Time-series of GhG emissions estimation for the learning of inverse modeling

2021 ◽  
Author(s):  
Carlos Gómez-Ortiz ◽  
Guillaume Monteil ◽  
Marko Scholze
Keyword(s):  
Time Series ◽  
Inverse Modeling ◽  
Fossil Fuel ◽  
Transport Model ◽  
A Priori ◽  
Ghg Emissions ◽  
Atmospheric Concentration ◽  
Training Tool ◽  
Sources And Sinks ◽  
Atmospheric Inversions

<p>Inverse modeling is a commonly used method to infer greenhouse gases (GhGs) sources and sinks based on their observed concentrations in the atmosphere. It is a Bayesian framework and requires a priori fluxes of all the evaluated sources and sinks, atmospheric observations, an atmospheric transport model that relates the observations to the a priori fluxes and the uncertainties of both fluxes and observations. Various techniques exist to solve the inversion.</p><p>Atmospheric inverse modeling is and will become even more important in the future quantification of GhGs to monitor the compliance of the Nationally Determined Contributions (NDCs) under the Paris Agreement. Therefore, the scientific and educational communities are becoming more interested in using atmospheric inversions and this has risen a necessity of creating tools that facilitate understanding as well as training in these techniques.</p><p>Quantifying anthropogenic GhG emissions, such as CO<sub>2</sub> from fossil fuel burning or CH<sub>4 </sub>from human activities, from atmospheric concentration observations is difficult since the carbon from all sources, both natural and anthropogenic, is mixed in the atmosphere, making it necessary to use other signals or tracers to separate anthropogenic emissions from natural sources. For fossil fuel CO<sub>2</sub> emissions radiocarbon (<sup>14</sup>CO<sub>2</sub>) is an excellent emission tracer because, due to its radioactive decay (~ 5000 years), it cannot be found in fossil fuels, which have been deposited millions of years ago as organic material. We have developed a Jupyter Notebook based on Python for the quantification of multi-tracer GhGs fossil fuel emissions and its isotopes. The notebook solves for the emissions by applying atmospheric inversions within a practical two-box model. The inverse modeling notebook is based on the analytical maximum a posteriori (MAP) solution of the Bayes’ theorem and allows to assess the error in the state vector and its uncertainty.</p><p>This basic but powerful notebook is meant to be an educational and training tool for university students and new researchers in the field as well as for researchers interested in the estimation of long-term (>centennial) time-series of GhG emissions since it is built as a modular algorithm to be easily modified, coupled or expanded to other approaches or models depending on the application. The notebook was initially developed for the inverse modeling of CO<sub>2</sub> and <sup>14</sup>CO<sub>2</sub> simultaneously and it is being expanded for additional GHG such as CH<sub>4</sub> and <sup>13</sup>CH<sub>4</sub>.</p>

scholarly journals An attempt at estimating Paris area CO<sub>2</sub> emissions from atmospheric concentration measurements

2014 ◽  
Vol 14 (7) ◽  
pp. 9647-9703 ◽  
Author(s):  
F. M. Bréon ◽  
G. Broquet ◽  
V. Puygrenier ◽  
F. Chevallier ◽  
I. Xueref-Rémy ◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Transport Model ◽  
Atmospheric Transport ◽  
Current System ◽  
A Priori ◽  
Temporal Correlation ◽  
Atmospheric Concentration ◽  
Sources And Sinks ◽  
Paris Area ◽  
Concentration Measurements

Abstract. Atmospheric concentration measurements are used to adjust the daily to monthly budget of CO2 emissions from the AirParif inventory of the Paris agglomeration. We use 5 atmospheric monitoring sites including one at the top of the Eiffel tower. The atmospheric inversion is based on a Bayesian approach, and relies on an atmospheric transport model with a spatial resolution of 2 km with boundary conditions from a global coarse grid transport model. The inversion tool adjusts the CO2 fluxes (anthropogenic and biogenic) with a temporal resolution of 6 h, assuming temporal correlation of emissions uncertainties within the daily cycle and from day to day, while keeping the a priori spatial distribution from the emission inventory. The inversion significantly improves the agreement between measured and modelled concentrations. However, the amplitude of the atmospheric transport errors is often large compared to the CO2 gradients between the sites that are used to estimate the fluxes, in particular for the Eiffel tower station. In addition, we sometime observe large model-measurement differences upwind from the Paris agglomeration, which confirms the large and poorly constrained contribution from distant sources and sinks included in the prescribed CO2 boundary conditions These results suggest that (i) the Eiffel measurements at 300 m above ground cannot be used with the current system and (ii) the inversion shall rely on the measured upwind-downwind gradients rather than the raw mole fraction measurements. With such setup, realistic emissions are retrieved for two 30 day periods. Similar inversions over longer periods are necessary for a proper evaluation of the results.

scholarly journals Characterizing uncertainties in atmospheric inversions of fossil fuel CO<sub>2</sub> emissions in California

2019 ◽  
Vol 19 (5) ◽  
pp. 2991-3006 ◽  
Author(s):  
Kieran Brophy ◽  
Heather Graven ◽  
Alistair J. Manning ◽  
Emily White ◽  
Tim Arnold ◽  
...  
Keyword(s):  
Fossil Fuel ◽  
Transport Model ◽  
Atmospheric Transport ◽  
Temporal Distribution ◽  
Potential Effect ◽  
Spatial And Temporal Distribution ◽  
Transport Models ◽  
Total State ◽  
Pseudo Data ◽  
Atmospheric Inversions

Abstract. Atmospheric inverse modelling has become an increasingly useful tool for evaluating emissions of greenhouse gases including methane, nitrous oxide, and synthetic gases such as hydrofluorocarbons (HFCs). Atmospheric inversions for emissions of CO2 from fossil fuel combustion (ffCO2) are currently being developed. The aim of this paper is to investigate potential errors and uncertainties related to the spatial and temporal prior representation of emissions and modelled atmospheric transport for the inversion of ffCO2 emissions in the US state of California. We perform simulation experiments based on a network of ground-based observations of CO2 concentration and radiocarbon in CO2 (a tracer of ffCO2), combining prior (bottom-up) emission models and transport models currently used in many atmospheric studies. The potential effect of errors in the spatial and temporal distribution of prior emission estimates is investigated in experiments by using perturbed versions of the emission estimates used to create the pseudo-data. The potential effect of transport error was investigated by using three different atmospheric transport models for the prior and pseudo-data simulations. We find that the magnitude of biases in posterior total state emissions arising from errors in the spatial and temporal distribution in prior emissions in these experiments are 1 %–15 % of posterior total state emissions and are generally smaller than the 2σ uncertainty in posterior emissions. Transport error in these experiments introduces biases of −10 % to +6 % into posterior total state emissions. Our results indicate that uncertainties in posterior total state ffCO2 estimates arising from the choice of prior emissions or atmospheric transport model are on the order of 15 % or less for the ground-based network in California we consider. We highlight the need for temporal variations to be included in prior emissions and for continuing efforts to evaluate and improve the representation of atmospheric transport for regional ffCO2 inversions.

scholarly journals The importance of transport model uncertainties for the estimation of CO<sub>2</sub> sources and sinks using satellite measurements

2010 ◽  
Vol 10 (20) ◽  
pp. 9981-9992 ◽  
Author(s):  
S. Houweling ◽  
I. Aben ◽  
F.-M. Breon ◽  
F. Chevallier ◽  
N. Deutscher ◽  
...  
Keyword(s):  
Transport Model ◽  
Atmospheric Transport ◽  
A Priori ◽  
Model Errors ◽  
Tracer Transport ◽  
Satellite Measurements ◽  
Transport Models ◽  
Sources And Sinks ◽  
The Impact ◽  
Ppm Level

Abstract. This study presents a synthetic model intercomparison to investigate the importance of transport model errors for estimating the sources and sinks of CO2 using satellite measurements. The experiments were designed for testing the potential performance of the proposed CO2 lidar A-SCOPE, but also apply to other space borne missions that monitor total column CO2. The participating transport models IFS, LMDZ, TM3, and TM5 were run in forward and inverse mode using common a priori CO2 fluxes and initial concentrations. Forward simulations of column averaged CO2 (xCO2) mixing ratios vary between the models by σ=0.5 ppm over the continents and σ=0.27 ppm over the oceans. Despite the fact that the models agree on average on the sub-ppm level, these modest differences nevertheless lead to significant discrepancies in the inverted fluxes of 0.1 PgC/yr per 106 km2 over land and 0.03 PgC/yr per 106 km2 over the ocean. These transport model induced flux uncertainties exceed the target requirement that was formulated for the A-SCOPE mission of 0.02 PgC/yr per 106 km2, and could also limit the overall performance of other CO2 missions such as GOSAT. A variable, but overall encouraging agreement is found in comparison with FTS measurements at Park Falls, Darwin, Spitsbergen, and Bremen, although systematic differences are found exceeding the 0.5 ppm level. Because of this, our estimate of the impact of transport model uncerainty is likely to be conservative. It is concluded that to make use of the remote sensing technique for quantifying the sources and sinks of CO2 not only requires highly accurate satellite instruments, but also puts stringent requirements on the performance of atmospheric transport models. Improving the accuracy of these models should receive high priority, which calls for a closer collaboration between experts in atmospheric dynamics and tracer transport.

scholarly journals Characterizing Uncertainties in Atmospheric Inversions of Fossil Fuel CO<sub>2</sub> Emissions in California

2018 ◽  
Author(s):  
Kieran Brophy ◽  
Heather Graven ◽  
Alistair J. Manning ◽  
Emily White ◽  
Tim Arnold ◽  
...  
Keyword(s):  
Fossil Fuel ◽  
Transport Model ◽  
Atmospheric Transport ◽  
Temporal Distribution ◽  
Temporal Variations ◽  
Potential Effect ◽  
Spatial And Temporal Distribution ◽  
Transport Models ◽  
Pseudo Data ◽  
Atmospheric Inversions

Abstract. Atmospheric inverse modelling has become an increasingly useful tool for evaluating emissions of greenhouse gases including methane, nitrous oxide and synthetic gases such as hydrofluorocarbons (HFCs). Atmospheric inversions for emissions of CO2 from fossil fuel combustion (ffCO2) are currently being developed. The aim of this paper is to investigate potential errors and uncertainties related to the spatial and temporal prior representation of emissions and modelled atmospheric transport for the inversion of ffCO2 emissions in the U.S. state of California. We perform simulation experiments based on a network of ground-based observations of CO2 concentration and radiocarbon in CO2 (a tracer of ffCO2), combining prior (bottom-up) emission models and transport models currently used in many atmospheric studies. The potential effect of errors in the spatial and temporal distribution of prior emission estimates is investigated in experiments by using perturbed versions of the emissions estimates used to create the pseudo data. The potential effect of transport error was investigated by using three different atmospheric transport models for the prior and pseudo data simulations. We find that the magnitude of biases in posterior state-total emissions arising from errors in the spatial and temporal distribution in prior emissions in these experiments are 1–15 % of posterior state-total emissions, and generally smaller than the 2-σ uncertainty in posterior emissions. Transport error in these experiments introduces biases of −10 % to +6 % in posterior state-total emissions. Our results indicate that uncertainties in posterior state-total ffCO2 estimates arising from the choice of prior emissions or atmospheric transport model are on the order of 15 % or less for the ground-based network in California we consider. We highlight the need for temporal variations to be included in prior emissions, and for continuing efforts to evaluate and improve the representation of atmospheric transport for regional ffCO2 inversions.

scholarly journals Importance of fossil fuel emission uncertainties over Europe for CO<sub>2</sub> modeling: model intercomparison

2011 ◽  
Vol 11 (13) ◽  
pp. 6607-6622 ◽  
Author(s):  
P. Peylin ◽  
S. Houweling ◽  
M. C. Krol ◽  
U. Karstens ◽  
C. Rödenbeck ◽  
...  
Keyword(s):  
Inverse Modeling ◽  
Fossil Fuel ◽  
Transport Model ◽  
Carbon Sink ◽  
Temporal Distribution ◽  
Co2 Concentration ◽  
Emission Inventories ◽  
Transport Models ◽  
Diurnal Cycles ◽  
Fossil Fuel Emission

Abstract. Inverse modeling techniques used to quantify surface carbon fluxes commonly assume that the uncertainty of fossil fuel CO2 (FFCO2) emissions is negligible and that intra-annual variations can be neglected. To investigate these assumptions, we analyzed the differences between four fossil fuel emission inventories with spatial and temporal differences over Europe and their impact on the model simulated CO2 concentration. Large temporal flux variations characterize the hourly fields (~40 % and ~80 % for the seasonal and diurnal cycles, peak-to-peak) and annual country totals differ by 10 % on average and up to 40 % for some countries (i.e., the Netherlands). These emissions have been prescribed to seven different transport models, resulting in 28 different FFCO2 concentrations fields. The modeled FFCO2 concentration time series at surface sites using time-varying emissions show larger seasonal cycles (+2 ppm at the Hungarian tall tower (HUN)) and smaller diurnal cycles in summer (−1 ppm at HUN) than when using constant emissions. The concentration range spanned by all simulations varies between stations, and is generally larger in winter (up to ~10 ppm peak-to-peak at HUN) than in summer (~5 ppm). The contribution of transport model differences to the simulated concentration std-dev is 2–3 times larger than the contribution of emission differences only, at typical European sites used in global inversions. These contributions to the hourly (monthly) std-dev's amount to ~1.2 (0.8) ppm and ~0.4 (0.3) ppm for transport and emissions, respectively. First comparisons of the modeled concentrations with 14C-based fossil fuel CO2 observations show that the large transport differences still hamper a quantitative evaluation/validation of the emission inventories. Changes in the estimated monthly biosphere flux (Fbio) over Europe, using two inverse modeling approaches, are relatively small (less that 5 %) while changes in annual Fbio (up to ~0.15 % GtC yr−1) are only slightly smaller than the differences in annual emission totals and around 30 % of the mean European ecosystem carbon sink. These results point to an urgent need to improve not only the transport models but also the assumed spatial and temporal distribution of fossil fuel emission inventories.

scholarly journals Importance of fossil fuel emission uncertainties over Europe for CO<sub>2</sub> modeling: model intercomparison

2009 ◽  
Vol 9 (2) ◽  
pp. 7457-7503 ◽  
Author(s):  
P. Peylin ◽  
S. Houweling ◽  
M. C. Krol ◽  
U. Karstens ◽  
C. Rödenbeck ◽  
...  
Keyword(s):  
Inverse Modeling ◽  
Fossil Fuel ◽  
Transport Model ◽  
Carbon Sink ◽  
Temporal Distribution ◽  
Co2 Concentration ◽  
Transport Models ◽  
Annual Variations ◽  
Diurnal Cycles ◽  
Fossil Fuel Emission

Abstract. Inverse modeling techniques used to quantify surface carbon fluxes commonly assume that the uncertainty of fossil fuel CO2 (FFCO2) emissions is negligible and that intra-annual variations can be neglected. To investigate these assumptions, we analyzed the differences between four fossil fuel emission maps with spatial and temporal differences over Europe and their impact on the model simulated CO2 concentration. Large temporal flux variations characterize the hourly fields (~40% and ~80% for the seasonal and diurnal cycles, peak-to-peak) and annual country totals differ by 10% on average and up to 40% for some countries (i.e., The Netherlands). These emissions have been prescribed to seven different transport models, resulting in 28 different FFCO2 concentrations fields. The modeled FFCO2 concentration time series at surface sites using time-varying emissions show larger seasonal cycles (+2 ppm at the Hungarian tall tower (HUN)) and smaller diurnal cycles in summer (−1 ppm at HUN) than when using constant emissions. The concentration range spanned by all simulations varies between stations, and is generally larger in winter (up to ~10 ppm peak-to-peak at HUN) than in summer (~5 ppm). The contribution of transport model differences to the simulated concentration std-dev is 2–3 times larger than the contribution of emission differences only, at typical European sites used in global inversions. These contributions to the hourly (monthly) std-dev's amount to ~1.2 (0.8) ppm and ~0.4 (0.3) ppm for transport and emissions, respectively. First comparisons of the modeled concentrations with 14C-based fossil fuel CO2 observations show that the large transport differences still hamper a quantitative evaluation/validation of the emission inventories. Changes in the estimated monthly biosphere flux (Fbio) over Europe, using two inverse modeling approaches, are relatively small (less that 5%) while changes in annual Fbio (up to ~0.15 Gt C/yr) are only slightly smaller than the differences in annual emission totals and around 30% of the mean European ecosystem carbon sink. These results point to an urgent need to improve not only the transport models but also the assumed spatial and temporal distribution of fossil fuel emission maps.

scholarly journals Diagnostic methods for atmospheric inversions of long-lived greenhouse gases

2016 ◽  
Author(s):  
Anna M. Michalak ◽  
Nina A. Randazzo ◽  
Frédéric Chevallier
Keyword(s):  
Inverse Problems ◽  
Greenhouse Gases ◽  
Atmospheric Chemistry ◽  
Transport Model ◽  
Ghg Emissions ◽  
Surface Fluxes ◽  
Diagnostic Methods ◽  
Clear Understanding ◽  
Atmospheric Observations ◽  
Atmospheric Inversions

Abstract. The ability to predict the trajectory of climate change requires a clear understanding of the emissions and uptake (a.k.a. surface fluxes) of long-lived greenhouse gases (GHGs). Furthermore, the development of climate policies is driving a need to constrain the budgets of anthropogenic GHG emissions. Inverse problems that couple atmospheric observations of GHG concentrations with an atmospheric chemistry and transport model have increasingly been used to gain insights into surface fluxes. Given the inherent technical challenges associated with their solution, it is imperative that objective approaches exist for the evaluation of such inverse problems. Because direct observation of fluxes at compatible spatiotemporal scales is rarely possible, diagnostics tools must rely on indirect measures. Here we review diagnostics that have been implemented in recent studies, and discuss their use in informing adjustments to model setup. We group the diagnostics along a continuum starting with those that are most closely related to the scientific question being targeted, and ending with those most closely tied to the statistical and computational setup of the inversion. We thus begin with diagnostics based on assessments against independent information (e.g., unused atmospheric observations, large-scale scientific constraints), followed by statistical diagnostics of inversion results, diagnostics based on sensitivity tests and analyses of robustness (e.g., tests focusing on the chemistry and transport model, the atmospheric observations, or the statistical and computational framework), and close with the use of synthetic data experiments (a.k.a. observing system simulation experiments (OSSEs)). We find that existing diagnostics provide a crucial toolbox for evaluating and improving flux estimates, but, not surprisingly, cannot overcome the fundamental challenges associated with limited atmospheric observations or the lack of direct flux measurements at compatible scales. As atmospheric inversions are increasingly expected to contribute to national reporting of GHG emissions, the need for developing and implementing robust and transparent evaluation approaches will only grow.

scholarly journals Retrieving global aerosol sources from satellites using inverse modeling

2008 ◽  
Vol 8 (2) ◽  
pp. 209-250 ◽  
Author(s):  
O. Dubovik ◽  
T. Lapyonok ◽  
Y. J. Kaufman ◽  
M. Chin ◽  
P. Ginoux ◽  
...  
Keyword(s):  
Inverse Modeling ◽  
Transport Model ◽  
A Priori ◽  
Time Variability ◽  
Aerosol Transport ◽  
Space And Time ◽  
Aerosol Emission ◽  
Coarse Mode ◽  
Aerosol Distribution ◽  
Aerosol Sources

Abstract. Understanding aerosol effects on global climate requires knowing the global distribution of tropospheric aerosols. By accounting for aerosol sources, transports, and removal processes, chemical transport models simulate the global aerosol distribution using archived meteorological fields. We develop an algorithm for retrieving global aerosol sources from satellite observations of aerosol distribution by inverting the GOCART aerosol transport model. The inversion is based on a generalized, multi-term least-squares-type fitting, allowing flexible selection and refinement of a priori algorithm constraints. For example, limitations can be placed on retrieved quantity partial derivatives, to constrain global aerosol emission space and time variability in the results. Similarities and differences between commonly used inverse modeling and remote sensing techniques are analyzed. To retain the high space and time resolution of long-period, global observational records, the algorithm is expressed using adjoint operators. Successful global aerosol emission retrievals at 2°×2.5 resolution were obtained by inverting GOCART aerosol transport model output, assuming constant emissions over the diurnal cycle, and neglecting aerosol compositional differences. In addition, fine and coarse mode aerosol emission sources were inverted separately from MODIS fine and coarse mode aerosol optical thickness data, respectively. These assumptions are justified, based on observational coverage and accuracy limitations, producing valuable aerosol source locations and emission strengths. From two weeks of daily MODIS observations during August 2000, the global placement of fine mode aerosol sources agreed with available independent knowledge, even though the inverse method did not use any a priori information about aerosol sources, and was initialized with a "zero aerosol emission" assumption. Retrieving coarse mode aerosol emissions was less successful, mainly because MODIS aerosol data over highly reflecting desert dust sources is lacking. The broader implications of applying our approach are also discussed.

scholarly journals The added value of satellite observations of methane forunderstanding the contemporary methane budget

2021 ◽  
Vol 379 (2210) ◽  
pp. 20210106
Author(s):  
Paul I. Palmer ◽  
Liang Feng ◽  
Mark F. Lunt ◽  
Robert J. Parker ◽  
Hartmut Bösch ◽  
...  
Keyword(s):  
Atmospheric Chemistry ◽  
Transport Model ◽  
A Priori ◽  
Added Value ◽  
Atmospheric Methane ◽  
Sources And Sinks ◽  
Methane Budget ◽  
Spatial Coverage ◽  
New Research ◽  
The Tropics

Surface observations have recorded large and incompletely understood changes to atmospheric methane (CH 4 ) this century. However, their ability to reveal the responsible surface sources and sinks is limited by their geographical distribution, which is biased towards the northern midlatitudes. Data from Earth-orbiting satellites designed specifically to measure atmospheric CH 4 have been available since 2009 with the launch of the Japanese Greenhouse gases Observing SATellite (GOSAT). We assess the added value of GOSAT to data collected by the US National Oceanic and Atmospheric Administration (NOAA), which have been the lynchpin for knowledge about atmospheric CH 4 since the 1980s. To achieve that we use the GEOS-Chem atmospheric chemistry transport model and an inverse method to infer a posteriori flux estimates from the NOAA and GOSAT data using common a priori emission inventories. We find the main benefit of GOSAT data is from its additional coverage over the tropics where we report large increases since the 2014/2016 El Niño, driven by biomass burning, biogenic emissions and energy production. We use data from the European TROPOspheric Monitoring Instrument to show how better spatial coverage and resolution measurements allow us to quantify previously unattainable diffuse sources of CH 4 , thereby opening up a new research frontier. This article is part of a discussion meeting issue ‘Rising methane: is warming feeding warming? (part 1)’.

scholarly journals The zonal structure of tropical O<sub>3</sub> and CO as observed by the Tropospheric Emission Spectrometer in November 2004 – Part 1: Inverse modeling of CO emissions

2009 ◽  
Vol 9 (11) ◽  
pp. 3547-3562 ◽  
Author(s):  
D. B. A. Jones ◽  
K. W. Bowman ◽  
J. A. Logan ◽  
C. L. Heald ◽  
J. Liu ◽  
...  
Keyword(s):  
Inverse Modeling ◽  
Transport Model ◽  
A Priori ◽  
Systematic Errors ◽  
A Posteriori ◽  
Chemical Transport Model ◽  
Australian Region ◽  
Equatorial Africa ◽  
Modeling Analysis ◽  
The Tropics

Abstract. We conduct an inverse modeling analysis of measurements of atmospheric CO from the TES and MOPITT satellite instruments using the GEOS-Chem global chemical transport model to quantify emissions of CO in the tropics in November 2004. We also assess the consistency of the information provided by TES and MOPITT on surface emissions of CO. We focus on the tropics in November 2004, during the biomass burning season, because TES observations of CO and O3 and MOPITT observations of CO reveal significantly greater abundances of these gases than simulated by the GEOS-Chem model during that period. We find that both datasets suggest substantially greater emissions of CO from sub-equatorial Africa and the Indonesian/Australian region than in the climatological emissions in the model. The a posteriori emissions from sub-equatorial Africa based on TES and MOPITT data were 173 Tg CO/yr and 184 Tg CO/yr, respectively, compared to the a priori of 95 Tg CO/yr. In the Indonesian/Australian region, the a posteriori emissions inferred from TES and MOPITT data were 155 Tg CO/yr and 185 Tg CO/yr, respectively, whereas the a priori was 69 Tg CO/yr. The differences between the a posteriori emission estimates obtained from the two datasets are generally less than 20%. The a posteriori emissions significantly improve the simulated distribution of CO, however, large regional residuals remain, and are likely due to systematic errors in the analysis. Reducing these residuals and improving the accuracy of top-down emission estimates will require better characterization of systematic errors in the observations and the model (chemistry and transport).

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