## 1. Introduction

Ensemble-based filters or data assimilation methods, including the ensemble Kalman filter (EnKF; Evensen 2003) and ensemble square root filters such as the ensemble transform Kalman filter (ETKF; Bishop et al. 2001) and the ensemble adjustment Kalman filter (EAKF; Anderson 2001), provide accurate statistical estimation of a geophysical system combining a forecast model and observations. These methods quantify the uncertainty of the system using an ensemble that samples the information of the system. Geophysical systems are complex and of high dimension and thus require enormously huge computational costs for long time integration. Thus, the ensemble-based methods are indispensable tools for data assimilation as the methods allow computationally cheap and low-dimensional state approximation. Because of the simplicity and efficiency of the ensemble-based filters, these methods are widely applied to various fields of geophysical science such as numerical weather prediction (Kalnay 2003).

Despite their successful applications in geophysical applications, ensemble-based filters suffer from small ensemble size due to the high dimensionality and expensive computational costs [frequently referred as “curse of dimensionality” (Snyder et al. 2008) or “curse of small ensemble size” (Majda and Harlim 2012)], which can lead to filter divergence. Sampling errors due to insufficient ensemble size and imperfect model errors often yield underestimation of the uncertainty in the forecast and thus filters trust the forecast with larger confidence than the information given by observations. Inaccurate uncertainty quantification in the forecast fails to track the true signal and thus filter performance degrades, which is called filter divergence (Majda and Harlim 2012). Also insufficient ensemble size can lead to spurious overestimation of cross correlations between otherwise uncorrelated variables (Hamill et al. 2001; Whitaker et al. 2009; Sakov and Oke 2008) which also affects filter performance. Covariance inflation, which inflates the prior covariance and pulls the filter back toward observations, is one method to remedy the filter divergence (Anderson 2001). For the overestimation of cross correlations between uncorrelated variables, localization, which multiplies the covariances between prior state variables and observation variables by a correlation function with local support, is a powerful method to correct the overestimated cross correlations (Houtekamer and Mitchell 2001).

Catastrophic filter divergence (Harlim and Majda 2010; Gottwald and Majda 2013) is another important issue hindering the applications of the ensemble-based methods to high-dimensional systems especially in the case with sparse and infrequent observations and small observation errors. Catastrophic filter divergence drives the filter predictions to machine infinity although the underlying system remains in a bounded set. In data assimilation of geophysical systems in the ocean, for example, observations are often sparse and infrequent. In observations of ocean dynamics such as sea surface temperature, observations become accurate using various techniques such as tropical moored buoys, ocean reference status, and surface drifting buoys. But observations are still inadequate and sparse to sample over the vast surface and the interior of the ocean.

It is shown rigorously in Kelly et al. (2015) that catastrophic filter divergence is not caused by numerical instability, instead the analysis step of filters generates catastrophic filter divergence. Although the standard covariance inflation and localization stabilize filters and improve accuracy, they cannot avoid catastrophic filter divergence. In Harlim and Majda (2010), it is demonstrated that ensemble-based methods with constant covariance inflation still suffer from catastrophic filter divergence. In this study we also see that covariance localization decreases the occurrence of catastrophic filter divergence but does not prevent catastrophic divergence.

To avoid catastrophic filter divergence, a judicious model error using linear stochastic models was studied in Harlim and Majda (2010) with skillful results in some parameter regimes. Recently a simple remedy of catastrophic filter divergence without using linear stochastic models has been proposed through rigorous mathematical arguments and tested for the Lorenz-96 model in Tong et al. (2016). The approach in Tong et al. (2016) adaptively inflates covariance with minimal additional costs according to the distribution of the ensemble. The strength of inflation is determined by two statistics of the ensemble: 1) ensemble innovation, which measures how far predicted observations are from actual observations; and 2) cross covariance between observed and unobserved variables [see (12) and (13) in section 3, respectively]. If the filter is malfunctioning based on these two statistics, inflation is triggered and becomes larger when filters stray further into malfunction. Note that there are other adaptive covariance inflation techniques. Anderson (2007) proposed a method that uses the Bayesian update theory to the inflation parameters. Li et al. (2009) use the Kalman filter update to the inflation parameters based on Gaussian assumptions for the innovation statistics. Instead of using Bayesian update for the inflation parameters, Luo and Hoteit (2013) use the innovation statistics directly to determine the filter performance and trigger the inflation [see Luo and Hoteit (2014) for nonlinear observations]. The method in Tong et al. (2016) is different from these methods in that Tong et al. (2016) use the cross covariance between observed and unobserved variables in addition to the innovation statistics, which are derived from a rigorous mathematical theory to stabilize filters. We will show below that the cross covariance is essential, besides the innovation, in preventing catastrophic filter divergence in baroclinic turbulence (see Fig. 7).

In this study we demonstrate catastrophic filter divergence of ensemble-based filters in the two-layer quasigeostrophic equations, which are classical idealized models for geophysical turbulence (Salmon 1998). The adaptive inflation method is then applied for this two-layer system to avoid catastrophic filter divergence. Both a coarse-grained ocean code, which ignores the subgrid-scale parameterization, and stochastic superparameterization (Majda and Grooms 2014; Grooms et al. 2015b), which is a seamless multiscale method developed for large-scale models without scale gap between the resolved and unresolved scales, are applied to generate forecasts with a coarse spatial resolution

Using both the ocean code and stochastic superparameterization, various kinds of covariance inflation with or without localization are compared. We verify that proper adaptive covariance inflation can effectively stabilize the ensemble-based filters uniformly without catastrophic filter divergence in all test regimes. Furthermore, stochastic superparameterization achieves accurate filtering skill with localization while the ocean code performs poorly even with localization.

The structure of this paper is as follows. In section 2 we briefly review an ensemble method, the EAKF (Anderson 2001) with covariance inflation and localization. The adaptive inflation method to prevent catastrophic filter divergence is described in section 3 including how to choose parameters of the adaptive method. In section 4 the two-layer quasigeostrophic equation with baroclinic instability is described and two coarse-grained forecast models—the ocean code and stochastic superparameterization—are explained. Numerical experiments with various inflation strategies with or without localization are reported in section 5 along with stabilized and improved filtering results using the adaptive inflation method. In section 6 we conclude this paper with a discussion.

## 2. Ensemble filtering

*n*th observation time. We consider a linear observation of

*σ*independent in different times and space grid points. Thus, we have a natural decomposition of

*k*th column is given by the ensemble perturbation

*k*th column of the ensemble perturbation matrix

## 3. Adaptive additive inflation

*d*while its rank cannot exceed

*K*is the ensemble size. Thus, in adaptive additive inflation, we use the inflated prior covariance (10) to calculate the posterior mean while the posterior covariance does not change. That is, instead of (6), the posterior mean is defined as

As

## 4. Model equations and forecast models

In atmosphere and ocean science, quasigeostrophic equations are widely used as classical idealized models of geophysical turbulence (Salmon 1998). In this study we use a two-layer quasigeostrophic equation as the model equation to observe catastrophic filter divergence in high-dimensional data assimilation and test the adaptive additive inflation to prevent catastrophic filter divergence. The system is maintained by baroclinic instability imposed by vertical shear flows and shows interesting features in geophysical turbulence such as inverse cascade of energy and zonal jets. After describing the model equation in section 4a, two coarse-grained forecast models—an ocean code that ignores the subgrid scales and another forecast method with stochastic parameterization of the subgrid scales—are explained in section 4b.

### a. Two-layer quasigeostrophic equations

*r*is a linear Ekman drag coefficient at the bottom layer of the flows, and

*ν*, which is commonly used in turbulence simulations. To maintain nontrivial dynamics of (20) by baroclinic instability, a large-scale zonal vertical shear is applied with equal and opposite unit velocities that are related to the terms

Following the experiments in Grooms et al. (2015a) and Lee et al. (2017), we test three different regimes corresponding to low-, mid-, and high-latitude ocean models by changing the *β*-plane effect *r* (see Table 1 for the parameter values of the three test regimes). While the deformation wavenumber *ν* is set to 1.28 × 10^{−15} and we use a pseudospectral space discretization while the time integration uses a fourth-order semi-implicit Runge–Kutta method by incorporating an exponential integration for the linear stiff dissipation term. The time step is fixed at 2 × 10^{−5} for all test regimes.

In the high-latitude case (or the *f*-plane case), the quasigeostrophic equation is dominated by spatially homogeneous and isotropic flows (see Fig. 1 for snapshots of the upper- and lower-layer streamfunction). In the mid- and low-latitude cases, which have the *β*-plane effect, the flows organize into inhomogeneous and anisotropic structure such as zonal jets.

### b. Forecast models with and without stochastic parameterization

As a forecast model in data assimilation of the true signal given by (20), we consider two forecast models on low-resolution

^{−4}.

We consider another forecast model called stochastic superparameterization, which uses randomly oriented plane waves for the parameterization of the subgrid scales. The subgrid scales are generally not zero and influence the evolution of the resolved scales. Especially in quasigeostrophic turbulence, which includes regimes with a net transfer of kinetic energy from small to large scales (Charney 1971), it is important to accurately model the effects of the underresolved eddies to obtain accurate properties of the system such as energy spectrum. Stochastic superparameterization is developed as a multiscale model for turbulence without scale gap between the resolved and unresolved scales (Grooms and Majda 2014; Majda and Grooms 2014). Among various versions of stochastic superparameterization, we use the most recent version developed in Grooms et al. (2015b) to deal with arbitrary boundary conditions using finite-difference numerics for the large scales.

Time-averaged total kinetic energy (KE) spectra by direct numerical reference (black), stochastic superparameterization (blue), and ocean code (red).

Citation: Monthly Weather Review 145, 2; 10.1175/MWR-D-16-0121.1

Time-averaged total kinetic energy (KE) spectra by direct numerical reference (black), stochastic superparameterization (blue), and ocean code (red).

Citation: Monthly Weather Review 145, 2; 10.1175/MWR-D-16-0121.1

Time-averaged total kinetic energy (KE) spectra by direct numerical reference (black), stochastic superparameterization (blue), and ocean code (red).

Citation: Monthly Weather Review 145, 2; 10.1175/MWR-D-16-0121.1

## 5. Catastrophic filter divergence and numerical experiments

In this section we demonstrate catastrophic filter divergence for all three test regimes regardless of the two forecast models—the ocean code and stochastic superparameterization—with sparse high-quality observations that are infrequent in time. Catastrophic filter divergence is effectively prevented using the adaptive additive inflation for both forecast methods. Stochastic superparameterization achieves accurate filtering skill with localization while the ocean code fails to achieve accurate skill even with localization.

### a. Filtering setup

For EAKF, we use a sequential update of observations used in Anderson (2001), which avoids explicit computation of the SVD in (8) by processing observations individually. The true signal is given by a fine-resolution solution of (20) using ^{−5}. The two forecast models use the same coarse resolution ^{−4}.

We observe only the upper-layer streamfunction, analogous to observation of sea surface height, on a sparse *Z* is the time-averaged total enstrophy ^{−4}, this observation interval requires 16 time integrations for each forecast step. The ensemble size is 17 smaller than the dimension of the forecast model, which is general in real data assimilation.

Constant and adaptive inflation parameters

### b. Filter experiments—Catastrophic filter divergence and stabilization

If no inflation is applied, EAKF has catastrophic filter divergence for both forecast models. Figure 3 shows a sequence of snapshots of the low-latitude case upper-layer streamfunction by the ocean code without inflation and localization (observation points are marked with black circles). At the 570th cycle, the filter still captures the meridional structure of the low-latitude case but as more cycles continue, instability develops at unobserved grid points, which eventually diverges to machine infinity after the 600th cycle. The first row of Fig. 4 shows time series of the RMS errors by the forecast methods when they suffer from catastrophic filter divergence. The RMS errors increase gradually but they eventually diverge to machine infinity. The two forecast models run slightly longer with localization but localization fails to prevent catastrophic filter divergence. The second row of Fig. 4 shows time series of RMS errors with the constant + adaptive inflation where the cycles at which adaptive inflation is triggered is marked with dots. In the ocean code case with no localization, the adaptive inflation is triggered at the beginning period and then stops although the filter still degrades. Inflation is triggered again when the filter fails to capture the true signal. The ocean code with localization triggers the adaptive inflation most of the time and obtains a stable result but also fails to achieve accurate filtering skill. In the stochastic superparameterization case with the adaptive inflation and localization, adaptive inflation is triggered only 99 times out of 1000 cycles where most of the adaptive inflation is triggered at the beginning and infrequently triggered later as the filter is performing well.

Low-latitude case. Snapshots of posterior upper-layer streamfunctions by the ocean code at the 570th, 580th, 590th, and 600th cycles. Observation points are marked with circles. Catastrophic filter divergence is invoked after the 600th cycle.

Citation: Monthly Weather Review 145, 2; 10.1175/MWR-D-16-0121.1

Low-latitude case. Snapshots of posterior upper-layer streamfunctions by the ocean code at the 570th, 580th, 590th, and 600th cycles. Observation points are marked with circles. Catastrophic filter divergence is invoked after the 600th cycle.

Citation: Monthly Weather Review 145, 2; 10.1175/MWR-D-16-0121.1

Low-latitude case. Snapshots of posterior upper-layer streamfunctions by the ocean code at the 570th, 580th, 590th, and 600th cycles. Observation points are marked with circles. Catastrophic filter divergence is invoked after the 600th cycle.

Citation: Monthly Weather Review 145, 2; 10.1175/MWR-D-16-0121.1

Low-latitude case. Time series of upper-layer RMS error. The cycles, at which adaptive inflation is triggered, are marked with filled circles. Standard deviation of the streamfunction is shown by the dashed line.

Citation: Monthly Weather Review 145, 2; 10.1175/MWR-D-16-0121.1

Low-latitude case. Time series of upper-layer RMS error. The cycles, at which adaptive inflation is triggered, are marked with filled circles. Standard deviation of the streamfunction is shown by the dashed line.

Citation: Monthly Weather Review 145, 2; 10.1175/MWR-D-16-0121.1

Low-latitude case. Time series of upper-layer RMS error. The cycles, at which adaptive inflation is triggered, are marked with filled circles. Standard deviation of the streamfunction is shown by the dashed line.

Citation: Monthly Weather Review 145, 2; 10.1175/MWR-D-16-0121.1

The occurrence percentage of catastrophic filter divergence out of 100 different runs is in Table 3. With no localization and inflation, the filter suffers from catastrophic filter divergence more than 75% for both the ocean code and stochastic superparameterization. The constant inflation stabilizes the filter slightly but it does not prevent catastrophic filter divergence perfectly. The CI with no localization has a higher percentage of divergence than the no inflation case for the stochastic superparameterization forecast model. Through stochastic parameterization of subgrid scales, stochastic superparameterization has more variability than the ocean code and thus additional constant inflation is not necessary.

Occurrence percentage of catastrophic filter divergence out of 100 different runs with and without localization. No inflation (noI), constant (CI), adaptive (AI), and constant + adaptive (CAI) inflation methods.

AI with and without localization significantly decreases the number of occurrence of catastrophic filter divergence but the ocean code fails to prevent catastrophic filter divergence entirely. For the CAI, all methods are stable even without localization. Note that for stochastic superparameterization, both AI and CAI work well preventing catastrophic filter divergence while the ocean code fails to prevent the divergence in the AI case. As we discussed before, stochastic superparameterization has enough ensemble spread through stochastic parameterization of the subgrid scales and thus when adaptive inflation is already applied, constant inflation plays a marginal role in improving filter skill.

For the stabilized filters with the CAI, we compare the filter performance using the time-averaged posterior RMS errors and pattern correlations (the performance difference between the AI and CAI is marginal when there is no catastrophic filter divergence). In the low-latitude case (shown in Table 4), both the ocean code and the superparameterization methods fail to achieve accurate filtering skill without localization. The RMS errors are larger than the standard deviation of the streamfunction and both forecast methods do not capture the correlation with the true signal. When localization is combined with the adaptive inflation, it helps to increase filtering skill for both methods. The superparameterization has significantly improved results; RMS error is smaller than 50% of the standard deviation of the streamfunction and pattern correlation is larger than 90% for both layers. Although the lower-layer streamfunction is completely unobserved, the adaptive filter achieves accurate filter skill. The ocean code result is improved using localization but it still suffers from standard filter divergence with RMS errors larger than the standard deviation of the streamfunction.

Low-latitude case. Streamfunction estimation for both layers. Posterior RMS errors and pattern correlations are in parentheses.

In the midlatitude case, the superparameterization still has meaningful filtering skill and is superior to the ocean code although the performance is slightly degraded compared to the low-latitude case as the midlatitude is more turbulent than the low-latitude case. The RMS error by superparameterization with the adaptive inflation and localization is about 30% smaller than the standard deviation and pattern correlations are larger than 75% (see Table 5 for the midlatitude case RMS errors and pattern correlations). On the other hand, the ocean code does not show any significant skill even with the adaptive inflation and localization. In the midlatitude case, the ocean code using adaptive inflation displays comparable results with and without localization, and both fail to achieve meaningful filtering results. For the superparameterization, on the other hand, significant improvement in filter skill can be achieved using localization (see the second row of Fig. 5 for the time series of RMS errors with the adaptive inflation). As the RMS errors are more fluctuating than the low-latitude case, the adaptive inflation is triggered most of the time for all combination of inflation and localization.

Midlatitude case. Streamfunction estimation for both layers. Posterior RMS errors and pattern correlations are in parentheses.

Midlatitude case. Time series of upper-layer RMS error. The cycles, at which adaptive inflation is triggered, are marked with filled circles. Standard deviation of the streamfunction is shown by the dashed line.

Citation: Monthly Weather Review 145, 2; 10.1175/MWR-D-16-0121.1

Midlatitude case. Time series of upper-layer RMS error. The cycles, at which adaptive inflation is triggered, are marked with filled circles. Standard deviation of the streamfunction is shown by the dashed line.

Citation: Monthly Weather Review 145, 2; 10.1175/MWR-D-16-0121.1

Midlatitude case. Time series of upper-layer RMS error. The cycles, at which adaptive inflation is triggered, are marked with filled circles. Standard deviation of the streamfunction is shown by the dashed line.

Citation: Monthly Weather Review 145, 2; 10.1175/MWR-D-16-0121.1

The last test regime, high-latitude case (Fig. 6), is the most difficult test case as it is strongly turbulent and dominated by homogeneous and isotropic vortical flows with no spatial structure. In this test regime, stochastic superparameterization with CAI and localization still achieves a smaller RMS error and a larger pattern correlation than the ocean code. The observed upper-layer RMS error is 10% smaller than the standard deviation while the unobserved lower-layer RMS error is only 5% smaller than the standard deviation (Table 6).

High-latitude case. Time series of upper-layer RMS error. The cycles, at which adaptive inflation is triggered, are marked with filled circles. Standard deviation of the streamfunction is shown by the dashed line.

Citation: Monthly Weather Review 145, 2; 10.1175/MWR-D-16-0121.1

High-latitude case. Time series of upper-layer RMS error. The cycles, at which adaptive inflation is triggered, are marked with filled circles. Standard deviation of the streamfunction is shown by the dashed line.

Citation: Monthly Weather Review 145, 2; 10.1175/MWR-D-16-0121.1

High-latitude case. Time series of upper-layer RMS error. The cycles, at which adaptive inflation is triggered, are marked with filled circles. Standard deviation of the streamfunction is shown by the dashed line.

Citation: Monthly Weather Review 145, 2; 10.1175/MWR-D-16-0121.1

High-latitude case. Streamfunction estimation for both layers. Posterior RMS errors and pattern correlations are in parentheses.

For the stochastic superparamterization case with localization and the CAI, the time series of the two statistics,

Time series of

Citation: Monthly Weather Review 145, 2; 10.1175/MWR-D-16-0121.1

Time series of

Citation: Monthly Weather Review 145, 2; 10.1175/MWR-D-16-0121.1

Time series of

Citation: Monthly Weather Review 145, 2; 10.1175/MWR-D-16-0121.1

## 6. Conclusions

Ensemble-based filtering methods are indispensable tools in atmosphere and ocean science as they provide computationally cheap and low dimensional ensemble state estimation for extremely high dimensional turbulent systems. But these methods can suffer from catastrophic filter divergence, which drives the forecast predictions to machine infinity especially when the observation is sparse, accurate, and infrequent although the underlying true signal remains bounded. Using an idealized model for the geophysical turbulence of the ocean, the two-layer quasigeostrophic equation with baroclinic instability, and a sparse observation network, which is general in real applications, we were able to see catastrophic filter divergence of the ensemble adjustment Kalman filter, which is one of the most stable and accurate ensemble methods.

The constant covariance inflation and localization, which are widely used methods to account for the sampling errors due to insufficient ensemble size and model errors from imperfect forecast models, stabilize the filter but fail to prevent the catastrophic filter divergence. Increasing the observation size or ensemble number can help to prevent catastrophic filter divergence but this approach is practically prohibitive and sometimes impossible as it requires enormous amount of financial and computer resources to cover the vast surface of the ocean. Instead we followed the adaptive inflation approach of Tong et al. (2016) to prevent catastrophic filter divergence. The adaptive approach requires a minimal additional computational cost compared to the standard ensemble based methods and uses only two low-order statistics of the ensemble—the ensemble innovation and cross covariance between observed and unobserved variables.

We tested the adaptive inflation using two forecast models—the ocean code without parameterization of the subgrid scales and stochastic superparameterization—which parameterizes the subgrid scales by modeling them as randomly oriented plane waves. Although both forecast models are stabilized with the adaptive inflation, stochastic superparameterization displays filtering skill superior to the ocean code. When the ensemble method is combined with localization and adaptive inflation, stochastic superparameterization achieves RMS errors smaller than the climatological error while the ocean code still suffers from the standard filter divergence with RMS errors comparable to the climatological error.

As we have shown in this study, covariance inflation is an important and useful technique in ensemble based methods to improve filtering skill. There are another class of adaptive inflation techniques such as Anderson (2007) and Ying and Zhang (2015). Although the adaptive inflation in Tong et al. (2016) is based on rigorous mathematical arguments, it would be interesting to test other adaptive inflation methods to avoid catastrophic filter divergence like the blended filter (Majda et al. 2014; Qi and Majda 2015) that combines a particle filter in a low-dimensional subspace and efficient Kalman filter in the orthogonal part. As it is investigated in Harlim and Majda (2010) through a linear stochastic model for the forecast, a judicious model error could be alternative to prevent catastrophic filter divergence.

## Acknowledgments

The research of A. J. Majda is partially supported by Office of Naval Research Grant ONR MURI N00014-12-1-0912 and DARPA 25-74200-F4414. Y. Lee is supported as a postdoctoral fellow by these grants. D. Qi is supported as a graduate research assistant by the ONR grant. We thank the three anonymous reviewers for their comments, which significantly improved the manuscript.

## APPENDIX

### Stochastic Superparameterization

*A*is a tunable parameter that determines the strength of the small-scale contributions,

*θ*represents the orientation of the small-scale plane waves and is modeled by a Weiner process that is independent at different coarse-grid points

*σ*is a parameter that determines the decorrelation time of the small scales and thus this parameter can be estimated from the small-scale decorrelation time without tuning (in our experiments,

*A*is hand-tuned by matching the kinetic energy of stochastic superparameterization which yields 6750, 1700, and 350 for the high-, mid-, and low-latitude cases, respectively. When the large-scale equation is solve by a low-order method such as the finite-difference method used in this paper, the subgrid-scale parameterzation accumulates strongly at the coarse grid scale. The smoothing operator

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