The Indianapolis Flux Experiment aims to utilize a variety of atmospheric measurements and a high-resolution inversion system to estimate the temporal and spatial variation of anthropogenic greenhouse gas emissions from an urban environment. We present a Bayesian inversion system solving for fossil fuel and biogenic CO_{2} fluxes over the city of Indianapolis, IN. Both components were described at 1 km resolution to represent point sources and fine-scale structures such as highways in the a priori fluxes. With a series of Observing System Simulation Experiments, we evaluate the sensitivity of inverse flux estimates to various measurement deployment strategies and errors. We also test the impacts of flux error structures, biogenic CO_{2} fluxes and atmospheric transport errors on estimating fossil fuel CO_{2} emissions and their uncertainties. The results indicate that high-accuracy and high-precision measurements produce significant improvement in fossil fuel CO_{2} flux estimates. Systematic measurement errors of 1 ppm produce significantly biased inverse solutions, degrading the accuracy of retrieved emissions by about 1 *µ*mol m^{–2} s^{–1} compared to the spatially averaged anthropogenic CO_{2} emissions of 5 *µ*mol m^{–2} s^{–1}. The presence of biogenic CO_{2} fluxes (similar magnitude to the anthropogenic fluxes) limits our ability to correct for random and systematic emission errors. However, assimilating continuous fossil fuel CO_{2} measurements with 1 ppm random error in addition to total CO_{2} measurements can partially compensate for the interference from biogenic CO_{2} fluxes. Moreover, systematic and random flux errors can be further reduced by reducing model-data mismatch errors caused by atmospheric transport uncertainty. Finally, the precision of the inverse flux estimate is highly sensitive to the correlation length scale in the prior emission errors. This work suggests that improved fossil fuel CO_{2} measurement technology, and better understanding of both prior flux and atmospheric transport errors are essential to improve the accuracy and precision of high-resolution urban CO_{2} flux estimates.

## Introduction

Changes in climate have increased due to the impact of greenhouse gas (GHG) emissions on the Earth’s radiative budget over recent decades. The carbon dioxide produced from fossil fuel combustion (CO_{2}ff) is the most important cause of the increase in atmospheric CO_{2} concentration. Atmospheric CO_{2} concentrations have risen by 40% since pre-industrial times and are now at their highest level for the past 800,000 years at a minimum (Lüthi et al., 2008). During 2002–2011, global carbon emissions from fossil fuel combustion and cement production averaged 8.3 ± 0.7 GtC yr^{–1} (1 GtC = 1 Gigatonne of carbon = 10^{15} grams of carbon) (Boden et al., 2016), with over 70% of CO_{2}ff emissions attributable to urban areas (EIA, 2013). Quantitative estimation of anthropogenic CO_{2} emissions from urban areas is a high research priority for the formulation and implementation of policies to mitigate climate change and ensure urban sustainability (Hutyra et al., 2014).

Estimation of anthropogenic carbon emissions to the atmosphere has generally been performed via two complementary approaches: “bottom-up” and “top-down” methods. Bottom-up methods aggregate together source-specific CO_{2}ff flux estimates to form a total emission inventory based on activity data (such as energy consumption, population density, traffic data and local air pollution reporting) and emission models (*e.g*. a building energy consumption model) (Gurney et al., 2012). Inventories can be highly resolved in both space and time (Gurney et al., 2009), but they are prone to systematic errors and their uncertainties are not well known (Andres et al., 2014). Top-down methods infer quantitative information on surface CO_{2} fluxes from variations in atmospheric CO_{2} concentrations through inverse modeling with atmospheric tracer transport models (Ciais et al., 2011), and may include isotope composition measurements to identify fossil fuel sources (Levin et al., 2003; Miller et al., 2012; Turnbull et al., 2015; Basu et al., 2016). Uncertainties in atmospheric transport models (Peylin et al., 2002; Lauvaux et al., 2009; Peylin et al., 2011; Isaac et al., 2014), limited density of atmospheric measurements (Gerbig et al., 2009; Lauvaux et al., 2012; Turner et al., 2016) and uncertainties in prior fluxes (Peylin et al., 2005; Carouge et al., 2010; Lauvaux et al., 2016) all constitute sources of error in this method (Engelen et al., 2002).

With the increasing interest in monitoring and verifying surface CO_{2} exchange, several studies have been conducted to invert for biogenic (Peters et al., 2007; Schuh et al., 2010; Lauvaux et al., 2012; Ogle et al., 2015) and anthropogenic CO_{2} fluxes (Lauvaux et al., 2013; Bréon et al., 2015; Staufer et al., 2016; Lauvaux et al., 2016; Verhulst et al., 2017). The Indianapolis Flux Experiment (INFLUX, http://sites.psu.edu/influx/) was proposed to develop, test and improve methods to estimate anthropogenic GHG emissions from cities, using Indianapolis as a test bed (Davis et al., 2017). This project uses aircraft (Cambaliza et al., 2014; Heimburger et al., 2017) and a high-density surface tower network (Miles et al., 2017; Richardson et al., 2017) combined with high-resolution atmospheric modeling (Deng et al., 2017; Sarmiento et al., 2017) to infer CO_{2}ff emissions at 1 km spatial resolution (Lauvaux et al., 2016). Figure 1 shows the distribution of instrumented towers and daytime average surface CO_{2} fluxes during the first 10 days of September 2013. The availability of a high-resolution emission inventory (Gurney et al., 2012) and a high-precision atmospheric transport model (Deng et al., 2017; Lauvaux et al., 2016) enables us to test the possible improvements and limitations of an urban atmospheric CO_{2} inversion system.

Quantification of urban CO_{2} fluxes is limited by several challenges. Of particular importance is the separation of CO_{2}ff emissions and biogenic CO_{2} (CO_{2}bio) exchange (Pataki et al., 2003, 2007; Briber et al., 2013; Hardiman et al., 2017). Measurement of the radioactive carbon isotope (^{14}C) is a highly effective approach to isolate the ^{14}C-free CO_{2}ff emissions given the depletion of radiocarbon in extremely old fossil fuels (Turnbull et al., 2009). In addition, carbon monoxide (CO) can be used as a tracer for CO_{2}ff, relying upon an empirical emission ratio of CO to CO_{2}ff from incomplete combustion of hydrocarbons (Silva et al., 2013). CO measurements are more readily available, while flask measurements of ^{14}C are expensive and discontinuous, although CO is a less accurate tracer of CO_{2}ff than ^{14}C (Levin and Karstens, 2007). Another challenge is minimization of uncertainty in high-resolution atmospheric transport models used to simulate trace gas transport in an urban setting, where complex boundary layer structures may be formed due to the land-use/land-cover change and intensive human activities (Wang et al., 2011; Sarmiento et al., 2017; Gaudet et al., 2017). In addition, the inverse estimation of CO_{2}ff emissions is constrained by atmospheric CO_{2} measurements, and the trade-off between measurement density and quality is an important emerging debate for urban GHG monitoring (Wu et al., 2016; Shusterman et al., 2016; Turner et al., 2016; Martin et al., 2017). Lastly, uncertain spatial structures in prior flux errors influence the precision of inverse flux estimates (Saide et al., 2011; Lauvaux et al., 2016). Therefore, studying the impacts of CO_{2}bio fluxes, atmospheric transport errors, observation deployment strategies and prior flux error structures on CO_{2}ff flux estimates are important considerations for advancing our understanding of uncertainties in the estimation of anthropogenic carbon emissions in an urban environment.

An Observing System Simulation Experiment (OSSE) (Figure 2), designed to examine the ability of synthetically generated measurements (pseudo-data) to retrieve the assumed “true” fluxes within a Bayesian synthesis inversion framework, is a useful approach for quantifying the impacts of different inversion system configurations and error characteristics on flux estimates and their uncertainties (Law et al., 2002; Carouge et al., 2010; Gourdji et al., 2010; Chatterjee et al., 2012). Using an OSSE to evaluate uncertainties in the urban CO_{2} inversion system has three advantages. First, the presupposed true fluxes make it possible to evaluate the impact of different inversion scenarios on the ability to infer CO_{2}ff emissions. Second, since the synthetic CO_{2} measurements are generated from surface CO_{2} fluxes within the domain of interest, there is no need to consider inflow at the boundary (*i.e*. CO_{2} from outside of the study area), which avoids possible biases from incorrect estimation of boundary conditions for limited-domain inversions although this is an important source of error in real inversions (Schuh et al., 2013; Lauvaux et al., 2012, 2016). Third, the atmospheric transport can be known perfectly (*i.e*. no bias) because the same transport matrix is used to create the synthetic measurements and to estimate fluxes in the inversion system.

In this study, we conduct a series of OSSEs to evaluate the sensitivity of urban-scale flux estimates to various observational and inversion system configurations over the city of Indianapolis. The primary objectives of this study are threefold. First, we test the impact of prior flux errors on the inferred CO_{2}ff flux uncertainties. Second, we demonstrate a method to estimate the impacts of different observational configurations and CO_{2}bio fluxes on the ability to infer CO_{2}ff emissions. Third, we investigate the impacts of atmospheric transport errors and synthetic CO_{2}ff measurements on the accuracy and precision of inferred CO_{2}ff emissions.

## Inverse theory

Atmospheric inverse modeling of CO_{2} sources and sinks is a process to infer a set of statistically optimal fluxes (posterior fluxes), which assimilates all available information sources (measurements and prior fluxes) within their respective uncertainties. Solving this inverse problem requires (1) a set of atmospheric CO_{2} mole fraction measurements, (2) a priori estimation of CO_{2} fluxes, and (3) a linear operator representing the atmospheric transport linking prior CO_{2} fluxes to simulated CO_{2} mole fractions at the location of observations. Knowledge of these three elements, together with their associated uncertainties, allows one to reduce the errors in prior CO_{2} fluxes and improve the estimation of CO_{2} sources and sinks (Ciais et al., 2011).

A Bayesian synthesis inversion (Enting, 2002; Tarantola, 2004) is an algorithm used to maximize posterior conditional probability or minimize posterior variance by minimizing a cost function (*F*):

where *x* is an m × 1 vector of the discretized unknown surface CO_{2} fluxes, *x*_{0} is the prior state vector of surface CO_{2} fluxes with m × 1 elements, and *y* is an n × 1 vector of atmospheric CO_{2} mole fraction measurements. *H* is a known n × m matrix describing the sensitivity of CO_{2} mole fractions to surface CO_{2} fluxes. *B*(m × m) is the flux error covariance matrix that represents the uncertainties in prior state and *R*(n × n) is the observation error covariance matrix describing the error magnitude of discrepancies between observed (*y*) and modeled (*Hx*) CO_{2} mole fractions caused by measurement and atmospheric transport errors.

The inverse (or posterior) fluxes (*x _{a}*) and their uncertainties (

*A*) are derived from minimizing the cost function (

*F*) with respect to

*x*:

Gain (*G*) and error reduction (*ER*) are two metrics used to quantitatively evaluate the inverse flux estimates (mean) and their uncertainties (standard deviation) (Lauvaux and Davis, 2014).

where *x _{ai}, x_{ti}* and

*x*

_{0i}are the posterior flux, the true flux and the prior flux at the i grid respectively. The

*σ*and

_{ai}*σ*are the standard deviations (corresponding to variances at the diagonal of

_{bi}*A*and

*B*matrixes) in posterior state and prior state at the i grid. The gain metric represents the improvement of flux magnitude after inversion. And the error reduction metric represents the increase of confidence from prior state to posterior state. These two metrics complement each other to comprehensively assess the inversion performance.

## Data and Methods

### Urban fossil fuel CO_{2} emissions

Indianapolis was the 14th largest city in the U.S. in 2013 with a population of ~835,000 and an area of ~963.5 km^{2}. The city is surrounded by agricultural areas (primarily cropland) and is located far from other metropolitan areas, so changes in GHG concentrations from the city can be isolated with relative ease. In addition, the flat terrain makes the meteorological conditions relatively simple to simulate. The Hestia Project is the first effort to use bottom-up methods to quantify hourly CO_{2}ff emissions for an entire urban landscape down to the scale of individual buildings, road segments, and industrial/electricity production facilities at ~200 m resolution (Gurney et al., 2012). Hestia shows that traffic, utility and industry are the main sectors contributing to anthropogenic CO_{2} emissions in Indianapolis. Figure 3A is the spatial distribution of daytime CO_{2}ff emissions average from 13 to 19 local standard time (LST) during the first 10 days of September 2013.

### Vegetation CO_{2} fluxes

The CO_{2}bio fluxes over the city of Indianapolis were simulated hourly at 1 km resolution using the Vegetation Photosynthesis and Respiration Model (VPRM) coupled to the Weather Research and Forecasting (WRF) model. In the WRF-VPRM system, VPRM uses meteorological fields from WRF and high-resolution satellite indices to simulate the CO_{2}bio fluxes with spatiotemporal patterns (Ahmadov et al., 2007). Specifically, VPRM simulates gross ecosystem exchange (GEE) for different vegetation categories using (1) shortwave radiative flux (SWDOWN) and temperature at 2 meters (T2) provided by the WRF simulation; (2) enhanced vegetation index (EVI), which represents the fraction of shortwave radiation absorbed by leaves; and (3) the land surface water index (LSWI), which reflects changes in both leaf water content and soil moisture (Xiao et al., 2004). Respiration fluxes are estimated as a linear function of T2. To account for the abundant soybean and corn fields surrounding Indianapolis and the different photosynthesis and respiration of these two crops (Lokupitiya et al., 2009), we added an extra vegetation category into the WRF-VPRM implementation from the United States Department of Agriculture National Agricultural Statistics Service Cropland Data Layer (USDA-NASS-CDL) to distinguish corn fields, and the remaining croplands were treated as soybean fields.

The net ecosystem exchange (NEE) measured by two eddy covariance flux towers from AmeriFlux network were used to optimize four user-estimated parameters in VPRM (Schmid et al., 2000; Mahadevan et al., 2008). Morgan Monroe State Forest (US-MMS: 39.32 N, 86.41 W) and Fermi National Accelerator Laboratory – Batavia (US-IB1: 41.86 N, 88.22 W) are two closest stations to the study area with available data for the ecosystems of interest (Ehman et al., 2002; Matamala, 2016). US-MMS flux measurements from 2013 were used to represent broadleaf forest. US-IB1 flux measurements from 2008 and 2009 were used to represent corn and soybean, respectively, based on the crops grown at the site during those years. Therefore, we used these flux data to optimize parameters for three vegetation categories (deciduous broadleaf forest, corn and soybean), which together account for more than 95% of the total area in the simulated domain. We optimized these parameters simultaneously using an unconstrained nonlinear optimization method (Nelder and Mead, 1965). Figure 3B shows the daytime (13 to 19 LST) average CO_{2}bio fluxes during the first 10 days of September 2013.

### Atmospheric transport model

This study used the WRF model with a slightly modified chemistry module (WRF-Chem) and the Lagrangian Particle Dispersion Model (LPDM) (Uliasz, 1994) to simulate CO_{2} footprints (*i.e*. influence functions, *H* matrix in Equation 1) (Lauvaux et al., 2016). The simulation domain is centered on Indianapolis and covers an area of 87 km × 87 km at 1 km spatial resolution and hourly temporal resolution in the LPDM. The National Centers for Environmental Prediction North American Regional Reanalysis (NCEP-NARR) gridded meteorological data were used as the initial conditions to drive the WRF-Chem modeling system (Mesinger et al., 2006), which continuously assimilated meteorological observations using a Four-Dimensional Data Assimilation (FDDA) system to produce more accurate meteorological conditions (Deng et al., 2009), similar to the WRF-WMO-FDDA case described in Deng et al. (2017). The wind field, potential temperature, and turbulent kinetic energy from the WRF-Chem simulations were used as input variables to drive the particle backward motions from the tower locations (Figure 1) in the LPDM. At each tower location, 6300 particles were released every hour for 12-hour back-trajectories. Since the simulation of atmospheric transport during nighttime may have large errors due to difficulty in simulating the stable boundary layer, this study utilizes CO_{2} footprints during 7 daytime hours (13–19 LST) in the first 10 days of September 2013 to conduct pseudo-data inversion experiments. Although we do not use synthetic nocturnal observations, the influence functions used to interpret daytime observations do extend into the nighttime (12 hours before the synthetic daytime observations), and hence the current system has some sensitivity to nocturnal emissions.

Quantitative estimation of uncertainties in atmospheric transport is a critical element in urban inversions. Limited model resolution, imperfect atmospheric initial conditions, and imprecise model physical parameterizations can all lead to significant errors in the simulated CO_{2} mole fractions. These uncertainties are difficult to quantify. The urban environment is challenging since the underlying surface is heterogeneous, potentially leading to complex sub-grid scale flows. Additionally, the high-resolution atmospheric simulation tends to introduce highly spatio-temporally correlated errors within the urban domain, which are complicated to characterize and could influence the inverse flux estimates and their uncertainties (Lauvaux et al., 2009). Our objective is to focus primarily on the interaction of CO_{2}ff and CO_{2}bio fluxes. We make the simplifying assumption that transport errors are uncorrelated, which means *R* matrix is a diagonal matrix. We do vary the assumed magnitude of uncertainty in atmospheric transport (*i.e*. random error) to evaluate the impact of improvements to atmospheric transport model.

### Observing system simulation experiment

We set up a series of observing system simulation experiments by assuming that the daytime average CO_{2}ff emissions (*x _{f}*) from the Hestia Project and CO

_{2}bio fluxes (

*x*) from the WRF-VPRM system are the true fluxes (

_{b}*X*). After combining the true fluxes with the linear transport matrix (

_{t}*h*), the synthetic “perfect” CO

_{2}mole fraction measurements (

*Y*) at each site were produced. We use two different inversion schemes to simulate atmospheric CO

_{p}_{2}measurements. One inverse system (scheme 1) utilizes only total CO

_{2}mole fraction measurements (CO

_{2}tt,

*y*). The other inverse system (scheme 2) utilizes both CO

_{t}_{2}tt and CO

_{2}ff (

*y*) mole fraction measurements. These two schemes are achieved by reconstructing the transport matrix (

_{f}*H*) as follows:

To illustrate the impacts of biogenic CO_{2} fluxes and observational network on anthropogenic CO_{2} flux estimates, our experiments are based on three different scenarios (Figure 4). Scenario 1 (S1), a reference case, includes only CO_{2}ff emissions and synthetic CO_{2}tt mole fraction measurements. Since there are no CO_{2}bio fluxes, S1 conceptually corresponds to the winter when the CO_{2}bio exchange between land and atmosphere is assumed to be negligible compared to CO_{2}ff emissions. Scenario 2 (S2) includes CO_{2}ff emissions, CO_{2}bio fluxes and only CO_{2}tt mole fraction measurements (scheme 1). This scenario conceptually represents summer conditions, but a more limited atmospheric observing system. Scenario 3 (S3) has both CO_{2}ff and CO_{2}bio fluxes (like S2), but includes both CO_{2}tt and CO_{2}ff mole fraction measurements (scheme 2). The comparison of S1 and S2 illustrates the impact of CO_{2}bio fluxes on the inverse estimate of CO_{2}ff emissions. The impact of adding CO_{2}ff measurements on the inversion performance is evaluated by comparing S2 and S3. Additionally, we also vary the assumed uncertainties in the prior fluxes, atmospheric transport, and atmospheric observations to test the sensitivity of inverse CO_{2}ff flux estimates to these characteristics of the system.

Evaluating our ability to reduce prior flux errors is the primary objective of this study. Among prior flux errors, the most important challenge is to remove biases. Thus we respectively add mean biases of 3 *µ*mol m^{–2} s^{–1} and –2 *µ*mol m^{–2} s^{–1} to form prior CO_{2}ff and CO_{2}bio fluxes, which are about 60% of the average flux signals for each component. These biases represent systematic errors in the prior CO_{2} fluxes (Figure 3C and 3D). All of the following experiments include these prior flux biases. Random errors in the prior fluxes, atmospheric transport, and atmospheric measurements also confound our ability to retrieve the true CO_{2} fluxes. The magnitudes of these errors vary according to the quality of our instrumentation, atmospheric transport and prior flux models. Therefore, we impose a range of assumed random errors, which are combined with different scenarios, to provide a comprehensive evaluation of the inversion system.

Our first cases explore random errors in the prior flux estimates. The random error magnitude, or Root Mean Square Error (RMSE), represents the magnitude of flux error at each grid point corresponding to diagonal elements of *B* matrix in Equation 1. The spatial coherence in the flux error is approximated with an exponentially decaying function of the distance between two grid points. The Spatial Correlation Length (SCL) at which the correlation between two separated grid points is less than 0.5 is defined to characterize the spatial correlation in the prior flux error structures corresponding to off-diagonal elements in *B* matrix (Houweling et al., 2004; Peters et al., 2005; Saide et al., 2011; Wu et al., 2011). Neither the random error magnitude nor the spatially correlated error structures are well known. We use S1 with 2 *µ*mol m^{–2} s^{–1} RMSE (~40% of the average CO_{2}ff fluxes) and 5 km SCL as the default case (Figure 3C). The RMSE is varied to be 1 *µ*mol m^{–2} s^{–1} or 4 *µ*mol m^{–2} s^{–1} (*i.e*. half of or double the default case) to test the sensitivity of the flux error reduction to the prior flux error magnitude, and the SCL is varied to be 2 km or 8 km to explore the influence of different prior flux error structures on the posterior flux uncertainties (Table 1). Both the Degree of Freedom in the Signal (DFS) and the averaging kernel sensitivity are tested to evaluate the impact of the correlation structures on the solutions (Rodgers, 2000; Bocquet, 2009).

. | . | RMSE-B (µmol m^{–2} s^{–1})
. | SCL(km) . |
---|---|---|---|

Scenario 1 | case R^{a} | 2 | 5 |

case A | 1 | 5 | |

case B | 4 | 5 | |

case C | 2 | 2 | |

case D | 2 | 8 |

. | . | RMSE-B (µmol m^{–2} s^{–1})
. | SCL(km) . |
---|---|---|---|

Scenario 1 | case R^{a} | 2 | 5 |

case A | 1 | 5 | |

case B | 4 | 5 | |

case C | 2 | 2 | |

case D | 2 | 8 |

^{a}The default case.

In addition, we use S1, S2 and S3 to investigate the impacts of CO_{2}bio fluxes, different observational configurations (*i.e*. density, accuracy and precision of observations) and the use of CO_{2}ff measurements on posterior CO_{2}ff flux estimates and their uncertainties. This study generated synthetic CO_{2} mole fraction measurements for 7 daytime hours (13–19 LST) during the first 10 days of September 2013 at each tower location, and varied the magnitude of observation error to represent different accuracy and precision of atmospheric measurements. For example, 1 ppm observation error means that we set 1 ppm standard deviation to generate hourly random noise for the entire observation period (10 days with 7 hours per day), and then add it to the model-data mismatch at each site. We first estimate flux error reduction under S1 and S2 for four different observation cases (Table 2): (1) 5 sites (towers 1, 2, 3, 5 and 9) with 1 ppm observation error; (2) 12 sites with 1 ppm observation error; (3) 12 sites with 3 ppm observation error; (4) 5 sites (towers 1, 2, 3, 5 and 9) with 1 ppm observation error and the other 7 sites with 3 ppm observation error. The comparison of case 1 and case 2 indicates the effect of increasing the number of observation sites, and the impact caused by different observation precision is evaluated in the comparison of cases 2, 3 and 4. To explore the impact of observation biases on inversion performance, we set another case (case 5) as 12 sites with 1 ppm bias and 1 ppm RMSE (Table 2). These random errors and biases could be caused by either imperfect atmospheric CO_{2} measurements or by atmospheric transport errors. The use of CO_{2}ff measurements is tested in S3 for 5 sites and 12 sites, respectively. Since using ^{14}C to infer CO_{2}ff mole fractions introduces additional measurement errors (Turnbull et al., 2015), the CO_{2}ff observation errors are increased 1 ppm compared to the CO_{2}tt mole fraction measurements (Table 2).

. | . | Number of sites . | RMSE-R (ppm) . | Bias (ppm) . |
---|---|---|---|---|

Scenario 1 Scenario 2 | case 1 | 5 | 1 | 0 |

case 2 | 12 | 1 | 0 | |

case 3 | 12 | 3 | 0 | |

case 4 | 5 & 7 | 1 & 3 | 0 | |

case 5 | 12 | 1 | 1 | |

Scenario 3 | case 1 | 5 | 1 (CO_{2}tt)^{a}/2 (CO_{2}ff)^{b} | 0 |

case 2 | 12 | 1 (CO_{2}tt)/2 (CO_{2}ff) | 0 |

. | . | Number of sites . | RMSE-R (ppm) . | Bias (ppm) . |
---|---|---|---|---|

Scenario 1 Scenario 2 | case 1 | 5 | 1 | 0 |

case 2 | 12 | 1 | 0 | |

case 3 | 12 | 3 | 0 | |

case 4 | 5 & 7 | 1 & 3 | 0 | |

case 5 | 12 | 1 | 1 | |

Scenario 3 | case 1 | 5 | 1 (CO_{2}tt)^{a}/2 (CO_{2}ff)^{b} | 0 |

case 2 | 12 | 1 (CO_{2}tt)/2 (CO_{2}ff) | 0 |

^{a}Total CO_{2} mole fraction.

^{b}Fossil fuel CO_{2} mole fraction.

Finally, the effect of improved atmospheric transport modeling is explored by decreasing the magnitude of random error in the observation error covariance matrix (*R* matrix in Equation 1). Richardson et al. (2017) demonstrated that the instrument error from continuous measurements of CO_{2}tt mole fractions using wavelength-scanned cavity ring-down spectroscopy (WS-CRDS, Picarro Inc.) is approximately 0.1 ppm. Atmospheric transport error is not as well defined, but has been estimated to be much larger (approximately 2 to 5 ppm in the U.S. Great Plains) depending on the atmospheric conditions and the scale of interest (Lauvaux et al., 2012). Due to the combination of a high-resolution transport model and a meteorological data assimilation system, this study approximates current random atmospheric transport error to be 1 ppm (~30% of the daytime average urban CO_{2} enhancement) (Deng et al., 2017). With unbiased synthetic measurements at 12 instrumented towers, the impact of different atmospheric transport models is explored by setting random errors in simulated CO_{2}tt mole fractions to be 1.0, 0.5 and 0.1 ppm, corresponding to cases that considering reducing and essentially eliminating atmospheric transport errors (Table 3).

. | . | RMSE-R (ppm) . |
---|---|---|

Scenario 1 Scenario 2 | case I | 0.1 |

case II | 0.5 | |

case III | 1.0 | |

Scenario 3 | case I | 0.1 (CO_{2}tt)^{a}/1.0 (CO_{2}ff)^{b} |

case II | 0.5 (CO_{2}tt)/1.5 (CO_{2}ff) | |

case III | 1.0 (CO_{2}tt)/2.0 (CO_{2}ff) |

. | . | RMSE-R (ppm) . |
---|---|---|

Scenario 1 Scenario 2 | case I | 0.1 |

case II | 0.5 | |

case III | 1.0 | |

Scenario 3 | case I | 0.1 (CO_{2}tt)^{a}/1.0 (CO_{2}ff)^{b} |

case II | 0.5 (CO_{2}tt)/1.5 (CO_{2}ff) | |

case III | 1.0 (CO_{2}tt)/2.0 (CO_{2}ff) |

^{a}Total CO_{2} mole fraction.

^{b}Fossil fuel CO_{2} mole fraction.

## Results

We first present the impact of prior flux errors on the precision of posterior flux estimates. Figure 5 shows spatial distributions of error reduction for five cases in Table 1, using 12 towers with 1 ppm observation error at each site. There is little difference in the spatial structure of error reduction corresponding to the change of prior flux error (RMSE-B) (Figure 5R, 5A and 5B). However, the change of SCL causes an obvious difference in the estimation of flux error reduction, which is consistent with to a previous study (Saide et al., 2011). In the case with 2 km SCL (Figure 5C), prior flux errors are reduced less than 20%, and only close to the tower locations. About 50% of prior flux errors are removed in the vicinity of the towers in the case with 8 km SCL (Figure 5D) and the error reduction area expands relative to the 2 km case. Since larger SCL means that uncertainties in the prior fluxes are correlated in a larger spatial area, more flux errors can be removed using the same number of observation sites. We find that DFS and averaging kernel sensitivity, additional measures sometimes used to evaluate the spatial structure of inverse flux estimates, provide little information for the range of SCLs we have studied. The DFS is nearly constant across the range of SCLs that we examine, and only decreases as the SCL approaches and exceeds the spacing between our towers (Figure S1). The DFS decreases for very small SCL values (less than 2 km) (Figure S1), which is related to a singularity of the Continuum Limit (Bocquet, 2005). Similarly, maps of the averaging kernel sensitivity show very small changes across the range of SCLs we have examined (Figure S2). The metric of error reduction yields more information about the change in sensitivity of the solution to the assumed SCL.

We next explore the impact of different observational networks (*i.e*. number of towers and quality of measurements) on correcting flux errors. Figure 6 shows error reduction for different observational configurations in S1 (Table 2). With the increase of observations from 5 sites to 12 sites (Figure 6.1 and 6.2), the area of error reduction is expanded and the magnitude of error reduction in the center of the city is increased from ~20% to ~40%, which indicates that it is beneficial to increase the density of observations in a high-resolution urban CO_{2} inversion system. In addition, the increase of observation error from 1 ppm to 3 ppm significantly increases uncertainties in the posterior flux estimates (Figure 6.3). Since the daytime average urban CO_{2} enhancement in Indianapolis ranges from 0.3 ppm to 2.9 ppm (Miles et al., 2017), high-precision measurements are important to remove prior flux errors. For the mixed configuration (Figure 6.4, case 4), flux error reductions in the vicinity of the towers increase to ~30% from ~10% in case 3 (Figure 6.3), but the error reduction is still not comparable to case 2 (Figure 6.2). With the existence of CO_{2}bio fluxes (S2), uncertainties in the posterior CO_{2}ff flux estimates are obviously increased, as demonstrated by the reduced flux error correction and the shrinkage of error reduction area (Figure 7). Even for the case with the highest observational density and the most precise measurements (12 sites with 1 ppm observation error), the error reduction in S2 is decreased to less than 20% (Figure 7.2) from ~40% in S1 (Figure 6.2). The presence of CO_{2}bio fluxes significantly weakens our ability to reduce CO_{2}ff flux errors by limiting our ability to distinguish fossil fuel emissions from biogenic fluxes.

To further test the use of biased sensors and CO_{2}ff measurements to infer CO_{2}ff flux estimates, we compared the gain, error reduction and flux bias averaged across the urban domain for different observational configurations (Table 2). The gain is negative when using 12 biased sensors (case 5) in S1 (S1_c5 in Figure 8), meaning that the posterior fluxes have a higher bias than the prior state (S1_c5 in Figure 9). It indicates that high-accuracy measurements are necessary to remove systematic errors in the prior CO_{2}ff flux estimates. As expected, both gain and error reduction are small (less than 0.2 and 8%, respectively) for S2 compared to S1 for all cases with unbiased observations (case 1 to case 4 in Figure 8). The comparison of S1 and S2 indicates that the presence of CO_{2}bio fluxes decreases the gain, and increases random and systematic errors in the estimation of CO_{2}ff emissions. Including 12 CO_{2}ff measurement sites (S3_c2 in Figure 8) increases the spatially averaged gain to 0.40 from 0.19 in the scenario without CO_{2}ff measurements (S2_c2 in Figure 8), corresponding to the obvious correction of flux bias in the posterior state (S3_c2 in Figure 9). This implies that high-density CO_{2}ff mole fraction measurements can partially compensate for the interference from CO_{2}bio fluxes.

Figure 10 shows the absolute difference between posterior CO_{2}ff flux estimates and true CO_{2}ff fluxes corresponding to different observation errors (Table 3). The variety of observation error represents different atmospheric transport model errors, assuming that high-precision instruments are used. The prior flux errors (default error setting for CO_{2}ff and CO_{2}bio flux components) and number of measurement towers (12 sites) are constant for all cases. The difference between the posterior fluxes and the true fluxes decreases continuously as the transport error decreases. The existence of CO_{2}bio fluxes (S2) causes more flux differences around the urban boundary (middle column in Figure 10). Using CO_{2}ff mole fraction measurements (S3) yields reduced flux differences compared to S2, and spatial patterns (right column in Figure 10) are more similar to S1. As expected, the most significant error reduction occurs in the scenario without CO_{2}bio fluxes and with the smallest observation error (S1 with 0.1 ppm RMSE-R in Figure 11). The worst case is the one with CO_{2}bio fluxes and the largest observation error but no CO_{2}ff measurements (S2 with 1.0 ppm RMSE-R in Figure 11), in which the error reduction is about 10% limited to the area immediately around the tower locations. The use of CO_{2}ff measurements expands the area of significant error reduction (right column in Figure 11). In addition, more precise atmospheric transport model (*i.e*. smaller random error) significantly enhances the magnitude of error reduction around tower locations from ~40% (1 ppm error) to ~80% (0.1 ppm error), and expands the error reduction area for the three scenarios.

Figure 12 shows the spatially averaged gain and error reduction for CO_{2}ff and CO_{2}bio flux components corresponding to different atmospheric transport errors (Table 3). The reduction of atmospheric transport errors (*i.e*. RMSE-R decreases from 1 ppm to 0.1 ppm) enhances gain and error reduction. Comparing S2 and S3 with the same observation error criterion shows that gain and error reduction are improved by including CO_{2}ff measurements. For the CO_{2}ff flux component, it is interesting to note that S1 with 0.5 ppm error (S1-cII-ff in Figure 12) is equivalent to S3 with 0.1 ppm error (S3-cI-ff in Figure 12), which implies that having precise CO_{2}ff mole fraction measurements and small atmospheric transport error can partially compensate for the interference caused by the CO_{2}bio fluxes. Spatially averaged flux bias is shown in Figure 13. Without CO_{2}bio fluxes (S1), the atmospheric inversion can remove about 70% of the prior flux bias, reducing the bias from 3 *µ*mol m^{–2} s^{–1} in the prior state to less than 1 *µ*mol m^{–2} s^{–1} in the posterior state (S1-cIII-ff in Figure 13). There are still large posterior CO_{2}ff flux biases in S2 (with the presence of CO_{2}bio fluxes but no CO_{2}ff measurements), especially in cases where the observation errors are 0.5 ppm and 1 ppm (S2-cII-ff and S2-cIII-ff in Figure 13). The use of CO_{2}ff measurements (S3) also improves the correction of systematic errors as compared to S2. However, the S3 case with the smallest observation error (S3-cI-ff in Figure 13) is equivalent to the case in S1 with the largest observation error (S1-cIII-ff in Figure 13), which indicates that the influence of CO_{2}bio fluxes is significant for the correction of biases in whole-city CO_{2}ff emissions estimates.

## Conclusions and Discussion

Based on a series of observing system simulation experiments, we demonstrated that high-accuracy and high-precision measurements are necessary to achieve high levels of accuracy and precision in urban CO_{2} flux estimates. Within the bounds of the Indianapolis environment and our assumed prior error structures, random observation errors of 1 ppm or less can reduce systematic flux errors to less than 1 *µ*mol m^{–2} s^{–1} and remove more than 30% of prior random flux errors in the center of the city. A systematic observation error of 1 ppm increased the posterior flux bias over the prior state. In addition, the presence of uncertain biogenic CO_{2} fluxes significantly weakens our ability to invert for anthropogenic CO_{2} emissions, but assimilating continuous high-precision (less than 1 ppm hourly random measurement errors) fossil fuel CO_{2} measurements partially compensates for the degraded performance caused by biogenic CO_{2} fluxes. Moreover, increasing the number of measurement sites from 5 towers to 12 towers enhances the magnitude of error reduction from ~20% to ~40% in the center of the city, and expands the error reduction area. Systematic and random flux errors can be further reduced by reducing model-data mismatch errors caused by atmospheric transport uncertainty. Finally, the precision of the inverse flux estimate is highly sensitive to the correlation length scale in the prior emission errors.

It is important to note that real data inversions are subject to more complexity than synthetic data experiments (Gourdji et al., 2010), but pseudo-data experiments provide a baseline to compare the constraint on fluxes achieved by various inversion setup choices and to illuminate the best achievable performance of real-data inversions. That is, if an approach for quantifying urban emissions fails in a synthetic data experiment, it is unlikely to succeed given the added complication of a real measurement deployment. In order to relate the results of this synthetic data study to a real urban measurement network and inversion system, we discuss three important issues: the atmospheric measurement network, prior flux error structures and atmospheric transport errors.

This study shows that sensor quality (*i.e*. accuracy and precision) must be relatively high to ensure accurate and precise urban inverse flux estimates. Some recent studies found that sensors with lower measurement quality, given sufficient numbers, serve as potentially useful tools for the detection of urban CO_{2} emissions (Wu et al., 2016; Turner et al., 2016; Shusterman et al., 2016; Martin et al., 2017). These studies, however, only considered random error in the sensors, not sensor bias. We demonstrate that even moderately biased sensors (*e.g*. 1 ppm) introduce systematic errors in the posterior flux estimates which can degrade the posterior fluxes to a point that is worse than the prior flux estimates. Even with state-of-the-science instruments, minimization of sensor bias requires extensive inter-calibration (Richardson et al., 2017), and calibration efforts require significant resources which can counter the apparent benefit of sensors that might have a lower initial capital cost. Lower-cost sensors (Stephens et al., 2011) were considered for Indianapolis and ruled out because greater calibration requirements were estimated to cost more to deploy and operate over time than more expensive, but more stable instruments. We also note that Indianapolis is a medium-sized city where the daytime average, city-center CO_{2} enhancement is about 3 ppm (Miles et al., 2017). It is likely that the threshold for sensor quality is related to the magnitude of the urban CO_{2} enhancement.

Sensor development and new analytic methods, particularly for CO_{2}ff measurements, are needed to capitalize on the inversion methods outlined in this study. Our study shows that additional CO_{2}ff measurements can partially compensate for the interference from CO_{2}bio fluxes and improve the inversion performance for CO_{2}ff emissions. We assumed, however, that continuous CO_{2}ff measurements with 1 ppm precision were available. This measurement capability has not yet been demonstrated, but continuous lower-precision measurements are available via a combination of periodic ^{14}C and continuous CO measurements (Levin and Karstens, 2007). During the INFLUX experiment, ^{14}C measurements directly related to CO_{2}ff are collected weekly at 5 towers using a flask sampling system (Turnbull et al., 2012). Continuous measurements of CO could be expanded to all 12 towers. The accuracy and precision of the inference of continuous CO_{2}ff from this potentially observational system (integrating ^{14}C and CO measurements at 12 towers) have not yet been quantified, and are complicated by CO/CO_{2}ff ratios that vary as a function of emission source, and by photochemical CO production. Additional research is needed in pursuit of continuous, accurate and precise CO_{2}ff measurements.

Uncertainty in the spatial structure in prior emission errors greatly limits our ability to map CO_{2} emissions at high resolution with confidence. Prior flux errors are likely to be correlated as a function of emission sectors (*e.g*. traffic, utility, industry). For example, errors in fuel efficiency estimates are probably correlated along highways. A few studies have addressed this problem using hyper-parameter optimization (Desroziers et al., 2005) which provides a direct constraint on the prior emission error structures. For example, Wu et al. (2013) optimized the length scale of Gaussian error structures in a mesoscale inversion system. Similar techniques could be implemented to constrain the spatial structures of emission errors at the urban scale. Direct assessment could also be conducted via the input data and equations used to construct the prior flux estimates (Ogle et al., 2010).

Finally, this study demonstrates that reducing the random errors introduced by uncertainties in atmospheric transport is an effective approach to improving inversion performance. Multiple elements in the atmospheric transport model (*e.g*. parameterization schemes, boundary and initial conditions, and spatial resolution) complicate the assessment of transport errors (Isaac et al., 2014). Evaluation and minimization of transport errors can be achieved by improving model parameterizations (Sarmiento et al., 2017) and by assimilating site-specific meteorological observations (Deng et al., 2017). We note also that our study makes the simplifying assumption of uncorrelated transport errors, whereas, in reality, transport errors are likely to be correlated, especially at the spatiotemporal scales characteristic of an urban study. Our simplifying assumption of uncorrelated transport errors yields the maximum error reduction for a given observational network and assumed error structures. Thus, the current study represents a best-case scenario for the level of error reduction that could be achieved by improving atmospheric transport. The effect of correlated transport errors on inversion performance for urban CO_{2} emissions is an important topic for future studies.

## Data Accessibility Statement

The Hestia inventory is available on the website (http://hestia.project.asu.edu/), and other data from this study can be made available upon request.

## Acknowledgments

The authors would like to thank the editor and two anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.

## Funding information

This study is funded by the National Institute of Standards and Technology (Project # 70NANB10H245).

## Competing interests

The authors have no competing interests to declare.

## Author contributions

Contributed to conception and design: TL KJD KW

Contributed to acquisition of data: TL AD ILC KRG RP

Contributed to analysis and interpretation of data: KW TL KJD

Drafted and/or revised the article: KW KJD TL ILC

Approved the submitted version for publication: KW KJD TL

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