A multiresolution spatial parameterization for the ... · A multiresolution spatial parameterization for the estimation of fossil-fuel carbon dioxide emissions via atmospheric inversions

Geosci. Model Dev., 7, 1901–1918, 2014www.geosci-model-dev.net/7/1901/2014/doi:10.5194/gmd-7-1901-2014© Author(s) 2014. CC Attribution 3.0 License.

A multiresolution spatial parameterization for the estimation offossil-fuel carbon dioxide emissions via atmospheric inversions

J. Ray1, V. Yadav2, A. M. Michalak 2, B. van Bloemen Waanders3, and S. A. McKenna4

1Sandia National Laboratories, P.O. Box 969, Livermore, CA 94551, USA2Carnegie Institution for Science, Stanford, CA 94305, USA3Sandia National Laboratories, P.O. Box 5800, Albuquerque, NM 87185-0751, USA4IBM Research, Smarter Cities Technology Centre, Bldg 3, Damastown Industrial Estate, Mulhuddart, Dublin 15, Ireland

Correspondence to:J. Ray ([email protected])

Received: 28 December 2013 – Published in Geosci. Model Dev. Discuss.: 6 February 2014Revised: 2 July 2014 – Accepted: 18 July 2014 – Published: 3 September 2014

Abstract. The characterization of fossil-fuel CO2 (ffCO2)emissions is paramount to carbon cycle studies, but the useof atmospheric inverse modeling approaches for this pur-pose has been limited by the highly heterogeneous and non-Gaussian spatiotemporal variability of emissions. Here weexplore the feasibility of capturing this variability using alow-dimensional parameterization that can be implementedwithin the context of atmospheric CO2 inverse problemsaimed at constraining regional-scale emissions. We constructa multiresolution (i.e., wavelet-based) spatial parameteriza-tion for ffCO2 emissions using the Vulcan inventory, and ex-amine whether such a parameterization can capture a realis-tic representation of the expected spatial variability of actualemissions. We then explore whether sub-selecting waveletsusing two easily available proxies of human activity (im-ages of lights at night and maps of built-up areas) yieldsa low-dimensional alternative. We finally implement thislow-dimensional parameterization within an idealized inver-sion, where a sparse reconstruction algorithm, an extensionof stagewise orthogonal matching pursuit (StOMP), is usedto identify the wavelet coefficients. We find that (i) the spa-tial variability of fossil-fuel emission can indeed be repre-sented using a low-dimensional wavelet-based parameteriza-tion, (ii) that images of lights at night can be used as a proxyfor sub-selecting wavelets for such analysis, and (iii) thatimplementing this parameterization within the described in-version framework makes it possible to quantify fossil-fuelemissions at regional scales if fossil-fuel-only CO2 observa-tions are available.

1 Introduction

The characterization of fossil-fuel CO2 (ffCO2) emissionsis paramount to carbon cycle studies. ffCO2 emissions arethe largest net carbon flux at the atmosphere–surface in-terface (Friedlingstein et al., 2006) and spatially disaggre-gated (or gridded) ffCO2 emissions form a critical inputinto general circulation and integrated assessment models(Andres et al., 2012). An understanding of fossil-fuel emis-sions is clearly necessary for characterizing the anthro-pogenic climate impact. In addition, a process-level under-standing of the terrestrial carbon sink requires the quantifi-cation of terrestrial biospheric fluxes at fine spatiotemporalscales, which, in turn, requires the differentiation betweenanthropogenic and biospheric fluxes at those scales.

Gridded inventory estimates of ffCO2 emissions can be de-rived using socio-economic data (Oda and Maksyutov, 2011;Rayner et al., 2010), and such “bottom-up” estimates havebeen proposed as a means of monitoring international agree-ments aimed at mitigating ffCO2 emissions (Pacala et al.,2010). Gridded inventory estimates are derived from ffCO2budgets and produced by a few institutions; seeAndres et al.(2012) for a list. These budgets are compiled from nationaland provincial statistics on fossil-fuel production and con-sumption. These large-scale estimates can then be down-scaled to finer spatiotemporal scales using easily observedproxies of human activity (and consequently ffCO2 emis-sions) such as images of lights at night (henceforth “night-lights”), population density, etc. (Oda and Maksyutov, 2011;Rayner et al., 2010; Doll et al., 2000). More sophisticated

Published by Copernicus Publications on behalf of the European Geosciences Union.

1902 J. Ray et al.: A spatial parameterization for fossil-fuel carbon dioxide emissions

approaches to the fine-scale bottom-up estimation of ffCO2emissions have also begun to emerge, including, for exam-ple, the Vulcan inventory that includes estimates for the US ata 10 km and hourly resolution for 2002 (http://vulcan.project.asu.edu; Gurney et al., 2009). Such approaches rely on de-tailed reporting and monitoring data, which are not currentlyavailable for many regions of the world.

Although inventory estimates provide a key tool in theunderstanding of anthropogenic CO2 emissions, their accu-racy at large scales depends on the accuracy of reportednational consumption data, e.g., the error in ffCO2 emis-sions from China lies in the 15–20 % range (Gregg et al.,2008). When evaluated at finer spatiotemporal scales, theiraccuracy also depends on the method used to disaggregatenational/provincial ffCO2 emission budgets to finer spatialscales.Pregger et al.(2007) showed that two 0.5◦ invento-ries for Western Europe, for 2003, differed at the grid-celllevel by 20 %, with a standard deviation of 40 %; at finer res-olutions, the disagreement worsened. Sources of errors in in-ventories are discussed in detail inAndres et al.(2012), Rau-pach et al.(2009) andRayner et al.(2010). These uncertain-ties lead to frequent corrections of sub-national inventories ofcombustion products (Streets et al., 2006) and model predic-tions of CO2 concentrations that disagree with observationslocally (Brioude et al., 2012).

Given both the benefits, and the limitations, of inventory-based estimates, interest has emerged in the development of“top-down” estimates of ffCO2 emissions. These estimatesrely on attributing the observed variability in CO2 and re-lated trace gas concentrations in the atmosphere to the under-lying fossil-fuel emissions, through the application of statis-tical atmospheric inversion methods. Some of the proposedapproaches, (e.g.,Turnbull et al., 2011) have relied on obser-vations of114CO2 measurements or other non-CO2 tracers.One challenge with these approaches, however, is a combina-tion of the limited number of available observations and theneed to understand emission ratios for any co-emitted tracers.Atmospheric inversions relying on atmospheric CO2 mea-surements, on the other hand, have primarily targeted bio-spheric CO2 fluxes, often by first pre-subtracting the fossil-fuel influence calculated from an inventory; seeCiais et al.(2010) for a review of atmospheric inversion methods. Themeasurements then consist of CO2 concentration obtainedfrom in situ or remote-sensing observations, and estimatesare obtained at a variety of spatiotemporal resolutions for ei-ther global or regional domains. The statistical approachesapplied typically rely on Gaussian assumptions for flux resid-uals from prior estimates or other spatiotemporal patterns. In-vestigations aimed at the estimation of ffCO2 emissions areless common because (1) measurements of ffCO2 concen-trations, (e.g., using114CO2) are expensive and not com-prehensive and (2) the statistical assumptions used in inver-sions aimed at understanding biospheric fluxes are ill-suitedto the highly heterogeneous and non-Gaussian spatiotempo-ral variability of ffCO2 emissions. However, some estimates

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Figure 1. Differences in the spatial distribution of biospheric (top) and fossil-fuel (bottom) CO2 fluxes.The biospheric fluxes are stationary, whereas ffCO2 emissions are non-stationary and correlated withhuman habitation. The fluxes/emissions cover June 1 - June 8, 2002. The biospheric fluxes are obtainedfrom CASA-GFED (http://www.globalfiredata.org/index.html). The post-processing steps to obtain thefluxes as plotted are described in Gourdji et al. (2012). The units of fluxes/emissions are µmols−1 m−2

of C. The ffCO2 emissions are calculated by spatiotemporal averaging of the Vulcan inventory. Note thedifferent colormaps; ffCO2 emissions can assume only non-negative values.

33

Figure 1. Differences in the spatial distribution of biospheric (top)and fossil-fuel (bottom) CO2 fluxes. The biospheric fluxes are sta-tionary, whereas ffCO2 emissions are non-stationary and correlatedwith human habitation. The fluxes/emissions cover 1–8 June 2002.The biospheric fluxes are obtained from CASA-GFED (http://www.globalfiredata.org/index.html). The post-processing steps to obtainthe fluxes as plotted are described inGourdji et al.(2012). The unitsof fluxes/emissions are µmol s−1 m−2 of C. The ffCO2 emissionsare calculated by spatiotemporal averaging of the Vulcan inventory.Note the different color maps; ffCO2 emissions can assume onlynon-negative values.

of ffCO2 emissions at urban scales are beginning to emerge,including for Salt Lake City (McKain et al., 2012), Indi-anapolis (Gurney et al., 2012), and Sacramento (Turnbullet al., 2011).

Geosci. Model Dev., 7, 1901–1918, 2014 www.geosci-model-dev.net/7/1901/2014/

http://vulcan.project.asu.edu

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J. Ray et al.: A spatial parameterization for fossil-fuel carbon dioxide emissions 1903

The goal of the work presented here is to address the sec-ond limitation above by exploring the possibility of defin-ing an inversion framework that is specifically targeted atthe characteristics of the spatiotemporal variability of ffCO2emissions at regional scales. We will model spatial variabilityat 1◦

× 1◦ resolution. Such a framework would require,among other things, a low-dimensional spatial parameteriza-tion of ffCO2 emissions, given the data limitations associatedwith any atmospheric inversion system. We explore this topicthrough a sequence of three specific objectives:

1. Identification of a low-dimensional parameterizationfor ffCO2 emissions. ffCO2 emissions are strongly non-stationary (see Fig.1), and any parameterization mustbe able to represent such variability. Wavelets, whichare an orthogonal basis set with compact support, arewidely used to model non-stationary fields, e.g., im-ages (Strang and Nguyen, 1997; Chan and Shen, 2005).We will examine a number of wavelet families to iden-tify the wavelet type that can represent ffCO2 emissionsmost efficiently, i.e., with minimum error if only a lim-ited number of wavelets were to be retained. The ffCO2emissions will be obtained from the Vulcan inventory(for the US only) as a realistic example of what the vari-ability of true ffCO2 emissions is likely to be. This ob-jective will ultimately answer the question of whethera low-dimensional parameterization is possible for thetype of spatial variability expected in real ffCO2 emis-sions.

2. Evaluation of the use of a low-dimensional parameter-ization in combination with easily available proxies ofanthropogenic emissions.For most areas of the world,fine-scale estimates of ffCO2 emissions are based ondownscaling of national inventories using easily ob-served proxies of human activity, such as maps of night-lights or of built-up areas (BUA). In this second objec-tive, we will use the wavelet types selected in the firstobjective, and sub-select them using these two proxiesof human activity for the United States. We will thenevaluate the degree to which the remaining wavelets canbe used to represent the complexity of spatial patterns inffCO2 emissions. The Vulcan inventory will be used forthis purpose too. This objective will answer the ques-tion of whether an easily observable proxy can be usedto reduce the dimensionality of a wavelet-based spatialparameterization for ffCO2 emission fields. The set ofwavelets selected in this manner form a random fieldmodel, which we will refer to as the multiscale randomfield (MsRF) model.

3. Evaluation of the parameterization in an atmosphericinversion, using sparse reconstruction.In the third ob-jective, we will use the reduced basis identified in Ob-jective 2 within an idealized synthetic-data atmosphericinversion aimed at characterizing the spatiotemporal

variability of US ffCO2 emissions. A new sparsity-enforcing optimization method that preserves the non-negative nature of ffCO2 emissions will be used to solvethe inverse problem. (The termssparsityandsparsity-enforcementare defined in Sect.2.) The new sparse re-construction method is used to ensure an unique solu-tion and to guard against overfitting (fitting to obser-vational noise). The optimization procedure will iden-tify the subset of wavelets in the MsRF that can actu-ally be estimated from the observations, while “turningoff” the rest. In doing so, it will ensure that the MsRF,as designed, has sufficient flexibility to extract informa-tion on ffCO2 from the observations. For simplicity, thesynthetic-data inversion will focus only on ffCO2 emis-sions. We recognize that an ultimate application withreal data would require a combination of methods tocapture both the biospheric and fossil-fuel signals, orwould require the pre-subtraction of the influence ofbiospheric fluxes on observations. For the purposes ofthe work presented here, however, the question that weaim to answer is whether an inversion approach basedon a low-dimensional parameterization is feasible, evenunder idealized conditions.

We view this investigation as a methodological first stepin the development of an inversion scheme for ffCO2 emis-sions. To that end, we focus on algorithmic and parameter-ization issues, and demonstrate our solution in an idealizedsetting. We use synthetic-data collected from a measurementnetwork sited with an eye towards biospheric CO2 fluxes, asa network optimized for ffCO2 measurement does not ex-ist. We also ignore emissions outside the US. Further, weassume that the errors incurred by the transport model aresmall, uniform across time and all measurement locations.Thus the method would need to be extended to be used ina realistic setting; we identify some of these adaptations, aswell as potential starting points for such investigations. Theproposed method, by construction, addresses two issues pe-culiar to ffCO2 emissions: (1) it is insensitive to underreport-ing of country-wide ffCO2 emissions; and (2) it can (approx-imately) capture intense regions of ffCO2 emissions, evenwhen the spatial parameterization is deficient.

The paper is structured as follows. In Sect.2, we reviewthe use of wavelet modeling in inverse problems. In Sect.3,we investigate families of wavelets for modeling ffCO2 emis-sions and construct two MsRF models, based on nightlightsand maps of BUA. In Sect.4, we describe the inverse prob-lem and the numerical method used to solve it. In Sect.5we perform inversion tests with synthetic data. In Sect.6 wediscuss the idealizations adopted in this paper and how theymay be relaxed. Conclusions are in Sect.7.

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2 Wavelet modeling in inverse problems

Wavelets are a family of orthogonal bases with compact sup-port (Williams and Amaratunga, 1994; Walker, 2008). Theyare generated using a scaling functionφ′ that obeys the re-cursive relationship:φ′(x) =

∑i ciφ

′(2x − i). A waveletφis generated from the scaling function by taking differences,e.g.,φ(x) =

∑i(−1)ic1−iφ

′(2x − i). The choice of the fil-ter coefficientsci and φ′ determine the type of the result-ing wavelets. The simplest type is the Haar, which are sym-metric, but not differentiable. Wavelets can be shrunk andtranslated, i.e.,φs,i = 2

s2 φ(2sx − i), wheres is the dilation

scale andi refers to translation (location). This allows themto model complex, non-stationary functions efficiently. Foreach increment in scale, the support of the wavelet halves.Wavelets are defined on dyadic (power-of-two) hierarchicalor multiresolution grids.

Consider a domain of sizeD, discretized by a hierarchy ofmeshes with resolutions1D/D = {1,1/2,1/22, . . .1/2M

}.Wavelets are defined on each of the levels of the hierarchi-cal mesh and can be positioned at any even-numbered grid-cell 2i,0 ≤ 2i ≤ 2s

− 1, on any scales of the hierarchicalmesh. An arbitrary 1-D functiong(x) can be represented as

g(x) = w′φ′(x)+∑M

s=1∑2(s−1)

−1i=0 ws,iφs,i(x), where the co-

efficients (or weights)ws,i andw′ are obtained, via projec-tions ofg(x), using fast wavelet transforms. In 2-D, a func-tion g(x,y), defined on aD × D domain with a hierarchi-cal 2M

× 2M mesh, can be subjected to a wavelet transformby applying 1-D wavelet transforms repeatedly, e.g., first byrows and then by columns. Wavelets of scales have a sup-port 2M−s

×2M−s,0 ≤ s ≤ M and can be positioned (in 2-Dspace) at location(i,j),0 ≤ (i,j) < 2s . A 2-D wavelet trans-form results in 2M × 2M wavelet coefficients. In general,

g(x,y) = w′φ′(x,y) +

M∑s=1

W(s)∑i,j

ws,i,jφs,i,j (x,y), (1)

whereW(s), |W(s)| = (4s

− 4s−1), is the set of(i,j) indicesof wavelet coefficients on scales. |W(s)

| is the size of theset, i.e., the number of wavelets inW(s). A large number offine-scale (i.e., highs) wavelets model fine spatial details.

2.1 Wavelet-based random field models

Random field (RF) models (Cressie, 1993) provide a system-atic way of generating multi-dimensional fields based on val-ues assumed by the model’s parameters. The parameters areindependent and can assume random values, thus generatingrandom fields. Any structure that a field may need to obey,e.g., smoothness, is encoded into the model. RF models maybe used for dimensionality reduction, e.g., one can generatefields on a fine grid by varying a handful of parameters. Al-ternatively, one can design a RF model to generate fields sys-tematically, by independently varying parameters that controlstructures at different spatial scales. The spatial parameteri-

zation that we construct in Sect.3 is an instance of the lattertype of RF model.

Wavelets are often used to represent fields, most com-monly, to represent images, e.g., the JPEG2000 stan-dard (Taubman and Marcellin, 2002). Most fields/images arecompressible in a judiciously chosen wavelet basis, i.e., mostof the wavelet coefficients are small and can be discarded toform a reduced-rank approximation of the field (Welstead,1999). In Auger and Tangborn(2002), a reduced-rankwavelet model was developed for global, time-variant CH4concentration, which were updated with limited observationsusing a Kalman filter. The selection of wavelets to form therandom field (RF) model was done empirically, by decimat-ing the fine-scale wavelets. The construction of the RF modelcan be performed more rigorously if a prior model is avail-able. InJafarpour(2011) wavelets were used in the recon-struction of permeability fields from limited measurementsof flow through a porous medium. An ensemble of perme-ability field realizations, drawn from the prior distribution,were used to develop a multivariate Gaussian prior distri-bution for the wavelet coefficients. The RF model was cre-ated by discarding wavelet coefficients with small means.In Romberg et al.(2001), the authors constructed a hid-den Markov model to encode the relationship between thewavelet coefficients on adjacent scales of the wavelet tree.This RF model was used to reconstruct compressively sensedsignals (Duarte et al., 2008) and images (He and Carin,2009). Thus the use of wavelet-based RF models to parame-terize and reconstruct complex fields from limited measure-ments is quite common.

The RF model need not be constructed offline usinga prior; it can also be constructed during the inversion, ina data-driven manner. This occurs in the compressive sens-ing (CS) of signals and images (Candes and Wakin, 2008).In this approach, all the wavelets in a field’s representationare retained and the ones that cannot be estimated from avail-able observations are identified and removed. Letg be an im-age of sizeN that can be represented sparsely usingL � N

wavelets. Letg′, of size Nm, L < Nm � N , be its com-pressed measurement, obtained by projectingg onto a set ofrandom vectorsψ i , i.e.,g′

=9g =98w. Here the rows of9 consist of the random vectorsψj and columns of8 con-sist of waveletsφi .8 is aN ×N matrix while9 is Nm ×N .The bulk of the theory that allows reconstruction of the im-age/field with such observations was established inCandesand Tao(2006), Donoho(2006), andCandes et al.(2006).In CS, the reconstruction ofg (alternativelyw) can be per-formed using a number of methods. It is posed as an opti-mization problem

minimizew∈RN

‖w‖1, subject to‖g′− Aw‖2 < ε2, A =98. (2)



Thus we enforce sparsity inw with its `1 norm (|| · ||1),under the constraint that the2 norm (|| · ||2) of the mis-fit betweeng′ and the image reconstructed fromw is keptwithin a bound. Some of the methods to solve this problemare basis pursuit (Chen et al., 1998), matching pursuit (Mal-lat and Zhang, 1993), orthogonal matching pursuit (Troppand Gilbert, 2007) and stagewise orthogonal matching pur-suit (StOMP;Donoho et al., 2012). We will refer to the pro-cess of enforcing sparsity as “sparsification” and the result ofthe process will be called the “sparsified” or sparse solution.

2.2 Wavelets and sparsity in inverse problems

Sparsity is often used to solve inverse problems in physics,with the9 operator representing the physical process. InLiand Jafarpour(2010), sparsity was used to estimate a per-meability field (represented by wavelets) from fluid trans-port observations. A good review of the use of sparsity inpermeability field reconstruction is available inJafarpour(2013). Seismic tomography, which estimates subsurface ge-ologic facies, also has exploited sparsity for reconstruction.This has been demonstrated with wavelet representations ofthe subsurface and nonlinear forward models (Loris et al.,2007; Simons et al., 2011). Gholami and Siahkoohi(2010)used a split Bregman iteration (Goldstein and Osher, 2009)to solve a seismic tomography problem, imposing sparsityvia soft thresholding (Donoho, 1995). The authors also ex-panded the imposition of sparsity from the wavelet space tothe finite-difference space, i.e., they used an`1 norm to spar-sify deviations of the solution from a “best-guess”, in the ab-sence of constraining observations. In this manner, both theprior/guessed solution and sparsity are used to regularize theinversion.

To summarize, wavelet-based RF models are routinelyused in inverse problems. Their dimensionality can be re-duced a priori using prior information. Data-driven dimen-sionality reduction can also be performed by enforcing spar-sity during the inversion. This has been demonstrated withnonlinear problems too.

3 Constructing a multiscale random field model

We seek an approximate representation of ffCO2 emissions,which is low dimensional or sparse, i.e., many of thews,i,j

in Eq. (1) may be set to zero. For this purpose, we use ffCO2emissions from the Vulcan inventory, coarsened to 1◦ reso-lution and averaged over a year to yieldfV (see Fig.1 fora plot). The emissions are described on a 2M

× 2M grid,M = 6. The finest wavelets, on scales = 6, have a supportof 2◦

× 2◦; the ones ons = 5 are a factor of two bigger, i.e.,they have a support of 4◦

× 4◦. The rectangular domain ex-tents are given by the corners 24.5◦ N, 63.5◦ W and 87.5◦ N,126.5◦ W. ffCO2 emissions are restricted toR, the lower 48states of the US.

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3.1 Choosing a wavelet

We investigate a number of wavelet families (Haar,Daubechies, Symlet, and Coiflet) in order to determine whichprovides the sparsest representation offV . fV is first subjectedto a wavelet transform using a chosen wavelet. At each scales, we remove wavelets that contribute little tofV . We set“small” wavelet coefficients (“small” is defined as the ra-tio |ws,i,j/wmax,s | ≤ 10−3, wherewmax,s is the wavelet co-efficient on scales with the largest magnitude) to zero. Werefer to the fraction of zero wavelet coefficients at scales

as itssparsity. Figure2 plots the sparsity at scales = 4,6for a large combination of wavelet families and orders. Wesee that the Haar wavelet (also called the Daubechies, or-der 2 wavelet) provides the sparsest representation, makingit a candidate for developing a low-dimensional MsRF forffCO2 emissions. This is a consequence of the spatial distri-bution of fV – the map offV (Fig. 1b) is largely empty westof 100◦ W, which manifests itself as small coefficients for thewavelets whose support lie in that region. This favors simpler(non-smooth) and low-order wavelets.

Next we investigate the variation of the magnitude ofwavelet coefficients, as a function of the type of waveletsused to modelfV . We select five wavelet types, e.g., Haar,Daubechies, order 4 and 8, and Symlet, order 4 and 6, andperform a wavelet transform offV . At each scales, we setthe “small” wavelet coefficients to zero. In Fig.3 we plotthe average and standard deviation of the non-zero waveletcoefficients. The means of the wavelet coefficients at thefiner scales are small, regardless of the wavelet type. Wesee that while Haar provide the sparsest representation, thenon-zero wavelet coefficients tend to have large magnitudes.In contrast, smoother wavelets with broader support, e.g.,



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Figure 3. We plot the average value of the non-zero coefficients(solid lines) and their standard deviation (dashed line), at differentscaless, when fV is subjected to wavelet transforms using Haar,Daubechies 4 and 6, and Symlet 4 and 6 wavelets. We find that whileHaar may provide the sparsest representation (Fig.2), the non-zerovalues tend to be large and distinct.

Daubechies, order 8, have more non-zero wavelet coeffi-cients, but with smaller wavelet coefficients. This is a con-sequence of the spatial distribution offV (Fig. 1) which hassharp gradients, placing smooth wavelets at a disadvantage.In Fig. 3, we also see that the means and standard deviationsshrink, especially after scales = 3; further, the distributionsof wavelet coefficients arising from the different wavelettypes begin to resemble each other. This arises from the factthat there are sharp boundaries around the areas where ffCO2emissions occur; when subjected to a wavelet transform, theregion in the vicinity of a sharp boundary gives rise to largewavelet coefficients down to the finest scale. Thus the fewnon-zero wavelet coefficients at the finer scales assume sim-ilar values, irrespective of the wavelet type.

Finally, we check the accuracy of a Haar representation offV at various levels of sparsity. Again, we define a “small”wavelet as|ws,i,j/wmax,s | ≤ α. We perform a wavelet trans-form of fV using Haar wavelets and sparsify (set smallwavelets to zero) using 10−6

≤ α ≤ 1. We then performan inverse transform to reconstruct a “sparsified”fV

′. In

Fig. 4, we plot overall sparsity, reconstruction errorε =

‖fV′− fV‖2/‖fV‖2 and the Pearson correlation between the

true and reconstructedfV as a function ofα. We define thePearson correlation betweenfV

′andfV as

ρ(fV

′, fV)

=cov(fV, fV

′)

σfVσfV

′

,

whereσ 2fV

andσ 2fV

′ are the variances of the true and recon-

structed fluxes and cov(Z1,Z2) is the covariance betweentwo random variablesZ1 andZ2. Here‖ · ‖2 denotes the2norm. We find that forα < 10−2 there is practically no error

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Figure 4. Variation of sparsity, reconstruction errorε, and thePearson correlation between the true and reconstructedfV , i.e.,ρ(fV , fV

′) as a function ofα.

(as measured by these metrics) though we achieve a sparsi-fication of about 80 %. Even at a sparsity of around 90 %,the error is less that 10 %. Thus a small collection of Haarwavelets have the ability to reproducefV with an acceptabledegree of error. This low-dimensional character of a Haarrepresentation offV can be invaluable in an inverse prob-lem predicated on sparse observations, and henceforth, wewill proceed with Haar wavelets as the basis set of choice forrepresenting ffCO2 emissions. Figure5 shows the decompo-sition of fV on aM = 6 hierarchy of Haar wavelets.

We will use wavelets selected using the (single) nightlightand BUA maps to estimate weekly ffCO2 emissions. Ourtests above show that they model annually averaged Vulcanemissions adequately, and we assume that while emissionsmay wax and wane with time, their spatial distribution doesnot vary sufficiently to require a new wavelet selection. Webase this assumption on ffCO2 emissions’ correlation withhuman activities and static sources like powerplants which donot display large spatial dislocations with time. Note that thelocation and strengths of intense sources of ffCO2 emissions,such as powerplants, can be found at the CARMA (CarbonMonitoring for Action; websitehttp://carma.org). Note, also,that CARMA isnotpeer-reviewed and only provides data fora limited number of years (version 3.0 has data only for 2004and 2009).

3.2 Constructing a random field model

We seek a spatial parameterization for ffCO2 emissions, ofthe form

f = w′φ′+

M∑s=1

∑i,j

ws,i,jφs,i,j , {s, i,j} ∈ W (s), (3)

where W (s) is a set containing a small number ofHaar bases. We will investigate the usefulness of an


http://carma.org


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Figure 5. Annually averaged Vulcan emissions fV are modeled using Haar wavelets on scales 1, 2, 4, 5and 6. The figure at the top (left) plots fV , and the rest its decomposition across wavelet scales. Note thatwe have displayed only the relevant part of the dyadic 2M × 2M grid on which wavelets are described.

37

Figure 5. Annually averaged Vulcan emissionsfV are modeled using Haar wavelets on scales 1, 2, 4, 5, and 6. The figure at the top (left)plotsfV , and the rest its decomposition across wavelet scales. Note that we have displayed only the relevant part of the dyadic 2M

×2M gridon which wavelets are described.

easily observed proxyX of human activity (whichcorrelates with ffCO2 emissions) to select the com-ponents of W (s). Radiance calibrated nightlights(http://www.ngdc.noaa.gov/dmsp/download_radcal.html;Cinzano et al., 2000) have been used for constructingffCO2 inventories (Doll et al., 2000) and are an obvi-ous choice forX. However, nightlight radiances are also

affected by economic factors (Raupach et al., 2009),and we will explore maps of built-up area as an alterna-tive (http://www.sage.wisc.edu/atlas/maps.php?datasetid=18&includerelatedlinks=1&dataset=18; B. Miteva, personalcommunication, 2013). The map of BUA uses nightlightradiances in its computations, and so these arenot inde-pendent proxies; however, the BUA map also includes


http://www.ngdc.noaa.gov/dmsp/download_radcal.html

http://www.sage.wisc.edu/atlas/maps.php?datasetid=18&includerelatedlinks=1&dataset=18

http://www.sage.wisc.edu/atlas/maps.php?datasetid=18&includerelatedlinks=1&dataset=18


information from IGBP (International Geosphere–BiosphereProgramme,http://www.igbp.net/) land-cover map. The twochoices forX will be compared with respect to (1) sparsity,(2) the correlation betweenX and fV , and (3) the ability ofW (s) to capturefV .

In Fig.6 (top row), we plot maps of the two proxies, coars-ened to 1◦ resolution. Comparing with Fig.1 (bottom), wesee that they bear a strong resemblance tofV . We subjectX toa wavelet transform and set all wavelet coefficients|ws,i,j | <

δ to zero, whereδ is a user-defined threshold. The bases withnon-zero coefficients are selected to constituteW (s). We re-constitute a “sparsified” proxy,X(s), using just the bases inW (s), and compute the correlation betweenX(s) andfV . Fi-nally, we projectfV onto W (s), obtain its “sparsified” form

fV(s)

, and compute the errorεf = ‖fV(s)

− fV‖2/‖fV‖2. InFig. 6 (middle row), we plot the sparsity, correlation andεf

for various values ofδ. We do so for both nightlights andBUA. For nightlights, we achieve a sparsity of around 0.75for δ < 10−2, i.e., we need to retain only a quarter of the Haarbases to represent nightlights. The nightlights so representedbear a correlation of around 0.7 withfV , and achieve an errorεf of around 0.1. Note that this measure of error reflects theinability of the MsRF to represent fine-scale details, and notspatially aggregated quantities, which are represented moreaccurately. In contrast, using BUA as a proxy, we see thatwhile the sparsity achieved is similar, the correlation betweenX(s) andfV is slightly higher. The behavior ofεf is similar,except the error increases faster withδ, as compared to night-lights. However both nightlights and BUA maps show signif-icant correlation withfV and the sparsified set of Haar basesthat they (i.e., the proxies) provide (usingδ = 10−2 in boththe cases) allow us to construct a low-dimensional parame-terization of ffCO2 emissions.

Finally, we useX(s) to create a “prior model”fpr = cX(s)

for ffCO2 emissions,f. c is computed such that

∫R

fVdA =

∫R

fprdA = (4)

c

∫R

(w′

(X)φ′+

∑l,i,j

w(X),s,i,jφl,i,j

)dA, {l, i,j} ∈ W (s),

whereR denotes the lower 48 states of USA andw(X) =

{w(X),s,i,j } are coefficients from a wavelet transform ofX.This implies thatc is calculated such that bothfV and fprprovide the same value for the total emissions for the US.c

is the ratio of the aggregate total of ffCO2 emissions to theaggregate total of radiances (for the nightlights) or percent-ages of built-up areas. In Fig.6 (bottom row), we plot theerror (fpr − fV). (The Supplement contains a scatter plot offpr vs. fV .) We see that neither nightlights nor the BUA mapprovide afpr that is an accurate representation offV , thoughthey share similar spatial patterns, i.e.,fpr may be used to

provide a guess forf in an inverse problem, but, by itself, isa poor predictor, regardless of the proxyX used to create it.

4 Formulation of the estimation problem

In this section, we pose and solve an inverse problem to esti-mate ffCO2 emissions using the MsRF developed in Sect.3.The method is new, and uses a sparse reconstruction methodthat is summarized in Sects.4.1 and 4.2; full details arein Ray et al.(2013). The inversion technique is most relevantin situations where accurate, finely gridded ffCO2 invento-ries are unavailable, and one has to take recourse to easilyobservable proxies for information on the spatial pattern offfCO2 emissions.

The inverse problem is predicated on synthetic observa-tions,yobs, of CO2 concentrations measured at 35 towers (anetwork that existed in 2008). These are plotted as markersin Fig. 8; seeRay et al.(2013) for their precise locations andnames. The measurements are related to ffCO2 emissions de-scribed on a finely gridded domain as

yobs= y + ε = Hf + ε, (5)

whereH is the transport or sensitivity matrix, obtained froma transport model,y is the CO2 concentration predictedby the atmospheric model, which differs from its measuredcounterpart by an errorε. The ffCO2 emissionsf are definedon a grid, and are assumed to be non-zero only withinR.

The estimation of CO2 fluxes, typically biospheric (Nassaret al., 2011; Chatterjee et al., 2012; Gourdji et al., 2012), isusually posed as the minimization of an objective functionJ ,

J = (yobs−Hf)TR−1

e (yobs−Hf)+(f−fm)TQ−1(f−fm), (6)

where fm are “prior” (or guessed) fluxes andR−1e is a di-

agonal matrix containing the standard deviation of Gaussiannoise used to model measurement error. The discrepancybetween the “true” and prior fluxes is modeled as a multi-variate Gaussian field, whose covarianceQ is calculated of-fline. In contrast, in our ffCO2 inversion,f will be modeledusing the MsRF rather than a multivariate Gaussian field.Further, the second term in Eq. (6) is omitted and the ef-fect of the “guessed” or “prior” emissionsfpr is introducedin a manner that is amenable to sparse reconstruction (seeSect.4.1). The calculation of the sensitivitiesH is describedin detail inGourdji et al.(2012); we have reused them in ourwork. The elements of theH matrix are calculated using theStochastic Time-Inverted Lagrangian Transport Model (Linet al., 2003), with wind fields from the Weather Research& Forecasting model (Skamarock and Klemp, 2008), ver-sion 2.2, driven by 2008 meteorology. Details of the WRFsettings and the nested grid used for the wind fields to cal-culateH are in Gourdji et al. (2012). Concentration foot-prints (or sensitivities) were calculated at 3 h intervals by


http://www.igbp.net/


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Figure 6. Top row: Maps of nightlight radiances (left) and BUA percentage (right), for the US. Middlerow: The sparsity of representation, the correlation between X and fV and the normalized error εf be-tween the Vulcan emissions fV and the sparsified form obtained by projecting it on X. These values areplotted for nightlights (left) and the BUA maps (right). Bottom row: Plots of (fpr− fV) obtained fromnightlights (left) and BUA maps (right).

38

Figure 6.Top row: maps of nightlight radiances (left) and BUA percentage (right), for the US. Middle row: the sparsity of representation, thecorrelation betweenX andfV and the normalized errorεf between the Vulcan emissionsfV and the sparsified form obtained by projecting

it on X. These values are plotted for nightlights (left) and the BUA maps (right). Bottom row: plots of(fpr − fV) obtained from nightlights(left) and BUA maps (right).

integrating the trajectories over a North American 1◦× 1◦

grid as described inLin et al. (2003). The sensitivity ofthe CO2 concentration at each observation location due tothe flux at each grid cell (the “footprint”) is calculated inunits of ppmv µmol−1 m2 s (ppmv: parts per million by vol-ume). ffCO2 emissions were averaged over 8-day intervalsand the sensitivity ofy to the 8 day-averaged emissions were

obtained from the 3 h sensitivities described above by sim-ply adding the 8×24/3 = 64 sensitivities that span the 8-dayperiod. Thereafter, the grid cells outsideR were removed toobtain theH matrix used in this study. The size of theH ma-trix is (KsNs) × (NRK), whereKs is the number of towermeasurements per year,Ns is the number of sensors/towers,NR is the number of grid cells inR, the part of the domain



covered by the lower 48 states of the US andK is the numberof 8-day periods that constitute the duration over which theemissions are estimated.

4.1 Posing and solving the inverse problem

We denote the spatial distribution of emissions during an ar-bitrary 8-day periodk asfk. The 8-day period was chosen tominimize aggregation error. We seek emissions over an en-tire year, i.e., we seekF = {fk},k = 1. . .K. We will modelemissions on the 2M × 2M ,M = 6 mesh with wavelets:

fk = w′

kφ′+

M∑s=1

∑i,j

ws,i,j,kφs,i,j , {s, i,j} ∈ W (s)

=8wk. (7)

Note that8 comprises of only those wavelets selectedusing X and contained inW (s). For the entire year,the expression for emissions becomesF = {f1, f2, . . . fK} =

{8w1,8w2, . . .8wK} = 8w. Since8wk models the emis-sions over all grid cells, i.e., over the rectangular region givenby the corners (24.5◦ N, 63.5◦ W) and (87.5◦ N, 126.5◦ W),and not justR, F contains emissions over the lower 48 states,as well as the region outside it (where we have assumedthat the emissions are non-existent). We separate out the twofluxes by permuting the rows of8

F =

(FRFR′

)=

(8R8R′

)w,

where8R and8R′ are(NRK)×(LK) and(NR′K)×(LK)

matrices, respectively. HereL is the number of wavelets inW (s) andNR′ is the number of grid cells inR′, the regionoutsideR but inside the rectangular domain. The modeledconcentrations at the measurement towers, caused byFR,can be written asy = HFR. For arbitraryw, FR′ , the emis-sions in the region outsideR, are not zero. Consequently, itwill be necessary to specifyFR′ = 0 as a constraint in theinverse problem.

Specifying the constraintFR′ = 0 directly is not very ef-ficient since it leads toNR′K constraints. In a global inver-sion, or at resolutions higher than 1◦

×1◦, this could get verylarge. Consequently, we adapt an approach from compressivesensing to enforce this constraint approximately. ConsideraMcs× (NR′K) matrixR, whose rows are direction cosinesof random points on the surface ofNR′K-dimensional unitsphere. This is called a uniform spherical ensemble (Tsaigand Donoho, 2006). The projection of the emission fieldFR′

onR, i.e.,RFR′ compressively samplesFR′ . Mcs is the num-ber of such projections or compressive samples. Setting thisprojection to zero during inversion allows us to enforce zeroemissions outsideR. However, to do so, we add onlyMcsconstraint equations. The computational savings afforded byimposing theFR′ = 0 constraint in this manner is investi-gated inRay et al.(2013).

The equivalent of Eq. (5) is written as

Y =

(yobs

0

)≈

(H 8RR8R′

)w = Gw. (8)

We incorporate the spatial patterns inX into the esti-mation procedure by usingw(X) to normalizew. Other,less effective, methods were investigated and discarded inRay et al.(2013). We rewrite Eq. (8) as

Y ≈ Gdiag(w(X)

)diag

(w−1

(X)

)w = G′w′

=

(H 8

′

RR8

′

R′

)w′, (9)

wherew′= {ws,i,j/(c w(X),s,i,j )}, {s, i,j} ∈ W (s), is the nor-

malized set of wavelet coefficients,8′

R = 8R diag(w(X))

and8′

R′ = 8R′ diag(w(X)).The underdetermined system Eq. (9) is solved using opti-

mization. Given the small number of towers (35) and their lo-cation (the towers were sited with biospheric fluxes in mind),it may not be possible to estimate all the elements ofw′, es-pecially those that contribute to the fine-scale details ofFR.Further, a priori, we do not know the identity of these “un-estimatable” wavelet coefficients inw′. Consequently, weemploy a sparse reconstruction method, based on`1 mini-mization, that identifies and estimates the elements ofw′ thatcan be constrained byyobs, while setting the rest to zero. Wecast the optimization problem as

minimizew′∈RN

‖w′‖1, subject to‖Y − G′w′

‖22 < ε2. (10)

This is of the same form as Eq. (2) and is solved usingstagewise orthogonal matching pursuit (StOMP) (Donohoet al., 2012).

4.2 Imposing non-negativity on ffCO2 fluxes

Estimates ofw′ calculated by StOMP do not necessarilyprovide non-negative estimates ofFR = 8Rw. In practicenegative ffCO2 emissions occur in only a few grid cellsand are usually small in magnitude. We devised an iterativemethod to impose non-negativity as a post-processing step.We present a summary here; details are inRay et al.(2013).

We use the StOMP solution to generateF = 8w, discardFR′ and manipulate the emissionsE = {Ei}, i = 1. . .NRK

in R directly. We start with a guessedE (= |FR|) and at themth iteration calculate an increment1E(m−1) to the currentiterateE(m−1)

yobs− HE(m−1)

= 1y ≈ H1E(m−1). (11)

This is an underdetermined problem, and we seek thesparsest set of increments1E(m−1) using StOMP. Theincrement is used to calculate a correctionξ = {ξi}, i =

1. . .NRK, |ξi | ≤ 1 and updateE(m−1)

ξi = sgn

(1E

(m−1)i

E(m−1)i

)max

(1,

∣∣∣∣∣1E(m−1)i

E(m−1)i

∣∣∣∣∣)

, (12)

E(m)i = E

(m−1)i exp(ξi).



The iteration is stopped when‖yobs−HE‖2/‖yobs

‖2 ≤ ε3for a small value ofε3.

5 Numerical tests

Numerical tests are performed for the domain between thecorners 24.5◦ N, 63.5◦ W and 87.5◦ N, 126.5◦ W. It is dis-cretized by a 2M × 2M , M = 6 mesh, with 4096 grid cells.Of these,NR = 816 cells lie insideR, while the rest,NR′ =

4096− 816= 3280 lie outside inR′. We estimate emissionsoverk = 1. . .K,K = 45, i.e., for 45×8 = 360 days (approx-imately a year). We generate synthetic observationsyobs us-ing the ffCO2 emissions in Vulcan, which provides them onlyin R. Hourly Vulcan fluxes are coarsened from 0.1◦ resolu-tion to 1◦, and averaged to 8-day periods. These fluxes aremultiplied byH to obtain ffCO2 concentrations at theNs =

35 measurement towers. Observations are available every 3 hand span a full year, i.e., we collectKs = 24/3×360= 2880observations per tower. A measurement errorε ∼ N(0,σ 2) isadded to the concentrations to obtainyobs, as used in Eq. (8).The sameσ is used for all towers and is set to a very lowvalue of 0.1 ppmv. Although such a value is unrealisticallysmall for real-data inversions, it is used here to isolate theimpact of the proposed parameterization and inversion ap-proach.

We solve Eq. (10) and enforce non-negativity onFR toobtainE. The coefficientsw(X) used in Eq. (9) are obtainedfrom a wavelet decomposition offpr based on nightlights(Sect.3). The constantc in Eq. (4) is obtained by using fluxesfrom the Emission Database for Global Atmospheric Re-search (EDGAR,http://edgar.jrc.ec.europa.eu; Olivier et al.,2005) for 2005, i.e., instead of using emissions from Vul-can to calculatefV , we use EDGAR. EDGAR emissions ag-gregated overR are 7.1 % higher thanfV , resulting in acorrespondingly higherc. The RMSE between the two is0.035 µmoles m−2 s−1 of C and the Pearson correlation co-efficient is 0.726. Also, sincec is an aggregate total overR,it reflects the I.E.A country total. The following parametersare used in the inversion process (Sect.4.2): ε2 = 10−5,ε3 =

5.0× 10−4,Mcs = 13 500, i.e., 300 compressive samples foreach 8-day period. The numerical values ofε2 andε3 wereset by reducing them till the solutionw′ became insensitive tothem. The setting forMcs is more involved and is describedin Ray et al.(2013). The initial guess forw′ in Eq. (10) iszero.

In Fig. 7 we plot the cumulative distribution function(CDF) of ffCO2 emissions before and after the enforcementof non-negativity. We see the existence of a few grid cellswith negative fluxes, but their magnitudes are not very large.Thus the sparse reconstruction scheme provides a good start-ing guess for the imposition of non-negativity via the iterativemethod described in Sect.4. In Fig. 8, we plot the true andreconstructed emissions for the 33rd 8-day period (k = 33).We also plot the estimation error(Ek − fV,k), averaged over

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Figure 7.CDF of emissions inR, before and after the imposition ofnon-negativity, as described in Sect.4. We see that the CDF of theemissions without non-negativity imposed contains a few grid cellswith negative fluxes; further, the magnitude of the negative emis-sions is small. Thus the spatial parameterization, with sparse recon-struction, provides a good approximation of the final, non-negativeemissions.

the 32-day period 33≤ k ≤ 36. We see that the reconstruc-tion in the NE quadrant is qualitatively similar to the trueemissions. In contrast, the reconstruction on the west coastcontains significant inaccuracies. For example, we see thatthe Los Angeles–San Diego region (southwest quadrant) isestimated incorrectly. The estimated emissions in the centerof the country (Continental Divide and Great Plains, in thewestern quadrants) show similar errors, as well as far morestructure than the true ffCO2 emissions. The region aroundthe Gulf of Mexico is also not well estimated. The qualityof the reconstruction in the various regions correlate with thedensity of observations towers, though the wind fields alsoplay an important part. In the regions where the observa-tions are not very informative, the impact of normalizationby fpr is clear as some of its structure is retained in the es-timated emissions. These errors are almost entirely at finespatial scales.

In Fig. 9 (top) we plot a time-series of errors defined asa percentage of total, country-level Vulcan emissions. Per-cent errors in reconstructed emissions andfpr are calculatedusing Eq. (13).


http://edgar.jrc.ec.europa.eu

1912 J. Ray et al.: A spatial parameterization for fossil-fuel carbon dioxide emissions Discussion

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Figure 8. Reconstruction of the ffCO2 emissions from the 35 towers (plotted as diamonds). The trueemissions are on top and the reconstructions in the middle. The figures represent emissions for k = 33(end of August). At the bottom, we plot the estimation error, (Ek − fV,k), averaged over 33≤ k ≤ 36.We see that the large scale structure of the emissions have been captured. The west coast of the US hasfew towers near heavily populated regions and thus is not very well estimated. On the other hand, due tothe higher density of towers in the Northeast, the true and estimated emissions are qualitatively similarand estimation error are low. Emissions have units of µmolm−2 s−1 of C (not CO2).

40

Figure 8. Reconstruction of the ffCO2 emissions from the 35 tow-ers (plotted as diamonds). The true emissions are on top and thereconstructions in the middle. The figures represent emissions fork = 33 (end of August). At the bottom, we plot the estimation error,(Ek − fV,k), averaged over 33≤ k ≤ 36. We see that the large-scalestructure of the emissions have been captured. The west coast of theUS has few towers near heavily populated regions and thus is notvery well estimated. On the other hand, due to the higher density oftowers in the northeast, the true and estimated emissions are qual-itatively similar and estimation error are low. Emissions have unitsof µmol m−2 s−1 of C (not CO2).

Errork (%) =100

K

K∑k=1

Ek − EV,k

EV,k

,

whereEk =

∫R

EkdA (13)

andEV,k =

∫R

fV,kdA,

Errorpr,k (%) =100

K

K∑k=1

Epr − EV,k

EV,k

,

whereEpr =

∫R

fprdA.

Here,fV,k are Vulcan emissions averaged over thekth 8-day period andEk are the non-negativity enforced emissionestimates in the same time period. A positive error denotes anoverestimation by the inverse problem. We see 25 % errorsin fpr. The large error is a consequence not only of the dis-agreement between EDGAR (from 2005) and Vulcan (from2002), but also the manner in which they account for emis-sions. As can be seen, assimilation ofyobs reduces the errorsignificantly vis-à-visfpr. The least accurate reconstructionsare during spring (k = 10–15). In order to check the accuracyof the spatial distribution ofEk, we calculate the Pearson cor-relationsρ(Ek, fV,k) andρ(fpr, fV,k). We see that data assim-ilation results in a clear increase in the correlation. When theemissions are aggregated/averaged over 32-day periods, thecorrelation increases to about 0.85, whereas the “prior” cor-relation was around 0.7. Thus the ffCO2 emissions obtainedusing a nightlight proxy are substantially improved by the in-corporation ofyobs. Only about half the wavelet coefficientscould be estimated; the rest were set to zero by the sparsereconstruction technique (Ray et al., 2013).

We next investigate the effect of using BUA maps, in-stead of nightlights, as the proxy. Changing the proxy resultsin a different set of wavelets being chosen (nightlights re-sulted in aW (s) of 1031 wavelets; the corresponding num-ber for BUA was 1049); further, one wasnot a strict sub-set of the other. It also results in a different normalizationin Eq. (9). The inversion was performed in a manner iden-tical to that adopted for the nightlight proxy. In Fig.9 (top)we see that the ffCO2 emissions developed using nightlightsand BUA as proxies are similar, as measured by reconstruc-tion error (Eq.13), though the BUA reconstruction errortends to be slightly smaller. The aggregated error between thetrue and “prior” fluxes remains unchanged (nightlights vs.BUA) since it just reflects the difference between EDGAR(in 2005) and Vulcan (in 2002) inventories. In Fig.9 (bot-tom) we plot the spatial correlation between the true, recon-structed and “prior” fluxes. The correlation between true andreconstructed emissions (from BUA) tends to be worse thanthe nightlight reconstruction. The correlation offpr with true


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Figure 9. Comparison of reconstruction error and correlations. Top: We plot the error between the recon-structed and true (Vulcan) emissions in black (using nightlights as priors) and in blue (using BUA priors).We plot the error between fpr and Vulcan emissions using dashed lines – black for nightlights and bluefor BUA. We see that assimilation of yobs leads to significantly improved accuracy vis-à-vis fpr. Bottom:We plot the accuracy of the spatial distribution of the reconstructed emissions. The Pearson correlationsρ(Ek, fV,k) and ρ(fpr, fV,k) show that incorporating yobs marginally improves the spatial agreement ofestimated emissions vs. the true one when using nightlights, though the results are less clear for BUApriors. If the emissions are averaged over 32 day periods, rather than 8 day periods, the correlation withtrue (Vulcan) emissions rises to around 0.85, irrespective of the prior used.

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Figure 9.Comparison of reconstruction error and correlations. Top:we plot the error between the reconstructed and true (Vulcan) emis-sions in black (using nightlights as priors) and in blue (using BUApriors). We plot the error betweenfpr and Vulcan emissions usingdashed lines – black for nightlights and blue for BUA. We see thatassimilation ofyobs leads to significantly improved accuracy vis-à-vis fpr. Bottom: we plot the accuracy of the spatial distribution ofthe reconstructed emissions. The Pearson correlationsρ(Ek, fV,k)

andρ(fpr, fV,k) show that incorporatingyobs marginally improvesthe spatial agreement of estimated emissions vs. the true one whenusing nightlights, though the results are less clear for BUA priors.If the emissions are averaged over 32-day periods, rather than 8-dayperiods, the correlation with true (Vulcan) emissions rises to around0.85, irrespective of the prior used.

emissions from Vulcan are different for nightlights and BUAreflecting the distinct spatial difference between them as seenin Fig. 6. This results in the difference between the twodashed lines. Averaging over 32-day intervals improves thecorrelation and makes them almost indistinguishable fromthose obtained using nightlights.

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Figure 10.Top: emission reconstruction error in the NE (blue) andNW (black) quadrants, when performed with BUA (line) and night-lights (symbols) as proxies. We see that the NW quadrant is verybadly constrained and the BUA-based estimates have very large er-rors. The errors in the NE quadrant are far smaller and very similarwhen generated using the competing proxies. Bottom: the compari-son of correlations between true and reconstructed emissions showssimilar trends.

In Fig. 10 we investigate the differences between thenightlight- and BUA-based reconstructions at the quadrantlevel. We see in Fig.10 (top) that the difference betweennightlight- and BUA-based reconstruction errors in the NEquadrant are smaller than those for the NW quadrant. Thus,while the fpr from nightlights and BUA are quite different(see the last row of Fig.6), the estimated emissions arewell informed byyobs in the NE quadrant and the impactof the proxies is small. This is not the case for the NWquadrant, where the reconstruction based on BUA is clearlymuch worse than the nightlight-based reconstruction. This isnot surprising given the paucity of towers there (see Fig.8),which increases the impact offpr. In Fig. 10 (bottom) weplot the correlation of the reconstructed and true emissions



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Figure 11. Top: Comparison of emission estimates developed using fpr constructed from nightlight radi-ances and BUA maps. We plot the difference between the two estimates. We see that differences are notlocalized in any one area. Bottom: Prediction of ffCO2 concentrations at 3 measurement locations, usingthe true (Vulcan; plotted with symbols) and reconstructed emissions (blue lines) over an 8 day period(Period no. 34). Observations occur every 3 h. We see that the concentrations are accurately reproducedby the estimated emissions.

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Figure 11.Top: comparison of emission estimates developed usingfpr constructed from nightlight radiances and BUA maps. We plotthe difference between the two estimates. We see that differencesare not localized in any one area. Bottom: prediction of ffCO2 con-centrations at three measurement locations, using the true (Vulcan;plotted with symbols) and reconstructed emissions (blue lines) overan 8-day period (Period no. 31). Observations occur every 3 h. Wesee that the concentrations are accurately reproduced by the esti-mated emissions.

in the NE and NW quadrants. We see that there is little tochoose between the correlations generated using nightlight-vs. the BUA-based emission estimates. Again, due to thelarger density of towers in the NE, the correlations are higherthere. Thus, while Fig.6 (middle row) showed that BUA hada slightly better correlation with true (Vulcan) emissions, itslarger errors, as seen in Fig.6 (bottom row) lead to a lessaccurate reconstruction. This result is also a testament to theinadequacy ofyobs over the whole country for constrainingffCO2 emissions; had there been sufficient data to informE,the impact offpr would have been minimal.

Next, in Fig. 11 (top) we compare the estimated emis-sions for the 34th 8-day period developed from the twocompeting prior models. We plot the difference between the

two estimates; it shows differences spread over a large area,though their magnitudes are not very big. Thus the “prior”model has a measurable impact on the spatial distributionof the emissions. In Fig.11 (bottom) we ploty predictedby the reconstructed emissions (from nightlights as priors)at 3 towers. The towers were chosen to represent the rangeof the ffCO2 signal strengths encountered in our test cases.We see that the ffCO2 concentrations are well reproducedby the estimated emissions. Further, note that the measure-ment noise (σ = 0.1 ppmv) is relatively large compared tosome of the observations. Thus, the lack of fidelity at thesmaller scales (seen in Fig.8) does not substantially impactthe measurements. This is due to the weak strengths of the er-roneous emission sources (while a few may be intense, theyare present only over a small area) and their distance fromthe towers.

6 Discussion

The numerical results in Sect.5 show that the MsRF andsparse reconstruction techniques can solve the inverse prob-lem as formulated in Sect.4, conditioned on limited mea-surements of ffCO2 concentrations. The solution reproduceslarge-scale spatial patterns of the true flux field, and someof the finer ones. The rough spatial nature of the emissionfield is preserved in the estimates. Furthermore, the methodis insensitive to underreporting of ffCO2 emissions by coun-tries which are used to construct inventories such as EDGAR.Inventories are used only to calculatec in Eq. (4) which ap-pears as a normalization constant in Eq. (9). The accuracy ofthe estimates (the constraint||Y − G′w′

||22 < ε2 in Eq. (10))

is unaffected by the value ofc. The chief source of errors inthe estimates is the paucity of observations sensitive to fossil-fuel-emitting regions. Regions with low tower density, e.g.,the western quadrants in Fig.8, have large errors due to thefaint ffCO2 signal at existing observational sites. One limi-tation of the deterministic estimation method presented hereis that it does not provide any measure of the uncertainty inthe estimates. The numerical parametersε2, ε3 andMcs arenot significant sources of uncertainty since they were set atvalues where the solution of the inverse problem became in-sensitive to them.

Given our focus on the algorithmic issues in the esti-mation of ffCO2 emissions under realistic conditions, theinverse problem that we constructed is idealized and em-bodies a number of simplifications. We have used a sen-sor network (that existed in 2008) that was sited with aneye towards estimating biospheric CO2 fluxes. This networkis therefore not optimized for constraining ffCO2 emissionsources, leading to a faint ffCO2 signal (< 2 ppmv). Thismade the use of a small model–data mismatch error neces-sary for the synthetic-data experiments presented here (σ =

0.1 ppmv), a value that would not be realistic for inversionsusing real data. The experiments conducted here also assume



the availability of observations that isolate the ffCO2 sig-nal, which would either require observations of a ffCO2ns-specific tracer, or the pre-subtraction of the influence of bio-spheric fluxes from observations. Some of the other simplifi-cations used in the setup, on the other hand, are common tosynthetic-data inversion experiments focusing on biosphericfluxes and reported elsewhere, e.g.,Gourdji et al.(2010). Forexample, we have ignored emissions outsideR; in a real-istic ffCO2 estimation problem, emissions outsideR wouldhave to be modeled as boundary conditions to the examineddomain, as is done for regional biospheric inversion studies.Furthermore, we have also assumed a constant data–modelmismatch (σ ) across all sensors, and site- and seasonallyvarying model–data mismatch statistics would be requiredwhen real data are used.

The use of proxies to construct the MsRF for ffCO2 emis-sions can be a source of estimation errors and consequently,in Sects.3 and5, we investigated nightlights and BUA mapsto explore the impact of using such proxies for sub-selectingthe wavelets to be used in the inversion. Errors in the prox-ies themselves (i.e. inaccuracies in the nightlights and BUAdata themselves) are unlikely to be a large source of estima-tion errors in the inversion, as these proxies are used onlyto select wavelets, whereas the wavelet coefficients are ob-tained in the inversion step. Rather, inversion errors can stemfrom the fact that nightlights correlate with energy consump-tion and not energy production. This can lead to two types oferrors: (1) when a fine-scale wavelet covering a region witha strong ffCO2 source and little human habitation is omit-ted from the MsRF and (2) when we choose a “superfluous”fine-scale wavelet in a region with much human activity andlittle emission. An example of the first type of error is largepowerplants, which are usually sited far from densely popu-lated areas. In such a case, the point-source is modeled by thecoarse-scale wavelet that covers the area in question, leadingto a “smeared” reconstruction. Such large point-sources offfCO2 emissions could instead be obtained from databasessuch as CARMA and incorporated directly into the inver-sion. Alternatively, one could augment the wavelets in theMsRF with those chosen using a second proxy, e.g., thermalimages, where large emitters of heat can be easily detected.The second type of error, that of the “superfluous” wavelet, isrectified when it is simply removed by the sparse reconstruc-tion scheme in the inversion step. An exception can occur ifthe superfluous wavelet contains a measurement tower in itssupportand is far from all other towers. Since measurementtowers are very sensitive to fluxes in their vicinity (Gerbiget al., 2009), it could lead to the estimation of a spuriousemission source.

ffCO2 emissions, averaged over 8-day intervals and pred-icated on 3-hourly measurements, were estimated as inde-pendent variables, i.e., without imposing a temporal correla-tion or modeling their temporal evolution in any way. Thereason is as follows. Estimation of ffCO2 emissions overa 8-day interval requires the calculation of 1031 wavelet

coefficients (when using the nightlights-derived MsRF) from35× 8× 8 = 2240 measurements. This is not an underdeter-mined problem, even though a sparse reconstruction methodwas required to remove fine-scale structures (wavelets) in theemission field that did not affect the measurements. We wereable to constrain the coefficients of the remaining waveletswithout imposing a temporal correlation structure. Such cor-relations could be used if ffCO2 fluxes were to be estimatedat finer temporal resolution.

The spatial parameterization and the sparse reconstructionmethod can also be used in observation system simulationexperiments (OSSEs) to inform the design of measurementnetworks targeted for ffCO2 emissions. The approach can beused to decide locations of towers, the frequency at whichffCO2 measurements are to be made, and the fidelity requiredin measurements and the transport model. The trade-offs andcosts of various ffCO2 measurement technologies can also bestudied in such a setting. In addition, OSSEs can reveal theimportance of a more accurate MsRF, e.g., one augmentedusing thermal imagery, vs. the errors introduced in the esti-mates due to limited measurements.

7 Conclusions

We have devised a multiresolution parametrization (alsoknown as a multiscale random field or MsRF model) formodeling ffCO2 emissions at 1◦ resolution. The MsRF mod-els emissions in the lower 48 states of the US and is designedfor use in atmospheric inversions. The parameterization em-ploys Haar wavelets which provide a sparser representationthan other smoother wavelets with wider support. This is thefirst “abstract” parameterization, i.e., a RF model for spa-tially resolved ffCO2 emissions.

The dimensionality of the MsRF was reduced by judi-ciously selecting its component Haar wavelets using proxiesof human activity, and therefore indicative of ffCO2 emis-sions. We developed two MsRFs based on images of lights atnight and maps of built-up areas. The former had a slightlylower dimensionality but was not a strict subset of the latter.The MsRF models were also used to develop two approxi-mate emission models that differed in their fine spatial de-tails.

The MsRF model was tested in a synthetic-data inversion.Time-dependent ffCO2 emissions, averaged over 8-day pe-riods, were estimated for a 360-day period from measure-ments of ffCO2 concentrations at 35 towers. These obser-vations were sufficient only for estimating about half thewavelets retained in the MsRF model. We used a sparse re-construction technique, namely Stagewise Matching Orthog-onal Pursuit (StOMP), to identify and estimate wavelet coef-ficients in MsRF that could be informed by the available data.The StOMP estimates were not necessarily non-negative (asffCO2 emissions are required to be) and we devised an it-erative, post-processing procedure to impose non-negativity.



Furthermore, the MsRF actually models emissions in a rect-angular region and constraints had to be imposed during theinversion to ensure that emissions were restricted to the lower48 states. To our knowledge, this is the first instance of usinga sparse reconstruction method in atmospheric CO2 inver-sions. The algorithmic details of the inversion procedure arein Ray et al.(2013). This is also the first use of an MsRFmodel in ffCO2 emission estimation.

The inversions using the competing MsRF models(nightlights- vs. built-up-area-based) resulted in emission es-timates that differed in their fine-scale details. Country-levelestimates of emissions, and their correlation with the trueemissions, differed little when developed using either of thetwo MsRFs. Further, emissions obtained by dividing the USinto quadrants showed large errors vis-à-vis true emissions.This is a testament to the fact that the current observationsnetwork was designed to focus on biospheric rather thanfossil-fuel CO2 fluxes.

The spatial parameterization and its use in reconstructingan emission field was demonstrated in an idealized setting.A number of adaptions are required before it can be used ina realistic scenario to infer ffCO2 emissions. These includeaspects that are standard to any regional inversion, such as ac-counting for fluxes outside the modeled domain via bound-ary conditions, and accommodating errors incurred by thetransport model. Results using even our idealized setup showthat constraining fossil-fuel emissions at regional to nationalscales would require a measurement network better targetedto that goal. In addition to the prototypical application pre-sented here, our approach can also be used in OSSEs for de-signing such a measurement network – determining optimallocations of towers, intervals between measurements, and thefidelity required in the transport model and the measurementsthemselves.

Our inversion formulation suffers from two drawbacks.One is the need for observations of ffCO2 concentrationsat the measurement towers, which could be obtained eitherby making use of co-emitted tracers or by pre-subtractingthe influence of biospheric fluxes from observations. Nei-ther of these, however, is easily achievable given currentobservational constraints and uncertainties associated withbiospheric fluxes. The second drawback is the determinis-tic nature of the reconstruction, which does not provide errorbounds on the estimates of the MsRF parameters (the waveletcoefficients).

In conjunction with this paper, we are also providing theMATLAB ® code required to (1) reduce the dimensionalityof the MsRF using nightlights and (2) perform the inversionusing synthetic observations at our website (Ray, 2013). Thewebsite also contains links to the (free) MATLAB toolkitsthat our code depends on, along with a user’s manual.

The Supplement related to this article is available onlineat doi:10.5194/gmd-7-1901-2014-supplement.

Acknowledgements.Thanks are due to Sharon Gourdji, KimMueller and Abhishek Chatterjee for fruitful discussions regardingalgorithms, the challenges of working with field observations aswell as some of the data used in this work. This work was supportedby Sandia National Laboratories’ LDRD (Laboratory DirectedResearch and Development) funds, sponsored by the GeosciencesInvestment Area. Sandia National Laboratories is a multi-programlaboratory managed and operated by Sandia Corporation, a whollyowned subsidiary of Lockheed Martin Corporation, for the U.S.Department of Energy’s National Nuclear Security Administrationunder contract DE-AC04-94AL85000.

Edited by: A. Stenke

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A multiresolution spatial parameterization for the ... · A multiresolution spatial parameterization for the estimation of fossil-fuel carbon dioxide emissions via atmospheric inversions