High-quality, long-time-series measurements of atmospheric greenhouse gases show interannual variability in the measured seasonal cycles. These changes can be analyzed to better understand the carbon cycle and the impact of climate drivers. However, nearly all discrete measurement records contain gaps and have noise due to the influence of local fluxes or synoptic variability. To facilitate analysis, filtering and curve-fitting techniques are often applied to these time series. Previous studies have recognized that there is an inherent uncertainty associated with this curve fitting, and the choice of a given mathematical method might introduce biases. Since uncertainties are seldom propagated to the metrics under study, this can lead to misinterpretation of the signal. In this study, we use an ensemble-based approach to quantify the uncertainty of the derived seasonal cycle metrics. We apply it to CO

Ongoing in situ measurements of the atmospheric CO

Curve-fitting methods are often used to preprocess atmospheric time series for analysis. Three examples are found in the commonly used software packages, HPspline

Diagram showing how the skewness of the seasonal cycle can influence the estimation of the CUP based on the ZCD. The three seasonal cycles have similar seasonal cycle maxima, minima and downward ZCDs but very different upward ZCDs.

Metrics derived from CO

We first use the EFD method to confirm that the CO

We use the discrete CO

Observational sites of NOAA/ESRL network used in this study.

Mean seasonal cycle of CO

The time series of CO

Flow diagram explaining the processes of curve fitting (purple boxes) and ensemble generation (blue boxes).

First, we separate the long-term trend and mean seasonal cycles (

CCGCRV is a curve-fitting method developed by Kirk Thoning and Pieter Tans (Global Monitoring Laboratory (GML), NOAA) in the late 1980s. The method fits a combination of polynomials and annual harmonics to the data to approximate the long-term variation and seasonal cycle. The short-term and interannual variability are retained by filtering the residuals from the fit using a low-pass filter. A detailed description of the routines used for fitting the data and filtering of residual can be found in

Further, for uncertainty estimation, we generate 500 bootstrap samples from the curve-fitted data. For this, we calculate the difference between the smoothed data and the observational data, which gives the new set of residuals for generating bootstrap samples. These residuals are resampled (with replacement) and added to the initial fitted curve, producing a resampled time series. The resampled time series is processed as described in the preceding sections to obtain a continuous and smooth data set with daily values. The residual resampling and further processing are iterated 500 times to create an ensemble of 500 slightly different de-trended time series (bootstrap samples) which are all consistent with the observations (Fig.

The ensemble of fitted curves is used to constrain the uncertainty in seasonal cycle metrics estimates. If the estimated metrics differ largely across the bootstrap samples, it indicates that the metric estimate is influenced by the inherent uncertainty in extracting a definitive seasonal cycle, by curve-fitting the discrete data. Hence, interpreting these metrics without accounting for this uncertainty can be misleading.

At high-latitude measuring stations, the CUP extends from the seasonal cycle maximum in spring to the seasonal cycle minimum in late summer

Schematic diagram showing the timing of the CUP as determined by the first derivative method. The timing is marked by a threshold, defined in terms of the first derivative of the CO

For each ensemble member, we calculate the first derivative of the time series as a proxy for the rate of CO

To determine the onset and termination of the CUP from CO

For the EFD method, we first optimize the threshold as described in Sect.

Standard deviation (among ensemble members) in the onset

Box plot showing the distribution of the

We estimate the duration of the CUP for each year using different approaches: (1) the difference between the seasonal cycle maximum and minimum times, (2) the difference in the ZCD and (3) using the EFD method. Figure

Box plots showing the bootstrap standard deviation (i.e., the standard error of estimate) in the timing of the seasonal cycle maximum (

When using the EFD method, the CUP estimates have least uncertainty across the ensemble members (Table

The interquartile range of the standard deviation in the CUP duration across all years as described in Fig.

CUP duration estimated from the loess-fitted residual bootstrap samples using the timing of the ZCD (

Here we show that using the EFD method, the uncertainty in the CUP estimate is reduced across all the studied sites. Previous studies

Box plot showing the distribution of the

To further test the robustness of the CUP estimates based on the loess-fitted residual bootstrap method, we compared them against the CUP estimates from an ensemble using the CCGCRV curve-fitting method. Comparable results were obtained when the same CUP estimation method (ZCD/EFD) was applied to the ensemble members using the two different curve-fitting methods (Fig.

The CUP duration approximated using the ZCD shows larger spread for sites like MLO (with an interquartile range of 16 d for CCGCRV fitted data and 43 d for loess fitted data), irrespective of the curve-fitting method used. This is attributed to variability in the UZCD due to the skewness of the seasonal cycle during periods of net release and is similar in both the curve-fitting methods. Furthermore, we find that using the EFD method of CUP estimation resulted in smaller spread across the bootstrap samples for both the curve-fitting methods (Fig.

In this study, we quantify the uncertainty in the CO

The shape of the CO

Fitted bootstrap samples (thin yellow lines) representing the seasonal cycle of a year at

At lower-latitude stations like MLO, MID and NWR, we find that the ZCD can vary across ensemble members as shown in Fig.

We find that in addition to having a larger annual uncertainty, the range of CUP values over the study period for the ZCD approach is much larger than that of the EFD approach for some sites (Fig.

Bootstrap samples representing the seasonal cycle of 2 years (red and blue) with largely different CUP timing when estimated with the two methods (a case taken from Fig.

To test the sensitivity of the different methods to changes in the CUP, we performed a synthetic data experiment. We compare the EFD and ZCD methods and additionally an approach where we determine the CUP using a 25 % threshold as in

Sensitivity of the EFD (orange), ZCD (blue) and an approach similar to that of

As seen from the experiment, the ZCD is the least sensitive to changes in the flux. Hence, years with extreme CUP approximated by the ZCD as in Fig.

The EFD and the approach in

The synthetic experiment presented here used simulated data from only 1 year, thus controlling for the influence of inter-annual variability in atmospheric transport. In reality, atmospheric transport plays an important role in explaining a significant portion of observed CO

Here, we discuss a method for estimating the timing, duration and uncertainty of the CUP and related metrics from a discrete time series of CO

The code is made freely available under a Creative Commons-BY license, along with the documentation in this paper (

The coding and analysis were performed by TK with contributions of MR. The study was conceptualized by JM, AB and MR with contributions from WP. JM, AB, WP, MR and PT contributed with expert knowledge. The original manuscript was drafted by TK, which was reviewed and edited by WP, JM, AB, MR and PT.

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.

We thank NOAA/ESRL for the measurement and maintenance of the CO

The article processing charges for this open-access publication were covered by the Max Planck Society.

This paper was edited by Abhishek Chatterjee and reviewed by two anonymous referees.