2.1. Times series decomposition

Since the focus of our forecast study is on short-time predictions, it is crucial that the data used does not exhibit periodicity due to seasonal variability. Additionally, since the data covers a large time period, we removed the underlying trend corresponding to the interannual variability to only keep the high-frequency variability, the SLA residuals (Mann, Reference Mann1945; Kendall, Reference Kendall1975) see supplementary materials). As we got rid of the seasonal and multi-annual variability here, SLA dynamics will now refer to the sub-seasonal spatial patterns and gradients of SLA variability across our ocean regions.

In Figure 2 we present the image of SLA residuals at time t (t = 05/09/1993 Figure 2a), its evolution for two steps (t + 5 and t + 10 Figure 2bc); the difference SLA(t)−SLA (t + dt) (Figure 2ef) represents the transformation of the SLA field due to the dynamic.

Figure 2. (a) SLA residuals (t = 24/10/1993) and its evolution 5 (b) and 10 (c) days after. SLA evolution at 5 (e) and 10 (f) days compared to the previous state: $ SLA\left(t+k\right)- SLA(t),k=5(10) $ . (d) Standard deviation of the SLA over the Gulf Stream region estimated using the 20 years.

2.2. Methodology

We sought a neural approach that would enable us to predict SLA residuals given the knowledge of SLA and SST residuals time series. To accomplish this, we separated our data into three independent datasets for training, validation and testing. The same datasets were used for all experiments presented in this study. Twenty days were removed from the learning dataset at the beginning of 1994 and at the end of 2017 to separate the test from the learning phase. The three datasets cover different periods and are normalized separately:

• • training set: 1994–2014 (7412 images)

• • validation set: 2015–2017 (1275 images)

• • test set: 1993 2018 2019 (873 images)

Deep convolutional networks are designed for image processing, using convolution operations to extract spatial context. In this study, we have chosen the U-net architecture (Ronneberger, Reference Ronneberger, Fischer and Brox2015) for its robustness and flexibility, which make it suitable for a wide range of image processing problems. The aim is to reduce the input spatial resolution by passing it through various layers to get image patterns. As the first part reduces the input matrix size, it filters unnecessary information like noise to focus on patterns. The input information is propagated through these layers without losses via skipping connections. Moreover, we added Batch Normalization (Ioffe, Reference Ioffe2015) to the convolution layers of the original U-net to limit the overflow issues. Finally, we have made a slight modification inspired by the residual connections process (Goodfellow, Reference Goodfellow2016). Essentially, this involves an identity mapping of the input $ X $ , added to the output $ F(X) $ of U-net transformations, resulting in $ F(X)+X $ (refer to Figure 3).

Figure 3. SLA-Res-U-net Architecture: A typical U-net architecture comprises two main components. Initially, the encoder reduces spatial resolution to capture patterns and incorporates feature channels for context propagation. Then, the decoder expands the resolution features from the encoder, and its output is combined with the input image to act as a residual unit. Batch normalization is added between the 2D convolutions during descent and after SiLU activation during ascent. Hyper-parameters like activation, depth, and learning rate are determined via Bayesian optimization (S0).

The nature of the task is changed such that instead of predicting an image, the network predicts an additive transformation over the last time step. This method has already been successfully applied in various domains, such as medical image processing (Guan, Reference Guan2020). In the following, we denote this architecture as SLA-Res-U-net. We tried different architectures, including the classical U-net and found that SLA-Res-U-net outperforms them in every case (Zeng, Reference Zeng2015).

Due to the high correlation between SLA and SST (González-Haro, Reference González-Haro2020), the SST images are provided as inputs along with the associated SLA images. Thus, the inputs are the images of SLA and SST residuals taken at successive times equally spaced, and the targets are the following SLA and SST residual images1.

(1) $$ IM\left(t+ dt\right)=G\left[ IM(t), IM\left(t- dt\right), IM\left(t-2 dt\right),\dots IM\left(t- ndt\right)\right] $$

where IM represents the (SLA, SST) images and dt the time lag between two images, the function G is our architecture (He, Reference He2015).

Accurate predictions must provide sufficient information on the phenomenon under study and its dynamic. We performed preliminary tests to determine the optimal number of time steps and the time lag between two-time steps. These were selected based on various criteria, such as the accuracy of the prediction, geophysical considerations, and computation time.

The time evolution of the SLA is driven by ocean eddies whose return time is approximately 20 days in the Gulf Stream area (Kang and Enrique, Reference Kang and Enrique2013; Martin and Synthesizing, Reference Martin, Manucharyan and Klein2023; Chelton et al., Reference Chelton, deSzoeke, Schlax, Naggar and Siwertz1998). Therefore, we sampled the eddy at four time steps (t, t−dt, t−2dt, t−3dt) separated by a time lag dt of dt = 5 days, resulting in a total sampling period of 20 days. These four-time step images of (SLA, SST) were chosen as inputs of the U-net model 3. The objective was to predict the SLA and SST images at the next time step forward (a time step corresponds to 5 days). We denote SLA-SST/SLA-SST the SLA-Res-U-net model obtained at the end of the learning phase. To understand the contribution of each variable, we trained two other SLA-Res-U-net by modifying the inputs and the outputs. The first model (called SLA/SLA) uses as input the SLAs at four specific time steps backward and estimates SLA at the next time step forward (5 days later). The second model (denoted SLA-SST/SLA) uses the two-time series (SLA, SST) to estimate the next SLA only, the cost function minimizes only the prediction of the SLA.

We focused the study on the determination of the SLA only since SLA altimetric resolution is much weaker than SST (Dufau et al., Reference Dufau, Orsztynowicz, Dibarboure, Morrow and Traon2016). SST helps to retrieve SLA at high resolution. The second part of the study analyzes the number of time steps for which the forecast is accurate (prediction horizon).

We first tested the prediction given by the three models (SLA/SLA, SLA-SST/SLA, SLA-SST/SLA-SST,) by iterating each model in time. This involved introducing the estimated outputs at time t + ndt as input to estimate the image (SLA, SST) at time t + (n + 1)dt without new learning. For the SLA-SST/SLA, the true SST was introduced as input at each iteration since no prediction of SST was made. Therefore the model SLA-SST/SLA is not a forecast but rather a model of SLA evolution driven by SST. The SLA predicted by the model is reintroduced as an input to obtain the SLA values 10, 15, and 20 days later.

The second experiment consists in forcing the learning of the dynamics of the SLA fields with the model. First, we computed the output of the model at time t + dt. The model then received the preceding output as input and computed the output at the next step t + 2dt. This process was repeated, training the network to build the outputs at time t + 2dt from inputs at time t. We only iterated this process once more to estimate t + 3dt using the estimations at time t + 2dt, due to computation time constraints. In this way, the network can learn input conditions not seen during normal training.

The performances of the predictions were estimated at times t + dt, t + 2dt, t + 3dt, and t + 4dt (note that the prediction at t + 4dt was not learned). The flow chart of the procedure is shown in Figure 4. We denote this new architecture as DY-SLA-SST/SLA-SST. All the different configurations tested are shown in Table 1.

Figure 4. Training scheme: During the training phase, the network is fed with both SLA prediction and SST observations which extends the horizon of SLA. By performing backpropagation over several time steps, the model is forced to learn the dynamics of SLA.

Table 1. Input/output configurations and naming conventions for the various setups. Noted that for each setup, there are four input fields spaced five days apart, used to generate the subsequent five-day field

Moreover, to show the impact of SST on the trajectory of SLA, we did the same procedure with SLA-SST/SLA and trained the network by introducing the true SST at each step. We denote DY-SLA-SST/SLA in this model. In doing so, we learned the trajectory of the SLA image with the true SST information and consequently improved the SLA prediction at t + 4dt. This procedure allowed us to estimate SLA, whose observations are difficult to obtain in the near future, using SST which can be obtained almost instantaneously. The method was inspired by teacher-forcing (Goodfellow, Reference Goodfellow2016).

The learning process is performed (as shown in Figure 4) using classic backpropagation. During training, we randomly select whether to compute outputs at time t + dt, t + 2dt, or t + 3dt. The backpropagation is performed over the loop, meaning that we update the weights for all time step predictions. It is computationally heavier because for each time step the amount of calculation to run back the propagation increases proportionally for each time step.

2.3. Evaluation metrics

We consider the following metric to evaluate SLA predictions: the root-mean-square error computed on the test set (see Equation 2). Additionally, we use persistence performances as a threshold to assess our model’s performances. Persistence refers to a prediction that assumes either zero uniform velocities or velocities that cancel each other out. Additional metrics can be found in the supplementary materials.

(2) $$ RMSE=\sqrt{\frac{\Sigma_n{\left({y}_{pred}-{y}_{target}\right)}^2}{N}},n\in \left[ testpixel\right] $$