We describe an approach for determining limited information about the vertical distribution of carbon monoxide (

We present here the Temporal Atmospheric Retrieval Determining Information from Secondary Scaling (TARDISS) retrieval algorithm. TARDISS uses several simultaneously obtained total column observations of the same gas from different absorption bands with distinctly different vertical averaging kernels. The different total column retrievals are combined in TARDISS using a Bayesian approach where the weights and temporal covariance applied to the different retrievals include additional constraints on the diurnal variation in the vertical distribution for these gases. We assume that the near-surface part of the column varies rapidly over the course of a day (from surface sources and sinks, for example) and that the upper part of the column has a larger temporal covariance over the course of a day.

Using measurements from the five North American TCCON sites, we find that the retrieved lower partial column (between the surface and

Remote sensing retrievals of atmospheric gas abundances are used to diagnose the sources, sinks, and fluxes at local, regional, and global scales
(Connor et al., 2008, p. 2; Deeter, 2004; Kerzenmacher et al., 2012; Wunch et al., 2011). Compared with in situ measurements, these retrievals, which
are used in carbon cycle science investigations, are less influenced by nearby point sources or sinks and rapidly changing meteorological conditions
that would lead to erroneous flux calculations (Keppel-Aleks et al., 2012). Because the column represents the integral of a gas from the surface to
the top of the atmosphere, flux estimates from column amounts are less sensitive to errors in the assumed vertical transport than those using surface
measurements (Keppel-Aleks et al., 2011, 2012). In contrast, since signals of

Profile retrievals can, in principle, ameliorate these issues and thereby enable more direct information on surface processes. Theoretical analysis
shows that two to three vertical degrees of freedom (DoF) can be achieved in

Several operational

In our approach, we do not retrieve profile information directly from the spectra. Instead, we utilize the vertical and temporal domains to infer partial column dry mole fraction (DMF) values. We fit partial column scalar values to match TCCON-retrieved total column DMF that are (i) quality controlled and (ii) individually tied to World Meteorological Organization (WMO) trace gas standard scales, which mitigates a number of errors in the forward radiative transfer model, including those arising from errors in the spectroscopy. We use the extant multiple total column measurements from spectral windows with different line intensities and hence different shapes of column averaging kernel. We extract the vertical information from the diurnally varying differences in these total column values and additional a priori information about the expected temporal covariance in the different partial columns based on known atmospheric behavior. This method allows us to extract information focused on the lower atmosphere where the trace gas DMFs are most sensitive to surface exchange.

The uncertainty of this new method for retrieving partial column values is evaluated using comparisons with in situ vertical profile measurements. Section 2 describes the theory and parameters chosen for our retrieval and the data used for the retrieval, validation, and comparison. Sections 3.1 through 3.3 present our validation data and a sensitivity study of the retrieval parameters. Section 3.4 presents an error and information content analysis. Finally, Sect. 3.5 gives examples of the data retrieved using this approach.

The Total Carbon Column Observing Network (TCCON) is a ground-based network of solar-viewing Fourier transform spectrometers equipped with InGaAs and
Si detectors that gather spectra for the 3900 to 15 500

Column abundances of atmospheric species are computed from the measured spectra using a nonlinear least-squares fitting algorithm, GFIT, which
minimizes the residuals between a measured spectrum and one calculated by uniformly scaling a priori vertical profiles for the fitted atmospheric
species, yielding the optimal VMR (volume mixing ratio) scaling factors (VSF) of the fitted gases. The a priori profiles scaled by the VSF are
integrated to calculate the total column abundance of a species. The retrieved scaled column abundances are converted to column dry mole fraction
(DMF) by multiplying by 0.2095 and dividing by the column of

Vertical sensitivities of the total column retrievals from GFIT used in our algorithm for both

For each window and for each spectrum fit by GFIT, an associated column averaging kernel is computed that describes the sensitivity of the total column to changes in species abundance at each altitude (shown in Fig. 1). A perfect column averaging kernel would have values of 1 for all altitudes. More commonly, the kernels will vary slowly with altitude with a pressure-weighted average value close to 1. Values higher (lower) than 1 mean that the retrieval is more (less) sensitive to trace gas changes at that altitude. These sensitivities vary with solar zenith angle (SZA) as the spectral absorption deepens at higher SZA. The vertical sensitivity of each window is a result of its spectral properties. Optically thin spectral regions (windows) tend to be more sensitive to the upper troposphere and the stratosphere while optically thick windows tend to be more sensitive to the lower troposphere. Since information about the stratosphere comes only from near the line center as a result of diminished collisional broadening, if the absorption at the line center is saturated (nearly zero transmission), the spectrum will contain little information about the stratosphere, and hence the kernel will be low there. The differences in column averaging kernel shapes are the main source of information used in the Temporal Atmospheric Retrieval Determining Information from Secondary Scaling (TARDISS) algorithm. The outputs of the VSF values, a priori profiles, total column DMF values, and vertical averaging kernels from standard TCCON processing are used as input for the TARDISS algorithm.

We will refer to the spectral retrievals as being the TCCON retrievals and the temporal partial column retrievals as the TARDISS fit. We also use the terms retrieval and fit interchangeably to refer to the TCCON or TARDISS methodology.

Flowchart illustrating the steps performed by of the TARDISS retrieval. The input to the TARDISS retrieval is the output of the spectral fitting done by the GGG2020 software suite represented by the green row. The setup of the components of the TARDISS algorithm from the output of the TCCON spectral fits is shown in Eqs. (11) through (14) and in the middle row. The TARDISS retrieval is performed using Eq. (16), the output partial column DMF values are calculated using Eq. (17), and the information content is calculated by Eqs. (18) and (19) as shown in the bottom row.

Traditional profile retrievals fit spectra by adjusting the abundance of the trace gases at multiple vertical levels to determine the vertical
distribution of a specific atmospheric species (Pougatchev et al., 1995; Roche et al., 2021). Here, we describe the Temporal Atmospheric Retrieval
Determining Information from Secondary Scaling (TARDISS) algorithm that optimizes the scaling of the profile of our target gas separated into two
layers, one near the surface and the other at and above the typical well-mixed surface boundary layer. This is illustrated by the flowchart in
Fig. 2. The algorithm minimizes the cost function (Eq. 1) by finding the maximum a posteriori solution for a state vector containing upper and lower column scale factors for all TCCON observations in a given day. That is, if a day has

We use the notation and concepts of Rodgers and Connor (2003) with vectors represented with bolded lower-case letters and matrices represented with
bolded upper-case letters. We start in the vertical domain where Eqs. (3) through (9) are used for each spectral window, each TCCON measurement, and
each species retrieved (

To derive the values used in the Jacobian matrix,

Here,

The next step is to rearrange this equation so that our observed quantity is on the left-hand side, and the right-hand side is a linear combination of
the two scaling factors. Subtracting

Here, we assign

Since Eq. (5) is linear, we then group terms, reducing the right side of Eq. (5) to

Defining two new variables,

Equation (7) is applicable to all spectral windows for each spectrum measured. For example, for our

While Eq. (7) can be set up and solved for each spectrum using the total column value from each spectral window used in the TCCON fit, the TARDISS
retrieval uses an entire day's worth of TCCON retrievals in order to increase the signal-to-noise ratio and to utilize the information from the temporal
variation in the kernels. Fitting over an entire day of TCCON retrievals reduces the retrieved partial column error values compared to fitting
individual measurements using Eq. (7). Section S1 in the Supplement shows the influence of including
multiple observations on the retrieved partial column errors. Let

With

Since Eq. (10) is linear, we can apply a basic linear least-squares method to solve for the partial column scalars:

while the linear least-squares method provides a useable solution to our retrieval, it also has partial column error values on the order
of 10

We use the maximum a posteriori (MAP) approach (Rodgers, 2008) to calculate the most probable state vector from the given models and a priori
information. In line with the assumptions of the MAP approach, we assume our problem is linear and follows a Gaussian distribution. The MAP solution
can take a few equivalent forms. In this work we use

Once we have calculated the most likely solution for the partial column scalars as a vector in temporal space,

The MAP retrieval allows us to calculate the information content of the retrieval. In particular, we compare the degrees of freedom for our retrieval
calculated by taking the trace of the averaging kernel of the fit, calculated as the follows:

Generally, profile retrieval averaging kernels represent the sensitivity of a specific level of a profile to the rest of the levels in the profile. The averaging kernel for the TARDISS inversion is a temporal averaging kernel relating how each partial column calculation relates to every other measurement during a day. The DoF value for a day of the retrieval represents how many individual pieces of partial column information we can infer over the day of measurements. We either report the number of degrees of freedom from the fit over a day or normalize the degrees of freedom by the number of measurements in each day for a more comparative understanding of the TARDISS degrees of freedom with respect to a traditional profile retrieval and between days with a large variation in the number of measurements.

To evaluate the accuracy of our partial column retrieval, we use the smoothing calculation shown in Eq. (3) of Wunch et al. (2010), altered to use the
terminology of this work, to determine the value of the partial columns of the TCCON total columns used as input:

In order to compare the partial column retrievals to in situ profiles for validation purposes, we calculate the vertical sensitivities of the TARDISS
fit (shown in Fig. 8) using the gain matrix,

Since

For our TARDISS comparisons, we use an adjusted version of Eq. (20) to determine the value the inversion would return if it were using the true
profile instead of the scaled TCCON priors:

Example of an a priori covariance matrix color coded by the magnitude of the value. The axes represent the relationship of the contribution of each measurement to each partial column and each other measurement. The upper-right and lower-left quadrants are dark blue and represent zero assumed correlation between the upper and lower partial columns over a day of measurements, respectively. The diagonal is scaled to constrain the fit and the lower-right quadrant shows the assumed correlation between upper partial column scalar values over a day of measurement. The lower partial column has an a priori covariance that is a scaled identity matrix, the upper partial column has an a priori covariance that decays over one-third of the measurement day, and the cross covariances between the upper and lower partial columns are assumed to be zero.

An example of the profiles used in the direct comparison calculations using data from the Park Falls site on 27 July 2018. The profile above 10

Finally, the error for the retrieval is made up of model parameter error, smoothing error, and retrieval noise (Rodgers, 2008). There are no model
parameters in the state vector of the TARDISS retrieval, so the model parameter error is zero. The smoothing error is the square root of the diagonal
of the following matrix:

In order to report an error for our retrieval that reflects the performance of the retrieval in the validation comparisons in Sect. 3.1, the retrieval
output errors are multiplied by a scalar calculated from the one-to-one comparisons. Using the multiplier ensures that we are reporting a conservative
estimate of the error in the retrieval. We use the one-to-one comparisons to scale our error values to the point where at least 50 % of the comparison
points are within the 1 standard deviation error range of the one-to-one line. We calculate the scalar values as follows:

We begin by preprocessing the TCCON fits. We take the TCCON a priori profile and scale it by the median value of the TCCON output scalar values for each spectrum from the windows used so that our TARDISS fit is centered around the median TCCON a posteriori profile for each measurement point. The a posteriori errors from each window are not included in this calculation but are included in the formation of the measurement covariance matrix. This assumes that the true column-averaged VMR of a species is some linear combination of the VMRs calculated from the windows used in the TARDISS fit. We then calculate the a priori partial columns by integrating the scaled a priori profiles over the respective pressure levels for the upper and lower partial column. Finally, we assemble the necessary matrices for the fit described by Eq. (16).

The different components of Eq. (16) reflect where a priori information can be used in the algorithm and several additional choices can be made to improve the fit. The following describes our standard input for these components. We present tests of the retrieval's sensitivity to these choices in Sect. 3.2.

For the a priori covariance matrix,

The measurement error covariance matrix,

The primary information content used in our algorithm is derived from the fact that the total column abundances retrieved from different spectral windows of the same species will differ due to differences in their kernels unless the shape of the a priori profile is perfect. Accordingly, for this method to have sufficient information, windows with different vertical averaging kernels are needed, such as those shown in Fig. 1. Preferably, the TARDISS retrieval would use a window that is more sensitive to the lower atmosphere and a window that is more sensitive to the upper atmosphere so that a larger amount of information is contained between them. While it is imperative to use windows that have differing averaging kernel profiles, it is also necessary to use windows that have sufficiently low error in the TCCON fit.

For the partial column

Unlike the

Other windows output by TCCON retrievals were considered for the partial column calculations for both species. However, they had high levels of error in the TCCON fit or had fits that were particularly sensitive to changes in temperature.

We chose the lower partial column to integrate from the surface through the first five vertical layers of the GEOS meteorological fields. Using this
criterion, a site at sea level has a lower partial column from sea level to 2

Location, dates of measurement, and DOIs of the TCCON sites used in this work.

In this study, we use data from the five TCCON sites located across North America (Iraci et al., 2022; Wennberg et al., 2022a, b, c). The data record extends from as early as 2011 to as recent as 2021 (Table 1). These sites are located at Park Falls, Wisconsin; NASA Armstrong, Edwards Air Force Base, California; Lamont, Oklahoma (the Department of Energy Southern Great Plains Atmospheric Radiation Measurement site); the California Institute of Technology (Caltech); in Pasadena, California, and East Trout Lake, Saskatchewan.

Park Falls, WI, hosts the first operational TCCON site (July 2004–present). The site is in a rural, heavily forested area and generally far from
anthropogenic influence. The Fourier transform spectrometer (FTS) does not have an InSb detector, so we are able to only retrieve partial column values for

We use similar data from the TCCON site located at NASA's Armstrong Flight Research Center (formerly the Dryden Flight Research Center) in California,
which has been operational since July 2013. We report

The Lamont, OK, TCCON site is surrounded by farmland. It has been operational since July 2008, and an InSb detector was installed in October 2016. We
focus on data from Lamont obtained after 2011 after an issue with the instrument laser was resolved. We report

The TCCON site on the Caltech campus in Pasadena, CA, has been operational since July 2012 with an InSb detector measuring since October 2016. We
report

The East Trout Lake, SK, TCCON site is located in a remote, heavily forested area in the middle of Saskatchewan. The instrument uses an
InSb detector allowing us to retrieve partial column

We use in situ data from multiple aircraft programs and AirCore flights between 2008 and 2020 (Cooperative Global Atmospheric Data Integration Project, 2019; Baier et al., 2021) to evaluate our partial column retrieval.

The aircraft

We use AirCore profiles from July and August of 2018 at the Armstrong, Lamont, and Park Falls sites (Baier et al., 2021). The AirCore sampling system is composed of coiled
stainless-steel tubing that is open on one end while ascending on a balloon to

Finally, we use

The TARDISS algorithm is very quick – taking only a minute of processing time per year of data for each species – because it does not repeat the
spectral fitting. This speed enables the validation comparisons to be performed using many different model choices. Thus, we evaluated the sensitivity
of the TARDISS inversion by varying different forward model choices. The set of choices that we have designated as the operational setup for

The covariance matrix,

The value of the a priori scalar for the lower and upper partial column scalar (

For the

a covariance matrix,

an ideal a priori partial column scalar (

We vary two aspects of the algorithm and observe the differences in the validation comparisons. The results of these tests are discussed in Sect. 3.2 and represented in Tables 2 and 3.

We compare retrieved partial column values from three of the five sites presented in this work using measurements from the same set of in situ data
used to evaluate and derive the “in situ scaling factor” of the TCCON retrievals (Wunch et al., 2011). For

We also compare the partial columns calculated from the TCCON individual windows to further contextualize the performance of the TARDISS algorithm in Sect 3.3.1 and summarized in Table 4. The comparisons of the TCCON individual windows are performed in the same way as the TARDISS comparisons using Eq. (20) to calculate the smoothed in situ partial columns.

The comparison profiles were measured by aircraft-based instruments or AirCore measurements as described in Sect. 2.5 and Table S2. We revert to the
TCCON priors for parts of the profile not measured by in situ methods. For the errors associated with the aircraft measurements, we use the reported
measurement error for the measured parts of the profile, and for the unmeasured parts of the profile we use the average reported measurement error. To account for the errors involved with estimating the parts of the profiles not measured by in situ methods, we add in quadrature twice the
standard deviation of the measured profile in the respective partial column. For the errors associated with the AirCore measurements, we use the same
approach as for the aircraft measurement and include an extra error term to conservatively account for atmospheric variability as captured by
duplicate AirCores launched at approximately the same time. The error for AirCore from atmospheric variability is 0.6

We compare the TARDISS retrievals from spectra obtained within 1 h of the in situ profile to the integrated, smoothed, and in situ partial columns
calculated using Eq. (24). We report linear fits between the partial column retrievals and the integrated, smoothed, and in situ partial columns. Since
our retrieval is designed to be linear, we use fits with

We use these validation comparisons to perform sensitivity tests of our algorithm parameters and determine an operational set of parameters. We then
use these optimal parameters for the

Several terms in our retrieval do not have unambiguously correct values. To evaluate the sensitivity our retrieval to the choices made for these parameters, we have run our retrieval with alternate values and report the degrees of freedom and comparison to in situ data (specifically, the retrieval comparison error, slope of the zero-forced linear fit, and the mean ratio deviation value of the linear fit) for each test. We tested changes to two terms: the TARDISS a priori scale factors and the a priori covariance matrix scaling.

Variations in

Variations in

Comparisons of the TARDISS partial column retrieval to the partial column comparisons of the fits of the TCCON spectral windows from TCCON used as input for the TARDISS algorithm. The data in the TARDISS row uses the operational parameters for the fit that are identified in Tables 2 and 3 by an asterisk.

To test the sensitivity of the retrieval to the partial column scalar prior, we compare the changes in the validation when using

We also test the sensitivity of the retrieval to how the a priori covariance matrix is scaled. This term changes how strongly the retrieval is
constrained to the prior. In Tables 2 (

The agreement between the in situ and TARDISS retrievals for

We compare the validation performance of the TARDISS partial column retrievals to the partial column validations of the TCCON individual windows used
in the retrieval to demonstrate that TARDISS provides additional information about vertical distribution compared to the TCCON retrieval. We compute a
partial column from the TCCON output by integrating the posterior TCCON

The direct comparisons between the partial column DMF values retrieved from the TARDISS fit and the integrated, smoothed in situ partial columns for

The comparisons show that the TARDISS retrieved partial columns for

These comparisons suggest that, for

For

Finally, we compare the performance of the total column values calculated from the TARDISS scaled partial columns to the total column validations of the TCCON individual windows. The comparisons are shown in Fig. S3 in the Supplement and summarized in Table S3 in the Supplement. The total column comparisons show similar trends to the upper column comparisons. This is likely due to the upper partial column vertical sensitivity being much larger than the lower partial column sensitivities, as is discussed in Sect. 3.4.1.

East Trout Lake site direct comparisons between the partial column DMF values retrieved from the TARDISS fit and the integrated and smoothed aircraft partial columns for lower column

In addition to the aircraft and AirCore validation data that include profile measurements at altitudes in the upper troposphere and lower
stratosphere, we compare to aircraft data obtained as part of the NOAA GGGRN aircraft program at the Lamont and East Trout Lake sites. These
measurements were made more frequently but do not include enough high-altitude measurements to compare with our retrieved upper partial column values,
so we use them as an independent comparison to our validation data for our lower column

Similar to the validation comparison, we revert to the a priori profile for altitudes not measured by in situ methods. To account for the errors in
using the a priori profile, we add twice the standard deviation of the partial column that is measured to the average measurement error in
quadrature. Given the lower altitudes measured by the GGGRN program, the errors associated with the parts of the profile that use the a priori profile
are higher, and therefore the errors in the long-term comparative measurements tend to be much higher than the validation measurements, as shown in
the

Despite the larger error values, the consistency of the statistical parameters (summarized in Table S4 in the Supplement) using the larger number of measurements in the long-term comparisons further motivates the use of the extended validation
dataset. Some of the in situ profile comparisons occur during times with larger

Lamont site direct comparisons between the partial column DMF values retrieved from the TARDISS fit and the integrated and smoothed airborne partial columns for lower column

Validation comparison DoF, error, validation slope, mean ratio deviation, and site VEM values for lower and upper column

TARDISS uses an a priori covariance matrix with temporal covariance between upper partial column scalars over the course of a day of measurement, as
shown in Fig. 3. To determine how this constraint influences the retrievals, we compare the data above to the validation comparison from a

Vertical sensitivities of the lower partial column

The temporal covariance impacts our validation comparison through the partial column vertical sensitivities described in Eq. (22) via the gain matrix (Eq. 21). To assess the importance of the choice of a priori covariance matrix, we compare the vertical sensitivities for a temporally constrained upper column and a temporally unconstrained upper column (shown in Fig. 8) for a representative day (27 July 2018, at the Lamont site).

Without the temporal constraint, the upper column sensitivities are on the same order as the lower column sensitivities with values between

With the temporal constraint, the altitude of the maximum sensitivities with respect to SZA remains similar but the upper column sensitivities are roughly twice the value and the lower column sensitivities are half the value of the temporally unconstrained values. The imposed temporal covariance constrains the upper column to vary slowly over the span of a measurement day so that a change in the column at one measurement point induces changes at other measurement points, thereby increasing the vertical sensitivities in the upper column over the entire day. This constraint is also stringent enough that it propagates into the sensitivity of the lower column scalar. Since our goal is to retrieve a lower partial column, it seems counterintuitive that using sensitivities with an order of magnitude difference provides a better validation comparison. However, for this method we assume that we know the shape and behavior of the upper column fairly well and that most of the change occurs near the surface. Given these assumptions, constraining the upper column more heavily by introducing expected daily patterns through the a priori covariance matrix allows for the lower column retrieval to have improved comparisons with in situ data despite the decreased vertical sensitivities.

While we test retrievals simply with and without temporal covariance, the possible choice of a priori covariance matrix shape could be much more complex. Future study could include using model-generated or back-trajectory-based temporal covariances to include outside information in the retrieval dynamically. For an operational retrieval product, we will include the temporal covariance in the a priori covariance matrix as an operational parameter.

Errors in the

Using the information from the validation comparison, we can evaluate the errors of the entire dataset from each of the five sites. The output of the
retrieval is the partial column scalar and the error retrieved is the standard deviation of the partial column scalar calculated from the retrieval
variance and represented as another scalar value. To convert our partial column scalar error to a dry air mole fraction, we multiply the error scalar
value by the a priori partial column mixing ratio (

The retrieval error values range from 1.16 to 1.41

Because the model parameter error goes to zero in our implementation, the current total retrieval error is the square root of the sum of the smoothing
error (Eq. 25) and the retrieval noise (Eq. 26). The smoothing error is 94.0 % to 96.5 % of the total retrieval error on average for

Using the operational setup for our TARDISS fit, we calculate the site-specific VEM values using Eq. (27) (Tables 5 and 6). These values are used to
scale the error of the TARDISS fit for all the comparisons in this work. The VEM-scaled errors serve as a conservative estimate for the retrieval
errors and should be reevaluated with additional in situ profile measurements as they become available. For

Since the TARDISS retrieval cannot fully optimize the shape of the partial profile, the site-to-site differences in VEM are likely due to the
variation in the accuracy of the TCCON priors, which by design do not capture the local source, sink, and transport complexities. For

The total error for each site is determined by multiplying the retrieved errors by the site and partial column respective VEM values. After
implementing the VEMs, the errors for the lower partial column

Since the overall biases are small with validation slopes close to 1, the errors are sufficiently small that the TARDISS retrievals have skill in
evaluating

The information content of the retrieval is determined by the DoF and Shannon information content (H) of the retrieval, each calculated from the averaging kernel. The DoF values represent the independent pieces of information that can be retrieved from a measurement. We report our DoF values normalized by both the number of measurements made in a day and the daily overall DoF. Since the DoF values are calculated as the trace of the averaging kernel, we isolate and report the DoF from the upper and lower column separately along with the total. The Shannon information content is a single value to represent the effectiveness of the retrieval when recovering information from the model with respect to the variance in the data. Higher Shannon information content values correspond to a retrieval with a higher possible effectiveness.

Degrees of freedom per measurement (and per day) for the lower column, upper column, and total retrieval, in addition to the Shannon information content separated by site for the

The information content is summarized for each site in Table 7. The overall average lower column DoF per measurement across all sites and collected
data is 0.047 for

Ideally, DoF values greater than 1 are desired for traditional profile retrievals. However, the temporal aspect of our retrieval complicates the
discussion. If we consider the

The information content shown in the DoF is mirrored in the Shannon information content. Similar to the DoF, Park Falls has the lowest and Armstrong
the highest Shannon information content on average for

The same comparison shown in Fig. 7 is shown here without error bars and color coded by the DoF per measurement for the comparison day retrieval. The dotted–dashed blue line above the black one-to-one line is the linear fit of the data with the

Time series plot of the monthly median lower

The informational content of the retrieval assists in evaluating the TARDISS algorithm, but also serves as a diagnostic of the effectiveness of the
retrieval for each day of measurement. Figure 9 shows the long-term comparisons between the retrieved lower partial column and the smoothed,
integrated, and in situ data at the Lamont site color coded by the DoF per measurement for each point. The comparisons with higher DoF per measurement
generally sit closer to the one-to-one line as expected and suggest that days with higher DoF per measurement have lower associated VEM. Figure S7 shows the VEM calculated after removing days that have DoF per measurement values below a
specific threshold. The VEM calculated for the long-term comparison data decreases consistently with increasing DoF filters until it reaches 1 at

Time series plot of the monthly median lower

The TARDISS algorithm is applicable to any spectra reported as TCCON data with the correct detector requirements (InGaAs for

Figure 10 shows the monthly mean lower and upper partial column data retrieved from spectra obtained over the last decade at the North American TCCON
sites. These upper columns reflect the global seasonal patterns in

Figure 11 shows the monthly median retrieved lower and upper partial column

The TARDISS retrieval algorithm enables partial column information to be derived from the TCCON total column observations of

Using measurements from the five North American TCCON sites, we compare our retrieved partial columns of

We use the comparison data to calculate validation error multiplier (VEM) values to scale retrieved errors to be representative of the in situ
comparisons. The average VEM scaled errors for the lower partial column

The Bayesian TARDISS algorithm enables the informational content of the retrieval to be estimated. The average DoF for the lower partial column
retrievals are 8.89 and 27.4 degrees of freedom so that

Future implementations of the retrieval could use the DoF values to create dynamic VEM to provide error values that are more precise than the static
VEM. Similarly, future work could improve the effectiveness of the retrieval of lower partial column

The data used in this study are made up of TARDISS retrieval products from five TCCON stations. The retrieval data are publicly available through CaltechDATA (

The supplement related to this article is available online at:

HAP wrote the TARDISS algorithm following an approach suggested by POW. HAP retrieved the data with it and prepared the paper with thorough feedback from the other co-authors. JLL developed the theoretical framework for the TARDISS algorithm. CMR retrieved the TCCON data using GGG for the Lamont, Caltech, and Park Falls sites. GCT gave input on the retrieval algorithm. DW gave input on the validation data and method. LTI and JRP maintain the Armstrong site. KM and BCB provided insights and in situ data for the validation. All authors contributed to the review and editing of the work.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors thank the ObsPack team and data providers for the in situ profile data used for validation. The data were downloaded from

This work was funded via grant no. 80NSSC22K1066 for supporting the retrievals from the TCCON stations. A portion of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (grant no. 80NM0018D0004). Bianca C. Baier is supported by NASA (grant no. 80NSSC18K0898).

This paper was edited by Thomas von Clarmann and reviewed by two anonymous referees.