Data assimilation (DA), the statistical combination of computer models with measurements, is applied in a variety of scientific fields involving forecasting of dynamical systems, most prominently in atmospheric and ocean sciences. The existence of misreported or unknown observation times (time error) poses a unique and interesting problem for DA. Mapping observations to incorrect times causes bias in the prior state and affects assimilation. Algorithms that can improve the performance of ensemble Kalman filter DA in the presence of observing time error are described. Algorithms that can estimate the distribution of time error are also developed. These algorithms are then combined to produce extensions to ensemble Kalman filters that can both estimate and correct for observation time errors. A low-order dynamical system is used to evaluate the performance of these methods for a range of magnitudes of observation time error. The most successful algorithms must explicitly account for the nonlinearity in the evolution of the prediction model.

Ensemble data assimilation (DA) is one of the tools of choice for many earth system prediction applications, including numerical weather prediction and ocean prediction. DA is also applied for a variety of other earth system applications like sea ice (Zhang et al., 2018), space weather (Chartier et al., 2014), pollution (Ma et al., 2019), paleoclimate (Amrhein, 2020), and the earth's dynamo (Gwirtz et al., 2021). While DA was originally applied to generate initial conditions for weather prediction, it is also used for many related tasks like generating long-term reanalyses (Compo et al., 2011), estimating prediction model error (Zupanski and Zupanski, 2006), and evaluating the information content of existing or planned observing systems (Jones et al., 2014).

DA can also be used to explore other aspects of observations. An important part of many operational DA prediction systems is estimation and correction of the systematic errors (bias) associated with particular instruments (Dee and Uppala, 2009). Estimating the error variances, comprised of both instrument error and representativeness error (Satterfield et al., 2017), associated with particular observations is also possible (Desroziers et al., 2005) and can be crucial to improving the quality of DA products. DA methods have also been extended to explore problems with the forward operators, the functions used to predict the value of observations given the state variables of the prediction model. These techniques can focus on particular aspects of forward operator deficiencies (Berry and Harlim, 2017) or attempt to do a more general diagnosis that can improve arbitrary functional estimates of forward operators, for instance, an iterative method that can progressively improve the fit of the forward observation operator to the observations inside the data assimilation framework (Hamilton et al., 2019). Here, DA methods for estimating and correcting errors in the time associated with particular observations are explored.

Most observations of the earth system being taken now have precise times associated with them that are a part of the observation metadata. However, this is a relatively recent development for most applications. Even for the radiosonde network, which is one of the foundational observing systems for the mature field of numerical weather prediction, precise time metadata have only been universally available for a few decades (Haimberger, 2007). Before the transition to current formats for encoding and transmitting radiosonde observations, many radiosonde data did not include detailed information about ascent time or the time of observations at a particular height. Even the exact launch time was not always available in earlier radiosonde data that are a key part of atmospheric reanalyses for the third quarter of the 20th century (Laroche and Sarrazin, 2013).

This lack of time information is also a problem for surface-based observations, especially those taken before the radiosonde era, which relied on similarly limited encoding formats. Ascertaining the time of observations becomes increasingly problematic as one goes further into the past. As an example, coordinated time zones were not defined in the United States until the 1880s, resulting in local time uncertainty of minutes to hours in extreme cases. In fact, the major push for establishing coordinated time was motivated by the need for consistent atmospheric observing systems (Bartky, 1989). Similar issues were resolved earlier or later in other countries and not resolved globally until the 20th century.

As historic reanalyses extend further back in time (Slivinski et al., 2019), the lack of precise time information associated with observations can become an important issue. There is also a desire to use less quantitative observations taken by amateur observers and recorded in things like logbooks and diaries. An example is the assimilation of total cloud cover observations from personal records in Japan (Toride et al., 2017). While individual observers might have rigorous observing habits, the precise time at which their observations were taken often remains obscure. Curiously, the problem of time error may be less for observations used for historical ocean reanalyses (Giese et al., 2016). This is because a precise knowledge of time was required for navigation purposes. Nevertheless, observations obtained from depth can involve unknown delays, and failures to record the exact time associated with observations can remain (Abraham et al., 2013).

Even older observations, for instance, those associated with paleoclimate, can have greater time uncertainty. Here, the fundamental relationship between the observations and the physical state of the climate system is poorly known, and identifying the appropriate timescales is crucial to improved DA (Amrhein, 2020). Observations related to the evolution of the geosphere can have even more problematic time uncertainty. Initial work on using DA to reconstruct the evolution of the earth's geodynamo highlights the problems associated with specifying the time that should be associated with various observations (Gwirtz et al., 2021).

Failing to account for errors in the time associated with an observation can lead to significantly increased errors in DA results. This is especially true if time errors are correlated for a set of observations since they can result in consistently biased forward operators. Section 2 briefly describes the problem of observation time error, while Sect. 3 discusses extensions to ensemble DA algorithms that can explicitly use information about some aspects of time error. Section 4 describes several algorithmic extensions of ensemble DA that can provide estimates of time error distributions. Section 5 describes an idealized test problem, while Sect. 6 presents algorithms combining the results of Sects. 3 and 4 to produce a hierarchy of ensemble DA algorithms that both estimate and correct for observation time error. Section 7 presents results of applying these algorithms, and Sect. 8 includes discussion of these results and a summary.

The vector

The time errors involved with many real measurements could be distinctly non-Gaussian. For instance, there is reason to believe clock errors may be skewed. For real application, it would be important to involve input from experts with detailed knowledge on the expected time error distributions. The case where time error is non-Gaussian can be approached using the same arguments as in Sect. 4 but is not explored further here.

The observations have an error

Algorithms are described to extend ensemble Kalman filters (Burgers et al.,
1998; Tippett et al., 2003) to use information about the time offset of
observations. Suppose the time offset at the current analysis time,
conditioned on the value of the current observation, is distributed as

Two methods of obtaining the prior mean and observation error variance are
explored in the following subsections. In both cases, assume that an
ensemble of prior estimates of the true state,

Define

A prior estimate of the observations for each ensemble can be obtained by
linearly interpolating the values of the state to time

The previous section has presented algorithms to extend ensemble Kalman
filters to cases where an estimate of the distribution of time offset

The distribution is (incorrectly) assumed to be

The distribution from which the time offset is drawn is used,

This algorithm assumes that the difference between the observation and the
truth at the analysis time,

Assuming the system has locally linear behavior near time

We want to find the relative likelihood of a particular time offset

In real applications, the difference between the observation and the truth
at the analysis time cannot be computed, but the difference between the
observation and the prior ensemble mean,

Defining the difference of the observation error

As for the interpolation method in Sect. 3.2, assume that an ensemble of
prior estimates of the true state,

On the other hand, recall that the relative likelihood that

A set of assimilation methods described in the next section are applied to
the 40-variable model described in Lorenz and Emmanuel (1998), referred to
as the L96 model. The model has 40 state variables

A fourth-order Runge–Kutta time differencing scheme is applied with a
non-dimensional time step of

Results are explored for five different simulated observing systems that differ
by the analysis period,

For a given analysis period, the L96 model is integrated from an initial condition of 1.0 for the first state variable and zero for all others to generate truth trajectories. A total of 11 initial conditions are generated by saving the state every 1100 analysis times. The first initial condition is used to empirically tune localization and inflation, and the other 10 are used for 10 trials using the tuned values.

For each observing system, several values of the standard deviation of the
observation time offset

At each analysis time

List of observing system cases explored. For each of the five analysis periods, a number of different values for the time offset standard deviation were explored.

Figure 1 shows a short segment of the trajectory of the truth,

Figure 1 also shows the value of the state linearly extrapolated from the
analysis time to the observed time as a blue vector and teal “

A short segment of the truth for a state variable and the
observation generation process from the case with an analysis period of 0.6 and
time error standard deviation of 0.2. The true trajectory is indicated by the
small grey asterisks every 0.01 time units. The black asterisks indicate the
true value at each analysis time. The blue circles are the truth at the
actual observed time (the analysis time plus the observation time offset for
that analysis time). The yellow crosses are the actual observations that are
assimilated and are generated by adding a random draw from

Five assimilation methods were tested for each L96 case. All applied a standard ensemble adjustment Kalman filter (EAKF; Anderson, 2001) with 80 members using a serial implementation (Anderson, 2003) to update the ensemble with observations. All but the first method made adjustments to the prior observation ensemble and/or the observation error covariance to deal with the observation time offset.

Observations were assimilated with a standard EAKF. This is consistent with
the assumption made in Sect. 4.1 that the time offset is

This method assumed time offset

This method used Eqs. (13) and (14) to compute the mean and variance of
the time offset. This distribution for

A naive application of this method was not successful in any of the L96
cases. The tuned assimilations worked successfully for some number of
analysis times, but the RMSE of the ensemble mean always began to increase
with time before 1100 analysis times, and results were worse than for
NOCORRECTION. The magnitude of the estimate of the mean value of the time
offset

This occurred because of the statistical challenge of separating observation time offset from prior model error. Suppose this method was applied to a model with only a single time-varying variable that is observed. The prior ensemble mean will almost always have an error. If, for example, that error has the same sign as the time tendency of the model at the analysis time, the linear correction method will attribute part of that error to a time offset in the observation and will not correct the error as strongly as it would if no time offset were assumed. This means that the forecast at the next analysis time is likely to be consistent with the model state at a time later than the analysis time. Again, the algorithm will attribute some of this error to a time offset in the observation. The net result is that the estimated model state is likely to drift further and further ahead of the true trajectory in time.

To avoid this problem, estimates of the time offset that were (nearly)
independent of the error for a given state variable were needed. This was
accomplished using a modified version of Eq. (13) to compute a separate
value of

A subset of the components of the vector

This method used Eqs. (10) and (11) to compute the mean and variance of
the time offset. This distribution for

The nonlinear algorithm in Sect. 4.5 was used to estimate the most likely
value of the time offset

In addition to estimating the model state, each of the five methods also
estimated the value of the time offset,

Figure 2 shows the results for the five methods applied to all cases. For each case, the RMSE of the prior ensemble mean is plotted for each of the 10 trials done with each assimilation method. The results for different methods are distinguished by the color of the markers and the horizontal offset of the plot columns. Note that ranges of both axes vary across the figures and that the horizontal axis is logarithmic (with the exception of the value for no time offset).

The blue markers (leftmost) are the results of the NOCORRECTION method, which
ignores the time offset and performs a standard ensemble adjustment Kalman
filter with

The LINEAR method is shown in teal (second from left). For almost all cases, it generally produces smaller RMSE than NOCORRECTION, with the relative improvement being largest for analysis period 0.1 and 0.15 and larger time error standard deviation. LINEAR produces larger RMSE than NOCORRECTION for all cases with an analysis period of 0.6. The poor performance for the cases with time error standard deviation greater than 0.15 is due to errors in the linear tangent approximation for the evolution of the L96 state trajectories (see examples for the 0.6 analysis period in Fig. 1). LINEAR applies the same increment to the observational error variance as VARONLY. It performs better than VARONLY for most cases. However, VARONLY is better than LINEAR for cases with period 0.6, showing that the additional linear correction to the prior ensemble is clearly inappropriate for these cases.

Additional insight into the performance of LINEAR can be gained from the results for IMPOSSIBLE, shown in black (second from right) in the figure. Not surprisingly, since it has access to the truth when estimating the offset, it always produces smaller RMSE than LINEAR (except for cases with no time error). For the 0.05, 0.1, and 0.15 analysis period cases, the RMSE for IMPOSSIBLE is nearly independent of the time error standard deviation. This is not the case for analysis periods 0.3 and 0.6, where the error increases as the time error standard deviation increases. The cause of this error increase is that the linear tangent approximation becomes inaccurate as the time error increases. However, especially for analysis period 0.6, IMPOSSIBLE does not produce significantly better RMSE than NOCORRECTION, even for smaller time error standard deviation, where the linear tangent approximation should normally be accurate. Apparently, the larger prior error resulting from infrequent observations dominates the errors introduced by the time error in these cases.

The NONLINEAR method plotted in yellow (rightmost) has additional information about the distribution of the time offset and almost always performs significantly better than NOCORRECTION. The relative importance of nonlinearity in the prior truth trajectories is revealed by comparing the RMSE for IMPOSSIBLE and NONLINEAR. For time error standard deviations smaller than 0.1, IMPOSSIBLE is almost always significantly better, but for time error standard deviation of 0.1 and 0.2, NONLINEAR is always better.

All methods also produce an estimate,

RMSE of the ensemble mean over 1000 analysis time steps for cases with an analysis period of

For the analysis period of 0.1 (Fig. 3a), the estimates from all methods are always less than the specified time error standard deviation and become smaller fractions of the specified value as the value increases. This is because it is easier to detect time error when that error is relatively larger compared to the observation error. LINEAR and VARONLY have smaller RMSE than NOCORRECTION for larger time error standard deviations, with LINEAR being slightly better than VARONLY. The RMSE for NONLINEAR is much larger than for NOCORRECTION for smaller time error standard deviations. This is because the possible offset estimates are selected from the discrete set of times for which the truth and prior ensemble are computed (see Eq. 15), which are spaced 0.01 time units apart. The time offset estimates for all other methods can take on any real value. For the case with time error standard deviation 0.1, the nonlinearity is large enough that the NONLINEAR estimate of the offset is comparable to that produced by VARONLY and is better than NOCORRECTION.

For the larger analysis period of 0.3 (Fig. 3b), the estimate from LINEAR is not better than NOCORRECTION, while VARONLY is better than LINEAR for larger time error standard deviations. In this case, NONLINEAR still has the largest RMSE for cases with time error standard deviation of 0.025 and 0.05 but has by far the smallest RMSE for cases with 0.1 and 0.2.

The RMSE of the estimate of the time offset for cases
with an analysis period of

A number of simplifying assumptions were made in the algorithms described
here. These include assuming that every state variable is observed directly,
that all observations share the same time offset, that the observation error
covariance matrix

It is straightforward to deal with some of these issues. The assimilation
problem can be recast in terms of a joint phase space, where an extended
model state vector is defined as the union of the model state variables and
prior estimates of all observations (Anderson, 2003). Then, all observed
quantities are model state variables by definition. However, for methods
that use linear extrapolation via Eq. (1), the model equations are no longer
sufficient. One can either develop equations for the time tendency of
observations or simply use finite difference approximations to compute

Since a serial ensemble filter is being used for the actual assimilation, it is possible to partition the observations into subsets that are themselves assimilated serially. All observations that share a time offset can be assimilated as a subset, including a subset for those observations with no time offset.

All of the methods for estimating the offset at a given analysis time except
NOCORRECTION make explicit use of

The methods also assume that the observation error covariance matrix

The methods described have a range of computational costs. The VARONLY method only requires a single evaluation of Eqs. (2) and (3) at each analysis time step and has an incremental cost that is a tiny fraction of the NOCORRECTION base filter. The LINEAR method requires an evaluation of Eq. (16) for every observation, and Eq. (16) requires the computation, storage, and inversion of a prior ensemble covariance matrix. However, this matrix could be reduced in size to only include the subset of observations that is used to compute the offset for each observation. It would be application-specific to determine this size. For example, a radiosonde would make a large number of observations, e.g., temperature and wind at a number of levels, but many of these are correlated in time. We can capture most of the information about time offset from a smaller subset of the observations and just do the inversion on those to make the matrix small compared to the total model size.

The NONLINEAR method involves a large amount of additional computation. The
prior ensemble needs to be available over a range of times covering the
possible offsets. In the idealized cases here, that meant that ensemble
forecasts were required to extend to the second analysis time in the future,
doubling the forecast model cost. Then Eq. (15) must be evaluated for each
of the available times. The dominant cost in Eq. (15) is computing the prior
covariance matrix for the observations that share an offset. This requires

The importance of accounting for observation time errors in many earth system DA applications remains unexplored. The range of methods discussed here have varying cost, but all could be applied for at least short tests in any application for which ensemble DA is already applicable. In particular, applications to atmospheric reanalyses for periods well before the radiosonde era seem to be especially good candidates for improvement. Future work will assess the algorithms presented here in both observing system simulation and real observation experiments with global atmospheric models and observing networks from previous centuries.

All code used to generate the data
for this study (written by Jeffrey L. Anderson), the generated data, and code for creating the figures
and figure files are available at

EG proposed the study, developed the mathematical methods, and implemented initial test code. JLA solved the problems with the possible linear method, ran final tests, and implemented figures. Both authors wrote the final report.

The contact author has declared that neither 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 would like to thank Peter Teasdale and Valerie Keeney of Summit Middle School and Paul Strode and Emily Silverman of Fairview High School for their commitment to science education and providing the authors with an opportunity to collaborate. The authors are indebted to NCAR's Data Assimilation Research Section team for providing guidance in developing the software used for this work.

This material is based upon work supported by the National Center for Atmospheric Research, which is a major facility sponsored by the National Science Foundation under cooperative agreement no. 1852977.

This paper was edited by Amit Apte and reviewed by Carlos Pires and one anonymous referee.