1. Introduction
A geophysical prediction is made by integrating the model in time from its initial condition. The quality of the forecast will rely on the quality of the initial condition, and the quality of the model, given by the implementation of the set of equations describing nature. Predictions evaluated beyond the deterministic limit of two weeks typical of numerical weather prediction are useful mainly when considering the statistical properties of the natural system over forecast ranges of several weeks, months, or perhaps years. Where the accuracy of numerical weather predictions are determined by error in the initial conditions, centennial climate projection evaluations are determined by boundary conditions such as atmospheric greenhouse gas concentrations, while the signature of the initial condition is lost (Hawkins and Sutton 2009). Seasonal-to-decadal prediction spans time horizons of up to approximately 10 years, falling between numerical weather prediction and centurial projections. Correct initialization of the model is known to improve forecast quality on horizons of several years (Doblas-Reyes et al. 2013). Such predictions require Earth system models that include coupled feedbacks between the atmosphere and the bio-, hydro-, and cryospheres across all time scales, and that incorporate greenhouse gases. The largest uncertainty on seasonal-to-decadal time scales is attributed to model error (Hawkins and Sutton 2009).
Models are affected by errors due to incorrect representation of dynamics, its projection onto a finite grid, and unresolved scales insufficiently represented through parameterizations. These errors affect the long-term statistical properties of the system (i.e., its climate). The result is a “model climate” different from the natural climate (e.g., in its variability and time mean). Initial conditions obtained directly from observations of the climate system often do not represent a feasible model state, so that forecasts drift toward the model’s own equilibrium. Drift is defined as the time evolution of the prediction error averaged over the initial conditions. Initialization shocks can be seen as specific cases of drift in which the imbalance of initial conditions leads to a rapid short-term readjustment of the system.
Full field initialization (FFI) makes use of the best possible available estimate of the real state. In this way FFI reduces the initial error, but the unavoidable presence of model deficiencies causes the model trajectory to drift away from the observations regardless of initial error size (e.g., Stockdale 1997). Anomaly initialization (AI) assimilates, in place of the observations, the observed climate anomalies on top of an estimate of the model mean climate. This initial state, at the expense of an initial error of the size of the model bias, is expected to be closer to the model’s own attractor (e.g., Smith et al. 2007), so that drift is reduced.
Comprehensive comparisons between FFI and AI using Earth system models for seasonal-to-multiyear time horizons have recently appeared (Magnusson et al. 2013; Smith et al. 2013; Hazeleger et al. 2013). These studies represent a first attempt to assess their respective performance using exactly the same observational and model setup and are therefore of central importance in guiding future development of initialized climate prediction systems. Results have been far from conclusive, varying regionally and among models. Carrassi et al. (2014) introduced a formalism based on data assimilation notation that helps identify universal error characteristics of either scheme, and illustrated the error features of FFI and AI using a low-order climate model.
Anomaly initialization has been devised by the climate prediction community in an ad hoc fashion to deal with problems related to drift (e.g., by allowing for more statistically robust bias correction techniques). Many scientists working in the community also dislike the idea of an unbalanced initial state, observing how the model readjusts toward equilibrium over the course of the prediction. The initial condition after AI is both close to the mean state of the model, and includes observational information. However, conclusive evidence associating AI with improved forecast quality is lacking.
In a pioneering study, Toth and Peña (2007) have introduced a conceptual framework that differentiates between initialization schemes according to two competing paradigms. The “fidelity” paradigm reduces the initial error of the system by starting the forecast as close to nature as possible. Its name derives itself from the goal of adhering to the truth. It represents the traditional approach of trusting the observational information as much as possible, and assumes that the model is perfect. The “mapping” paradigm recognizes that models are imperfect, and links corresponding states of nature on the model attractor by means of a mapping vector. A corresponding state on the model attractor imitates the natural state to the model’s best ability, without leaving the attractor of the model. A corresponding state can be interpreted as an “image” of nature on the imperfect model attractor. A mapping vector will have identified the images corresponding to their natural states, as far as they exist. Such images can represent “better” initial conditions for an imperfect model in terms of improved forecast skill (Toth and Peña 2007). Using this framework, we associate FFI and AI with practical realizations of the fidelity and mapping paradigms, respectively. The bias term added onto the observations in AI can be seen as a state-independent mapping vector. This interpretation allows for important inferences and becomes useful in understanding the merits and limitations of AI.
This study assesses the performance of AI as a function of its ability to conform to the mapping paradigm objective. We derive two simple indicators of the quality of a mapping scheme: the degree to which the initial conditions approximate the model attractor, and the sensitivity of the forecast skill toward random errors in the initial conditions. Using a hierarchy of low-order dynamical systems of increasing realism, the forecast quality of AI with respect to both indicators is assessed. We simulate model error through parameter misspecification, and evaluate, with respect to a target trajectory, initialized predictions using imperfect configurations. The degree of approximation of the model attractor is assessed by calculating the overlap of the AI initial conditions and model probability distribution function (PDF) using the Bhattacharyya coefficient. As a useful reference, we carry out the same analysis for FFI and compare the performance.
The paper is organized as follows. Section 2 formalizes AI and FFI. Section 3 connects AI to the mapping paradigm as introduced by Toth and Peña (2007). Section 4 describes the experimental setup, along with the hierarchy of low-order models. Results and the conclusions are given in sections 5 and 6, respectively.
2. Initialization methods
Note that (4) does not include a Kalman gainlike operator as is typical for data assimilation update equations (Kalnay 2002). This is because in standard FFI the observations are assimilated as if they were perfect (i.e., the model state is substituted by the observational values). FFI does not use a criteria of optimality, such as the least squares method at the basis of Kalman filtering (Kalman 1960) appearing in the form of a gain matrix. More sophisticated initialization procedures based on the data assimilation approach might account for the relative accuracy of model and observations and give weight to either accordingly (see Smith and Murphy 2007; Carrassi et al. 2014).
How the accuracy of the initial conditions will impact the prediction skill at seasonal-to-decadal time scales is unclear. Weather forecast practice with a horizon of two weeks often neglects model error and prioritizes the control of chaotic error growth (e.g., Palatella et al. 2013). In contrast, in seasonal-to-decadal prediction the bias caused by model deficiencies is more important. Forecasts initialized close to the observed state as in FFI will drift toward the model climate following a nonlinear state-dependent evolution, which can appear as an initial dynamical shock caused by the displacement of the model state onto the observed values lying outside the model attractor. At the expense of larger initial errors, the objective of AI is to keep the initial state close to the model attractor and reduce the drift. The mean forecast error is less dependent on lead time and, as argued by Magnusson et al. (2013), the use of standard a posteriori bias correction techniques is more robust. Anomaly initialization can reduce initialization shocks, but is unable to avoid shocks caused by inconsistencies (e.g., geographical mismatches) between the model climate and observed anomalies (Magnusson et al. 2013).
3. Anomaly initialization as a specific case of the mapping paradigm
a. Linking full field/anomaly initialization to the fidelity/mapping paradigms
We interpret FFI and AI using the general framework introduced by Toth and Peña (2007). They have proposed categorizing initialization schemes as belonging to the traditional fidelity paradigm or the mapping paradigm. The fidelity paradigm focuses on initializing the model as close to nature as possible. A premise of traditional thinking is that forecast errors are reduced by bringing the initial conditions closer to nature. The quality of the initial conditions is particularly important in chaotic systems like the atmosphere, because initial errors contribute to a loss of predictability. Given the degree of instability of the dynamics, the actual forecast time at which this loss of predictability occurs is determined by the size and structure of initial errors. The conventional paradigm thus consists of estimating the state of nature as precisely as possible (i.e., the analysis) and running numerical model forecasts from the analysis field. The systematic error in the numerical forecasts—which is state dependent—is assessed and removed in postprocessing.
We associate FFI with the fidelity paradigm because of its strict adherence to the observations. The analysis field is obtained according to (4). Systematic error is assessed by initializing many so-called hindcast experiments from past start dates and comparing such hindcasts with available reanalysis fields. The systematic error is removed a posteriori according to both the forecast start date within the year and the forecast horizon.
Toth and Peña (2007) recognize that the fidelity paradigm is justified only in the special case when perfect models are used for prediction, becoming progressively worse as models deviate from nature in practical applications. States obtained via the fidelity paradigm lead to a transient drift as they are not on the attractor of an imperfect model, which would not occur had an appropriate initial state on (or near) the model attractor been chosen.
The mapping paradigm explicitly recognizes the systematic difference between corresponding states of nature and a numerical model of it. It “maps” an initial state from nature close to the attractor of a similar system, which is a numerical model of nature. Mapping is thus defined as an operation linking each natural state to a state near the model attractor; such a corresponding state on the model attractor can be interpreted as an image of nature. Initialization using the mapped state (or image) leads to a model forecast that best reproduces the time evolution of nature. Before evaluation the forecast is remapped back to the nature attractor. The emphasis is laid on a “shadowing” (Toth and Peña 2007) of the nature trajectory within the numerical model. For example, initializing the model close to the observations and outside the model attractor can lead to a quasi-systematic excitation of model climate dynamics (i.e., the drift is projected onto the main climate internal modes, which is dissociated from the actual nature evolution intended to be captured). This is observed in experiments with Earth system models in which initialization of the Pacific Ocean nudged to reanalysis leads to an artificial sequence of El Niño/La Niña events in the model during the first 4 years of integration (Sanchez-Gomez et al. 2015). In such a case, a mapped state would avoid an excitation of the model dynamics and remapped forecasts would reproduce (i.e., “shadow”) the timing and structure of natural anomalies better. The proximity of the mapped initial conditions to the model attractor reduces the drift and associated forecast errors.
The mapping paradigm starts with an estimation of the mapping vector 
We attribute AI, as defined in (6), to the mapping paradigm, where the model bias in (5) plays the role of the mapping vector. Thus, AI becomes a specific case of a mapping scheme in which the mapping vector is a constant (given by the estimated bias) applied to all initial conditions indiscriminately. The bias correction at forecast evaluation can be seen as a remapping. Anomaly initialization as defined in (6) is qualitatively equivalent to the climate mean mapping described by Toth and Peña (2007) if the model bias is calculated based on the entire set of observations. In an Earth System model, the bias term is calculated for each forecast month of the year and for each start date of the year, taking into account the seasonal cycle of the bias. Thus, in AI the mapping vector is dependent on the month in which the forecast is issued, retaining some dependence of state.
Attributing AI to the mapping paradigm should not suggest that predictions initialized with AI do not drift. Anomaly initialization is far from an ideal mapping scheme able to identify the image of nature on the model attractor, and is subject to transient dynamics for reasons described in the following section (section 3b).
Finally, a common practice in hindcast experiments causes the implemented AI scheme to defer from the strict mapping procedure: the fields are first conventionally analyzed and then mapped onto the model, reversing the order of analysis and mapping.
b. Anomaly initialization: A state-independent mapping scheme
In this study we focus on a core limitation of AI in pursuit of the mapping paradigm objective: the actual difference 
A perfect forecast is achievable when both the model and the initial conditions are perfect. A perfect forecast using FFI would thus necessitate a perfect model, full observational coverage, and no observational error. A perfect forecast using AI would impose a slightly weaker requirement on the model: under the central condition that all else remains identical, the model can differ from nature in its mean climate.
Figure 1 is a schematic of the linear mapping principle behind AI using the attractor of the three-variable Lorenz model (Lorenz 1963). The pink line depicts the phase space evolution of the (defined) “nature” state, the green line depicts the evolution of an imperfect model of the nature, the black crisscross represents a perfect “observation” 
Schematic of the linear mapping principle behind AI using the attractor of the Lorenz (1963) model. Depicted are the phase space evolution of the nature state (pink line), the evolution of an imperfect model of the nature (green line), a perfect observation (black crisscross), and initial condition after AI (red crisscross). The nature trajectory has been obtained through the addition of a constant vector 
Citation: Monthly Weather Review 143, 11; 10.1175/MWR-D-14-00398.1
In Fig. 1 the model integration shadows the nature evolution in a different region of phase space. At the desired forecast horizon, the bias 
Figure 1 is useful in that it resolves an apparent contradiction of both paradigms: although the initial condition after AI has a large initial error, AI still profits from a reduction of the observational error. Here, a perfect forecast using AI is achievable only for zero observational error. Thus, the seemingly opposing challenges of an initialization scheme between reduction of initial error and a mapping onto the model attractor through addition of an error term are, at least in this hypothetical case, not contradictory. The mapping paradigm seeks a good representation of the true state on the model attractor, so far as such a representation exists, which itself will profit from better knowledge of the true state. In the case of AI in reality, this effect is very much “blurred” by the fact that imperfect models differ in many more ways from nature than only their mean climate, so that AI has been shown to profit less from observational improvements (Carrassi et al. 2014). The justification of the application of AI will therefore depend on how well the bias explains the difference between the model and nature attractors.
c. Diagnostic tools of a mapping scheme
Let 
The map
If
The variable
From properties 1 and 2 we can derive corresponding diagnostic tools with regards to the applicability of a given mapping scheme. The overlap of the mapped states and the model PDF reproduces the degree of approximation of the model attractor (property 1). This is done here using the Bhattacharyya coefficient as detailed in section 4. Then, the sensitivity to observational error is a necessary condition for the mapped states to be good corresponding states of nature (property 2).
Finally, note also that the fidelity paradigm corresponds to 
4. Experimental setup
Experiments are conducted under the framework of an observing system simulation experiment (OSSE; Bengtsson et al. 1981) test bed in which a nature is represented by a trajectory, a solution of the model (1), and targeted by predictions using imperfect configurations (2). A model of the nature is simulated by introducing error in the (model) equations describing the nature.
The objective is to assess the forecast quality after initialization of the imperfect configuration (using FFI/AI) compared to noninitialized trajectories (i.e., the control run of the imperfect configuration). To maximize the effect of initialization the full system is observed, so that the linear observational operator from (4) is equal to the identity, 
Figure 2 is a schematic of a “hindcast” setup. The nature is obtained after a spinup run, and comprises an integration time of 
Schematic of the hindcast setup. The observations (red crisscrosses) are sampled each month from the nature (green line) and used to initialize the forecasts (blue lines). The model control run is shown (orange line) systematically below the nature, indicating a model bias.
Citation: Monthly Weather Review 143, 11; 10.1175/MWR-D-14-00398.1
The three models used in this study are briefly described in the following. Table 1 gives an overview of the models and the experimental setup.
The three OSSEs as well as the associated erroneous parameters, the number of configurations, the hindcast period, the number of start dates/forecasts (i.e., sample size), and the observational error. The prediction horizon is 
a. Lorenz (1963) model
As listed in Table 1: 
b. Peña and Kalnay (2004) model
We define the parametric values of the nature to be 
Model error is simulated by altering the model parameters with respect to the nature. We have generated two distinct sets of parametric error: in the forcing parameter r and in the coupling parameters c and 
c. Vannitsem and De Cruz (2014) model
Our third OSSE is based on the 24-variable coupled model by Vannitsem and De Cruz (2014, the model is hereafter referred to as VD14). [See also e.g., Vannitsem (2014).] In spite of the reduced number of variables, the model possesses a high degree of realism and represents the most challenging framework of our analysis. The atmospheric component is a two-layer quasigeostrophic flow defined on a beta plane, developed by Charney and Straus (1980) and extended by Reinhold and Pierrehumbert (1982). It consists of 20 variables. The ocean component is based on the reduced-gravity quasigeostrophic shallow-water model (Vallis 2006), and consists of four variables. The atmosphere and ocean are coupled through momentum transfer at the interface only. For full details see Vannitsem and De Cruz (2014).
Here again model error is simulated through mismatch in the coupling δ and in the thermal forcing parameter θ. Based on Vannitsem and De Cruz (2014), θ has the range of validity, [0, 0.2], while δ (normalized by the Coriolis parameter) [0.0001, 0.01]. For the nature trajectory, the parameters are chosen to be 
The 16 configurations with erroneous forcing/coupling parameters 
A second-order numerical scheme known as the Heun scheme [see Kalnay (2002)] with a time step of 0.01 time unit is used. The dimensional time unit is equal to 0.112 15 days. An initial spinup of 
5. Results
In most subsequent figures the forecast horizon has been chosen to be the first forecast month, over which the RMSSS in (7) is averaged. The RMSSS is obtained for each OSSE from a sample size given in Table 1. Using the models of this study, the differences in forecast quality between AI and FFI are most distinguished in the initial forecast stage; a larger initial skill corresponds to a later (e.g., “seasonal”) forecast time at which all skill is lost. This is not true in general; the expectation behind AI is to improve skill at the seasonal-to-decadal time horizon by reducing drift, perhaps at the expense of larger error in the initial forecast stage. This behavior is not expected in OSSE 1 because the L63-variable model is uncoupled; it is neither observed in OSSE 2, although the slow ocean compartment modulates the fast tropical atmosphere (Peña and Kalnay 2004). In OSSE 3 we consider longer forecast horizons, because anomaly initialized forecasts show decadal skill in the ocean compartment.
a. OSSE 1: Lorenz (1963) model
We begin by showing results from the Lorenz (1963) system given by (10) (nature) and (11) (model configurations). The sample size is 360, and the number of configurations is 20, see OSSE 1 in Table 1. Recall that this first system is the closest to the idealized schematic in Fig. 1 in that the model attractor is different from the nature attractor mainly in its first moment (i.e., its mean), which is controlled by the tuning parameter 
The top panels in Fig. 3 show the first month average RMSSS as a function of the bias for the x, y, and z variables, respectively. The biases have been normalized by the respective nature variance for better comparison among variables. The black (red) lines refer to FFI (AI). FFI skill in the z variable is proportional to the bias (which itself is proportional to 
RMSSS averaged over the first forecast (top) month and (bottom) BC of FFI (black) and AI (red) as a function of the bias for the x, y, and z variables of (11) (OSSE 1). The bias is normalized by the respective variance (of the nature) for better comparison among variables. Displayed are 20 points corresponding with configurations characterized by a constant offset 
Citation: Monthly Weather Review 143, 11; 10.1175/MWR-D-14-00398.1
The bottom panels in Fig. 3 display the BC as a function of the bias, and are otherwise analogous. The BC [(9)] measures the overlap of the FFI/AI initial conditions distributions and the model PDF for each variable separately. Each BC value corresponds to a single configuration (making a total of 20 points). The constant BC about the value of one in the x and y variables (which have negligible bias) for FFI indicates the high coincidence of the observations with the model PDF in these variables (the bias is mainly in the z variable). For the z variable the BC of FFI decreases monotonously with the magnitude of the bias corresponding with decreasing overlap of the initial conditions and model PDF. On the other hand, the BC for AI (red line) remains constant as a function of the bias magnitude in all three variables. Thus, the AI initial conditions lie well within the space confined by the model attractor according to property 1 of section 3c.
Figure 3 illustrates two main points: the forecast skill decreases monotonously the further the initial conditions lie outside the model attractor, as is the case with FFI in the presence of this type of model error. Anomaly initialization initial conditions approximate the model attractor irrespective of 
Figure 4 displays the RMSSS averaged over the first forecast month as a function of the observational error for all three variables of (11). The results of FFI (black) and AI (red) of three configurations with increasing model error 
RMSSS averaged over the first forecast month as a function of the observational error for all three variables of (11). Displayed are FFI (black) and AI (red) of three configurations with increasing model error 
Citation: Monthly Weather Review 143, 11; 10.1175/MWR-D-14-00398.1
b. OSSE 2: Peña and Kalnay (2004) model
In the following we show results from the second system (12) characterized by error in the forcing parameter 
Figure 5 displays the (normalized) distributions of the first-, second-, third-, and fourth-order moments (from top to bottom) of the model PDFs of 40 configurations characterized by an erroneous forcing parameter 
(from top to bottom) Distributions of the first-, second-, third-, and fourth-order moments of the model PDF given for each variable of the tropical atmosphere (OSSE 2) for configurations characterized by an erroneous forcing parameter 
Citation: Monthly Weather Review 143, 11; 10.1175/MWR-D-14-00398.1
In analogy to Fig. 3, Fig. 6 shows results obtained from the tropical atmosphere of OSSE 2a in (12). The RMSSS (top) and BC (bottom) are given as a function of the bias of each variable. The biases have been normalized by the respective nature variance for better comparison among variables. Figure 6 shows 40 points corresponding to configurations characterized by erroneous forcing parameters 
RMSSS averaged over the first forecast (top) month and (bottom) BC of FFI (black) and AI (red) as a function of the bias of each variable (tropical atmosphere, OSSE 2a). The bias is normalized by the respective variance (of the nature) for better comparison among variables. Displayed are 40 points corresponding with configurations characterized by an erroneous forcing parameter 
Citation: Monthly Weather Review 143, 11; 10.1175/MWR-D-14-00398.1
The BC reveals a relationship similar to Fig. 3 with respect to all three variables. Here again, the BC related to AI is about the value of one for all three variables; the approximation of AI initial conditions toward the model attractor is successful, and results in better forecast quality. We can thus observe comparable forecast skill results (cf. Fig. 3) in the face of comparable changes in the imperfect model statistics, regardless of the origin of such a difference (i.e., parametric error 
OSSE 2b in Table 1 introduces error in the coupling parameters 
In analogy to Fig. 6, Fig. 7 shows results obtained from the tropical atmosphere of OSSE 2b in (12). Displayed are 109 points corresponding to configurations characterized by erroneous coupling parameters 
RMSSS averaged over the first forecast (top) month and (bottom) BC of FFI (black) and AI (red) given as a function of the bias of each variable (tropical atmosphere, OSSE 2b). The bias is normalized by the respective variance (of the nature) for better comparison among variables. Displayed are 109 points corresponding with configurations characterized by erroneous coupling parameters 
Citation: Monthly Weather Review 143, 11; 10.1175/MWR-D-14-00398.1
Comparing Figs. 6 and 7, we observe that AI performs badly when property 1 does not hold (i.e., when AI does not approximate the model attractor successfully). The statistical differences between the nature and model PDFs can apply to the moments of any order, a description based only on the bias (as implicitly assumed by AI) is satisfactory for OSSE 2a and unsatisfactory for OSSE 2b. In the latter case the fixed error term added onto the observational values (
Figure 8 illustrates why for some models an approximation of the attractor using AI is successful and for others it is not. The figure shows the PDFs of the FFI (black) and AI (red) initial conditions, as well as the PDFs of the model (blue) and the nature (green). The panels display the 
PDFs of FFI (black) and AI (red) initial conditions, and PDFs of the model (blue) and nature (green) attractors. (top) The 
Citation: Monthly Weather Review 143, 11; 10.1175/MWR-D-14-00398.1
The reason for the varying approximation of AI initial conditions toward the model attractor lies in a comparison of the nature and model PDFs: we observe that they differ from each other in their higher (than first) order moments, so that the first-order moment correction constituted by AI delivers a bad approximation. It is often overlooked in the implementation of AI that, being a linear correction of the observations with the goal of approximating the model attractor, the assumption of negligible higher-order differences between model and nature attractors must apply. This can be further understood in the light of the ideal experiment described in the introduction, in which we stated that AI imposes a weaker requirement on the model in that it can differ from nature in its mean climate under the condition that all else remains identical.
Figure 9 shows the RMSSS averaged over the first month of FFI (black) and AI (red) as a function of the BC. The panels correspond to the nine variables of (12), and the 109 points correspond to configurations defined by erroneous coupling parameters 
RMSSS averaged over the first forecast month of FFI (black) and AI (red) as a function of the BC. The panels correspond to the nine variables of (12) (OSSE 2b), and the 109 points correspond to configurations defined by erroneous coupling parameters 
Citation: Monthly Weather Review 143, 11; 10.1175/MWR-D-14-00398.1
Figure 10 displays, in analogy to Fig. 4, the RMSSS averaged over the first month as a function of the observational error for the 
RMSSS averaged over the first forecast month as a function of the observational error for the variables of the tropical atmosphere. Displayed are FFI (black) and AI (red) of two configurations corresponding to 
Citation: Monthly Weather Review 143, 11; 10.1175/MWR-D-14-00398.1
Figure 11 shows the drift after FFI of the 
Mean error over the initial conditions (normalized by the variable’s nature variance for comparison) of FFI (black) and AI (red) as a function of lead time in years for the 
Citation: Monthly Weather Review 143, 11; 10.1175/MWR-D-14-00398.1
Sanchez-Gomez et al. (2015) observe—in the Pacific region of an Earth system model after initialization of the ocean nudged toward reanalysis—a quasi-systematic excitation of ENSO warm events for all starting dates, followed by weak cold ENSO events in the second forecast year with spurious oscillatory behavior being progressively damped. The features of the drift in OSSE 2a seen in Fig. 11 appear to be in qualitative agreement with the behavior observed in Earth system models.
For the 
c. OSSE 3: Vannitsem and De Cruz (2014) model
Figure 12 is analogous to Fig. 9, but shows results from OSSE 3 in Table 1, which has 16 configurations and a sample size of 200. The first 20 panels (from top left) correspond to atmospheric variables, the last 4 correspond to ocean variables. The 16 points in each panel correspond to erroneous configurations as given in Table 2. A polynomial is fitted to the data using least squares, and the error bars are estimates of the standard deviation of the error in predicting the skill at a given BC value by the polynomial. FFI skill (black) clearly correlates with the BC, showing that the skill improves as the model improves. In the atmospheric variables AI skill (red) is consistently worse in comparison, which can be explained as in Fig. 7 by the fact that the BCs are similarly distributed to those of FFI, indicating that the approximation is unsuccessful. In many variables AI skill increases with increasing BC; in others, the linear regressions show opposite signs. However, the reader can compare with Fig. 9 to notice that for BCs close to one, the linear dependence is less clear and the distributions of points are wider. This is true for Fig. 12 as well; many configurations are close to the nature (seen in large BC values associated with FFI), so that a mapping scheme is less effective (with variable effects on skill). In the ocean a preference among schemes cannot be inferred, which will be looked at in the following.
RMSSS averaged over the first forecast month as a function of the BC. (from top left) The first 20 panels correspond to atmospheric variables and the last 4 correspond to ocean variables. Displayed are FFI (black) and AI (red), and the 16 points correspond to erroneous configurations of OSSE 3. A polynomial is fitted to the data using least squares, and the error bars are estimates of the standard deviation of the error in predicting the skill at a given BC value by the polynomial.
Citation: Monthly Weather Review 143, 11; 10.1175/MWR-D-14-00398.1
The top panels of Fig. 13 display the RMSSS of FFI (black) and AI (red) as a function of the lead time in months for the four ocean variables of a particular configuration corresponding to 
(top) RMSSS of FFI (black) and AI (red) as a function of the lead time in months for the four ocean variables of a configuration corresponding to 
Citation: Monthly Weather Review 143, 11; 10.1175/MWR-D-14-00398.1
In analogy to Figs. 4 and 10, in the bottom panels the RMSSS of FFI (black) and AI (red) as a function of the observational error for the same variables and configuration is shown. We can see that AI shows similar sensitivity to observational error as FFI for all four variables, explaining AI’s competitive performance. In the atmospheric variables, the sensitivity to observational error of AI is consistently worse than FFI (not shown), agreeing with the result that AI performs consistently worse than FFI in the atmosphere.
6. Conclusions
This study proposes to associate FFI to the fidelity paradigm and AI to an instance of the mapping paradigm upon which we focus. Anomaly initialization is interpreted as a state-independent mapping scheme, which intends to initialize the model on an image of nature on the model attractor. The observational information is used in order to seek out a model state—the image—corresponding to the nature state. Ideally, forecasts will no longer drift because they are initialized on feasible model states. The motivation behind this work is to move away from an ad hoc implementation of AI toward an approach based on considerations of the model at hand. To the best of our knowledge, this study is the first of its kind to research the circumstances under which AI initial conditions approximate the model attractor (which is often implicitly assumed), as well as the extent to which this characteristic might improve its forecast skill.
We have diagnosed AI according to two properties: how well the initial conditions approximate the model attractor, measured by calculating the overlap of the initial conditions PDF and the model PDF using the BC in (9) for each model variable; and how well the selected model state is a reflection of the nature state, indicated by the sensitivity of forecast skill to random (observational) error. The mapping hypothesis has been tested using a hierarchy of low-order models (see Table 1) of increasing complexity: L63, PK04, and VD14. Our results are based on the remapped fields given after a bias correction.
In OSSE 1, the tuning parameter 
In OSSE 2a, characterized by error in the forcing parameter 
In OSSE 2b, characterized by errors in the coupling parameters 
Finally, OSSE 3 shows a similar distribution of BC values of both AI and FFI in the atmospheric variables. Anomaly initialization performs worse than FFI, and has a larger spread in skill. Again, 
We have thus shown that the mapping paradigm is a useful concept that unifies complementary schemes such as AI and FFI into a single framework. The apparent contradiction between minimizing the initial error in FFI and adding an initial fixed error term onto the initial conditions in AI is solved when considering whether the model is close to perfect (FFI) or if the model PDF differs from the PDF of nature mostly in its first-order moment but is otherwise similar to nature (AI). The model PDF is a valuable source of information about the model error that can be estimated. The assumption that AI initial conditions approximate the model attractor must be applied cautiously under evaluation of the model and reanalysis statistics. Table 3 summarizes the prevailing schemes associated with each OSSE.
The prevailing initialization scheme corresponding to each OSSE.
It is clear that a successful estimation of the mapping vector is presently feasible only in highly idealized systems. Yet, only by being clear about the desired goal of initialization can we seek to improve it. Diagnosis tools such as those derived from the mapping framework in this study can a priori help assess situations in which implementing either FFI or AI is preferable. Using information from the control run of an Earth system model, we can in principle always analyze the differences in the statistics between the model and the reanalysis—avoiding many experiments—opting for either AI or FFI. In practice, measuring the approximation of the model attractor based on BC values and/or the skill sensitivity to random error of each model variable is likely to be a daunting task. The choice of either AI or FFI is a dichotomous question that would require analyzing a number of BC values as large as the number of model variables. Mixed initialization with AI/FFI simplifies such an analysis, although is likely to cause inconsistencies in the initial conditions. A more promising approach could rely on the evaluation of key (independent) variables or regions of the model.
We hope that this work can stimulate research toward initialization schemes of the mapping paradigm, which take into account the state dependence of the system. One logical continuation would be to use higher (than first) order moments in the design of the mapping vector. Another will require devising practical implementation strategies for evaluating model and reanalysis PDFs that can deal with the complexity of an Earth system model and/or the development of further properties of a successful mapping scheme along the line of properties 1 and 2. Investigating the emergence of skill at seasonal-to-decadal time horizons will be key, which we have only analyzed briefly (in anomaly initialized forecasts of the ocean compartment of VD14). The use of data assimilation algorithms that are now under consideration in the initialization of seasonal-to-decadal prediction is also expected to substantially improve initialization. This is an active field of research reflected by some recent studies (Counillon et al. 2014; Tardif et al. 2014) and is seen as a main area of development with several research initiatives worldwide.
Acknowledgments
A. Carrassi was financed through the IEF Marie Curie Project INCLIDA of the FP7. This work was supported by the EU-FP7 projects SANGOMA and SPECS under Grants 283580 and 308378, respectively.
APPENDIX
Significance Test
We have tested the significance of the difference in skill between AI and FFI in Fig. 13 by performing a one-way ANOVA test, which tests the hypothesis that the samples are drawn from populations with the same mean against the alternative hypothesis that the population means are not all the same. The populations were given by the distributions of the squared errors of either scheme at a defined forecast horizon. The hypothesis was rejected when the p value was above 0.05, meaning in such cases that the mean squared error of either scheme is not significantly different.
REFERENCES
Bengtsson, L., M. Ghil, and E. Källen, Eds., 1981: Dynamic Meteorology: Data Assimilation Methods. Springer Verlag, 330 pp.
Bhattacharyya, A., 1943: On a measure of divergence between two statistical populations defined by their probability distribution. Bull. Calcutta Math. Soc., 35, 99–110.
Carrassi, A., R. Weber, V. Guemas, F. Doblas-Reyes, M. Asif, and D. Volpi, 2014: Full-field and anomaly initialization using a low-order climate model: A comparison and proposals for advanced formulations. Nonlinear Processes Geophys., 21, 521–537, doi:10.5194/npg-21-521-2014.
Charney, J. G., and D. M. Straus, 1980: Form-drag instability, multiple equilibria, and propagating planetary waves in baroclinic, orographically forced, planetary wave systems. J. Atmos. Sci., 37, 1157–1176, doi:10.1175/1520-0469(1980)037<1157:FDIMEA>2.0.CO;2.
Counillon, F., I. Bethke, N. Keenlyside, M. Bentsen, L. Bertino, and F. Zheng, 2014: Seasonal-to-decadal predictions with the ensemble Kalman filter and the Norwegian Earth System Model: A twin experiment. Tellus, 66A, 21074, doi:10.3402/tellusa.v66.21074.
Doblas-Reyes, F., J. García-Serrano, F. Lienert, A. P. Biescas, and L. Rodrigues, 2013: Seasonal climate predictability and forecasting: Status and prospects. Wiley Interdiscip. Rev.: Climate Change, 4, 245–268, doi:10.1002/wcc.217.
Hawkins, E., and R. Sutton, 2009: The potential to narrow uncertainty in regional climate predictions. Bull. Amer. Meteor. Soc., 90, 1095–1107, doi:10.1175/2009BAMS2607.1.
Hazeleger, W., V. Guemas, B. Wouters, S. Corti, I. Andreu-Burillo, F. Doblas-Reyes, K. Wyser, and M. Caian, 2013: Multiyear climate predictions using two initialization strategies. Geophys. Res. Lett., 40, 1794–1798, doi:10.1002/grl.50355.
Janjić, T., and S. Cohn, 2006: Treatment of observation error due to unresolved scales in atmospheric data assimilation. Mon. Wea. Rev., 134, 2900–2915, doi:10.1175/MWR3229.1.
Kalman, R. E., 1960: A new approach to linear filtering and prediction problems. J. Fluids Eng., 82, 35–45.
Kalnay, E., 2002: Atmospheric Modeling, Data Assimilation and Predictability. Cambridge University Press, 364 pp.
Lorenz, E. N., 1963: Deterministic nonperiodic flow. J. Atmos. Sci., 20, 130–141, doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.
Magnusson, L., M. Alonso-Balmaseda, S. Corti, F. Molteni, and T. Stockdale, 2013: Evaluation of forecast strategies for seasonal and decadal forecasts in presence of systematic model errors. Climate Dyn., 41, 2393–2409, doi:10.1007/s00382-012-1599-2.
Palatella, L., A. Carrassi, and A. Trevisan, 2013: Lyapunov vectors and assimilation in the unstable subspace: Theory and applications. J. Phys. A: Math. Theor., 46, 254020, doi:10.1088/1751-8113/46/25/254020.
Peña, M., and E. Kalnay, 2004: Separating fast and slow modes in coupled chaotic systems. Nonlinear Processes Geophys., 11, 319–327, doi:10.5194/npg-11-319-2004.
Reinhold, B. B., and R. T. Pierrehumbert, 1982: Dynamics of weather regimes: Quasi-stationary waves and blocking. Mon. Wea. Rev., 110, 1105–1145, doi:10.1175/1520-0493(1982)110<1105:DOWRQS>2.0.CO;2.
Sanchez-Gomez, E., C. Cassou, Y. Ruprich-Robert, E. Fernandez, and L. Terray, 2015: Drift dynamics in a coupled model initialized for decadal forecasts. Climate Dyn., doi:10.1007/s00382-015-2678-y, in press.
Smith, D., and J. Murphy, 2007: An objective ocean temperature and salinity analysis using covariances from a global model. J. Geophys. Res., 112, C02022, doi:10.1029/2005JC003172.
Smith, D., A. Cusack, A. Colman, C. Folland, G. Harris, and J. Murphy, 2007: Improved surface temperature prediction for the coming decade from a global climate model. Science, 317, 796–799, doi:10.1126/science.1139540.
Smith, D., R. Eade, and H. Pohlmann, 2013: A comparison of fullfield and anomaly initialization for seasonal to decadal climate prediction. Climate Dyn., 41, 3325–3338, doi:10.1007/s00382-013-1683-2.
Stockdale, T., 1997: Coupled ocean–atmosphere forecasts in the presence of climate drift. Mon. Wea. Rev., 125, 809–818, doi:10.1175/1520-0493(1997)125<0809:COAFIT>2.0.CO;2.
Tardif, R., G. J. Hakim, and C. Snyder, 2014: Coupled atmosphere–ocean data assimilation experiments with a low-order climate model. Climate Dyn., 43, 1631–1643, doi:10.1007/s00382-013-1989-0.
Toth, Z., and M. Peña, 2007: Data assimilation and numerical forecasting with imperfect models: The mapping paradigm. Physica D, 230, 146–158, doi:10.1016/j.physd.2006.08.016.
Vallis, G., 2006: Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press, 745 pp.
Vannitsem, S., 2014: Stochastic modelling and predictability: Analysis of a low-order coupled ocean-atmosphere model. Philos. Trans. Roy. Soc. London, A372, 2018, doi:10.1098/rsta.2013.0282.
Vannitsem, S., and L. De Cruz, 2014: A 24-variable low-order coupled ocean-atmosphere model: OA-QG-WS v2. Geosci. Model Dev., 7, 649–662, doi:10.5194/gmd-7-649-2014.
