Carbon monoxide (CO) plays an important role in tropospheric chemistry and composition. Firstly, its reaction with the hydroxyl radical (OH) has a significant impact on the atmospheric oxidizing capacity. CO as well as non-methane volatile organic compounds (NMVOCs) are both responsible for positive climate forcing because they raise tropospheric ozone (O₃) and methane (CH₄) by creating a global sink for OH . The spatiotemporal variations in CO (and hence OH) can lead to changes in O₃ abundance depending on whether the concentration of nitrogen oxides (NO_x) is low or high . Regions of high NO_x experience O₃ production while clean areas with low NO_x experience the destruction of O₃ . Because OH is a photochemical sink of CO through oxidation, its global distribution determines the lifetime of CO and therefore, values of CO are lowest with the shortest lifetime in regions with the highest OH. Secondly, CO can also serve as a tracer of pollution transport given its medium-length lifetime (weeks to months) . Mainly a product of incomplete combustion of wood and fossil fuels, CO is co-emitted with nitrogen dioxide (NO₂), black carbon (BC), and carbon dioxide (CO₂). It is more concentrated in the atmospheric boundary layer (with less in the stratosphere) so observing CO can be useful as well to study the exchange of these gases and aerosols across the troposphere. Because CO is the main sink of OH , it is important for quantifying losses of CH₄ in the troposphere as well . The lack of constraints on OH spatiotemporal variability as well as the interannual variability of CH₄ have both made it difficult to accurately determine the global CH₄ budget . This means there is a need to reduce the uncertainty of primary drivers of OH, including CO, O₃, H₂O, NO_x, and NMVOCs .

Determining the spatial and temporal distribution of CO, along with the distribution of co-emitted species including NO₂, CO₂, CH₄ and aerosols, helps us to understand trends of combustion, biomass burning, and other sources of air pollution. For example, uses a combined mean and standard deviation approach for total column CO to classify local regions of Southeast and East Asia based on combustion activity into areas of intense urbanization, large-scale biomass burning, regions undergoing significant change from one type to another, and clean regions. utilized variations in enhancement ratios of elemental carbon (EC), CO, and NO₂ to investigate processes driving EC levels across the United States. demonstrated that local enhancement ratios of ΔCO/ΔNO2 can distinguish different combustion types, identifying whether or not a fire was in a flaming phase or smoldering phase while used ΔCO/ΔCO2 to identify combustion efficiency patterns in urban agglomeration. Building on this, combined ΔCO/ΔCO2 and ΔCO/ΔNO2 to better characterize regional-scale combustion process while used ΔCO/ΔNO2 and ΔCO/ΔSO2 to track the common combustion emission pathways across China. Recently, demonstrated the value of multi-species enhancement ratios (CO, CH₄, CO₂) along with multivariate regression methods to differentiate between regional and local influences, and identify the dominant processes driving these patterns.

However, determining accurate trends in air pollution is difficult in practice for many different reasons. While there has been an increasing availability of satellite measurements of trace gases and aerosols in the recent decade, most of these remote sensing retrievals cannot, in general, tell us where their source or sink was due to the diffusive nature of atmospheric transport . Another challenge is that space-based CO measurements often lack accuracy in areas with persistent cloud cover and near the surface, where thermal infrared sensors struggle due to interference from surface radiation and limited retrieval sensitivity . These significant sources of uncertainty require us to cross-validate satellites across different instruments as well as with ground and aircraft measurements during field campaigns, and are especially problematic over areas such as Sub-Saharan Africa, where there are very few air quality monitoring stations . Determining the background value of constituents is especially difficult on a global scale because it changes across time and space based on the sources and sinks for a given region and chemical regime . Because satellites measure the total amount of a constituent rather than the source of emission for a given region, it is difficult to tell whether local changes in the overall trend are due to anthropogenic or biogenic sources or the background or residual burden . As discussed in there are no overall trends in either CO or aerosol optical depth (AOD) in Central/Southern Africa despite the increasing trend in anthropogenic combustion in Central Africa as determined by which may be counteracting the global decrease in CO. have also noted that the increase in burning in the region as shown in may be potentially counteracting transport.

There are several complex challenges when determining emission inventories. As discussed by , emission inventories in China are low in comparison to in situ observations in the case of both forward and inverse models, and in particular there is an underestimation of CO despite having revised inventories as explained by an underestimation of residential coal combustion from either heating or cooling . Even though this underestimation of fossil fuel burning does explain the underestimation of Northern Hemisphere (NH) CO in global models , this is confounded by the large difference in modeled variability of the regional distributions of OH in the NH, which could offer an alternative explanation . used observations from the Korea-United States Air Quality (KORUS-AQ) field campaign to investigate this underestimate over East Asia and found that assimilation of CO improves the representation of OH in global chemical transport models by correcting OH / HO₂ partitioning. This also implies that assimilation of CO should help study coupled CH₄-CO-OH reactions, which allow for chemical feedback. Effectively distinguishing the signature of secondary CO from the influence of surface emissions – whether anthropogenic, biogenic, or from fires – is crucial for enhancing the reliability of source attribution studies, including inverse modeling for air quality assessments and chemistry-climate simulations . This separation also aids in the physical interpretation of observed air pollution patterns, with important implications for understanding the distributions of both OH and CH₄ .

To address some of these challenges, we attempt to answer the following questions in this study:

How can we use the spatial and temporal distribution of CO to detect the difference between anthropogenic and natural variability of CO?

We use the term “global background” to refer to a mathematical, non-physical quantity broadly represented as an average. This is different from terminology frequently used in literature. For example, defines a background concentration as the observed concentration at a site “not affected by local sources of pollution”. More recently, described background concentration as the collective contribution of sources from regional, anthropogenic, natural emissions, and long-range transport.

To attempt an answer to these questions, we use Empirical Orthogonal Function (EOF) analysis, which is a pattern analysis method to determine the dominant spatio-temporal patterns of a dataset . To avoid confusion with language, we clarify that the term EOF analysis refers to the usage of principal component analysis (PCA) (a statistical technique used to reduce dimensionality) when it is applied to the fields of climate science and geophysics. While the two methods are virtually identical, the use of EOF analysis implies that the principal components being found are not just linearly uncorrelated, but they are also being used to explain the maximum amount of variance possible in the original dataset.

EOF analysis has been extensively utilized in climatology, for instance performed an EOF analysis of geopotential height to generate what we now refer to as atmospheric teleconnections. It has had fewer applications to atmospheric chemistry, but for instance uses varimax rotated EOF analysis of de-seasonalized aerosol index to isolate major dust and biomass burning regions and dust transport. performed a similar analysis on AOD datasets. also verified the result of their classification of urban and biomass burning regions using EOF analysis. used EOF analysis to analyze the dominant patterns in summer O₃ pollution over Eastern China and their relationship to atmospheric circulations. EOF analysis has also seen applications to air quality monitoring. For instance, uses the principal components corresponding to rotated modes to analyze seasonality of surface ozone concentrations in the Eastern United States. used principal component analysis together with cluster analysis to examine monitoring sites and identify emission sources as well as cities with similar pollution behavior. They successfully applied this method to several species, including CO, O₃, and NO₂.

Our use of EOF analysis is especially motivated by studies that investigate the global CO budget through source attribution, for example, see e.g. . used three-dimensional-modeled CO to study the total CO budget by simulating the tagged contributions from fossil fuel, biomass burning, methane oxidation, and biogenic sources. In doing so, was able to reproduce global patterns in the CO seasonal cycle, most importantly the interhemispheric gradient observed in total CO measurements. The same gradient was also detected in the fossil fuel tag, which showed that the global CO seasonal cycle can be thought of as a convolution of the seasonal cycles for individual source terms. Our approach in this study is similar; however, we begin with satellite-based total column CO to separate it into patterns that represent the variability of its individual sources and sinks. This is precisely the class of problem that EOF analysis was designed to solve. Our focus in this paper is to explore the use of EOF analysis on satellite data of atmospheric composition in an attempt to separate anthropogenic and biogenic modes of variability from a global background field. Our results find that while EOFs have proven useful in climatological studies before, using it on atmospheric composition has significant limitations that require careful considerations, as detailed in our study.

We use total column CO data from the Measurements of Pollution in the Troposphere (MOPITT) instrument, which uses a correlation radiometer from the NASA Terra instrument. Terra flies at a normal altitude of 705 km in a Sun-synchronous polar orbit that passes over the equator at approximately 10:30 and 22:30 local time. The temporal resolution is daily, which has been averaged into 8 d periods, and the spatial pixel resolution is 22 km × 22 km at nadir, resampled to 1° from L3 products. The cross-track scanning angle is ±26° and yields a swath of approximately 640 km with global coverage every 3 d.

The radiometer performs gas correlation spectrometry for broadband measurements in thermal infrared (TIR) at approximately 2140 cm⁻¹ and near-infrared (NIR) around 4725 cm⁻¹ . This is done by modulating either pressure or length of a correlation cell filled with the gas of the target to determine spectral line differences. The TIR is measured using terrestrial thermal radiation, while the overtone band from NIR is measured from reflected solar radiation and enhances retrievals, especially near the surface. Products are available in thermal infrared only, near-infrared only, or joint TIR-NIR, although NIR signals are only available during the day and over land . For this study, we use Level 3 data using the V8 retrieval algorithm for TIR-NIR. Level 2 and Level 3 products for MOPITT for the V8 retrieval algorithms (and their most recent version, V9) are publicly available through repositories located at https://terra.nasa.gov/data/mopitt-data (last access: 20 November 2025).

The V8 product includes updates to MODIS cloud cover Collection version 6.1, spectral data for H₂O and N₂ as well as aircraft data from NOAA . This enables a radiance-based bias correction and accounts for both temporal bias drift and bias as a result of water vapor. Daytime retrievals of total column CO for V8 are stable and have a nominal drift of -0.015±0.061% relative to NOAA airborne flask sampling during the MOPITT mission. MOPITT measurements have been verified extensively using cross-validation with other satellites, in-situ measurements at the ground, as well as via aircraft, and more recently through ground-based solar Fourier transform infrared spectrometer measurements .

We note that satellite retrievals of tropospheric CO are also available using a multitude of other instruments, including the Atmospheric InfraRed Sounder (AIRS) onboard Aqua , the Tropospheric Emission Spectrometer (TES) on Aura , the Infrared Atmospheric Sounder Interferometer (IASI) on the MetOp platform , the Tropospheric Monitoring Instrument (TROPOMI) , and the Cross-track Infrared Sounder (CrIS) . These instruments are all sun-synchronous, capable of measuring CO in the infrared region, and have been shown to have consistent hemispheric CO variability when compared to MOPITT CO records from 2002 to 2018 (in the case of CrIS, from 2015 onward) . Because CrIS was launched in October 2011, it does not have a long or consistent enough period of measurement to be suitable for the determination of long-term trends of CO. The same applies to TROPOMI, which was launched in 2017. While CO measurements for IASI-A are long enough to determine trends, they have potential discontinuities since CO retrievals are not done using consistent humidity, temperature, and cloud information . Measurements from TES may be up to 2 orders of magnitude fewer than other instruments and are more difficult to verify due to not being collocated . Measurements from AIRS have a long and stable enough record for trend determination and have been shown to reproduce trends from MOPITT ; however, AIRS uses a cloud-clearing algorithm that increases global coverage at the expense of increasing ground pixel sizes to 45 km × 45 km, resulting in coarser resolution . Our choice of using MOPITT retrievals is because it has the longest available CO record with a verified history of reliable and stable observations and relatively fine spatial resolution.

Here, we briefly describe the source categories of CO, which include fossil fuel/biofuel burning, biomass burning, biogenic, non-methane hydrocarbon oxidation, and CH₄ oxidation as summarized in . Describing these categories is useful because surface emissions of CO can display unique signatures that may be linked to specific spatial and temporal patterns in observed CO abundance.

CO from anthropogenic combustion (fossil fuel/biofuel burning) is much larger in the NH compared to the Southern Hemisphere (SH), which is attributed to significantly more fossil fuel consumption in urban and industrial regions. We consider these areas of high emission to be mostly associated with eastern China, India, the eastern United States, and Western Europe. Historically, eastern China has contributed significantly to anthropogenic emissions because of a distinct reliance on coal and biofuel burning . In recent years, however, China has made significant progress in reducing emissions and improving air quality . This has been the case since 2014 when Premier Li Kequiang declared a “war on pollution”, and implemented widespread changes in regulatory policy .

Biomass burning is largest in the SH and a primary source of CO. Biomass burning consists of burning of savanna, forests, agricultural residue, fuelwood, and animal waste and is most concentrated in tropical and sub-tropical forests and grasslands . The largest contributions of fire emissions come from Africa and South America, although fires also occur in Indonesia, the Boreal forests of Canada, Alaska, and Russia, as well as Australia. African NH fires occur in the savanna south of the Sahara Desert and in tropical rainforests north of the equator, while SH fires begin burning in the western portion of the continent near Angola and then spread southeast down the eastern coast. Fires in South America happen on the southern edge of the Amazon rain forest, the cerrado grasslands, and drought-deciduous forests in the south . Another distinct feature of CO fire emissions is their timing . While long-term drivers such as land-use and land cover changes, as well as drought conditions, heavily influence fire occurrences, seasonal variations are primarily driven by agricultural burning practices and prevailing weather conditions. The anomalously high CO levels from these fires, coupled with their distinct spatial and temporal variability inherent to biomass burning, provide a more reliable signature for distinguishing fire emissions from other CO sources.

The contribution of CH₄ oxidation, as well as oxidation from non-methane hydrocarbons (NMHCs), represents secondary sources of CO that are most apparent in the SH. Oxidation from CH₄ represents roughly 28 % of the global budget while the biogenic component of NMHCs represents roughly 15 % . The oxidation of NMHCs includes a wide variety of constituents, which include isoprene, terpenes, acetone, as well as industrial and biomass NMHCs. Temperature-dependent biogenic emissions of isoprenes are considered the most dominant hydrocarbon emissions due to their large annual contribution (400–600 TgC globally ) and their high reactivity with tropospheric oxidants . Isoprenes are volatile organic compounds that are directly emitted by trees and shrubs through their leaves. They are also emitted by animals, including humans, and can be measured through various biological processes, including exhaled breath, perspiration, urine, feces, and tear production . Terpenes are produced mostly from coniferous forests and are released in higher amounts with warmer weather . The largest emissions of hydrocarbons occur in the tropics, which are primarily from isoprene and occur because of the large density of biomass and high temperature conditions. During the NH winter, there is a strong gradient resulting from biogenic emissions that are south of the equator, where the intertropical convergence zone is located, and includes emissions from South America, Southern Africa, and Australia. During the NH Summer, biogenic sources contribute more in the NH, resulting in a weaker but still observable gradient. This gradient is significantly stronger for surface observations but is observable at all levels of the atmosphere . CO production from CH₄ oxidation can be observed across the troposphere and tends to be spatially homogeneous given the longer lifetime of CH₄ (≈10 years). It does, however, exhibit some seasonality due to chemical loss with its reaction to OH.

The dominant source of oceanic CO is due to sunlight-initiated photolysis of chromophoric dissolved organic matter (CDOM) in the open ocean . CDOM represents the portion of decaying plants and microbes that absorbs light, degrading into CO and other products in the process. The secondary sources of oceanic CO include dark production (the production of CO from CDOM without the presence of sunlight) and emission from phytoplankton . While the physical processes that lead to dark production are not well understood, studies have observed it to be temperature dependent . The direct emission of phytoplankton is a similar source to CDOM in that CO is produced when exposed to photosynthetic radiation. The primary removal process for CO in the ocean is due to microbial consumption, and secondary removal comes from sea-air exchanges when the ocean releases CO into the atmosphere . A recent budget for oceanic CO in the Eastern Indian Ocean found that CDOM, dark production, and phytoplankton emission accounted for roughly 67 %, 30 %, and 3 % of oceanic emissions, respectively. Microbial consumption accounted for 94 % of the removal of oceanic CO while sea-air exchanges only contributed 6 %. Overall, the oceanic cycle of CO is not very well understood due to a general lack of observations and the fact that microbial consumption is difficult to predict due to its dynamically changing environmental conditions .

In addition to source signatures, the variability in CO (particularly its seasonality) is significantly influenced by its chemical loss with the OH radical. Notably, CO serves as the primary sink for OH, while CH₄ and NMVOCs both serve as secondary sinks. Reactions between CO and CH₄ have highly complex non-linear chemical feedback . This is due to the link between OH and CH₄ as well as the production of OH, which is tied to O₃. In addition to the primary production of OH through photolysis with O3, secondary production can also occur through photolysis with VOCs. These secondary reactions cause oxidation with either CO or VOCs to produce O₃ in the presence of NO_x, while also recycling OH in the process . The chemical lifetime of CO and, therefore, its abundance depends photochemically on the distribution of OH. Because oxidation increases with sunlight, OH is much more concentrated in the tropics and increases with latitude. This leads to the lifetime of CO being shortest in the tropics and longest near the poles, with an average lifetime of about 2 months.

Here we provide a simplified description of EOF analysis. A more complete description can be found in . The exposition presented here is detailed and specifically tailored for the use of EOF analysis in atmospheric chemistry datasets, especially because most references approach the subject from the perspective of meteorological applications. We present key equations relevant to our main discussion, while derivations are provided in Appendix A. We do this to emphasize the use of EOFs in atmospheric chemistry while keeping our discussion focused on their strengths and limitations rather than straying too far into technicalities.

Assume that we have a set of n observations in time of a gridded field with a total of m gridpoints. Let fij where i=1,2,…n j=1,2,…m represent the value of the field observed at time i and location j, then the data is represented by the n×m matrix F: 1F=f1,f2,…fnT=f11f12⋯f1mf21f22⋯f2m⋮⋮⋱⋮fn1fn2⋯fnm. Every row of F represents a 1×m vector corresponding to a single observation in time across all grid points.

We now construct the anomaly matrix F′ that represents the departure from normal climatology by removing from each grid point its corresponding temporal mean. Instead of constantly referring to F′ as the anomaly matrix, we will simply refer to it as F from now on.

The goal of EOF Analysis is to decompose the anomaly matrix into a matrix of orthogonal spatial patterns called the EOF modes C, and a set of uncorrelated time series called principal components (PCs) Φ such that 2F(x,t)=λ1ϕ1(t)c1(x)T+λ2ϕ2(t)c2(x)T+⋯λpϕp(t)cp(x)T=∑k=1pλkϕk(t)ck(x)T where p≤r, and r=rank(F) which satisfies: 3r≤min(n,m). The number of modes p to retain in the truncation is always chosen such that the maximum percent variance of the original dataset can be explained using the new set of basis functions. The data is compressed more when fewer modes of variability are kept in the expansion.

The EOF spatial patterns C correspond to the eigenvectors of the spatial covariance matrix (R=F′F) of F. We can similarly show that the PC time series Φ corresponds to the eigenvectors of the temporal covariance matrix (L=FF′) of F. It is important to note that because the covariance matrices are symmetric and positive semi-definite, they have non-negative eigenvalues with well-defined square roots as well as orthogonal eigenvectors. An equivalent formulation is to use singular value decomposition (SVD) on F, so we expand it into unitary matrices Φ and C and a matrix 4Σ=Λ1/2 of singular values such that: 5F=ΦΣCT.

Note that in this case that each column of F is divided by its standard deviation so that the covariance matrix then becomes a correlation matrix. We note, however, that our total column values are large in magnitude (they are measured in molec. cm⁻²), so we choose to use correlation matrices to avoid potentially having large differences in the variance, skewing our interpretations.

In Appendix A1, we detail exactly how the EOF patterns and PC time series can be computed numerically either using singular value decomposition (SVD) or by using eigendecomposition on the spatial or temporal covariance matrices and then projecting the resulting eigenvectors onto the original matrix F. In Appendix A2, we show that these two methods yield equivalent results in the context of our problem space. We note, however, that these methods are NOT in general the same. While SVD can be performed for rectangular matrices, eigendecomposition can only be done for square matrices . More importantly, while both methods are matrix factorizations for the original matrix F, SVD is generally preferred because it has better precision and is more stable computationally . This is because SVD uses a diagonal matrix of singular values (the square roots of the eigenvalues), which is always constructed to be non-negative. Unfortunately, SVD may require too much computing power in the case that F is very large because it requires us to compute eigenvectors of both the spatial covariance R and the temporal covariance L. Because F has m≫n (more gridpoints than observations in time), then dim(R)≫dim(L), so we decided to forego using SVD in favor of an eigendecomposition on L. On the other hand, if the number of observations had been large (n≫m), then dim(L)≫dim(R) and we would have instead used an eigendecomposition on R. Another alternative that is considerably faster for large matrices with no loss of stability is to use Lanczos iteration to solve the eigenvalue problem . Lanczos iteration is faster for larger matrices because it relies on Krylov subspaces to compute the largest eigenvalue first . While Lanczos iteration is more efficient, it is more complex to execute and therefore goes beyond the scope of our discussion.

Singular spectrum analysis (SSA) is a very powerful tool in time series analysis with a wide variety of applications, some of the more notable ones including forecasting, change point detection, filtering, and detection of spatio-temporal patterns. SSA also has applications to meteorology and air quality monitoring, for instance, uses SSA on the global surface temperature to separate a multi-decadal oscillation from a secular trend, and uses multichannel SSA to perform spatiotemporal analysis for atmospheric, continental hydrology, and non-tidal ocean changes to determine an annual signal for 16 sections depending on the climate zone. Another application is from , which uses the time derivative of a resulting SSA signal to detect air quality anomalies.

Broadly, our goal is to take a time series x=x1,x2,…,xN of length N and then decompose it into a sum of trends, periodic (or quasi-periodic) components, or noise, the sum of which will reconstruct the original series. A detailed description of the method is fairly technical, and we therefore refer the reader to summaries found in and for a more complete description. We perform a seasonal decomposition on the mean-centered and standardized time series for each grid point of our dataset A by using the trenddecomp function in Matlab with one mode of seasonal variability. This gives us 10A=LT+S+R, where LT is the long-term variation, S is the seasonal variation, and R is the remainder.

As we discussed in Sect. 2.1, we use total column CO data from MOPITT for our study. We use daily data averaged to 8 d periods to give us a sufficient resolution to observe CO during its lifetime while also reducing computation time and giving a more consistent overpass time. We also emphasize that for our specific application, daily data would not be sufficient because MOPITT does not have daily coverage, and EOF analysis fails to produce patterns if there are any gaps in the data. Even in using 8 d coverage, we needed to fill a small number of gaps using the inpaintn function offered by MATLAB. We use measurements for the 14-year duration of 2005–2018 to have sufficient data to study long-term patterns. Our data had long-term trends removed by taking the globally averaged time series and applying a Fourier fit to it. We note that because the trends in the total column data are a non-linear combination of the trends from individual sources, it is not immediately clear what sources may have had their long-term trends removed. Plots are shown below of the spatial (Fig. 1) and temporal (Fig. 2) means of de-trended CO for global data over our stated observation period. We now discuss the seasonal and interannual variability observed by MOPITT, focusing on its relationship to the sources and sinks of CO that we described in Sect. 2.2.

Reports of the interannual variability of CO can be identified as fluctuations in the observed gradient of its abundance between the NH and the SH. This gradient is driven by the significant differences in surface emissions between the hemispheres, along with the associated air mass exchange that occurs on timescales of a year or longer. To a first approximation, CO concentrations are lowest where OH abundance is highest, as CO's chemical lifetime is strongly dependent on reactions with OH. These areas are characterized by having high photochemical activity (solar radiation), as well as high concentrations of O₃ and NO_x, and correspond to continental regions during the summer. On the other hand, oceanic (remote) regions during the winter correspond to areas with the lowest OH and therefore the highest CO. In general, CO loading tends to reach maximum values in April and minimum values in September . However, the transport and mixing of surface emissions complicates this first-order relationship.

Seasonal differences in the distribution of OH and the imbalance of emissions in each hemisphere lead to a North-South gradient in CO that is clearly visible during the NH winter but is absent in the NH Summer. During the NH winter, low OH in the NH leads to high CO, while high OH in the SH leads to low CO. This, in turn, reinforces large NH anthropogenic emissions from Europe, North America, and Asia that peak during early springtime, which leads to a steep gradient during the NH winter. During the NH summer, we instead have low CO in the NH, but high CO in the SH, which balances a much lower budget of emissions of CO in the SH.

Significant changes in the annual cycle have been connected to fire emissions due to biomass burning, and limited studies have connected variability in the SH to climate indices where biomass burning is the primary driver. In the NH, fires from Boreal forests in North America and Eurasia have been shown to contribute to significant interannual variability. In the SH, while fires in southern Africa and South America generate overall high CO loadings, a primary driver of variability is due to fires in the Maritime Continent and Northern Australia. were able to use multilinear regression with four climate indices to explain the variability of CO across the SH and tropics, including Maritime SEA, two subregions in each of Australasia, Southern Africa, and South America. The indices used to explain variability included the El Niño Southern Oscillation (ENSO), the Indian Ocean Dipole (IOD), the Tropical South Atlantic Sea Surface Temperature Anomaly (TSA), and the Antarctic Oscillation (AAO) . They found that more than 50 % variability is explained over all regions, and more than 70 % variability was explained across maritime Southeast Asia and North Australasia. The fire intensity and burned area are connected to the amount, type, and dryness of fuel availability, which depend on climate-sensitive water availability and ecosystem responses. This indicates that SH interannual variability of CO loading has a clear but complex dependence on climate.

As discussed in Sect. 2.3.2, one of the first potential issues that needs to be addressed to produce physical EOF modes is whether or not the modes capture the local structure and if they are reproducible across differing subdomains. Before generating EOF modes, our data matrix of size (longitude, latitude, time) is reshaped into size (time, grid), and every grid point is mean-centered across time and then standardized. In Fig. 3, we show the first leading EOF mode for the dataset across the entire globe (a) and then recompute the first mode after subdividing the domain into North American (b), African (c), and Asian (d). To verify that these spatial patterns did not change over time, we recomputed the EOFs after segmenting the time series by year and determined that the patterns remained unchanged (they only showed negligible differences in magnitude). The spatial structure of the original pattern has been preserved in each of the three subdomains, showing us that the local and global structure for the pattern is the same, and therefore our dataset is independent of its domain.

In some problems, it is often a good idea to attempt varimax rotation to ensure that the resulting patterns are simple and easy to interpret , but in our case, there is not a real need to do so because our problem does not suffer from domain instability . To demonstrate that varimax does not improve our resulting modes, we show the eigenvalue spectrum of the first six unrotated EOF modes together with the spectrum of our varimax modes in Fig. 4. Each mode is plotted as a fraction of the percent variance, with the exact value shown within the text box in the middle of the plot. The varimax and unrotated modes are shown in blue and green, respectively, and the cumulative percent variance is shown in orange and purple. We compute the error bars using Eq. () for Δλk in Sect. 2.2.2 and plot the results in red. For our unrotated modes of the original MOPITT dataset in (a), the variance together with error bars (the length above/below each data value) are given by: 61.55±3.43%, 20.52±1.143%, 8.24±0.459%, 3.92±0.218%, 1.10±0.061% and 0.69±0.038%. After using varimax rotation, the percent variance of each mode was: 86.02±4.793%, 11.56±0.644%, 1.87±0.104%, 0.49±0.027%, 0.03±0.002% and 0.01±0.001%.

While our plots of percent variance for the varimax modes show better representation for mode 1 of every dataset compared to our unrotated modes, all of the remaining modes showed significantly worse representation (except the residual modes). None of these eigenvalues represent degenerate modes because their error bars do not overlap with adjacent neighbors. We note, however, that varimax modes 4, 5, and 6 have such a small percent variance that it is unlikely for them to rise above the level of noise and represent independent and important features. On the other hand, the higher modes of our unrotated EOFs have a much larger percent variance and are more likely to represent independent features, with modes 5 and 6 most likely representing noise. Comparing the cumulative percent variance for the first 3 MOPITT modes, we see our unrotated modes accounted for 90.3 % of the variance compared to 99.4 % for varimax. Because our third varimax mode only accounted for 1.87 % variance as opposed to 8.24 % for our unrotated mode, we could not guarantee that it would be well represented. We therefore decide to only analyze spatial patterns of the first 3 unrotated modes in our SSA decomposition, which comes at the cost of explaining 9.1 % less variance overall. We discuss the long-term, seasonal, and residual modes in greater detail in Sect. 3.3.

Finally, we note that our choice to display the first six modes does not represent an “optimal” representation of modes as could be determined using statistical tests for eigenvalues . This choice is made to ensure we display at least one potentially degenerate mode for both our varimax and unrotated EOFs. We also do this to show there is little motivation to keep more than 5 EOFs in our analysis, since each mode beyond the fifth adds a negligible (<1%) amount of variance, except the residual modes.

To visualize our SSA decomposition, rather than choosing a single grid point as an example, we plot a decomposition for the temporal mean of the dataset in Fig. 5 to show the average overall effect. Our results are plotted only for 2005–2009, as five yearly cycles were sufficient to show the effect of the decomposition. We show the decomposition for 4 different regions: Global (60° S–60° N), Northern Extratropics (25–60° N), Southern Extratropics (25–60° S), and Tropics (24° S–24° N).

The result is that the long-term component (red) has a yearly periodicity that reproduces the peaks of the original MOPITT series (blue), while the seasonal component (green) has a shorter temporal periodicity (6 months). The residual component (orange) shows the shortest periodicity, being on the order of 3 months. Note that in each case, the decomposition is phase shifted with reduced magnitude in comparison to MOPITT. When we compare the decomposition in each region, we see that the long-term component does the best job of reconstructing the original time series over the northern extratropics, as shown in Fig. 5b. The magnitude of the long-term component matches well with the original series, but peaks earlier in time. The same thing can be said regarding the southern extratropics, but with reduced magnitude relative to the original series. In Fig. 5d, we note that the long-term mode in the tropics is significantly shifted forward in time compared to the original series. Finally, we observe that the seasonal mode contributes the most to the reconstruction in the tropics since its magnitude relative to the original time series is largest.

By implementing SSA on each grid point, we produce 3 matrices, each with the same size as our original dataset, which are again mean-centered and standardized. We then conduct EOF analysis on the resulting matrices, displaying the first 3 global modes for MOPITT in Fig. 6 together with the first 3 modes of the long-term, seasonal, and remainder datasets. While there are some differences in magnitude and percent variance, we note that the first two long-term EOF modes reproduce the spatial pattern of the first two modes for the original MOPITT CO patterns. Examination of the third long-term EOF mode shows that the pattern appears to be noise, as indicated by the low percent variance and the complete lack of any spatial structure. We also note that there is a very similar structure between the third MOPITT EOF mode and the first Seasonal EOF mode; however, the first seasonal mode has a larger contrast between the positive and negative patterns, which is especially noticeable in Northeastern Asia and the east coast of the United States. The structure for seasonal modes 2 and 3, as well as all three residual modes, represents patterns distinct from our MOPITT EOF modes before we performed SSA on the data.

We also note that according to Fig. 6d, our long-term EOF mode 1 has the largest positive and negative anomalies in the extratropics. This observation is consistent with our time series in Fig. 5b and c, which showed that the long-term mode did the best job of matching the magnitude and phase of the original MOPITT time series in this region. Similarly, our seasonal EOF mode 1 in Fig. 6g showed the largest anomaly in the tropics. This is consistent with the time series of our seasonal mode in Fig. 5d, which had the largest magnitude relative to the original time series. A reduced influence of the long-term mode and a corresponding increase in the influence of the seasonal mode in the tropics is consistent with CO having a shorter lifetime in this region.

Our intention in performing EOF analysis was that each EOF mode would represent the variability from either a source or sink in CO; however, the patterns depicted in Fig. 6 do not correlate with physically separable sources. If each mode represented a source of CO, then we would expect to see modes that correlated with variability due to fossil fuel burning, biomass burning, biogenic oxidation, or CH₄ oxidation. However, none of the modes depicted in Fig. 6 show patterns of variability that can be directly connected with source distributions, nor do they appear to directly correspond with synoptic weather patterns or transport pathways.

As an example, let us attempt to connect MOPITT EOF 2 with fossil fuel burning. If we attempt the interpretation that blue areas represent areas with high contributions of fuel burning, this would be inconsistent with our knowledge of fossil fuel burning as a source of CO. While we do see significant areas of blue highlighted on the east coast of the United States which is consistent with combustion, the pattern shows too little activity over Eastern China and India and higher activity over Russia than expected. Similarly, when we examine EOF 3, we see a pattern that has some consistency with biomass burning in Africa and South America; however, the features are far too smeared to be a potential interpretation. For example, the pattern stretches too far across oceanic regions as well as the Sahara and Saudi Arabia to be classified as potential fire activity. For similar reasoning, it would not make sense for us to interpret this as a pattern of biogenic emission.

When examining EOF mode 1, we see a very clear boundary between the NH and SH as indicative of the annual cycle in CO caused by the combination of larger combustion activity and lower hydroxide radicals in the NH during the winter season. This interpretation is consistent with an observation we state in Sect. 3.3 on spectral analysis and time scales, which is that the time scale for EOF 1 is dominated by annual/semiannual variation.

The majority of the areas with the highest contrast in Residual EOF mode 1 are seen to be over oceanic regions, which suggests the possibility of a connection between Residual mode 1 and oceanic variability. However, these patterns do not match the variation of CO due to oceanic cycles, such as patterns as shown in . This could be because we would expect oceanic variability to be strongest near the surface, and patterns due to surface variations would be obscured by the changes in CO throughout other areas of the atmosphere, which we are accounting for by using total column data. This could also be because budgets for oceanic cycles of CO vary widely both spatially and temporally due to changing environmental conditions .

Examining the remaining seasonal and residual patterns, we see patterns that could be indicative of regional-scale transport; however, verification and further discussion of these modes is beyond the scope of our analysis at this time.

In Fig. 7, we display the principal component time series for the original MOPITT dataset as well as the long-term, seasonal, and residual components. We first note that the first two long-term principal components have very similar periodic patterns as the original 2 MOPITT modes, while the third mode is, of course, noise because, as we have already seen, the third long-term EOF pattern had no spatial structure. We also see that the first two seasonal components have a similar periodic pattern to the third MOPITT component, even though we saw from Fig. 6 that the second seasonal EOF has a very different spatial structure. The long-term modes have the largest period, while the seasonal components have shorter periodicity, and the residual modes have the shortest.

To determine the most dominant time scale for a given mode, we used Fourier transforms to compute the power spectral density of each principal component and then used a numerical trapezoid rule to determine the total cumulative power as a percentage, which is then plotted in Fig. 8 on a logarithmic scale. The percent of total power is labeled for a period of oscillation for 100 d, 1 year, and 1000 d. We note that in this context, having time scales longer than the lifetime of CO still makes physical sense because some variability of CO is driven by large-scale weather patterns. The intrahemispheric mixing time is about 6 months, while interhemispheric mass exchanges can happen across 1–2 years . Variability on this time scale could also be indicative of changes to climate drivers. For example, deforestation over several years may lead to long-term reduction of biogenic and fire emissions, while urbanization may lead to an increase in anthropogenic emissions. A dotted line and the value tmax indicate the period of oscillation that contributed the largest amount to the total power. For example, examining MOPITT Spectrum Mode 1, we see that 1.3 % of power is from periods of 100 d or smaller, 26.9 % of power is from periods of 1 year or smaller, and 43.2 % of power is from periods of 1000 d or smaller. We see that the first two MOPITT patterns correspond to variation dominated by an annual/semi-annual cycle, while the third mode is dominated by seasonal changes on the order of 3 months. Our long-term EOFs reproduce our original 2 modes, while our seasonal EOFs show dominant time scales on the order of 3 or 2 months, and our residual modes show dominant time scales on the order of 2 months or less.

As discussed in Sect. 2.2.3, one weakness of EOF analysis is that even though the principal component time series are constructed to be uncorrelated, it is not necessarily appropriate to assume that they represent independent processes, which complicates the interpretation of assigning physical meaning to the modes. Only in the special case that the data is Gaussian does it follow that uncorrelated random variables are then deemed independent. In Fig. 9, we plot the mean skewness field for MOPITT CO over the period of interest from 2005–2018. It is standard to consider values of absolute skewness less than 0.5 to have negligible skewness corresponding to a relatively symmetric distribution, while values between 0.5 and 1.0 are moderately skewed, and values larger than 1.0 are highly skewed. Contour lines are shown for skewness values of -1, -0.5, +0.5, and +1.0 to highlight areas that have a notable degree of skewness.

We note that these sharp gradients in skewness in Fig. 9 are driven by long-term seasonal changes in sources and sinks depending on the region. Regions of low skewness generally indicate places where pollution levels are relatively constant throughout the year, with few extreme highs or lows. High positive skewness values indicate a larger number of extreme pollution events are occurring in the area, and may be interpreted as local enhancement from long-term transport. For example, we interpret the high skewness we see in central Africa as transport due to biomass burning plumes which frequently move westward . Similarly, large skewness across South America may be indicative of the secondary contribution of isoprene oxidation near the equator. This observation is consistent with , who discuss how isoprene in South America is uplifted to sufficient levels to easily react with OH, where it is slowly transported far south and into the extratropics. The bands with high skewness across the SH are consistent with the north/south movement of the inter-tropical convergence zone due to exchange of CO through the Hadley circulation . The skewness patterns we see in the NH are more diverse and are indicative of the seasonal changes in fossil fuel usage, which vary by region and sector.

Overall, we see from Fig. 9 that our dataset has significant skewness in the SH, especially in South America, as well as over China, Indonesia, and East Asia. This indicates that we cannot consider our dataset to be approximately Gaussian, and this is a potential explanation for why our principal components show significant dependence in the distributions of their spectral responses. EOF analysis assumes that the dataset is stationary so that the mean and temporal covariance remain constant in time, which is done to ensure that the first and second moments (the mean and covariance) of random variables can be determined using a limit of simple summations (or integrals). In practice, this assumption is rarely realistic with physical data. An equivalent statement is that the temporal autocorrelation is constant as a function of lag time τ. If we consider the mean and covariance of CO, it would be realistic to assume that these statistics should be a function of time because their sources and sinks are correlated temporally. If we imagine a smokestack emitting CO at a fixed location, the CO measured in the future is highly correlated with the amount of CO in the present. This brings us to our next main point, which is that another reason why our modes cannot be interpreted in terms of individual sources and sinks lies in the non-stationarity of the MOPITT dataset, which remains unaccounted for in standard EOF analysis and is a fact we will now show.

As shown in Fig. 10, the autocorrelation for the original MOPITT time series, as well as each component from the seasonal decomposition, does not decay to zero but instead shows periodic variation that persists across time. The MOPITT series has a positive correlation that peaks for every year exactly and shows two negative peaks, both of which also recur at intervals of 1 year. We could therefore consider the MOPITT autocorrelation as being a sum of three periodic components. The long-term and seasonal autocorrelation are simpler functions, and both have a single periodic component that peaks exactly once per year and 6 months, respectively. The residual autocorrelation, like MOPITT, is a sum of periodic processes. It is important to note that we cannot force the dataset to become stationary by removing an annual cycle, because the autocorrelation represents a superposition of multiple periodic processes. We may therefore consider both the long-term and seasonal series to be wide-sense cyclostationary processes since they have a periodic temporal mean and a periodic temporal autocorrelation. Because the MOPITT series and residual series are composed of multiple periodic processes, we may consider them to be wide-sense polycyclostationary. We conclude this discussion by mentioning that in future studies, it may be worth exploring the use of cyclostationary EOF (CSEOF) analysis on either the original time series or on the seasonal modes. CSEOF analysis is a method that is well-suited for datasets that possess autocorrelation functions of this type . Unlike traditional EOF methods that assume stationary covariance, CSEOF can better capture evolving physical modes, a feature essential for analyzing atmospheric time series.