We use along-track altimetric data (L3 product) available between 1993 and 2018, as downloaded from the Copernicus Marine Environment Monitoring Service (CMEMS, product SEALEVEL_GLO_PHY_L3_MY_008_062). This embeds data from 12 satellite missions, organized in 21 data sets, as detailed in Table . For validation purpose, to keep independent observations that are not used in the estimation, we have divided this dataset in two subsets (A and B) as mentioned in the second column of the table.

The variable that is used from these datasets is the absolute dynamic topography. It is computed as the sum of the mean dynamic topography (obtained from a combination of gravimetric satellite missions, and interpolated along-track data) and the sea level anomaly (obtained from the satellite missions listed in Table ). In this product, corrections are applied to sea level anomaly to remove long wavelength errors (e.g. orbit errors), atmospheric errors and the ocean tide (internal and external).

For comparison purpose, we will also use mapped altimetric data (L4 product) which correspond to CMEMS product SEALEVEL_GLO_PHY_L4_MY_008_047. The maps are daily and have been produced by objective analysis using the along-track altimetric data described in Table .

As model data, we use the 30-member ensemble simulation of the Mediterranean described in . It is based on the NEMOMED12 configuration of the NEMO ocean model , implemented over the Mediterranean basin at 1/12° resolution and over 75 vertical levels, and driven by specified forcing between 1979 and 2020. The surface forcing is provided by the new ALDERA3 dataset , which consists in 3-hourly air-sea fluxes (wind stress, heat and freshwater fluxes), together with a Newtonian relaxation of sea surface temperature to maintain thermal stability (with a restoring coefficient equivalent to 40 W m⁻² K⁻¹). River runoff is prescribed at the river mouths and along the coastline as a surface freshwater flux, using a 1980–2017 climatological seasonal cycle derived from the interannual dataset of . Boundary forcing in the Gulf of Cadiz is applied in each ensemble member within a buffer zone, where the model temperature, salinity and SSH fields are relaxed towards their monthly climatological counterparts extracted from the ORAS4 global ocean reanalysis , with a relaxation time scale decreasing westwards. More details and references about the model configuration can be found in .

The simulation starts with a 20-year one-member spin-up (1979–1998). The 30 ensemble members are then initialized by this spun-up state, and subsequently driven by the same surface and lateral fluxes throughout the 39-year integration period (from January 1979 to December 2020). Ensemble dispersion is generated by tiny perturbations of the initial condition as explained by, so that the spread of the ensemble results only from the intrinsic variability of the nonlinear system. From this ensemble simulation, we use the daily maps of sea surface height and we apply a low-pass Lanczos filter to extract the low-frequency component of the signal (with periods longer than about 2.5 years). The first and last 2 years of filtered time series are discarded to avoid filtering side effects.

Then, we subsample the timeseries every month between 1993 and 2018, which corresponds to the observation period described above (chosen to be the largest intersection period between model and observations). The resulting low-frequency component of sea surface height is illustrated in Fig.  by the mean and standard deviation of the ensemble for January 1997 and January 2017. As observed in , the model simulation captures the main observed patterns of the Mediterranean circulation, and displays a large ensemble spread in the regions of intense mesoscale activity (Fig. , bottom panels). However, in most other regions, the ensemble spread, resulting here from intrinsic variability alone, is insufficient to explain the misfit with observations. This ensemble was indeed not specifically produced for this paper, but was dedicated to the specific study of intrinsic variability by . Important sources of uncertainties (e.g. in the atmospheric forcing and in the boundary conditions) were not taken into account; this is why we could not rely on the prior variance, and why an adjustment of the ensemble statistics is even more important.

These ensemble data are nonetheless important to solve our estimation problem because they can be expected to contain useful information about the low-frequency variability of the sea, in particular through a non-stationary, non-homogeneous and anisotropic description of the correlation structure. Non-stationarity of the ensemble statistics comes as a result of the time-varying external forcing that is applied to the model.

The vector of variables x that we want to estimate is going to include several components or subvectors xi, which are presented here successively. Our first objective is to reconstruct the low-frequency variability of the Mediterranean, as described here by the dynamic topography H only (even if other variables could in principle be included): 1x1=[H(x,y,t)]. The spatial resolution of these fields (grid where x, y and z are discretized) is the same as the model ensemble simulation (i.e. 1/12° on the horizontal), and the time resolution (for t) is set to 1 month, which is sufficient to resolve the low-frequency component of the signal. These fields contain interannual time scales, without the annual cycle, which is filtered out.

In addition, since the estimation will use data from a prior ensemble model simulation, we must account for a possible bias in these data, which we also include in the vector of variables to estimate: 2x2=[δH(x,y),δT(x,y,z),δS(x,y,z)]. This bias is here assumed constant in time. A more complex problem would be to allow for a slow variation of this bias over time, and a simplification would be to rely on available climatologies to compute the bias beforehand as the difference with the model climatology, which would exclude the bias x2 from the estimation vector. The complexity of the estimation problem needs to be tuned according to what is expected tractable with the available observations and methods.

Moreover, the estimation problem will require the covariance of the prior uncertainty and the covariance of the unresolved time scales (as the main contribution to the observation representativity error). Both can be obtained from the prior ensemble simulation, but again with possible biases (resulting for instance from an insufficient spread of the prior ensemble). This bias is introduced in the estimation vector, as two multiplicative factors applied to these two covariances: 3x3=[α(x,y),β(x,y)]. These factors α and β are here assumed to be constant in time, but again more complex assumptions would be possible. The introduction of such amplification factors in the estimation vector is what is usually done in adaptive Kalman filters, as proposed for instance by , , , and applied by to a square-root Kalman filter. It must be noted here that it may be challenging to estimate these two factors simultaneously, since it is always the sum of the prior and observation uncertainties that is observed. The joint estimate is still possible in principle (as in adaptive Kalman filters) if they have distinct correlation structures, but it is always possible that information is missing to obtain accurate estimates.

The vector of variables to estimate x is then built by aggregating the components (Eqs. –): x=[x1,x2,x3]. It is estimated using the Bayes theorem: 4pa(x)=px|yo∝pb(x)pyo|x where yo is the vector with all available observations, pb(x) is the prior probability distribution for x, p(y|x) is the conditional probability distribution for the observation vector y given the state x of the system (evaluated at y=yo to give the likelihood function of x), and pa(x) is the posterior probability distribution for x, given the observations yo. Our objective is to produce a sample of pa(x).

The reason why we needed to include x2 and x3 in the estimation vector x is that the ensemble model simulation described in Sect.  could not be assumed perfectly reliable. We know that there are substantial biases and that the spread is not always sufficient to be consistent with the observations. By including x2 and x3, we provide more freedom in the prior statistics to fit the observations, and we expect to obtain a more reliable description of the posterior uncertainty, as described by pa(x).

The prior statistical model is meant to embed all assumptions that are made in this study. These are the assumptions that need to be checked and evaluated. Our central assumption is to use a long-term ensemble model simulation as a prior distribution for the low-frequency interannual signal. The expectation is that the model simulation embeds useful information about this signal, in particular through the use of a realistic interannual atmospheric forcing. The prior statistics are thus nonstationary, since they can change in time according to how the model responds to a changing atmosphere. There is also the perspective that they can improve in the future when the model is made more and more reliable.

The first consequence of this choice is that only a small-size ensemble is available (30 members), which means that we can only have a simple prior model (in our case, described by the mean and the covariance, with localization), since no complex model can be inferred from a small sample. The second consequence is that we have to cope with possible systematic errors in the model simulation (bias and insufficient spread), by including additional degrees of freedom in the estimation vector (x2 and x3).

In our model, the prior distribution for the components (Eqs.  and ) of x (i.e. the bias and the spread adjustments) are assumed independent, while the distribution for x1 (i.e. the main target of the inverse problem) depends on x2 and x3: pb(x)=pb(x1|x2,x3)pb(x2)pb(x3). We will see that this makes the resulting prior statistical model for x non-Gaussian, because the statistics of x1 depends on x2 and x3, and because x3 is itself assumed non-Gaussian.

For component 1, the conditional distribution (for given x2 and x3) is assumed Gaussian, with mean x1b and covariance P1b. x1b and P1b are specified by the mean x1r and the covariance P1r of the low-frequency component of the reference ensemble simulation (described in Sect. ), with a bias correction given by x2 and a spread correction given by the component α of x3: 5x1b=x1r-x2andP1b=D(α)1/2P1rD(α)1/2 where D(α) is a diagonal matrix amplifying the variance of the reference ensemble simulation.

For component 2 (the bias estimate), the prior distribution is also assumed Gaussian, with zero mean and a covariance again obtained from the reference ensemble simulation: 6x2b=0andP2b=P2r but this time, P2r does not only include the interannual time scales, but also the annual cycle, which contains structures that can be expected in the model bias. The resulting prior variance is thus quite large to let the bias adjust almost freely to the observations.

For component 3 (the spread correction estimate), we assume a lognormal distribution, which is obtained as the exponential of Gaussian variables, with zero mean (x3b=0) and an analytically specified covariance P3b. In P3b, the variance is set to be quite large, to let the spread of the distributions be expanded or reduced almost freely to fit the observations, and the correlation structure is specified by a Gaussian function of the distance: γ3(r)=exp⁡(-r2/l32), where l3=200 km. We need this length scale to be small enough to let α and β adjust to the observations, and large enough to avoid having too many degrees of freedom in the estimation vector (which would not be controllable by the observations).

In addition, since the covariance matrices of components 1 and 2 are provided by a small size ensemble (the 30 members of the ensemble simulation described in Sect. ), they cannot be expected to represent adequately the correlation structure over the full Mediterranean basin and over the full time period (30 years). To cope with this difficulty, the classic approach is to localize the ensemble covariance by a Schur product with a correlation matrix Γloc: 7Pib,loc=Pib∘Γloc where the correlation structure in Γloc is here specified by a Gaussian function of the distance r and time difference Δt: 8γloc(r,Δt)=exp⁡-r2/lloc2×exp⁡-Δt2/τloc2 with lloc≃100 km and τloc≃6 months (tuned for our experiments). In this way, only the local ensemble correlation structure is used in the estimation problem, while the long-range ensemble correlations are dropped.

Observation error is assumed Gaussian, with zero mean (unbiased observations) and covariance R. It is described by the probability distribution: 9p(y|x)=1(2π)p|R|exp⁡-12y-Hx1TR-1y-Hx1 where H is the observation operator, which is here only a linear interpolation from the component x1 of the estimation vector (the only component that is directly observed) to obtain the observation equivalent (with dimension p). R is assumed diagonal and it is computed as the amplification of a reference error variance Rr by the factor β: 10R=D1/2(β)RrD1/2(β) where D(β) is a diagonal matrix amplifying the variance Rr obtained from the reference ensemble simulation. It is here included in the estimation vector to adjust for a possible bias in the estimation of Rr.

The observation cost function is then: 11Jo(x)=-log⁡pyo|x=12yo-Hx1TD1/2(β)RrD1/2(β)-1yo-Hx1+12∑j=1plog⁡βj+Cst where the last two terms result from the normalizing factor in Eq. (), and βj is the value of β corresponding to each observation [by interpolation of β(x,y)]. Since all matrices are diagonal, the computation of Jo can be written as a simple sum over all observations: 12Jo=12∑j=1pyjo-Hjx12βjRjr+log⁡βj which is straightforward to evaluate. The estimation of β thus results from a compromise between the two terms of Eq. (), the first one decreasing with β and the second one increasing with β, and it is coupled to the estimation of the other components through x1.

In the above problem, none of the distributions pb(x), p(y|x) or pa(x) are Gaussian, or even approximately Gaussian, so that linear estimation methods are not directly applicable. In the following, we will show that Monte Carlo Markov Chains (MCMC) iterative sampling methods can be made efficient enough to solve this problem of large dimension.

The method is based on the Metropolis/Hastings algorithm see e.g., which generates a sample of a target probability distribution [in our case, pa(x)] by constructing a stochastic process (a Markov chain) that is in equilibrium with this target distribution. In other words, this means that we construct a stochastic process whose “attractor” is characterized by the distribution pa(x) that we want to sample. In this algorithm, the next iterate of the Markov chain xK+1 is computed from the current iterate xK by sampling a perturbation of xK from a simple proposal distribution (usually Gaussian) and then accepting or rejecting the perturbation according to the variation of the cost function. Perturbations decreasing the cost function are always accepted, while perturbations increasing the cost function are only sometimes accepted to ensure the equilibrium condition (see Appendix  for more details).

This algorithm is quite general and usually efficient enough as long as the number of dimensions is not too large (typically a few hundreds) because the cost of sampling a large-dimension proposal distribution, even if Gaussian, can be substantial. For instance, a classic choice for the proposal distribution would be to compute the proposal perturbation as a linear combination of EOFs (Empirical Orthogonal Functions), with random Gaussian coefficients, as done for instance in or . However, if the dimension of the state vector (n) is large and if there are many relevant directions of perturbations (m), the cost of this linear combination (n×m operations) becomes quickly prohibitive (up to n2 for a perturbation with full rank covariance).

More specifically, in our practical example, even if we have only 30 members to define the prior probability distribution (which would make an acceptable value of m), we need to localize the prior covariance according to Eq. (). This makes the prior covariance a full-rank matrix, and thus demultiply the number of possible directions of perturbations. Prior uncertainty really occurs in a large dimension space, which needs to be explored. This is what makes the computation of the random perturbation by a linear combination prohibitively expensive and impracticable for our large-size problem.

To cope with this difficulty, we can actually take advantage of the localization of the ensemble covariance by Eq. () using the method proposed and illustrated in and applied in to ocean colour observations. In this method, the random perturbation of the state vector is not computed as a linear combination of basis vectors, but by a Schur product of several vectors randomly taken from a pre-computed set of vectors. This Schur product is designed in such a way that the covariance of the random perturbation is given by Eq. ().

Suppose for example that we have generated a sample of vectors x̃kloc, k=1, …, m with covariance Γ̃loc, where Γ̃loc is defined so that Γloc is the 6th Schur-power of Γ̃loc. In practice, to be consistent with Eq. (), this simply means that the correlation function corresponding to Γ̃loc must be: 13γ̃loc(r)=exp⁡-r2/6lloc2×exp⁡-Δt2/6τloc2 i.e. with a correlation length scale and a correlation time scale that are multiplied by 6.

Then, we use the general property that the covariance of Schur product is the Schur product of the covariance: if we take 6 different vectors from the x̃kloc, the covariance of their Schur product is the 6th Schur-power of Γ̃loc, and thus Γloc. If we further make the Schur product of this vector by a vector obtained from the prior ensemble (with covariance Pib), we obtain a pseudo-random perturbation vector with the localized covariance of Eq. (), at the cost of just 6 Schur products (i.e. 6n operations). This last Schur product can be viewed as the modulation of one member of the ensemble simulation (more precisely the anomaly with respect to the ensemble mean) by one large-scale pattern with covariance Γloc, as more extensively explained in for the MCMC sampler, or in in the framework of a localized ensemble Kalman filter.

The whole benefit comes from the fact that the sample x̃kloc, k=1, …, m, with covariance Γ̃loc needs to be computed only once. Then, the diversity of the possible directions of perturbations will only result from the combinatorial effect of the Schur products. For instance, with m=30 and 6 Schur products, the number of different directions of perturbation is: N=30×29×28×27×26×25×24≃1.026×1010, which is sufficient to span all relevant degrees of freedom of our system. Actually, the effective number of degrees of freedom (d) depends on the ratio between the size of the domain (in space and time) and the localization scales, and it is very tiny as compared to N. The idea behind this large value of N is to produce a quasi-random sample of this target space of dimension d, and thus ideally among the largest possible number of directions of perturbation (which has here to be discrete because of the finite samples). See for more details. It must also be mentioned that the size of the localizing ensemble could also be larger than the size of the original ensemble, which would also increase N. We used here the same size for simplicity and to limit the memory requirements.

A full technical description of the algorithm used in this paper is provided in the Appendix, where a description of the computational complexity of the algorithm is also included. It is shown that the method can be very easily implemented to be efficient on GPUs, and that we were able to obtain a gain of about 100 as compared to CPUs in the computation of the solution of our problem.

The first experiment (SSH) includes all available along-track altimetric observations. This experiment has been repeated twice (SSH-A and SSH-B) using the subsets of altimetric observations (A and B) described in Table . The characteristics of the experiments are given in Table .

As compared to the method described in Sect. , one simplification has been introduced in the practical implementation: we preferred to rescale the prior standard deviation to a constant value (set to a typical value of 4 cm) rather than using the model ensemble standard deviation, because it is much too small in several regions (especially along the coast). This constant standard deviation is then adaptively rescaled by the free factor α (in the estimation vector). This simplification is here needed because it turned out that, in this problem, the estimation of α from the observations is not easy (see Sect. ).

The cost of the experiments provided in Table  suggests that each member of the posterior ensemble can be obtained quite quickly and with moderate resources. It must be noted however that memory can also be a limitation in the size of the problem that can be afforded, because the whole prior ensemble must be in memory to keep the computation efficient. Since each GPU will load its share of the problem, a sufficient number of GPUs, each with enough memory, may thus be needed to solve a large-size problem.

As a first illustration of the solution of the problem, Fig.  shows the equivalent of Fig.  for the posterior ensemble of the SSH experiment, i.e. after the altimetric observation constraint has been applied. As an additional third line of panels, the figure also shows the corresponding low-frequency component of SSH obtained from the Aviso mapped (L4) dataset (but without a description of the associated uncertainty, see explanation at the end of this section). To obtain a comparable result, the same time filter as the one applied to the model data has been applied to the original Aviso maps to extract the low-frequency interannual signal only. From Fig.  (top panels), we can see that the posterior ensemble mean is substantially different from the ensemble mean of the model simulation (Fig. , top panels). A bias has been removed (see Sect. ), and the detailed structure has also been reshaped by the observations. The ensemble mean has become very similar to the Aviso counterpart, despite being obtained very differently. In Aviso, all observable time scales are mapped first and the time filter is applied afterwards, while we here directly estimate the low-frequency component from the observations. We can thus conclude that the two approaches made a rather consistent use of the same along-track observations.

Regarding the spread of the ensemble (second line of panels in Figs.  and ), we can see that it has been strongly reduced by the observation constraint. In 2017 (right panel), it is smaller than 1 cm in most of the Mediterranean (except in the Alboran Sea). At that date, there are indeed quite many altimetric satellites flying simultaneously, which provide enough data to mitigate uncertainty. The same figure for 1997 (left panel) displays more regions with higher values (up to 1.5 cm), especially in the gaps between the Topex/Poseidon ground tracks.

To illustrate the time variation of this residual uncertainty, Fig.  shows time series of a subset of 10 ensemble members for the prior ensemble (in black) and for the posterior ensemble (in blue). The ensemble means are represented as thicker lines, and the Aviso counterpart is in purple. This is shown at two close locations in the Ionian Sea: one at the points of high density in Topex/Poseidon and ERS ground tracks (18.5° E, 35° N), and the other one (17.1° E, 35° N) in a gap where the uncertainty is large in the early years of the altimetric measurements. In this figure, we can see the correction of the bias and the reduction of the spread in the updated ensemble as compared to the model ensemble. The updated ensemble also becomes more consistent with the Aviso time series, even if, in such local time series, there can be differences, which are mostly due to the smallest scales. At the Topex/Poseidon crossover location, the spread of the ensemble stays small over the full period, while at the gap location, the spread is larger at the beginning of the period and decreases with time as more satellite observations become available.

It must also be noted that, by producing an ensemble solution, we keep track of the residual uncertainty occuring at all scales (and, more generally, on any derived quantity). For instance, if one is interested in larger spatial scales, it is still possible to apply a smoothing operator on the ensemble members and see how the ensemble spread, and thus uncertainty, is decreased. Actually, in our solution, much of the spread displayed in Figs.  and  (and the misfit with Aviso) is due to the smallest scales being imperfectly observed. By contrast, in the current version of the Avsio mapped product, the residual uncertainty is not described by an ensemble, but by a map of the error standard deviation. From this information only, it is impossible to deduce the uncertainty on derived quantities, like the low-frequency component of the signal, which is why it could not be displayed in our figures.

In the MCMC method, the ensemble model simulation provides the local space and time correlation structure of the succesive random perturbations that are summed up to obtain every member of the updated ensemble. It is thus playing a central role in filling the gaps between observations, especially if they are sparse (in space and time), and in shaping the spread of the updated ensemble. It can vary in space and time according to the model dynamical behaviour. Length scales and time scales are not always and everywhere the same. Anisotropy can be forced by the presence of dynamical structures, like currents, or by topography, in particular along the coastline. Figure  illustrates the prior correlation structure for sea surface height with respect to two specific locations in the Mediterranean (9.65° E 37.45° N and 12.43° E 38.00° N) in January 2006. This corresponds to the model correlation structure, as obtained specifically for the low-frequency component of the signal, and with localization, using Eq. ().

In this figure, we can observe that the field decorrelates across strong currents or across coastlines, and the importance of localization to drop unreliable long-range correlations. It can also be asymmetric in time, which means that depending on where and when we are, future and past observations can have a different weight in the solution. Our assumption in this paper is that, at least for the low-frequency component of the signal, this information is best obtained by an interannual ensemble model simulation spanning the same period as the observations, with the perspective that it can be progressively improved, following future model enhancements.

A classic approach to assess the reliability of the ensemble solution is to build a rank histogram using independent observations . In our case, we can check the ensemble from experiment SSH-A using observations from subset B, and we can check the ensemble from experiment SSH-B using observations from subset A (see the definition of the subsets in Table ). To make the comparison with observations, we need to compute the ensemble equivalent to each individual observation, perturb the ensemble members by a random noise with the statistics of the observation error, and compute the rank of the observation in the resulting ensemble of values (sorted in ascending order). If the independent observation is consistent with the ensemble solution, then the probability distribution of the rank must be uniform. A necessary condition of reliability is thus that the rank histogram is flat.

Figure  (top panels) shows the rank histogram obtained for SSH-A and SSH-B. As compared to the observations, we can see that the ensemble solution is globally unbiased and only slightly overdispersive. The two bumps in the external regions of the histogram indicates that the detailed shape of the probability distribution is not perfectly consistent with the observations, with a bit too much probability both in the central regions and towards the extreme values. It must be noted however that what is checked here is the sum of posterior uncertainty and observation error, and that the amplitude of observation errors was also part of the estimation problem. It is thus for instance possible to obtain a flat histogram with underestimated posterior uncertainty and overestimated observation error. These two probability distributions cannot be verified separately using independent observations only.

To further check the consistency of the ensemble solution, we can compare experiment SSH (using all observations) to experiment SSH-A and SSH-B. Experiment SSH can indeed be viewed as the posterior ensemble obtained by conditioning the ensemble SSH-A to the observations of subset B or the ensemble SSH-B to observations of subset A. It should thus be consistent with the two of them. Figure  (bottom panels) shows the rank histogram of members of the ensemble SSH (at all grid points) in the ensemble obtained with experiment SSH-A (left panel) and experiment SSH-B (right panel). The result is averaged over members so that the numbers correspond to number of grid points in one member. What we can see is that the ensembles SSH-A and SSH-B are slightly underdispersive to be consistent with the ensemble SSH, which suggests that the spread (i.e. the posterior uncertainty) is presumably a bit too small in all posterior ensembles. This can directly result from oversimplified statistical assumptions, which may give too much confidence in the observations, for instance by assuming uncorrelated observation errors.

The first byproduct of the estimation procedure is an ensemble estimate of the bias in the prior ensemble simulation, as here directly obtained from the along-track altimetric observations. Figure  shows the ensemble mean (left panel) and standard deviation (right panel) of this bias estimate. In the ensemble mean, the most salient features are a too weak cyclonic circulation in the Balearic Sea and the South-Western Algerian basin (yellow anomaly), and too weak Iera-Petra and Pelops gyres in the Eastern basin (blue anomaly), as already mentioned in .

On the other hand, from the ensemble standard deviation, we can see that the uncertainty on the bias is quite small over the whole domain. This means that all Markov chains have converged towards a very similar map of the bias parameter to ensure best compatibility with observations (despite a zero-mean prior distribution for the bias, and with a large associated uncertainty). The largest residual uncertainty on the bias (Fig. , right panel) is along the coast (particularly in the Adriatic and Aegean seas), where the observation constraint is weaker.

Our experiments also embed an adaptive estimate of the prior ensemble standard deviation (through the correcting factor α in component x3). Figure  (top left panel) shows that the best estimate obtained with the MCMC procedure is always between 3 and 5 cm (i.e. quite close to the prior value of 4 cm). More precisely, what the figure shows is the corrected value of the standard deviation by an estimated factor computed as the exponential of the ensemble mean of the log of α (since the factor α was assumed lognormal).

On the other hand, the top right panel of Fig.  shows the standard deviation of the log of α, which gives an idea of the relative error on the estimate, thus here typically between 5 % and 30 %. This could look like an acceptable estimation, but actually, from this figure and from the variety of patterns displayed by the ensemble members, we must conclude that we were not able to provide an accurate adaptive estimate of the prior variance with this method. The various Markov chains just spread from their background value, without converging towards a common pattern, as it happens when a parameter is not controllable by the observations. This is why we had to start from a constant prior (4 cm) and use a moderate prior standard deviation for α (30 %) to keep the prior distribution within reasonable bounds.

As a more direct diagnostic of what the standard deviation of the prior ensemble should have been to be compatible with the observations, we can compute the standard deviation of the misfit between the prior and posterior ensembles (after the bias correction has been applied). The result is shown in Fig.  (bottom left panel), which confirms that the estimate obtained from the observations as a byproduct of the MCMC sampler (top left panel) is not reliable. The standard deviation of this misfit between prior and posterior ensembles (bottom left panel) can be compared to the actual spread of the prior ensemble displayed in Fig.  (bottom panels). The ratio between these two maps (i.e. the ratio between the required spread and the actual spread) is also displayed in Fig.  (bottom right panel). It is a kind of time average that is computed as the square root of the ratio between the time-averaged variances. This ratio informs us about the amount of missing uncertainties in the ensemble simulation to be compatible with the real world, and can thus serve as a guideline to track the regions where variance is missing in the model world.

From the figure, we can see that the prior ensemble simulation has enough variance in the deep basins (the ratio is about 1) where the intrinsic variability can be viewed as dominant. At some places (with a ratio lower than 1), it is even too large to be compatible with the observations. However, in many other regions, especially along the coasts and in shallow basins, the model largely underestimates its own uncertainty (the ratio is much larger than 1). In this case, improving the model reliability would require simulating explicitly the missing sources of uncertainty in the model, such as for instance uncertainties in the atmospheric forcing, at the boundaries (Gibraltar Strait, runoffs), or the effect of the unresolved scales.

The prior probability distribution that we used for the observation error standard deviation (σ0) is a lognormal distribution with the mean and standard deviation both set to 5 cm. This means that we assumed a large prior uncertainty on the statistics of observation errors, which leaves much freedom to adjust it as part of the estimation vector (factor β in component x3). This was useful in our system because observation errors not only contains the measurement error, but also all unresolved scales. In our case, this includes all high-frequency scales up to the annual cycle, so that the tuning of σ0 would have required information about the residual signal (which we excluded from our estimation problem). One possibility is then to assume that it is almost unknown and adjust σ0 as part of the estimation process.

Figure  illustrates the resulting estimate of σ0. The left panel displays the ensemble mean (actually using the exponential of the ensemble mean of the log of β), which shows that the estimated standard deviation of the unresolved scales is larger in the regions of intense mesoscale activity, for instance where eddies detach from the Algerian current or at the location of the Iera-Petra and Pelops gyres. This is where the misfit between the observations and the low-frequency signal that we estimate is the largest.

On the other hand, the right panel of Fig.  shows that the relative error on this estimate is low everywhere (about 1 % in most of the domain, and up to 4 % along the coast), which means that all Markov chains converge to about the same map for σ0. Moreover, we can verify that σ0 automatically adjusts so that the value of the observation cost function (the first term of Eq. ) becomes close to p/2 (where p is the number of observations), which is the expected value that it must have when the system is optimal .