In the GIOPS system, analysis increments are applied using an incremental analysis updating (IAU; Bloom et al., 1996) approach with increments applied evenly over a 1 d period. For the delayed-mode (GD) and real-time (GR) weekly cycles this is done over the last day of the 7 d assimilation window, and for the daily cycle (GU) this is done over the full 1 d window. To provide greater consistency in time, in RIOPSv2 a 7 d IAU is used for both RD and RR cycles inspired by the methodology of Benkiran and Greiner (2008). A linearly increasing ramp is used over the first day, followed by a constant increment for days 2–7. A decreasing ramp is applied for the first day of the following cycle. This approach is also applied in systems used by Mercator Océan International (Lellouche et al., 2013, 2018).

As mentioned above, the background error for RIOPSv2 is specified in terms of a series of error modes derived from a multiyear forced simulation over the period 2002–2011. Sub-monthly anomalies are constructed by removing low-frequency variations using a 30 d Hanning filter. The basic methodology used to calculate the error modes is the same as used for GIOPS apart from the smoothing used in the error modes. For RIOPSv2, 49 passes of a Shapiro filter are applied to temperature and salinity fields, resulting in a roughly 1∘ smoothing. Also, tides are removed from the SSH field using the online harmonic analysis described below prior to calculation of the sea surface height anomaly.

The resulting error modes provide a seasonally varying estimate of the background error. For a particular analysis cycle, error modes within a 90 d window around the Julian day of the analysis cycle from all 10 years of the forced simulation are used. As error modes are produced every 3 d, this provides roughly 300 error modes available for each analysis cycle.

While this approach does not provide an estimate of “errors of the day” as in an ensemble Kalman filter, it does nonetheless provide an estimate of the covariances in space and between model variables (i.e., SSH and three-dimensional fields of temperature, salinity, and zonal and meridional velocities). An example of the multivariate correlations is shown in Fig. 3. Correlations can vary considerably at different locations and between variables based on the underlying oceanographic conditions. For example, in the Gulf Stream (Fig. 3) strong correlations are present between SSH and temperature at short distances, representative of dynamic height variability in this region of strong mesoscale activity.

As a SEEK filter formulation is used, the model background error will determine the subspace within which the analysis is restricted. Here, this subspace is limited by the variability of the model fields over the 10-year period of the model simulation. As such, the spatial scales represented in the analysis are determined by the effective model resolution and the spatial filtering applied to the error modes.

Satellite altimetry observations contain a variety of signals including those produced by tides. Here, we have chosen to address the calibration of the modeled tides separately in advance and to use the altimetry observations to constrain geostrophic variability. This permits the use of standard AVISO Ssalto/Duacs altimetry products available for operational real-time applications as required by RIOPS. These products undergo a number of processing steps to remove tidal signals, as well as wet-tropospheric effects and longwave errors (e.g., Carrère et al., 2003; Dibarboure et al., 2011).

To provide the most accurate model equivalent possible, it is important to apply the same processing to model fields as has been done to satellite altimeter observations. As RIOPS includes tidal and atmospheric pressure forcing, the SSH observation operator must also be adjusted to filter these sources of variability as they have been filtered from the AVISO Ssalto/Duacs SLA observations that are assimilated. The inverse barometer effect can be calculated locally based on the hourly atmospheric pressure forcing applied to the model and accounted for directly. This will not capture nonlocal effects such as coastally trapped waves. As the SSH observation operator applies a 24 h mean, much of the remaining coastal variability will be removed. Moreover, as the SLA observations are assimilated with a larger error close to coasts these effects will not have a significant impact. The SSH observation operator is also modified to include the online harmonic analysis as described in Sect. 3.4.

As noted above, for both RIOPSv2 and GIOPSv3 the SST observations are assimilated in the SAM2 system using the CCMEP SST OI analysis. As this OI analysis uses covariances with e-folding length scales varying between 20 and 70 km (Brasnett and Colan, 2016) it is appropriate to apply a spatial filter to model SST fields to match the spatial scales present in the OI analysis. A Shapiro filter with 49 passes is used in RIOPS to provide a roughly 1∘ resolution field. This number of passes was chosen as it was found to provide a good match in the power spectral density of SST fields between the CCMEP analysis and RIOPS fields (not shown). Moreover, as a diagonal observation error covariance matrix is used in SAM2, the CCMEP analysis is decimated by 1 point out of 5 to reduce correlated variability. Figure 4 provides an example for 20 July 2016, showing the unfiltered 0.1∘ resolution CCMEP SST analysis, the CCMEP analysis decimated to 0.5∘ resolution, the model trial field (raw model equivalent) for the same date, and the model equivalent following application of the Shapiro filter. We can clearly see that the raw model fields (Fig. 4c) contain smaller scales than the CCMEP analysis, whereas the filtered model fields provide a more representative comparison to the CCMEP analysis. The innovation is then calculated as Fig. 4b minus d.

Given the model SSH time series Hn at the nth model time step at a particular point on the model two-dimensional grid, we aim to find the harmonic spectrum coefficient Xk for each kth frequency to provide the harmonic decomposition: 1An=EknXk,A∗n=Ek∗nX∗k, where the superscript * indicates complex conjugate, and An is the decomposition of the real time series Hn. The harmonic function base is defined as 2Ekn=Ckexp⁡(iωkτn), where Ck is a time-independent normalization constant factor, which could include the frequency-dependent initial phase term, ωk is the kth pre-chosen harmonic frequency, and τ is the harmonic analysis time step length. Note that here we use n and m as indexes of time step and k and j as the frequency index, k=0,1,2,…,K. In the RIOPSv2 system, we select K=33 and specify ω0=0 for the mean of time series H.

An is found by minimizing the following cost function:

3aJ(Xk)=12An-Hn∗WnmAm-Hm,3b=12EknXk-Hn∗WnmEjmXj-Hm, where Wnm is a real diagonal matrix of the time weights used in the sliding window; the indexes n and m take values from -N+1,…,-2,-1,0, where n=0 indicates the current step. N is the sliding-window length in units of model time steps and increases with model integration time. Here, we follow the Einstein summation convention, i.e., summing over the covariant and contravariant index pairs, unless noted otherwise.

According to Eq. (3b), the variation of J on X*k has 4δJδX∗k=Ek∗nWnmEjmXj-Hm=(Ek∗nWnmEjm)Xj-Ek∗nWnmHm. We denote Bkj=Ek*nWnmEjm with size (K+1)×(K+1) and Yk=Ek*nWnmHm with size (K+1)×1. If we set Eq. (4) to be equal to zero, we can obtain the minimization solution of J as follows: 5Xj=B-1kjYk, where B-1 is the inverse matrix of B. Hence, we can obtain the final solution of the tidal harmonic analysis as 6aAn=EjnB-1kjYk. Note that the diagonal matrix W guarantees that matrix B is Hermitian and J is real. From Eq. (6a) we can see that the constant C in Ekn (Eq. 2) can be canceled, which means the final solution A is independent of C. Therefore, we set C=1 without any loss of generality. A convenient feature of this approach when implemented as an online harmonic analysis in a numerical model is that we only need the value of A0 from the current time step. Following Eq. (2), Ej0 is equal to 1 so that we have 6bA0=∑j=0KB-1kjYk. Note that the mean of time series H is included in the above A0.

In the following, we will show how to use a rotation operator and a sliding-window approach to update the B and Y matrices for the current time step based on previous time steps B′ and Y′ (hereafter, the symbol ′ indicates the previous time step).

First, we split the summation index range of n (and m) into two sets, one for the current time step containing only step index 0 and another set for previous steps containing indexes -N+1,…,-2,-1; to differentiate, we will use indices p and q for the latter set. Therefore, considering that W is a diagonal matrix, we can rewrite the B and Y matrices as follows: 7aBkj=Ek∗nWnmEjm=Ek∗0W00Ej0+Ek∗pWpqEjq,7bYk=Ek∗nWnmHm=Ek∗0W00H0+Ek∗pWpqHq. According to Eq. (2), Ek0 and Ek*0 are simply equal to 1. If we denote W00 as α, which is specified as the ratio of time step length τ to a given restoring time length (e.g., 30 d in RIOPSv2; see Sect. 3.4.4 for discussion), then the first terms on the right-hand side (RHS) of Eq. (7a) and (7b) become α1kj and αVkH0; here 1kj and Vk are a matrix and a vector, respectively, with all elements equal to real number 1. In the second term on the RHS of Eq. (7b), we use n (or m) to replace p+1 (or q+1) and denote Hq=H′m. Here, H′m is the variable H at the mth time step referred to in the previous time step.

The sliding-window weight W can then be specified as a decreasing exponential function as follows: W-1,-1=(1-α)W00,W-2,-2=(1-α)W-1,-1,…Wn-1,m-1=(1-α)Wnm. In other words, Wpq=(1-α)Wnm. As such, the second term on the RHS of Eq. (7b) can be rewritten as 8Ek∗pWpqHq=∑p,q=-N+1-1Ek∗pWpqHq=exp⁡(iωkτ)∑p,q=-N+1-1exp⁡(-iωkτ)Ek∗pWpqHq=exp⁡(iωkτ)∑p,q=-N+1-1Ek∗p+1WpqHq=exp⁡(iωkτ)∑n,m=-N+20Ek∗n(1-α)WnmH′m=1-αexp⁡iωkτ∑n,m=-N+10Ek∗nWnmH′m+O(W-N+1,-N+1)=1-αexp⁡iωkτYk′+O(W-N+1,-N+1). Here, the term OW-N+1,-N+1 results from the change in the lower limit of the summation of n(m) from -N+2 to -N+1. Generally speaking, this term can be neglected when N is sufficiently large because W-N+1,-N+1 is very small, on the order of (1-α)N-1 and α(1-α)N-1 compared with W00 and Y, respectively. For example, in RIOPS with a 30 d restoring time length and after a half-year spin-up period, they are about e-6=2.479×10-3 and 1/(288⋅30)⋅e-6=2.869×10-7, respectively.

Therefore, using Eqs. (8) and (7b), we can obtain 9Yk=αVkH0+(1-α)Rk∗jY′j, where R is a diagonal time-independent rotation matrix, with its kth diagonal element taking the value 10Rkk=exp⁡(-iωkτ). The matrix R can be used to rotate a complex vector in the complex plane spanned by the kth complex harmonic function base. For instance, a complex vector eiθ could be rotated into a vector ei(θ-ωkτ) by operator Rkk. One exception is that when ω0=0, as is the case for the mean of time series of Hn, both R00 and E0n are equal to the real number 1 so that there is no complex plane spanned, and R only takes action in the one-dimensional real space. Similarly, we can rewrite the RHS of Eq. (7a) as 11aEk∗pWpqEjq=∑p,q=-N+1-1Ek∗pWpqEjq=exp⁡(iωkτ)exp⁡(-iωjτ)∑p,q=-N+1-1exp⁡-iωkτEk∗pWpqEjqexp⁡(iωjτ)=exp⁡(iωkτ)exp⁡(-iωjτ)∑p,q=-N+1-1Ek∗p+1WpqEjq+1=exp⁡(iωkτ)exp⁡(-iωjτ)∑n,m=-N+20Ek∗n(1-α)WnmEjm=1-αexp⁡iωkτexp⁡(-iωjτ)∑n,m=-N+10Ek∗nWnmEjm+O(W-N+1,-N+1)=1-αexp⁡iωkτBkj′exp⁡(-iωjτ)+O(W-N+1,-N+1)=1-αRk∗iBil′Rjl+O(W-N+1,-N+1). Combining Eqs. (7a) and (11a), we can write 11bBkj=α1kj+(1-α)Rk∗iB′ilRjl. Finally, based on Eqs. (6b) and (9)–(11b), we can get the final solution A0 for the current time step. Appendix A provides the derivation of the equivalent real-space version of these equations that is coded in RIOPSv2.

The main advantage of this sliding-window approach is that it is much cheaper to use than the traditional method. For instance, in RIOPSv2 the number of horizontal grid points is Nxy=1580×2198, and if we update tidal harmonics at each model step with a half-year spin-up window (or analysis time window), this implies a total number of time steps of Nstp=288×182. This means the size of Y is less than 2 GB (∼8×Nxy×(2K+1)) for our method and over 1 TB (∼8×Nxy×Nstp) for the traditional method. In addition, updating Y at each model time step using the traditional method by equation Yk=Ek*nWnmHm would require the computation of sine and cosine decomposition operators Nxy×(2K+1)×Nstp times. Alternatively, the method outlined here requires only the use of the multiplication operator Nxy×(2K+1) times. Moreover, to speed up computation in the traditional method one could save the sine and cosine decomposed values at each model step; however, the data size of Y will become huge, over 2K+1 TB. As such, the traditional method would be too slow for operational systems based on the present-day computing environments unless one made some modifications and/or simplifications. For instance, the cost could be lowered by reducing the resolution of the temporal and the spatial calculations. However, this could have negative consequences in coastal areas and areas of steep bathymetric slope as the local geometric structures have a strong influence on the tidal harmonics, as do high-frequency tidal constituents.

As we assume that the matrix B is homogeneous for the horizontal grids, the data size is very small ((K+1)×(K+1) in complex and (2K+1)×(2K+1) in real space, respectively), and there is no issue for B regarding differences of computational costs, memory, and I/O between the traditional method and online harmonic analysis with the sliding window.

Another advantage of the sliding-window approach is with respect to its spin-up. As we know, dynamic models and data assimilation systems require a certain time period to be spun up to equilibrium. When applying the online tidal harmonic analyses, we can allow all three parts (model, assimilation, and tidal filter) to be coupled in harmony during the common spin-up time window from the model cold start.

This sliding-window approach is implemented using the real-space version of the equations (Appendix A) such that Eqs. (A2)–(A4) are updated at every time step using 33 diurnal and semi-diurnal tidal constituents. Updating these equations less frequently (e.g., every four model time steps or hourly) was found to have a negative impact on the accuracy of the harmonic analysis and resulted in an increase in the tidal energy remaining in the SSH residual. The choice of 33 tidal constituents was based on the results of an offline harmonic analysis made using T_tide (Pawlowicz et al., 2002). Additionally, no harmonics with a period longer than 30 h were used to avoid contamination of the tidal signal by mesoscale variability.

The other free parameter in the sliding-window approach is the timescale used in the weighting function. Here a value of 30 d is used. This value must be large enough to permit an accurate fit of the different tidal constituents. Using a longer value reduces the ability of the system to adapt to seasonal (or longer) changes in tidal variability.

A comparison between the online harmonic approach and the well-known T_tide package is provided in Fig. 5. The unfiltered SSH and tidal residuals using both the online approach introduced here and T_tide are shown for a point in Ungava Bay (location indicated by a star in Fig. 6). This region is characterized by extremely large tidal amplitudes and seasonal ice cover. From Fig. 5 we can see that while both approaches provide similar tidal residuals (Fig. 5b), instantaneous differences of up to 0.5 cm may be present. This value corresponds to roughly 20 % of the observational error used for some satellite altimeters. Moreover, Fig. 5d shows that the amplitude of the high-frequency tidal variations in the tidal residual is larger for T_tide. While the period of analysis is too short to be conclusive, there does appear to be a seasonal cycle in the amplitude of the differences in tidal residuals (Fig. 5c), with larger differences in winter months potentially due to the presence of sea ice.

To illustrate the relative importance of the various filtering steps, the different components (tides, inverse barometer) have been separated and an example is shown in Fig. 6. The tidal signal is the most prominent source of variability and has a significant impact, with local variations exceeding 1 m in amplitude. In comparison, the inverse barometer effect is generally below 0.2 m. The residual SSH with tides and inverse barometer removed shows the main physical features well, including the Gulf Stream and Beaufort Gyre. Some tidal variability remains in coastal areas with strong tides (e.g., Bay of Fundy), but this is expected to have a negligible effect as SLA observation errors are amplified nearshore to account for representativity error.

The mean and rms innovation statistics for SLA are shown in Fig. 7. The largest innovations for both RIOPSv2 and GIOPSv3 occur over the Gulf Stream region due to the associated strong mesoscale variability. In GIOPSv3, rms differences exceed 25 cm from the east coast of North America eastward to 45∘ W. RIOPSv2 shows notably smaller rms differences over this region, with values closer to 20 cm. These improvements are due in part to the smaller values of MDT representation error used in RIOPSv2, together with the higher model grid resolution that can better represent mesoscale structures.

A second region of large SLA innovations can be found in the Arctic Ocean. In the northern Laptev and Beaufort seas, rms differences larger than 20 cm are present. These regions are also characterized by negative mean SLA innovations of greater than 10 cm in GIOPSv3 and RIOPSv2. As such, the differences may be associated with the CNES-CLS13 MDT field used. As this area is typically ice-covered, the MDT would have had a few years of open-water data to use in its production. These regions are also strongly affected by freshwater drainage from the Mackenzie River and rivers in Russia that may affect the SLA. For example, a negative SLA bias of 10 cm (positive values in Fig. 7c and d) at the mouth of the Lena River (135∘ W) is likely due to positive anomalies in actual river runoff compared to the climatological values used here by both systems. Note also that due to satellite orbits and ice coverage, far fewer observations are present over the Arctic Ocean, with fewer than 500 measurements per bin compared to over 2000 measurements per bin south of 66∘ N. As such, the statistics over the Arctic are less reliable and more seasonally dependent.

Finally, a region with slightly increased rms differences in RIOPSv2 is noted in Hudson Bay and the northern Labrador Sea. These regions show larger rms differences of up to 5 cm and are likely associated with residual tidal variability remaining after application of the online harmonic analysis due to fortnightly and monthly timescale tidal harmonics.

SST innovations between RIOPSv2 and GIOPSv3 are broadly similar (Fig. 8), with the largest errors occurring in the Gulf Stream and along the winter marginal ice zone in the Labrador Sea and Greenland–Iceland–Norwegian seas. The rms SST innovations in the Gulf Stream region decrease from upwards of 2.0 ∘C for GIOPSv3 to around 1.5 ∘C for RIOPSv2. The largest SST innovations for both systems occur along the tail of the Grand Banks around 45∘ W, likely due to the presence of strong SST gradients. Along the region of maximum ice extent in winter, errors of around 0.8 ∘C are present. In most other areas, errors in both systems are less than 0.5  ∘C, which is the nominal error of the CCMEP SST analysis (Brasnett and Colan, 2016). Similar to innovations of SLA (Fig. 7), a slight increase in rms SST innovations in RIOPSv2 can be seen in the central Labrador Sea.

Innovation statistics obtained from in situ temperature and salinity profiles averaged over the upper 500 m of the water column and over the range 500–2000 m are shown in Figs. 9–11. These observations are composed mainly of data from Argo profiling floats, with some additional profiles from field campaigns, moorings, voluntary observing ships, gliders, and marine mammals. Interestingly, the period of evaluation includes the Year of Polar Prediction (YOPP; Jung et al., 2016) for which a number of additional in situ ocean observations were deployed (Smith et al., 2019b). These include Argo and ALAMO floats (Wood et al., 2018) used seasonally during ice-free periods as well as ice-tethered profilers (ITPs; Toole et al., 2011). These, together with other additional ocean observations deployed during YOPP, provide an exceptional opportunity to evaluate water mass properties in RIOPS and GIOPS over the Arctic Ocean, for which a significant gap is usually present in the in situ ocean observing network.

As shown for SLA and SST, the largest innovations in profile data for both systems are found in the Gulf Stream region due to the strong mesoscale variability and important spatial variability in water mass properties (Figs. 9 and 11). Significant innovations in salinity are also found along many coastal regions. On the Canadian east coast, in the Gulf of St. Lawrence, and along the Labrador Coast, rms salinity innovations often exceeding 0.5 psu are present. Similar errors are present around the North Sea and along the Norwegian coastline, with slightly larger values in RIOPSv2. The larger biases in RIOPSv2 may be associated with a positive salinity bias in the upper 50 m of the water column (not shown). This bias may be due to a deficit in river runoff together with the associated reduction in vertical stratification, resulting in overly intense vertical mixing due to tides.

The Arctic observations are quite revealing, highlighting important positive biases in salinity (i.e., too salty) in the Beaufort Sea (Fig. 10c) associated with the halocline (centered at 50 m of depth) and Pacific water layer (200–300 m of depth; not shown). This bias is smaller in GIOPSv3 in the eastern Beaufort Sea, with a salinity bias of less than 0.2 psu, whereas RIOPSv2 shows values up to 0.4 psu. The source of this bias is under investigation and will be part of a study to investigate how to better constrain water masses in the Arctic using observations from YOPP.

Errors in other regions are generally quite small in both systems. In the Pacific Ocean, the 3D-Var bias correction scheme considerably reduces the biases in salinity over the upper 500 m. Without the bias correction scheme, errors of up to 0.5 psu occur over a broad region in the North Pacific Ocean along the Aleutian Islands (not shown).

As the Gulf Stream region shows the largest rms innovation errors for all fields, we now focus on a comparison of SLA anomalies and resolved spatial scales over this region. Figure 12 shows Hovmöller diagrams for a particular Jason altimeter track, illustrating the evolution of SLA anomalies over the evaluation period. The vertical axis is constructed using the SLA anomalies over the track from the 10 d repeat orbit with each row representing one track. The location of the track is shown in green in Fig. 12b. Both RIOPSv2 and GIOPSv3 capture the low-frequency variations in SLA due to the evolution of mesoscale ocean features in this highly dynamic region. Indeed, SLA variations are present with a range greater than 1 m, yet rms differences for RIOPSv2 and GIOPSv3 are only 12 and 14 cm, respectively (Table 3). Differences between the analysis system and the observations (Fig. 12d and f) tend to be quite localized in time. The differences occur less frequently in RIOPSv2, suggesting a better representation of the mesoscale eddy field. In addition, GIOPS shows a significant period of error related to a negative SLA anomaly that occurred around 62∘ W in mid-2017. The closer fit of RIOPS to observations is reflected in higher correlations and lower rms error between RIOPS and the SLA observations (Table 3).

Given the higher spatial resolution of RIOPSv2, one would expect to see finer scales present in the details of the along-track SLA anomalies. As this is not obvious from Fig. 12 we will now investigate the power spectral density (PSD) of the surface kinetic energy (KE) fields from RIOPSv2 and GIOPSv3 over the Gulf Stream region (Fig. 13) following the approach of Jacobs et al. (2019). First, we compute the Fourier transform of both zonal and meridional surface current fields over a box covering the Gulf Stream region (48.8–66.0∘ W, 32.3–45.0∘ N) using weekly instantaneous SLA fields from both RIOPSv2 and GIOPSv3 over the full 3-year evaluation period. The results are then averaged in time and between velocity components to provide the PSD of KE.

Overall, the surface kinetic energy for RIOPSv2 shows 1.7 times the power over the full spectrum, with higher power at all wavelengths. The PSD for both systems closely follows the k-3.4 line over the mesoscale band, in agreement with Jacobs et al. (2019), demonstrating a good representation of the energy cascade. The wavelengths for which the PSD deviates from the k-3.4 line at both the high and low wavelength limits are indicated in Fig. 13. Due to the higher model resolution, RIOPSv2 extends the mesoscale band down to about 35 km compared to 66 km for GIOPS. These scales provide an indication of the effective model resolution and are equivalent to roughly five and three model grid points for RIOPSv2 and GIOPSv3, respectively. The long-wavelength limits for RIOPSv2 and GIOPSv3 are broadly similar, with values of 178 and 208 km, respectively. Both systems show a peak in PSD around 416 km.