1. Introduction
In its simplest form, data assimilation (DA) is the process of optimally combining prior information, often in the form of ensemble forecasts, and observations to provide an accurate representation of the state of a physical system. Two methods that have been developed over the last several decades to perform sequential DA are the particle filter (PF; van Leeuwen 2009) and the ensemble Kalman filter (EnKF; Burgers et al. 1998). While the PF can represent any distribution, the scalability of this method is poor as model size increases (Snyder et al. 2015). The EnKF scales better than the PF for large geophysical models; however, it may not represent non-Gaussian distributions appropriately (Lei et al. 2010). The ideal method would incorporate the benefits of both PF and EnKF to work with arbitrary distributions and scale well with increasing geophysical model size.
Progress has been made over the last several decades to improve the representation of non-Gaussian prior distributions in the EnKF (Zupanski 2005; Hodyss 2012; Bocquet and Sakov 2012; Nino-Ruiz et al. 2018; Vetra-Carvalho et al. 2018) and to improve scalability of the particle filter with increasing model size (Poterjoy 2016; Penny and Miyoshi 2016; van Leeuwen et al. 2019). Additionally, studies have proposed to hybridize PFs with EnKFs; however, these methods usually require some parameter threshold or statistical measure of non-Gaussianity to determine the impacts each filter provides (Stordal et al. 2011; Frei and Künsch 2013; Chustagulprom et al. 2016; Grooms and Robinson 2021; Kurosawa and Poterjoy 2023). One extension of the EnKF algorithm is the rank histogram filter (RHF; Anderson 2010). The RHF can represent non-Gaussian prior distributions and arbitrary likelihoods associated with different observation types with similar computational costs to the standard ensemble filters (Anderson 2020). Anderson (2020) discusses how the RHF can more optimally treat bounded prior distributions with a Gaussian likelihood. One issue that arises is that Gaussian likelihoods are not appropriate for bounded observation error distributions. The RHF provides a way to pair appropriate likelihoods for the assimilated observation with the prior bounded probability density (Anderson 2022).
Sea ice provides an excellent application for testing the performance of different EnKF methods on bounded quantities. Sea ice is an important Earth system component to represent in climate models due to its link to Arctic amplification—rapid warming over the Arctic region (Serreze and Francis 2006; Rantanen et al. 2022; Screen and Simmonds 2010; Jenkins and Dai 2021). Multiple studies have highlighted the importance of sea ice initial conditions, especially related to initializing sea ice thickness, when trying to predict Arctic sea ice up to seasonal lead times (Msadek et al. 2014; Day et al. 2014; Dirkson et al. 2017). There have been numerous studies applying EnKF methods to sea ice problems to understand sea ice observing platform impacts in modeling systems. Approaches include observing system simulation experiments (OSSEs; Barth et al. 2015; Kimmritz et al. 2018; Zhang et al. 2018) and evaluation of real observations (Lisæter et al. 2003; Sakov et al. 2012; Massonnet et al. 2015; Fritzner et al. 2019). The different EnKF methods applied to different geophysical problems over the years have built-in Gaussian assumptions, dating back to the original Kalman filter formulation (Kalman 1960). However, if either the prior forecast distribution or observation error distribution is non-Gaussian, this can lead to optimal solutions estimated from the EnKF being erroneous when compared to the true distribution given by Bayes’ theorem. Non-Gaussian distributions are commonly found in sea ice fields due to having both double (e.g., sea ice concentration, i.e., 0–1) and single (e.g., sea ice thickness, i.e., 0–∞) bounded quantities. Riedel and Anderson (2023) found a negative bias in total sea ice area over the Arctic, which was related back to assimilating doubly bounded sea ice concentration observations using an EnKF method. The complexity of assimilating real sea ice observations is further increased because of the large uncertainties associated with satellite retrieval algorithms for obtaining the observations (Kwok and Cunningham 2008; Zygmuntowska et al. 2014; Tilling et al. 2016; Ricker et al. 2017). While previous literature has focused on the performance of applying the EnKF to sea ice problems, there are few previous studies adapting EnKF methods for the problems sea ice can pose.
Stochastic and deterministic EnKF formulation performance under non-Gaussian conditions has been the subject of prior studies. Although both formulations led to biased solutions, when compared to the true distribution from Bayes’ theorem, these studies suggested that the stochastic formulation is more stable (Lawson and Hansen 2004; Lei et al. 2010). The RHF was developed to remove the Gaussian assumptions in the prior distribution. Metref et al. (2014) developed an extension of the RHF that allowed non-Gaussian prior information and maintained a multivariate non-Gaussian structure. However, this RHF extension was significantly more computationally expensive than other EnKF methods. When working with sea ice applications, bounded prior distributions and also bounded observation error distributions are a concern. Employing a Gaussian likelihood while implementing a non-Gaussian distribution (e.g., truncated normal distribution) for observation errors will lead to erroneous observation updates, biasing the estimates of the state model fields (Pires et al. 2010; Fowler and Jan van Leeuwen 2013). Anderson (2022) offers an alternative approach to more appropriately represent the bounded prior state and the bounded observation errors by optimally combining distributions.
In this study, the performance of EnKF methods, some of which are adapted to address the challenges posed by sea ice, is investigated using multiple OSSEs. Throughout the investigation, a particular focus is placed on the performance of filters for handling commonly found non-Gaussian distributions, which has been shown to be challenging (Riedel and Anderson 2023). This study expands on those results by testing a filter that follows the newly developed framework laid out in Anderson (2022). This newly developed filter uses an RHF to represent the prior distribution and a truncated Gaussian distribution to represent the observation likelihood. This combination produces a posterior that closely resembles that of an RHF leading to an unbiased solution. We will demonstrate that using the newly developed filter will lead to accurate sea ice analyses by more optimally fitting sea ice observations with physical bounds.
2. Methodology and experimental setup
a. Icepack
Icepack is a single-column model (SCM) that uses column physical packages from the Los Alamos Sea Ice Model (CICE; Hunke et al. 2015), which is the sea ice component within the Community Earth System Model (CESM; Danabasoglu et al. 2020). Icepack includes options for simulating different sea ice processes related to thermodynamics and mechanical redistribution to represent area and thickness changes. Four different sea ice conditions are represented within Icepack: 1) open water, 2) slab ice, 3) ice thickness distribution plus snow, and 4) land. This work focuses on the Icepack setup that represents the evolution of sea ice thickness distribution plus snow since it most closely resembles the sea ice processes found in CICE5. Icepack v1.3.3 (Hunke et al. 2022) is used to simulate the evolution of sea ice with modifications to work with the DA interface software. Sea ice thickness is represented in Icepack as the quotient of sea ice volume and area. To represent the evolution of sea ice thickness, the sea ice pack distribution is partitioned into multiple thickness categories (Lipscomb 2001). Snow on top of the sea ice is also partitioned into the same five categories to represent snow depth. Five thickness categories are used in this study that have lower bounds of 0, 0.64, 1.39, 2.47, and 4.57 m, respectively. Icepack has three multicategory prognostic variables: sea ice area Aice,n, sea ice volume Vice,n, and snow volume Asnow,n, where n represents thickness categories 1–5. Additionally, an option that allows for prognostic sea surface temperatures to evolve with sea ice, based on a simple ocean mixed layer model, is turned on in all experiments (Knutson 2017). One unique challenge for the DA is respecting category bounds when updating sea ice area and sea ice volumes. If the category bounds are not respected, the code will return an error and the forecast will fail. Icepack needs two additional forcing components to simulate sea ice evolution: atmospheric forcing and ocean forcing data. The ocean forcing component, which mimics a slab ocean model, is represented by annually periodic, prescribed ocean forcing data (e.g., ocean heat fluxes and ocean currents). The atmospheric forcing data are provided from the Community Atmosphere Model, version 6 (CAM6)/Data Assimilation Research Testbed ensemble reanalysis (Raeder et al. 2021). Since Icepack represents a single grid point, the forcing data are gathered for a specific location over the central Arctic Ocean. The default namelist settings that come with Icepack are used in the study except for perturbing two input Icepack parameters. The perturbation of the Icepack parameters will be discussed later.
b. DART
The Data Assimilation Research Testbed (DART; Anderson et al. 2009) software was used to implement and test the different ensemble filter types applied to sea ice. An interface between Icepack and DART allows the user to use DA features within DART. The four different DART ensemble filter types are the ensemble adjustment Kalman filter (EAKF; Anderson 2001), EnKF with perturbed observations (Burgers et al. 1998; Anderson 2003), RHF (Anderson 2010), and bounded RHF (Anderson 2020, 2022). These four filters will assimilate observations that will update prior ensemble members in the observation space. The updates from observation space are projected to state space using linear regression when testing the different filter types. Due to the use of a single-column model with no neighboring grid points in space, we ignored horizontal localization. In addition, no localization was applied when updating the individual categories for each sea ice prognostic variable. Only the grid of interest is updated via regression from the observation space adjustments. Additionally, no prior inflation is applied to reduce the complexity when evaluating the impacts from different tested filters. Any assimilated observation type updates all the model state variables during the assimilation step, indicating that there is no cross-variable localization.
To become more familiar with the different ensemble filter types tested in this study, consider the example in which the physical variable is a doubly bounded quantity (Fig. 1). The five prior ensemble members (black stars) represent sea ice concentration, and all prior member values are between zero and one. For both EAKF and EnKF, the prior probability density functions (PDFs; black line) and the likelihood (red line) associated with the observation (gray star) are assumed to be Gaussian. The authors know that the EnKF is not a deterministic filter so the posterior ensemble shown in the figure is just one possibility. The prior PDFs for the RHF are computed where 1/(N + 1) probability is distributed uniformly in the interior portion of the distribution and N equals the total number of ensemble members. The unbounded regions on the tails are a partial normal (μ, σ2), where σ2 is the prior ensemble variance and μ is selected so that the cumulative density is equal to 1/(N + 1). The bounded RHF is similar to the RHF; however, the bounded ends do not have a Gaussian tail and the bins are treated like an interior bin. While the piecewise constant approximate likelihood for the RHF is assumed to be Gaussian, the bounded RHF has a piecewise constant approximate likelihood for the observation error distribution that the user specifies (truncated normal for this study; Anderson 2023). Regardless of the choice of observation likelihood, both the normal and bounded RHFs will result in a posterior that resembles an RHF. Comparing the posterior ensemble members (blue stars), each filter type pulls the prior members closer to the observation. Both the normal and bounded RHFs are better at pulling the largest prior member closer to the observation. However, the bounded RHF is the only filter that respects the bounded side while reducing the spread of the ensemble members. The EAKF, EnKF, and RHF do not respect the lower bound, which can result in some posterior ensemble members being less than zero as in the idealized example in Fig. 1.
Schematic showing the update of five SIC prior members (black stars) by an observation (gray star) using the EAKF, EnKF, RHF, and bounded RHF. The black lines or shading represents normal prior density or approximate piecewise continuous prior density. The red lines or asterisks represent normal observation likelihood or approximate piecewise continuous likelihood. The blue lines represent the normal posterior density (EAKF and EnKF) and piecewise continuous posterior (RHF and bounded RHF). The five posterior updated SIC members are shown by blue stars. The mean and standard deviation of the five ensemble members for the prior and posterior are provided in the legend. The observation value is provided in parentheses in the legend.
Citation: Monthly Weather Review 153, 4; 10.1175/MWR-D-24-0096.1
c. Perfect model OSSEs
The different filter types are tested in perfect model OSSEs to evaluate their performance on updating sea ice states. The setup of the OSSEs is similar to Riedel and Anderson (2023), except for the use of Icepack in this study. For each experiment, 80 Icepack members comprise the ensemble. The ensemble size is defined by the number of CAM6/DART reanalysis forcing files. The same ocean forcing is provided to each ensemble member. An Icepack parameter that impacts albedo (standard deviation of the dry snow grain radius Rsnw) and a parameter that impacts heat transfer through snow (thermal conductivity of snow ksnw) are each perturbed differently for each ensemble member to increase ensemble spread. Due to the less chaotic nature of sea ice, it takes many years to spin up a sea ice state in a fully coupled model. However, the resources that a user would need to spin up a fully coupled Earth system model would require a large amount of computational resources along with time that may not be available. Therefore, it is common in studies related to sea ice data assimilation to use parameter perturbations to drive sea ice spread (Zhang et al. 2018; Cheng et al. 2020; Zhang et al. 2021; Cheng et al. 2023; Riedel and Anderson 2023). In addition, Icepack is a single-column model that further limits the full interactions found in a fully coupled Earth system model. These parameters were chosen for this study because they drive sea ice variability in CICE5 throughout the year (Urrego-Blanco et al. 2016). The term Rsnw is one of the key parameters that determines snow albedo in the solar radiation parameterization (Briegleb and Light 2007), while ksnw directly affects the amount of heat that can be transferred through the snowpack (Sturm and Massom 2017). See the data availability statement for access to the perturbed parameter values used in this study. The CAM6 reanalysis 80-member ensemble provides 80 unique atmospheric conditions from which we can drive our sea ice ensemble. To do so, we start by repeatedly running CICE using 2011 atmospheric forcing from one CAM6 reanalysis member to obtain a climatological sea ice and snow state representing 2011. To build our own 80-member ensemble, we use each CAM6 member to provide unique forcing cycling CICE 10 times over 2011 to create our 80-member ensemble with atmospheric contributions built into the sea ice variability. Finally, to increase ensemble spread when generating free forecasts, we cycle each of our 80 members another five times over 2011 including a unique set of perturbed parameters (Figs. 2a,c,e). These forecasts can be used as a reference for comparing various DA experiments. To define the simulated truth, we randomly chose one member from these 80-member free forecasts.
Daily (a) sea ice area Aice, (c) sea ice volume Vice, and (e) snow volume Vsnow from Icepack free forecast simulations over the cycling period. Each gray line represents an individual ensemble member, the black line represents the ensemble mean, and the red line represents the truth member. Daily truth member (red line) and synthetic observations (blue line) over the cycling period for (b) SIC, (d) SIT, and (f) SNWD. The free forecast period is for 2011–19.
Citation: Monthly Weather Review 153, 4; 10.1175/MWR-D-24-0096.1
Four experiments were initialized on 1 January 2011 using the same initial conditions as the free forecasts and cycled for 9 years (31 December 2019). The observations are assimilated at a daily interval for all experiments. During the spinup process, information about the baseline variability of the ensemble is provided from the free forecasts (Figs. 2a,c,e).
d. Model verification
Different verification techniques are used to evaluate each experiment in both observation and state space. Rank histograms are generated by repeatedly recording the rank of the truth relative to the n-sized ensemble sorted from lowest to highest (Anderson 1996). The reliability of an ensemble can be determined by the shapes of the distribution of counts presented by the rank histogram (Hamill 2001). Riedel et al. (2021) provide more information on the interpretation related to the shapes of the rank histograms. Analysis increments (AIs) are the difference between the updated analysis after DA and the background state which here is a 1-day model forecast.
Observation space diagnostics are used to evaluate the performance of each tested filter type. Observation space diagnostics include rank histograms and AIs. In most cases, AIs are evaluated in state space after analysis increment updates are regressed to the state fields. In this study, we computed AIs after the filter assimilated each observation per cycle, which helped isolate the impacts of each filter in the observation space before the regression step updated the state variables. AIs in state space are accumulated updates from all observations, which obscure the influence of individual observations.
For state–space evaluation, AIs are computed to determine how observation updates are impacting model state variables. Pearson’s correlations are computed between AIs in observation space and state model fields to compare regression updates between experiments. A time series of daily forecast biases is computed for Aice, Vice, and Vsnow. The bias of forecast quantities is defined as the ensemble mean minus the truth. Last, RMSE is computed over the entire cycling period using the ensemble mean and truth for Aice, Vice, and Vsnow.
3. Experiment results
a. Simplified data assimilation experiment
A simplified DA experiment is performed to measure the performance of each filter type near a bound. This experiment is meant to mimic winter SIC over the pole, where SIC is generally constant and approximately near one (experimental truth value of 0.99). Synthetic observations are generated and assimilated over the cycling period using the observation error presented in section 2c. The initial 79-member ensemble distribution has a standard deviation of 0.0142 and a mean of 0.97. No inflation is applied in these experiments to mimic the Icepack experiments. Six mini experiments are completed that use each of the four different filter types along with two experiments that test different prior PDF and likelihood combinations for the RHF. Anderson (2022) shows that the RHF can be modified where either a bounded or nonbounded prior PDF or likelihood will result in an RHF posterior. Due to the flexibility of the RHF, it can be determined which aspect of the bounded RHF is most impactful when working with biased observations when the truth is close to the bound. Each experiment is cycled 5000 times. To generate the observations that are assimilated in these experiments, a truncated normal distribution is used to model the observation errors, which are then added to the truth. Due to the simplified nature of these experiments, there are no thickness categories like in Icepack so mapping from observation space to state space is linear.
The choice to mimic sea ice during wintertime was made to investigate the impacts related to observations that are generated when the truth is close to the bound. Due to the upper bound of SIC, the synthetic observations are negatively biased. This is observed in both the simplified experiment (red line in Fig. 3) and during the winter for the Icepack experiments (Fig. 2b). Over the cycling period, the mean of the prior ensemble moves toward the average observation value instead of the true value in both the deterministic and stochastic EnKF filters (Figs. 3a,b). Likewise for the RHF, the prior ensemble mean drifts from the truth (Fig. 3c). The drift of the prior ensemble mean toward the average observation value appears to be faster for both EAKF and EnKF. The bounded RHF is the only filter in which the prior ensemble mean is pulled toward truth over the cycling period (Fig. 3d). It is clear that the modifications made to the bounded RHF, which includes a bounded prior PDF and truncated normal likelihood, have a positive impact compared to the RHF and the other two EnKF filter types.
Prior ensemble mean (blue line) time series of SIC for experiments using (a) EAKF, (b) EnKF, (c) RHF, and (d) bounded RHF. The red line represents the average observation value over the cycling period. The black line represents the truth value. The blue shading represents the ensemble envelope. Experiments testing different combinations of (e) unbounded RHF prior and truncated normal likelihood along with (f) bounded RHF prior and normal likelihood are shown for the bounded RHF. These experiments contain 5000 cycling times.
Citation: Monthly Weather Review 153, 4; 10.1175/MWR-D-24-0096.1
The fact that the prior ensemble mean moves away from the true value demonstrates that our solution is biased when using the EAKF, EnKF, and RHF. Using an unbounded RHF to specify the prior PDF and a truncated normal for the likelihood results in the prior ensemble mean being pulled close to the truth (Fig. 3e). However, this setup leads to individual ensemble members above the upper bound of 1, meaning that our ensemble distribution is not bounded. Specifying a bounded RHF as the prior PDF and a normal likelihood results in the prior ensemble mean being pulled toward the biased observations over the cycling period (Fig. 3f). Thus, the truncated normal likelihood is impactful by pulling the ensemble closer to the truth; however, the bounded RHF prior PDF results in the updated ensemble distribution being bounded below one. Previous studies have highlighted that erroneous observation impacts, which can bias analysis estimates, are possible when applying a non-Gaussian distribution for observation errors with a Gaussian observation likelihood (Pires et al. 2010; Fowler and Jan van Leeuwen 2013). The bounded RHF allows for the correct specification of the observation likelihood matching that of the observation error distribution while keeping the ensemble distribution within the observational bounds.
b. Observation space evaluation
Evaluation in the observation space of each filter type helps determine how forecasts are being fitted to assimilated observations. Ensemble mean AIs are a metric that shows how much the observation is adjusting the background field, and over many cycling times, the desire is for this quantity to approach zero. One caveat is that the observation space AIs shown here are outputted after the filters have assimilated each observation type at each cycling time, meaning they are not assimilation window AIs. For SIC AIs, there is a sharp cutoff near zero on the positive side for all the filter types except for the bounded RHF during the refreeze season (Figs. 4a–d). Since the refreeze season is during wintertime, the low-biased SIC observations during this period (Fig. 2b) resulted in more negative AIs in the three filters that contain Gaussian assumptions. The bounded RHF has a more uniform distribution around zero for SIC AIs, indicating that the filter is weighting the biased SIC observations in a more appropriate way. The distributions of SIT AIs are similar between the EAKF, RHF, and bounded RHF (Figs. 4e,g,h). This is likely linked back to the fact that SIT thickness never approaches zero (Fig. 2d) so a normal distribution works in this situation. However, the distribution of SIT AIs is different for the EnKF, where there is a large spread in AI values (Fig. 4f). The different AI distributions between the assimilation of SIC and SIT observations for the EnKF could be related to the perturbed observations used in the EnKF and how AIs in observation space are projected back to the state via regression. Since SIC is assimilated first sequentially, the regression update occurs immediately before the assimilation of SIT observations, which could be linked to the large spread for SIT. Similar to the distributions for SIC AIs, there is a cutoff near zero on the negative side for SNWD AIs during the melt season (spring/summer) for all filter types except the bounded RHF (Figs. 4i–l). The cutoff found in some filter types is linked back to positive SNWD observation biases during the melt season (Fig. 2f). Since SNWD cannot be less than zero, we find the inverse problem compared to SIC near 1; thus, when SNWD observations are created near zero, they have the potential to be positively biased. During the refreeze, there are similar SNWD AI distributions between the EAKF, RHF, and bounded RHF. The distribution of SNWD AIs for the EnKF is similar to that found for SIT observations.
Histograms of (a)–(d) SIC, (e)–(h) SIT, and (i)–(l) SNWD AIs in observation space when using the (a),(e),(i) EAKF, (e),(f),(j) EnKF, (c),(g),(k) RHF, and (d),(h),(l) bounded RHF. Counts are shown for AIs found over the entire cycling period (black line), refreeze season (October–April; red line), and melt season (May–September; blue line). The median values of the AIs are computed for each evaluation time period represented by the number color. Counts of AI values falling above or below the bin ends are provided for each period represented by the number color.
Citation: Monthly Weather Review 153, 4; 10.1175/MWR-D-24-0096.1
Rank histograms allow users to evaluate in observation space both spread deficiencies and potential biases in a cycling ensemble system. Starting with SIC observations, the ensemble is underdispersive for all the filter types, besides the bounded RHF, regardless of the time of year (Figs. 5a–d). For the bounded RHF, there does appear to be a high bias (peak on the left side) and a slight low bias (right-side peak) for SIC. The slight low bias (right-side peak) occurs during the refreeze season, and a high bias (peak on the left side) occurs during both the refreeze and melt seasons. The low bias was likely linked to the truth member refreezing more rapidly than ensemble members even with DA updates. The high bias was likely due to sea ice in the truth member melting more rapidly than in the ensemble members. Because the ensemble spread is smaller during winter and no prior inflation was applied in this study, the weights associated with the observation updates were small, leading to a delay in sea ice melt. Transitions can occur abruptly for SIC in the single-column model, especially during sea ice refreeze (Fig. 2a). The right-hand peaks in counts for the bounded RHF also appear for the EAKF and RHF. All filter types are overdispersive for SIT, primarily during wintertime (Figs. 5e–h). The overdispersion for SIT will be discussed further below. During the melt season, the overdispersion disappears and there are low biases for EAKF and RHF. The overdispersion remains for the EnKF, and the bounded RHF has a relatively flat histogram. For snow depth, the histograms are similar across the DA algorithms for all seasons (Figs. 5i–l). The EnKF is overdispersive during the refreeze season, and all filters have a high bias with a peak on the left side of the histogram. Comparing the different filters, the bounded RHF has a small peak on the left side, indicating that the potential for a high bias is smaller. Overall, the bounded RHF is performing as well or better than the other filter types over different time periods.
Rank histograms evaluating (a)–(d) SIC, (e)–(h) SIT, and (i)–(l) SNWD in observation space when using the (a),(e),(i) EAKF, (e),(f),(j) EnKF, (c),(g),(k) RHF, and (d),(h),(l) bounded RHF. Counts are shown for AIs found over the entire cycling period (black line), refreeze season (October–April; red line), and melt season (May–September; blue line).
Citation: Monthly Weather Review 153, 4; 10.1175/MWR-D-24-0096.1
Additional analysis is required to gain an understanding about the overdispersive behavior of all filter types during the refreeze season (wintertime) for SIT. The role of Icepack parameters and their perturbations in SIT evolution could possibly be related to the overdispersion. Parameter perturbation is not an uncommon approach for inducing spread in atmospheric models (Murphy et al. 2004; Stainforth et al. 2005; Christensen et al. 2015; Orth et al. 2016; Berner et al. 2017). However, the impacts of perturbations on less chaotic systems, like sea ice, remain unclear and understudied. Correlations between SIT and the two perturbed Icepack parameters show larger and more significant correlations for the ksnw parameter compared to the Rsnw parameter for all the filter-type experiments (Fig. 6). Most of the significant correlations appear during winter and decrease during summer. Positive correlations mean that larger snow thermal conductivity values lead to thicker sea ice during winter. This connection is a result of increased energy flux through the snow–ice column due to the larger snow conductivity values, triggering an increase in basal sea ice growth during the winter (Blazey et al. 2013). Other studies have identified the connection between snow thermal conductivity values and sea ice evolution (Fichefet et al. 2000; Sturm et al. 2002; Lecomte et al. 2013; Holland et al. 2021). Due to this connection, the spread in sea ice thickness would be directly related to the spread in the perturbed ksnw parameter. Studies using perturbed parameters to increase spread in a sea ice ensemble need to be mindful of the amount of spread in perturbed parameters or could sample from a joint distribution to obtain multiple perturbed parameters.
Daily correlations between perturbed Icepack parameters and SIT for (a) EAKF, (b) EnKF, (c) RHF, and (d) bounded RHF. The two perturbed parameters are the standard deviation of the dry snow grain radius Rsnw and the thermal conductivity of snow ksnw. Correlations are computed using Spearman’s rank correlation method where both raw correlations (Raw) and significant correlations with confidence at 99% (Sig.) are shown.
Citation: Monthly Weather Review 153, 4; 10.1175/MWR-D-24-0096.1
c. State–space evaluation
Linear regression is used to pass the updates computed by the filter in observation space to the state–space model variables. Each filter assimilates the observations sequentially in the following order for each cycling time: 1) SIC, 2) SIT, and 3) SNWD. Correlations were computed for sea ice area and sea ice thickness observations to allow comparison between filter types (snow depth observations were omitted for conciseness; Fig. 7). Looking at the patterns and magnitudes of the correlations, the EnKF experiment has different patterns and magnitudes. The correlation patterns are similar for sea ice concentration and thickness observations in categories 1–3 for the EAKF, RHF, and bounded RHF. The magnitude of correlations is larger in categories 4–5 for the EnKF experiment. The physical processes associated with moving sea ice between the different thickness categories are nonlinear. Additionally, mechanical ridging is not represented in Icepack (Hunke et al. 2022), so the ice state is confined to the first three categories. Since the EnKF has added noise for the perturbed observation portion of the algorithm, there could be spurious correlations between the observation updates and the different categories (specifically for categories 4 and 5). A better choice might be a regression technique that can both respect the bounds and non-Gaussian aspects of the observed variable (Anderson 2023; Wieringa et al. 2023).
Daily correlations between observation space AIs and different category-based state model field variables over the cycling period. Correlations are computed for assimilated (a) SIC and (b) SIT observations. The category-based (Cat 1, Cat 2, Cat 3, Cat 4, and Cat 5) state model fields that are used in the correlation calculations are sea ice area Aice,n, sea ice volume Vice,n, and snow volume Vsnow,n.
Citation: Monthly Weather Review 153, 4; 10.1175/MWR-D-24-0096.1
Evaluation of model state variables helps determine how filter updates in observation space are projected to state space via linear regression. State–space AIs are one way to evaluate how this information is being projected. The observation space AIs are regressed to the different categories for state model variables; however, for simplicity, AIs are computed for the summed category quantities Aice, Vice, and Vsnow. First, the AI distributions for the EnKF are different from the other three filters for each state variable (Figs. 8b,f,j). The EnKF has a larger spread in AIs compared to the other filters. The differences in AI distributions for the EnKF could be related to the correlations between the observation space updates and the state variables. The AI distributions for the EAKF, RHF, and the bounded RHF are centered around zero for Aice, Vice, and Vsnow (Fig. 8). Over the refreeze season, only the bounded RHF has a positive median value for Aice AIs compared to the other filter types. This is similar to observation space AIs, where all the filters except the bounded RHF weighted the biased observations more during winter. Additionally, the distributions of Vice and SIT AIs are similar between each experiment (cf. Figs. 8e–h to Figs. 4e–h). This could mean that there is a strong connection between the SIT AIs in observation space and the different sea ice volume categories. In summer (melt season), only the bounded RHF has a negative median value for Vsnow AIs. Since SNWD observations would be biased during the summer when truth values would be close to zero, this means that the bounded RHF weights those positively biased observations less.
Histograms of (a)–(d) sea ice area Aice, (e)–(h) sea ice volume Vice, and (i)–(l) snow volume AI Vsnow in state space when using the (a),(e),(i) EAKF, (e),(f),(j) EnKF, (c),(g),(k) RHF, and (d),(h),(l) bounded RHF. Counts are shown for AIs found over the entire cycling period (black line), refreeze season (October–April; red line), and melt season (May–September; blue line). The median values of the AIs are computed for each evaluation time period represented by the number color. Counts of AI values falling above or below the bin ends are provided for each period represented by the number color.
Citation: Monthly Weather Review 153, 4; 10.1175/MWR-D-24-0096.1
To evaluate the experiment performance, daily biases of the ensemble mean Aice, Vice, and Vsnow are compared to the free forecasts throughout the cycling period (Fig. 9). For both the experiments and the free forecasts, biases are computed with respect to the truth. Each filter type keeps the ensemble mean closer to the truth compared to the free forecasts, especially for Vice and Vsnow. For Aice, the biases are largest during summer for all filter types (Fig. 9a). During summer periods where the truth SIC stays close to one (summers of 2013, 2015, 2016, and 2017), the EnKF has larger biases compared to the other filter types. This result was surprising since the stochastic formula of the EnKF has been shown to produce a less biased solution in non-Gaussian situations compared to deterministic ensemble Kalman filter formations (Lei et al. 2010). However, this summer bias could be linked back to the spurious correlations that are seen for the EnKF compared to the other filter types. All filter types perform well for Vice and Vsnow, keeping the ensemble mean closer to zero compared to the free forecasts (Figs. 9b,c). However, the EnKF biases for Vice and Vsnow are larger and noisier compared to the other filter types. Overall, the filters do not allow the different state variables to drift away from the truth over the cycling period.
Ensemble mean daily biases of (a) sea ice area Aice, (b) sea ice volume Vice, and (c) snow volume Vsnow from the free forecasts and experiments over the cycling period.
Citation: Monthly Weather Review 153, 4; 10.1175/MWR-D-24-0096.1
To further evaluate the analysis and forecast skill, RMSE is computed for state–space variables over different time periods for each filter type (Fig. 10). Analysis RMSE values are lower compared to the forecast RMSE values for all time periods and state variables, except for sea ice volume during the refreeze season for the EAKF, RHF, and bounded RHF (cf. Figs. 10a–c to Figs. 10d–f). The larger analysis RMSE values for sea ice volume during the wintertime could be linked back to the perturbed Icepack parameters, which were found to be significantly correlated (Fig. 6). For analyses, the bounded RHF has either the lowest or one of the lowest RMSE values for all the state model variables in the different seasons (Figs. 10a–c). It is a similar story for forecasts; however, the EAKF is just as skillful as the bounded RHF for sea ice area over both the entire cycling period and summer (Figs. 10d–f). This is not too surprising because the EAKF was the second-best algorithm in terms of statistics for sea ice areas for both the entire cycling period and summer. It is puzzling why the EnKF performed the best statistically among the algorithms during the refreeze season considering how different the observation space updates were for the EnKF. Additionally, all algorithms besides the EnKF had similar RMSE statistics (indicated by the stars) for sea ice volume, which is expected because distributions are more normal and values never really approach the zero bound throughout the experiment. Overall, this highlights that the performance of the bounded RHF is on par or better than the other filter types tested for sea ice variables.
(a)–(c) Analysis and (d)–(f) forecast RMSE for (a),(d) sea ice area Aice, (b),(e) sea ice volume Vice, and (c),(f) snow volume Vsnow. Ensemble mean RMSE is the top value in each cell, while the bottom value in each cell is the 95% confidence interval value. RMSE is computed over the entire cycling period, refreeze season, and melt season. Stars represent which filter or filters are significantly more skillful than the other filters.
Citation: Monthly Weather Review 153, 4; 10.1175/MWR-D-24-0096.1
4. Coupled data assimilation experiments
In most fully coupled Earth system frameworks, there is a mixture of different types of variables that a user will have to deal with in data assimilation. These different variables can have different distributions associated with them, including doubly bounded, singly bounded, and nonbounded distribution types. Therefore, it is important to understand and test the robustness of the bounded RHF when assimilating observation combinations with observation error distributions that are both bounded (e.g., SIC double bounded and SIT single bounded) and unbounded (SST unbounded). In all experiments, the namelist option for prognostic SSTs was turned on, allowing SSTs to evolve with sea ice. This allows for coupled DA experiments in which SST observations are assimilated in both weakly coupled (Lei et al. 2010) and strongly coupled (Kimmritz et al. 2018) DA frameworks. The coupled DA experiments tested only the bounded RHF when SIC, SIT, SNWD, and SSTs were assimilated. For the weakly coupled DA experiment, only sea ice and snow observations provided updates to the sea ice and snow state model variables, whereas the SST observations provided updates to the SST state model variable. For the strongly coupled DA experiment, the observation combinations of sea ice, snow, and SSTs provide updates via cross covariances to all sea ice, snow, and SST state model variables. Both coupled DA experiments were compared to the bounded RHF experiment, in which no SST observations were assimilated. These experiments are run for 1 year starting on 1 January 2011.
State–space daily biases are computed between the ensemble mean and truth for Aice, Vice, Vsnow, and SST. For all the state model variables, the differences between the biases for each experiment are small, except during the summer leading up to the refreeze (Fig. 11). While there are small Aice improvements when SSTs are assimilated in the coupled DA experiments, the improvements are the largest for Vice and SST. However, during the fall refreeze period, the assimilation of SSTs degrades the forecasts of Vice. The seasonal differences in Vice could be linked back to the impacts the perturbed ksnw parameters have on Vice, where it is the main driver for Vice evolution. This could help show that the impacts from the ocean on sea ice are more impactful compared to the impact on snow. Overall, the ensemble mean for the coupled DA experiments does not drift away from the truth or the experiments where SST observations are not assimilated.
Ensemble mean daily biases of (a) sea ice area Aice, (b) sea ice volume Vice, (c) snow volume Vsnow, and (d) SSTs from the different experiments over the cycling period. Experiments include no SSTs assimilated (bounded RHF-no SSTs) along with experiments assimilating SSTs: weakly coupled DA (bounded RHF-weakly) and strongly coupled DA (bounded RHF-strongly).
Citation: Monthly Weather Review 153, 4; 10.1175/MWR-D-24-0096.1
RMSE values are computed over the cycling period for sea ice, snow, and SST variables (similar to Fig. 10). For Aice, the strongly coupled experiment has the lowest RMSE over the entire cycling period and melt season (Fig. 12a). Interestingly, none of the experiments are significantly different from one another for Vice and Vsnow (Figs. 12b,c). However, the weakly coupled experiment has the lowest RMSE values for Vice and Vsnow (Figs. 12b,c). The strongly coupled DA experiments have the lowest RMSE values for SSTs for both the entire cycling period and during the melt season (Fig. 12d). Considering that Aice is part of the prognostic equation to evolve SSTs, it might not be surprising to see the strongly coupled DA experiment produce the lowest RMSE values for Aice and SSTs. Regardless, the bounded RHF is able to assimilate observations with and without bounded likelihoods without issues.
Forecast RMSE for (a) sea ice area Aice, (b) sea ice volume Vice, (c) snow volume Vsnow, and (d) SSTs. Experiments include no SSTs assimilated (bounded RHF-no SSTs) along with experiments assimilating SSTs: weakly coupled DA (bounded RHF-weakly) and strongly coupled DA (bounded RHF-strongly). Ensemble mean RMSE is the top value in each cell, while the bottom value in each cell is the 95% confidence interval value. RMSE is computed over the entire cycling period, refreeze season, and melt season. Stars represent which filter or filters are significantly more skillful than the other filters. If no stars are present, none of the experiments are statistically more skillful than each other.
Citation: Monthly Weather Review 153, 4; 10.1175/MWR-D-24-0096.1
5. Conclusions
Data assimilation methods, like the ensemble Kalman filter, provide a method to improve estimates of model state fields by utilizing information from observing systems in optimal ways. Even with DA advances, understanding how to handle non-Gaussianity is still an active area of research in the DA community. Sea ice fields have both doubly and singly bound quantities, which can lead to non-Gaussian distributions. Improving DA algorithms to better represent non-Gaussian distributions will allow for improved representation of sea ice states. This study used observing system simulation experiments (OSSEs) to investigate the performance of four different EnKF formulations on the generation of sea ice analyses. The different OSSEs provided insights into the benefits of the different EnKF formulations when handling non-Gaussianity that is commonly observed in sea ice fields.
A single-column sea ice model called Icepack was coupled to ensemble DA software called DART. The DART software provided a framework to test the performance of different ensemble Kalman filters in an OSSE configuration. Four different ensemble filter types were evaluated: EAKF, EnKF, RHF, and bounded RHF. Sea ice and snow observations were assimilated to evaluate the performance of the four different filter types.
A simple DA experiment was completed to test the performance of each filter type when the truth is near a bound. This simplified experiment mimicked wintertime SIC over the Arctic pole, where SIC values are expected to remain relatively constant near the upper bound of one. Through this exercise, we verified that performance is suboptimal for the EAKF, EnKF, and RHF filters near a bound, but the bounded RHF moves the prior ensemble mean toward the truth. The bounded RHF also constrains ensemble member SIC values to one. Using a truncated normal for the observation likelihood within the bounded RHF is key for pulling the ensemble mean toward the truth. One thing to note is that when the true sea ice concentration is equal to 1, observations will either be perfect or biased low because their values cannot exceed one (the same could be said for sea ice concentration values equal to 0). While bias correction away from the bounds could lead other DA algorithms (e.g., EAKF or EnKF) to produce results similar to the bounded RHF, this practice can be difficult in real-world applications. The bounded RHF algorithm provides an alternative that avoids applying bias correction to sea ice observations between or near bounds of 0 or 1.
In observation space, the bounded RHF produced SIC AIs that were more symmetric about zero during the winter. The other filter types had the tendency to produce mainly negative SIC AIs when SIC was near the bound in winter. Similar results were found when assimilating SNWD observations during summer when comparing the bounded RHF to the other filter types. While the ensemble was less underdispersive for the bounded RHF compared to the other filter types, potential biases were identified in the rank histograms from the peaks in the outside bins. Additionally, all filter types were overdispersive for SIT, which was largely present during the refreeze season. Additional analysis suggested that parameter perturbation could have driven the large spread for SIT.
Evaluation in state space was performed to investigate how the observation space updates were passed to the state model variables. Correlations between the observation space AIs and category-based sea ice variables were similar among the filter types, except for the EnKF. Since nonlinear processes control the distribution of sea ice variables in different categories, the additive noise in the EnKF could have created spurious correlations. The distribution of state–space AIs showed that the EnKF provided different updates compared to the other filter types. The distributions of state–space AIs were similar for the other filter types. One potential idea related to the different state–space AIs produced by the EnKF is related to the process of perturbed observations. Because the perturbed observations are sampled from a normal distribution, this may lead to unrealistic or nonphysical values, especially for SIC. Then, the perturbed observations lead to unrealistic AIs in observation space, which, in turn, leads to unexpected updates in state space via the regression step. Finally, the bounded RHF was the most skillful or one of the most skillful filters when evaluated in state space for both sea ice and snow variables. These results indicate that the bounded RHF, which is developed to work with bounded quantities, is a good filter for sea ice applications.
Coupled DA experiments were completed to test the robustness of the bounded RHF filter. The bounded RHF could successfully assimilate the combination of unbounded, singly, and doubly bounded observations while keeping the ensemble close to the truth. The differences between the coupled and noncoupled DA experiments were limited to summer. Using a strongly coupled DA setup with the bounded RHF produced skillful results for sea ice and SSTs when compared to the truth. More work is needed to fully determine why the SST observations did not provide statistically improved results for sea ice or snow volume.
Future work will further evaluate the robustness of the bounded RHF by testing it in a full sea ice model. While the results are positive in this study, understanding how observation space updates from the bounded RHF impact sea ice spatially is important. Determining alternative ways to create sea ice and snow ensemble spread besides the method of perturbing parameters within the Icepack is important to further investigate the bounded RHF impacts. Further investigation is required to understand the benefits of using a fully coupled Earth system modeling framework for generating ensemble spread and to determine how it might impact experiments using the bounded RHF. Last, further expansion of the coupled DA experiments is required to fully understand the impacts the SST observations have on improving sea ice and snow state model fields. These additional experiments would further help us understand how to properly use this newly developed filter for representing sea ice and snow in climate analyses.
Acknowledgments.
This material is based upon work supported by the NSF National Center for Atmospheric Research, which is a major facility sponsored by the National Science Foundation under Cooperative Agreement 1755088. Special thanks to the entire DART team for providing helpful input and source code support. I would also like to thank Cecilia Bitz and David Bailey for fruitful discussions regarding Icepack. We thank the editor and anonymous reviewers for constructive comments that helped improve the manuscript.
Data availability statement.
Icepack used for the experiments described here is publicly available for download from https://github.com/CICE-Consortium/Icepack (accessed in August 2022). The data assimilation software used here can be downloaded from https://github.com/NCAR/DART (accessed in August 2022). All data from the experiments described here are stored on NCAR’s Derecho computing system. The perturbed Icepack parameters used in this study are publicly available for download from https://doi.org/10.5281/zenodo.8206682.
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