cao et al. (1996). Other natural emissions, such as those from the ocean, termites, and mud volcanoes are
taken from the TransCom-CH_{4} inter-comparison experiment
(Patra
et al., 2011). The total emissions are taken as the truth for the OSSEs, and
the concentration simulated by MIROC4-ACTM will be referred to as synthetic
observations.

## 3.2 Prior flux preparation and LETKF setting

Based on our understanding of CH_{4} inverse modeling, the uncertainty in
regional flux estimation is found to be 30 % or lower
(Chandra et al., 2021). Therefore, we attempted to
reproduce the true flux by starting with a prior flux that is lower than the true flux by
30 % (prior flux has the same seasonal cycles as true flux).
The MIROC4-ACTM is initialized with a spin-up of 3 years (2007–2009)
with prior flux distribution. The initial CH_{4} distribution on 1 January 2007 was taken from an earlier simulation of 27 years. An initial
perturbation with standard deviation of approximately 6 %–8 % spread is
applied to the a priori flux as the initial ensemble spread, whereas no
ensemble perturbation was applied to the initial CH_{4} concentration. The
sensitivity of the initial ensemble spread to CH_{4} flux estimation is
discussed in Sect. 4.2. The uncertainty to perturb prior fluxes is
generated based on random positive values with normal distribution. The
monthly scale prior emission is linearly interpolated at 6-hourly intervals
to be used in the MIROC4-ACTM simulation for data assimilation. This study
performs two LETKF data assimilation experiments. In these experiments, we
provided initial perturbation on a regional basis over land (53 different land
regions; Chandra et al., 2021), and at every grid over
ocean, no spatial error correlation between grid points is considered among
ensemble members. However, in Sect. 4.2.5, we also discussed the
sensitivity of CH_{4} data assimilation by providing the initial ensemble
spread at every grid by considering the horizontal spatial error correlation
between grid points among ensemble members, with a global mean correlation
of 20 %.

## 3.3 Experiment 1: synthetic dense observation formulation

The OSSE setting with very accurate and dense observation surface data is an
attempt to demonstrate that the data assimilation system works reasonably in the
estimation of the true surface flux. Errors in the estimated flux could
arise due to the insufficient ensemble size and also the implemented
inflation methods to overcome the undersampling, along with a simplified
forecast process of the emissions. In real data assimilation, there are
additional sources of potential errors, such as atmospheric transport and
inappropriate prior or observation uncertainties. In our OSSEs, CH_{4}
fluxes as mentioned in Sect. 3.2 are used as “true” fluxes in generating
synthetic observations (CH_{4} concentrations). In Experiment 1, the
simulated surface layer CH_{4} concentrations at each grid for the entire
globe were used as synthetic observations. We added a constant
measurement uncertainty of 5 ppb, which is typically achieved by the
present-day measurement systems (e.g.,
Lan et al., 2022).

In this study, the CH_{4} observations are assimilated by applying the
observation error covariance localization (Kotsuki et al.,
2020) to reduce the spurious spatial correlation due to a smaller ensemble
size than the degrees of freedom of the system ($\mathbf{R}\leftarrow \mathbf{R}\times \mathrm{exp}(-\frac{\mathrm{1}}{\mathrm{2}}\mathit{\{}{\left({d}_{\mathrm{h}}/{\mathit{\sigma}}_{\mathrm{h}}\right)}^{\mathrm{2}}+{\left({d}_{\mathrm{v}}/{\mathit{\sigma}}_{\mathrm{v}}\right)}^{\mathrm{2}}\mathit{\left\}}\right)$), where *d*_{h} and
*d*_{v} denote the horizontal distance (km) and vertical
difference (log[Pa]) between the analysis model grid point and observation
location. The tunable parameters *σ*_{h} and
*σ*_{v} are the horizontal localization scale (km)
and vertical localization scale (log[Pa]), respectively. Using the spatial
localization technique, we have estimated the CH_{4} flux for each grid by
choosing the CH_{4} observations that influence the grid point using
an optimal cutoff radius ($\simeq \mathrm{3.65}{\mathit{\sigma}}_{\mathrm{h},\phantom{\rule{0.125em}{0ex}}\mathrm{v}}$; Miyoshi et al., 2007) with a horizontal covariance
localization (*σ*_{h}) of 2200 km and a vertical
covariance localization (*σ*_{v}) of 0.3 in the
natural logarithmic pressure (log[Pa]) coordinate. The localization is
performed to improve the signal-to-noise ratio of ensemble-based covariance.
Numerous sensitivity experiments have been performed by varying the
horizontal and vertical localization length in order to obtain the optimized
CH_{4} flux that best compares with the truth. The LETKF assimilates the
observations within the specified radius to solve the analysis state at each
grid point independently (Liu et al., 2016; Kotsuki
et al., 2020). The state vector of the analysis includes the atmospheric
CH_{4} concentration, which is the prognostic variable of forecast model,
and the state vector is further augmented by the surface CH_{4} flux, which is
not a model prognostic variable. This augmentation enables the LETKF to
directly estimate the parameter through the background error covariance with
observed variables (Baek et al., 2006). The state vector
augmentation is implemented similar to that used by Miyazaki
et al. (2011). This approach analyzes CH_{4} flux during the analysis
step. The purpose of the simultaneous CH_{4} emission and concentration
optimization is to reduce the uncertainty of the initial CH_{4}
concentrations on the CH_{4} evolution during the assimilation window and
to maximize the observations potential (Tian et al.,
2014).

The atmospheric CH_{4} concentration is changed during both the analysis
and forecast steps. A challenge of this scheme is that the analysis
increment is added to the model state at each analysis step, without
considering the global total CH_{4} mass conservation in the model but
consistent with the observed local CH_{4} abundance.

In this case, the surface flux at every model grid point is analyzed with an 8 d
assimilation window during the year 2010 with 100 ensemble members. The
ensemble size and assimilation window are chosen based on the CH_{4} flux
estimation accuracy calculated by performing a sensitivity experiment for
the ensemble size (60, 80, and 100) and assimilation window (3 and 8 d),
respectively (not shown).

## 3.4 Experiment 2: synthetic satellite observation formulation

One way to address the real-world CH_{4} flux estimation problem is to
first make the OSSE dataset like real observations. In this OSSE,
we have assimilated synthetic column-averaged CH_{4} concentrations with a
coverage mimicking GOSAT satellite observations. We prepared a model-simulated column-averaged CH_{4} concentration (XCH_{4}) dataset that
is spatiotemporally sampled with GOSAT observations as follows:

where XCH_{4} is the column-averaged model-simulated CH_{4}
concentration. XCH_{4(a priori)} is a priori
column-averaged concentration. CH_{4(ACTM)} and
CH_{4(a priori)} are the CH_{4} profile from ACTM
and a priori, respectively. *h*_{j} is the pressure weighting function (*j* is
the vertical layer index), and *a*_{j} represents the averaging kernel matrix
for the column retrieval, which is the sensitivity of the retrieved total
column at the various (“*j*”) atmospheric levels. In the next step, we added
the same retrieval (XCH_{4}) error as GOSAT to the XCH_{4} (ACTM-simulated) to make the OSSE more realistic and then attempt to estimate the
true fluxes.

In this case, the CH_{4} flux has been estimated for each grid by choosing
the CH_{4} observation with a cutoff radius ($\simeq \mathrm{3.65}\phantom{\rule{0.125em}{0ex}}{\mathit{\sigma}}_{\mathrm{h},\mathrm{v}}$), with a horizontal covariance localization (*σ*_{h}) of 5000 km and a vertical covariance localization
(*σ*_{v}) of 0.35 in the natural logarithmic
pressure (log[Pa]) coordinate. The optimal horizontal and vertical
covariance localization values are chosen based on a trial-and-error method
(those with the best fits to estimate the CH_{4} flux when compared with truth). A long
cutoff radius has been chosen due to sparse observational coverage of GOSAT.
Covariance localization is necessary to remove long-range erroneous
correlations and for mitigating sampling errors in the ensemble-based error
covariance with a limited ensemble size
(Miyoshi et al., 2007; Greybush et
al., 2011; Kotsuki et al., 2020). The surface flux is analyzed at every
model grid point with an 8 d assimilation window and 100 ensemble members; they are chosen based on the sensitivity experiments discussed in Sect. 4.2.

## 4.1 Experiment with dense OSSEs

The time series of normalized root mean square error (RMSE; $\sqrt{{\sum}_{i=\mathrm{1}}^{n}({x}_{i}^{\mathrm{a}}-{x}_{i}^{\mathrm{t}}{)}^{\mathrm{2}}/n}/{\stackrel{\mathrm{\u0303}}{x}}^{\mathrm{t}}$, where ${x}_{i}^{\mathrm{a}}$ and ${x}_{i}^{\mathrm{t}}$ are
the analysis and true state at *i*th model grid point, *n* is the total number of
grid points, and ${\stackrel{\mathrm{\u0303}}{x}}^{\mathrm{t}}$ represents the mean of true flux) in the
analyses over the global landmass region is shown in Fig. 2. The normalized
global RMSE is calculated using FM and RTPS inflation methods (Fig. 2) after
assimilating synthetic observation at every grid (Sect. 3.4). Noteworthy
is that the experiment with the FM inflation method shows 10 %–15 % larger error
in estimating the atmospheric surface CH_{4} flux compared to the RTPS
inflation method. One of the reasons of the better RMSE using the RTPS inflation
method is the higher number of degrees of freedom provided by relaxation
(*α*_{RTPS}) in the ensemble spread (Eq. 8) that could
nudge the ensemble of CH_{4} concentrations towards observations. The
initial flux analysis spread using RTPS and FM is shown in the Supplement (Fig. S1) and shows larger initial analysis flux spread over
Brazil, tropical America, and Asia in RTPS inflation compared to the FM
inflation method. We performed numerous sensitivity tests with the RTPS inflation
method and found that uniform relaxation is not substantial for some of the
regions. Figure 2 shows the RMSE for FM, fixed RTPS (*α*_{RTPS}=0.4; applied globally, the optimized value is obtained
by manual fine-tuning), and conditional RTPS (*α*_{RTPS}=0.3–0.7 applied different *α*_{RTPS} values regionally by manual fine-tuning). In the case of conditional
RTPS, the optimal values of *α*_{RTPS}, i.e., 0.6,
0.3, and 0.7 for the regions south of 20^{∘} S, 20^{∘} S–20^{∘} N, and
north of 20^{∘} N, respectively, were obtained from data assimilation
sensitivity calculations with varying *α*_{RTPS} values for
the three regions separately to best match the true states. We find that the
conditional RTPS method improves the accuracy by ∼ 5 %
compared to fixed RTPS and 10 %–15 % compared to FM. In the following, we
discuss the results obtained using the conditional RTPS and FM inflation
methods.

We have also shown the RMSE (not normalized) of the surface flux in the Supplement (Fig. S2). The flux RMSE has been estimated globally for both the
inflation methods and also for the region south of 20^{∘} N (by considering only
those land grids which fall in the region south of 20^{∘} N; Fig. S2) for
comparative purposes. It was noticed that (Fig. S2 in the Supplement), above north of 20^{∘} N, the flux estimation error is higher,
specifically during spring–summer when CH_{4} emissions peak over most of
the northern hemispheric regions (Fig. 3). The high uncertainty during
spring–summer (Fig. S2) in the flux estimation over these regions could
appear due to the attenuation of surface observations as a result of active
vertical mixing. The RMSE during autumn (Fig. S2) is comparable in the case of
the global region and the region south of 20^{∘} N, which indicates that the RMSE is arising from southern
hemispheric regions, likely over Brazil, as it peaks during autumn (Fig. 3).

Figure 3 shows a regional total flux seasonal cycle comparison of the
estimated fluxes for 15 terrestrial regions with the cycles of the prior and true
fluxes. The estimated flux retrieved using RTPS inflation method over
different regions agrees well with that of the true flux. We intend to show the
capability of LETKF-estimated fluxes over these regions using surface
observations to mimic the true fluxes in our understanding of the terrestrial
biosphere CH_{4} cycle. These results are consistent with Fig. 2, with
an annual global normalized mean bias (${\sum}_{i=\mathrm{1}}^{n}({x}_{i}^{\mathrm{a}}-{x}_{i}^{\mathrm{t}})/{\sum}_{i=\mathrm{1}}^{n}\left({x}_{i}^{\mathrm{t}}\right)$) of −0.04. It can also be noticed from Fig. 3 that estimated
fluxes converge to true fluxes over most of the regions after about 2–3 months.

To see the degree of similarity in the flux distribution between the estimated and true fluxes, we show monthly mean spatial flux distribution for June and November in Figs. 4 and 5, respectively, along with the bias in the prior and estimated flux. As shown in Figs. 4 and 5, the general spatial patterns of the true flux are estimated well. These results suggest that our LETKF system is capable of reproducing continental spatial flux patterns by using such idealized dense surface observational data. However, some clear differences in flux estimation could be noticed from the FM and RTPS inflation method (Figs. 4 and 5); e.g., over the Eurasian and American continent, analysis with RTPS shows clear improvement compared to the FM covariance inflation method. We calculated the global mean normalized bias with the RTPS and FM covariance inflation method, which is found to be −0.04 and −0.11, respectively, over land regions, and this showed that RTPS significantly improved the flux estimation compared to the FM covariance inflation method.

## 4.2 Experiment by mimicking the real satellite observational dataset

In this section we discuss the LETKF flux estimation by assimilation of
GOSAT synthetic CH_{4} concentration observations. Figure 6 shows the
model-simulated mean XCH_{4} concentration sampled spatiotemporally with
GOSAT observations during January and July for the year 2010 (sampling
method discussed in Sect. 3.4). In this case we have shown different LETKF
sensitivity experiments, such as LETKF sensitivity to (1) FM, RTPS, and adaptive
multiplicative inflation; (2) the assimilation window; (3) the ensemble size; (4)
the *χ*^{2} test; and (5) the prior ensemble spread. In the LETKF sensitivity
experiments from 1–4, the initial ensemble spread employed a similar method to
Experiment 1, and conditional RTPS inflation method is used. A conditional RTPS
method is also used in Sect. 4.2.6 for CH_{4} flux estimation.

### 4.2.1 LETKF sensitivity to FM, RTPS, and adaptive multiplicative inflation

This study mainly emphasizes FM and RTPS inflation methods used in
CH_{4} LETKF data assimilation. The annual average normalized RMSE
(absolute bias) with RTPS and FM covariance inflation is found to be 0.59
(0.18) and 0.64 (0.22), respectively. The RTPS inflation method performs
better than the FM inflation method overall. In addition to RTPS inflation,
a sensitivity test is also performed using an adaptive multiplicative inflation
method.

In the adaptive inflation, we need to provide an initial multiplicative
inflation factor at the beginning of data assimilation cycle (Cycle 1 in
Fig. 1). Following the method of Miyoshi (2011), the multiplication inflation factor information calculated in the previous cycle (i.e., Cycle 1 in Fig. 1) is used for the next data assimilation
cycle at every grid point (Cycle 2 in Fig. 1). We perform two sensitivity
experiments. In the first (second) case, we provided 50 % (40 %) initial
inflation in the beginning of Cycle 1 (Fig. 1). The normalized RMSE in the
both the adaptive inflation sensitivity experiments is comparable (0.65,
Supplement Fig. S3a) till July, but from the beginning of
August, the RMSE increases exponentially in the first experiment. However, in
terms of the *χ*^{2} distribution, CH_{4} flux estimation with the first
sensitivity adaptive multiplicative inflation experiment (50 % initial
inflation case) is better than with the second sensitivity experiment (Supplement Fig. S3b; *χ*^{2} test described in Sect. 4.2.4). To
identify the regions of high estimated CH_{4} flux error, we have shown
the background error spread in CH_{4} flux estimation over 15 regions
(Supplement Fig. S3c) and found that the spread over west and southeast Asia rises exponentially post-July, which indicates the rise of estimated
CH_{4} flux error over these regions in the first sensitivity adaptive
multiplicative inflation experiment. Our analysis suggests that CH_{4}
flux estimation depends on the initial inflation factor provided in the
beginning of the data assimilation cycle (Cycle 1, Fig. 1) in the adaptive
multiplication method. Also, we need to be very careful to monitor the
background error spread evolution with time to estimate the CH_{4} flux
with adaptive inflation; the *χ*^{2} distribution analysis is not sufficient.

In the case of RTPP inflation, we found the parameter *α*_{RTPP} is very difficult to fine-tune due to its very high
sensitivity to estimating the CH_{4} flux. We fail to obtain an optimized
*α*_{RTPP} value to estimate the CH_{4} flux.
Whitaker and Hamill (2012) also demonstrated the better
accuracy of the LETKF meteorological data assimilation with RTPS compared to
the RTPP covariance inflation method. They found the RTPP method produces very large
errors if the inflation parameter exceeds the optimal value.

### 4.2.2 Assimilation window

The LETKF data assimilation window length determines the time span of the
observations assimilated in each assimilation cycle. We have shown the
sensitivity of two assimilation window size configurations, 3 and 8 d, in the Supplement Fig. S4. Our sensitivity experiments with
window size configurations show that the 8 d long assimilation window
estimates the CH_{4} flux with better accuracy (∼ 10 %)
compared to the 3 d assimilation window because more observational
information is incorporated into the system with the 8 d long assimilation
window. This study uses an 8 d assimilation window for CH_{4} LETKF data
assimilation.

### 4.2.3 Ensemble size

Figure 7a shows the RMSE using different ensemble members. The RMSE
stabilizes gradually as the ensemble size increases from 60 to 80 to 100
ensemble members. The ensemble size dependency of flux estimation suggests
the further scope of the improvement in flux estimation by increasing the
ensemble members. In this study we stick to 100 ensemble members due to high
computational cost while solving large covariance matrices. The larger error
in flux estimation in the case of column-averaged synthetic GOSAT CH_{4}
observations assimilation compared to dense observations (Fig. 2) is likely
due to the weaker constraint on surface fluxes provided by satellite
observations and sparse observations.

### 4.2.4 *χ*^{2} test

We have carried out a *χ*^{2} test for the evaluation of background error
covariance matrix (Miyazaki et al.,
2012). For the *χ*^{2} test, the innovation statistics are diagnosed
from the observation minus forecast (*y*^{o}−*H***x**^{b}), the
estimated error covariance in the observation space
(*H***P**^{b}*H*^{T}+**R**), and
the number of observations *k* as

Using this statistic, the *χ*^{2} is defined as follows:

The performance of the background error covariance matrix is determined based on
the high and lower value of *χ*^{2}. The *χ*^{2} value should converge to
1; a value higher (lower) than 1 indicates underestimation (overestimation)
of the background error covariance matrices. Our results suggest that the background error covariance matrix is highly underestimated in both RTPS and
FM covariance inflation methods (Fig. 7b). However, the *χ*^{2} values'
convergence towards 1 is better in the case of RTPS compared to the FM
covariance inflation method, which indicates the improved representation of
background errors and then more appropriate data assimilation corrections in
the case of the RTPS inflation method. The *χ*^{2} distribution starts
saturating after the month of March. Post-March analysis shows the
background error covariance matrix underestimation is much higher
(*>*100 %) in the case of FM compared to the RTPS covariance inflation
method.

### 4.2.5 CH_{4} LETKF sensitivity to the initial ensemble spread

A test case for CH_{4} LETKF data assimilation has been performed, where
the initial spread is provided by considering the initial perturbation on
each model grid with spatial error correlation between grid points among
ensemble members, with a global mean correlation of 20 %. In this case, we
found that the analysis fluxes are extremely sensitive to the initial
ensemble spread if prior fluxes are perturbed with more than 5 % prior
uncertainty. Therefore, we used initial ensemble perturbation with only
2 % prior uncertainty. Reducing the initial ensemble spread reduces the
CH_{4} flux estimation sensitivity (*>*60 %). However, it also
poses a challenge to mitigate the underdispersed background error
covariance matrix. We performed LETKF data assimilations in this case with
the RTPS covariance inflation method (*α*_{RTPS}=0.9
optimized value is used here uniformly) with an 8 d long assimilation window
and 100 ensemble members and calculated the normalized RMSE between the analysis
and true fluxes (Supplement Fig. S5). It is noteworthy that the
estimated error between the analysis and true fluxes (Fig. S5) with this setting
(grid-wise initial ensemble spread) is still larger (25 %) than the case
when the region-wise initial ensemble spread is used (Fig. 7a; 100 ensemble
size). It suggests that initial ensemble spreads among ensemble members
need to be meticulously chosen so that they best represent CH_{4} variability
among ensembles to estimate the CH_{4} flux.

Note that the OSSEs used in this study did not consider the effects of
model errors other than CH_{4} fluxes, such as model transport errors. In
real situations, model errors can have a substantial impact on flux
estimates
(Locatelli et al.,
2013), which needs to be taken into account in background covariances.
Therefore, the optimal data assimilation setting can differ between the
OSSEs presented in this study and real observation cases. Further efforts,
e.g., by conducting a more comprehensive OSSE that accounts for various
model errors and by performing various sensitivity calculations in real
cases, would provide an improved understanding of the optimal inflation
settings to improve CH_{4} flux estimates in following study.

### 4.2.6 Estimated CH_{4} flux analysis

Figure 8 shows the regional flux seasonal cycle comparison for the estimated fluxes over 15 terrestrial regions with the cycles of the prior and true fluxes. We have also shown assimilation results in the case of the FM inflation method in the Supplement (Fig. S6), which shows the flux estimation disagreement over more regions compared to the RTPS inflation method, e.g., for tropical and North America, the whole African continent, and Australia–New Zealand.

We have shown the GOSAT observations in Figs. 6 and S7. We found very marginal flux estimation improvement over Central
Africa after May (Fig. 8), which could be associated with the lower GOSAT
coverage over this region (Fig. 6). On the other hand, over northern Africa,
no improvement in flux estimation is found. In the case of dense OSSEs too (Fig. 3), we did not find satisfactory flux estimation over northern Africa, which
is most probably related to the insufficient initial spread among ensemble
members over this region (we used the same initial ensemble spread in both
OSSE cases). Over Europe, GOSAT observations are remarkably fewer,
specifically for the first few months (January–April; Supplement
Fig. S7). Therefore, the flux update over Europe would be influenced by the
observations from neighboring regions falling under the chosen cutoff radius
that are mainly in northern Africa, where the flux estimation itself not
satisfactory. It could also be noticed that the retrieval error added in
this OSSE case is high over Europe (September–October; Supplement Fig. S7) and its adjacent sea (Mediterranean Sea; June–August),
which could also affect the surface CH_{4} flux estimation.

Figures 9 and 10 show spatial patterns of the true and estimated fluxes by
assimilating the column-averaged CH_{4} concentrations during June and
November (Fig. 6). It may be noticed that the RTPS covariance inflation method
is more able to estimate the true flux pattern compared to the FM covariance
inflation method. The spatial pattern shown using the RTPS inflation method
emphasizes the positive and negative bias in the estimated flux (Figs. 9 and
10) but generally agrees with the flux seasonal cycle plots shown in Fig. 8.

Our LETKF CH_{4} data assimilation experiment by assimilating GOSAT
synthetic observation with the implementation of the advanced RTPS
covariance inflation method better estimates the time-evolving surface
CH_{4} fluxes compared to the FM covariance inflation method. The difficulty
to estimate the surface CH_{4} flux over a few regions may be overcome by
applying additional methodologies, such as the assimilation of surface
observations simultaneously and the use of information about the CH_{4}
flux climatology. A correction factor derived based on empirical
formulation that could use CH_{4} flux climatology information is needed
to apply to maintain the CH_{4} mass conservation. This could be
implemented by checking the simulated CH_{4} burden gain between years
in comparison with the observed CH_{4} growth rates.

In this study, we have introduced a 4D-LETKF data assimilation system that
utilizes MIROC4-ACTM as a forward model for CH_{4} flux estimation. This
study has extensively tested both FM and RTPS inflation methods for the
LETKF CH_{4} flux estimation. We have conducted two experiments to
demonstrate the ability of LETKF system to estimate the CH_{4} surface
flux globally. In Experiment 1, we have assimilated the synthetic dense
surface CH_{4} observations, while in Experiment 2, synthetic GOSAT
CH_{4} observations are assimilated. Based on the results of the
sensitivity tests using FM and RTPS inflation methods in Experiment 1, we
have found that RTPS inflation produces significantly less normalized RMSE
(10 %–15 %) compared to the FM inflation method. In Experiment 2, we discussed
LETKF parameters, such as different inflation techniques, ensemble size,
assimilation window, initial ensemble spread sensitivity, and *χ*^{2}
test. The ensemble size (this study uses maximum 100 ensemble members)
sensitivity test suggests that more ensemble members could help to
accurately represent the covariance matrix with a higher number of degrees of freedom. The
assimilation window sensitivity test shows that an 8 d assimilation
window reduces the normalized flux RMSE by about 10 % compared to a 3 d
assimilation window in the case of GOSAT synthetic observations assimilation.

Our approach of assimilation with RTPS inflation could provide a higher number of degrees
of freedom to fit the ensemble of CH_{4} concentrations to the observed
ones, resulting in the improved analyzed fluxes. The RTPS inflation method is
capable of obtaining reasonable flux estimates with a normalized annual mean
bias of 0.04 and 0.61 in the case of dense surface synthetic observations and
GOSAT synthetic observations, respectively. We demonstrated in our
sensitivity OSSE with synthetic GOSAT observations that, over
American and African continents and also over Australia–New Zealand, the
LETKF data assimilation with the FM inflation method does not show much
improvement in the true flux estimation, but the RTPS inflation method
reasonably estimates the true flux over most of these regions. One of the
reasons for better flux estimates with the RTPS inflation method is the drastic prevention of analysis spread. In the CH_{4} LETKF flux
estimation, the surface CH_{4} flux is not a prognostic state vector in the
ACTM, which results in the continuous decay of spread in analysis steps.
The RTPS inflation method could mitigate such an underdispersed spread problem.
This study finds that spatially homogeneous relaxation is not sufficient. It
needs to be fine-tuned and applied conditionally.

The sensitivity of LETKF CH_{4} flux estimation to the initial ensemble spread
needs to be carefully dealt with when applied to real data assimilation
system. A future OSSE with an additive covariance inflation technique could be
interesting while applied with the RTPS inflation method for CH_{4} LETKF data
assimilation since in additive covariance inflation, initial estimated flux
error cannot propagate. The state vector augmentation technique used here
updates the flux after each data assimilation cycle, but it does not conserve
the total atmospheric CH_{4} amount, which is one of the limitations of
this work. A correction factor needs to be implemented to conserve the total
atmospheric CH_{4} amount after completion of a few data assimilation
cycles. We have not accounted for the transport error due to meteorological
fields in this work
(Patra
et al., 2011); in the case of real observation data assimilation, a week-long
window may introduce transport errors in CH_{4} analysis because of
the nonlinear growth of ensemble perturbations.

The LETKF source codes can be accessed from https://doi.org/10.5281/zenodo.7127658 (Bisht et al., 2022a). All the scripts for running the
LETKF data assimilation software and the input and output result data files are
available at https://doi.org/10.5281/zenodo.7098323 (Bisht et al., 2022b). The CH_{4}
ACTM simulation module coupled with MIROC4-AGCM can be accessed from
https://doi.org/10.5281/zenodo.7118365 (Bisht et al., 2022c). The source code of
MIROC4-AGCM is archived at https://doi.org/10.5281/zenodo.7274240 (Patra et al., 2022) with restriction because of the
copyright policy of the MIROC developer community. This work did not contribute to the MIROC4 source code development.

The supplement related to this article is available online at: https://doi.org/10.5194/gmd-16-1823-2023-supplement.

The LETKF data assimilation experiments were designed by JSHB. PKP, MT, and
TS helped to set up the LETKF code on MIROC4-ACTM for CH_{4} data assimilation.
The manuscript was prepared by JSHB, and analysis interpretation input and
feedback were provided by PKP, TS, and KM. All coauthors, KM, TS, PKP, NS, MT, and
YK, contributed to the writing and revision of the paper.

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 in published maps and institutional affiliations.

We acknowledge Ryu Saito for the initial setup of the LETKF code on MIROC4-ACTM for CH_{4}. We also thank Takemasa Miyoshi for his LETKF scheme, which served as the basis for the development of our data assimilation system.

This research has been supported by the Environment Research and Technology Development Fund (grant no. JPMEERF20182002) of the Environmental Restoration and Conservation Agency of Japan and the GOSAT-GW project fund.

This paper was edited by Shu-Chih Yang and reviewed by two anonymous referees.

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_{4}fluxes using an advanced 4D-LETKF method. The system was tested and optimized using observation system simulation experiments (OSSEs), where a known surface emission distribution is retrieved from synthetic observations. The availability of satellite measurements has increased, and there are still many missions focused on greenhouse gas observations that have not yet launched. The technique being referred to has the potential to improve estimates of CH

_{4}fluxes.

_{4}fluxes using an advanced 4D-LETKF method. The system was tested...