Ocean Currents Velocity Hindcast and Forecast Bias Correction Using a Deep-Learning Approach

Abstract

Today’s prediction of ocean dynamics relies on numerical models. However, numerical models are often unable to accurately model and predict real ocean dynamics, leading to a lack of fulfillment of a range of services that require reliable predictions at various temporal and spatial scales. Indeed, a numerical model cannot fully resolve all the physical processes in the ocean due to various reasons, including biases in the initial field and calculation errors in the numerical solution of the model. Thus, bias-correcting methods have become crucial to improve the dynamical accuracy of numerical model predictions. In this study, we present a machine learning-based three-dimensional velocity bias correction method derived from historical observations that applies to both hindcast and forecast. Our approach is based on the modification of an existing deep learning model, called U-Net, designed specifically for image segmentation analysis in the biomedical field. U-Net was modified to create a Transform Model that retains the temporal and spatial evolution of the differences between the model and observations to produce a correction in the form of regression weights that evolves spatially and temporally with the model both forward and backward in time, beyond the observation period. Using daily ocean current observations from a 2.5-year current meter array deployment, we show that significant bias corrections can be conducted up to 50 days pre- or post-observations. Using a 3-year-long virtual array, valid bias corrections can be conducted for up to one year.

1. Introduction

Today’s prediction of ocean dynamics relies on numerical models. The improvement and development of parameterization methods and the increase in spatial resolution along with computing power have improved the accuracy of numerical predictions. However, numerical models are often unable to accurately model and predict real ocean dynamics, leading to a lack of fulfillment of a range of services that require reliable predictions at various temporal and spatial scales. Indeed, numerical models cannot fully resolve all the physical processes in the ocean for various reasons, including biases in the initial field [1] and calculation errors in the numerical solution of the model. Thus, bias-correcting methods have become crucial to improving the dynamic accuracy of numerical model predictions. Currently, there are mainly two kinds of methods for numerical forecast product correction: traditional statistical methods and data-driven methods. Post-processing of numerical model outputs with statistical methods is the most common approach. Such methods consist of model output statistics (MOS) [2,3], Kalman filtering [4,5], and Bayesian probability decision [6], among others. Other methods involve correcting the bias within the numerical model if the source is known by creating a bias-forecasting model, as shown in the study by Chepurin et al. [7]. Observations can also be biased, and a bias removal algorithm can be included in the data assimilation scheme used in the forecast [8].

Data-driven approaches to numerical forecast product correction rely on sustained and reliable observations of the parts of the ocean where the predictions are made. Because oceans play a key role in global issues such as climate change, food security, and human health, efforts to create long-term repositories with global quality assurance and quality control standards have received significant support over the last decade. These efforts have come to fruition in the Global Ocean Observing System (GOOS) [9]. GOOS has opened a field of opportunity for new collaborations across regions, communities, and technologies, facilitating enhanced engagement in the global ocean-observing enterprise to benefit all nations. In the United States, these efforts led to the creation of the U.S. Integrated Ocean Observing System (IOOS), which through its eleven Regional Associations, implements regional observation systems covering all U.S. coasts and Great Lakes with activities spanning from the head of the tide to the U.S. exclusive economic zone [10]. Through these observatories, comprehensive ocean datasets, spanning several decades, have become available to the research community and have promoted the emergence of machine learning methods [11].

Machine learning algorithms are thus poised to determine patterns and structures in the increasing volume of observational and modeled data [12,13,14,15,16,17]. Deep learning-based bias-correction methods have also been developed to address systematic forecast errors of numerical ocean models. One of the applications has been minimizing biases in sea surface temperature (SST) prediction, for example, in coupled models, where SST biases can have significant impacts on climate predictions and heat flux calculations [18]. Another application has been on the bias correction in sea surface height (SSH) prediction, which is inherently biased due to sampling errors [19]. Yang et al. [20] used an ANN to correct non-stationary bias in SST forecast products of the U.S. National Center for Environmental Prediction Climate Forecast System (CFSv2), which are widely used in climate research and hurricane prediction for example. Choi et al. [21] used a deep learning model to restore real-time SST before assimilation by filling the data gaps with the deep learning model. Liu et al. [19] proposed a bias correction of a sea surface height model using an ML sequence-to-sequence model, which is a special class of recurrent neural network. Their model was applied to the Loop Current (LC) system in the Hybrid Coordinate Ocean Model–Navy-Coupled Ocean Data Assimilation (HYCOM-NCODA) Gulf of Mexico (GoM) simulation, which exhibits bias errors against the Gulf of Mexico Coastal Ocean Observing System’s (GCOOS) altimetry data, chosen here as the truth. Altimetry-derived products (SSH anomaly and geostrophic currents) have been shown to be as, and at times, more accurate than the HYCOM analysis, as well as data-assimilative models in general—also in relation to the LC patterns despite inherent sampling errors [22]. For the chosen prediction period of up to 15 days, Liu et al.’s [19] approach improves HYCOM’s skills by learning to forecast the systematic bias in the HYCOM-NCODA, outperforming persistence and improving the HYCOM forecast. However, according to Fei et al. [23], one of the weaknesses of the current methods for numerical forecast product correction is the lack of spatio-temporal relationships between the model and the observations.

In the study herein, we propose a new ML-based approach to conduct a bias correction on the 3D velocity field of ocean current models. This approach takes into account the joint temporal and spatial evolution of both the model and the observational fields, which is encapsulated in the Transform Model concept. To demonstrate our method, we take advantage of a recent and comprehensive set of observations in the GoM (Figure 1a) that provided current measurements of the full water column [24] over a large area of the GoM. In recent decades, the GoM has received and continues to receive a lot of attention from oceanographers. The GoM ocean dynamics is controlled by the pulsating LC, which is the most energetic circulation feature of the basin. Predicting the LC evolution is fundamental to almost all aspects of the GoM, including (1) anthropogenic and natural disaster response; (2) the prediction of short-term weather anomalies and hurricane intensity and trajectories; (3) national security and safety; and (4) ecosystem services [25]. The Deepwater Horizon oil spill in 2010 has bolstered a large amount of research toward improving our understanding and forecasting of the LC System in hope of mitigating further environmental and ecological damage. Furthermore, because of the reinforcing interaction between hurricanes, the LC and its eddies, long-term prediction of the LC states is becoming more and more relevant [26,27]. Thus, developing accurate and robust medium-term forecasting models of the LC System and its eddy formation is imperative and relies on the availability of ocean observations. Improved ocean observing systems are expected to reduce the uncertainty of ocean/weather forecasting and to enhance the value of ocean/weather information throughout the Gulf region [28].

The remainder of the paper is organized as follows. In Section 2 we provide an overview of the Transform Model concept. Section 3 describes the datasets, including the observations and the numerical model fields, the deep learning model and its implementation. Both free and data-assimilated simulations were used in this study. The metrics of the bias correction evaluation are also presented in this section. Section 4 presents the results of the bias correction and their evaluation against the testing datatset. Discussion and conclusion follow in Section 5.

2. Transform Model Concept

The rationale behind this concept lies in the fact that the dynamic history contained in the observations can be learned with a deep learning model, as shown by prediction exercises of the LC dynamics [29,30,31,32]. Deep learning allows computational models that are composed of multiple processing layers to learn representations of data with multiple levels of abstraction. These methods have dramatically improved the state-of-the-art in speech recognition, visual object recognition, object detection and many other domains such as drug discovery and genomics. Deep learning discovers intricate structures in large data sets by using the back-propagation algorithm to indicate how a machine should change its internal parameters that are used to compute the representation in each layer from the representation in the previous layer [33].

Numerical ocean models are mathematical approximations of the laws of physics that are used to estimate ocean states given a set of boundary conditions and constraints, such as the atmospheric and tidal forcing. When not constrained by data assimilation, the simulated states of the “free-run” are likely to be very different from the real ocean state, although the main features of the ocean dynamics will be captured and resolved. These features include the mesoscale dynamics, the water masses, and the eddy kinetic energy, among others, whose variability and magnitude is likely to be dissimilar to the observed ones, although present in both natural and virtual systems. Therefore, one can assume that both the simulated and real systems are on a parallel track although different in many ways.

Data assimilation techniques have been used in numerical predictions as a means of reducing the bias between the model’s field and observations at the time of the forecast when the background field is created. The primary objective is to optimally combine the observations and the model output. Observations assimilated in ocean forecasting systems now include altimetry, ocean color, surface velocities, sea ice and data from emerging platforms such as ocean gliders. However, direct measurements of surface current velocities are rare and subsurface current velocities even rarer. Many systems now employ multi-model approaches or ensemble modeling techniques. A variety of data assimilation are used in operational oceanography, and a detailed review of the data assimilation methods can be found in the study by Moore et al. [34]. Despite the large amount of research and the widespread use of data assimilation techniques in numerical model prediction, errors between assimilated models and real ocean dynamics remain significant. The useful forecast window of numerical ocean models is no longer than two days over a given area with the best initialization possible, as shown by Cooper et al. [35] in a dynamically active current system, such as the LC in the GoM. In this study, we show that remaining biases in data assimilated numerical models such as HYCOM can be reduced by our Transform Model when water column current velocities are used and the correction extended up to one year.

While deep learning has mostly been applied to a single data set as input for a variety of applications, we show in this study that it can be re-purposed to learn the differences between the simulated and real systems and their evolution over time and space. The acquisition of that knowledge between a given ocean numerical model simulation and a set of observations in the model domain over a time period concurrent to both enables the prediction of the differences and thus bias with a deep learning model. Therefore, the knowledge of the differences can be converted into a model field observations-based correction tool, called a Transform Model, with the advantage of predicting the correction of the model field beyond the period of observation. This method also accounts for the common history of the model fields and measurements in the correction process. We will show that the improvements to the model fields, namely the reduction in errors, are significant, despite the fact that no physical constraints were applied to the correction.

The concept of the Transform Model is depicted in Figure 1b. It is based on the availability of a long enough concurrent time series of model fields and observations. Both are jointly used in a deep learning model that creates a transformation tool of the numerical model field based on the observations. The Transform Model can then be used to correct the numerical model field at any time in the future and the past outside of the period of measurements.

3. Methods

3.1. Data Sets

For the demonstration of our Transform Model, we used three data sets that overlap in time, composed of two numerical model’s current velocity vector fields and one set of in-situ current measurements from an observing array in the GoM (Figure 1a). The model’s inherent dynamics may influence the effectiveness of the Transform Model, which will be demonstrated later on in the paper.

The first numerical model data set was obtained from the HYbrid Coordinate Ocean Model (HYCOM) Consortium. The data were generated by the HYCOM + NCODA Gulf of Mexico $1/{25}^{\circ }$ Reanalysis (GoMu $0.04$/expt_$50.1$), which spans from 1993 to 2012 and has a horizontal resolution of $4.4$ km. HYCOM + NCODA is a data-assimilative hybrid isopycnal-sigma-pressure (generalized) coordinate ocean model. The system uses the NCODA system [36,37] for data assimilation. NCODA uses the model forecast as a first guess in a three-dimensional variational scheme [37] and assimilates available satellite altimeter observations (along-track obtained via the NAVOCEANO Altimeter Data Fusion Center), satellite and in-situ sea surface temperature (SST) as well as available in-situ vertical temperature and salinity profiles from XBTs, ARGO floats and moored buoys. The Modular Data Assimilation System (MODAS) is used for downward projection of surface information [38]. The experiment includes tidal forcing. As noted here, no current vector flow information is used in the data assimilation process. Surface forcing values were obtained from the National Centers for Environmental Prediction (NCEP) Climate Forecast System Reanalysis (CFSR [39]).

The second model’s current vector fields were obtained from a free-running Massachusetts Institute of Technology (MIT) generalized circulation model simulation of the GoM circulation (MITgcm-GoM)—a z-level model. Surface forcing values were obtained from the National Centers for Environmental Prediction (NCEP)–National Center for Atmospheric Research reanalyses ([40]). The MITgcm–GoM model was originally developed for state estimation and prediction of the upper ocean circulation in the GoM, including LC evolution and eddy shedding. For these purposes, in its original version, satellite-derived ocean surface observations and subsurface in situ observations were assimilated using a four-dimensional variational (4DVAR) method [41]. For the data set used in this study, however, no data assimilation was performed. The model uses a telescopic grid with a horizontal resolution of 1/20° × 1/20° in the central GoM, which decreases to 1/10° × 1/10° toward the boundaries and western part of the domain. For this study, daily outputs from the simulation between 2009 to 2012 are used. No tidal or atmospheric pressure forcing was applied. More details on the simulation can be found in Morey et al. [42].

These two models are commonly used for forecasting and scientific studies. They were evaluated based on their basic characteristics of the deep GoM circulation [42]. Aspects of the models or their configurations were shown to impact the representation of major features of the deep circulation. Because of these differences and the fact that HYCOM simulation is data-assimilated and the MITgcm-GoM is not, we can estimate the relative difference in the bias correction of the two models’ simulations. The assimilated model field should require fewer corrections than the “free run” simulation.

The in-situ data were obtained from the Dynamics of the Loop Current in the U.S. Waters experiment (hereafter Dynloop)—a comprehensive observational study of the LC in the eastern GoM [24,43,44]. The observational array consisted of the instrumentation of nine tall moorings and seven short moorings, and an array of 25 pressure-equipped inverted echo sounders (PIES) (Figure 1a). The array system measured the water column velocity for 2.5 years (905 days), beginning in March 2009. Each tall mooring included water velocity measurements made from an upward-facing 75 kHz ADCP deployed near a depth of 450 m, as well as point current meters at five depths from around 600 to 3000 m, providing high-resolution water velocity data at depths between about 60 and 440 m and much lower resolution water velocity data below this. Additionally, each short mooring included a single point current sensor located 100 m above the sea floor, providing near-bottom water velocity data. The PIES array provided direct measurement of pressure and acoustic round-trip travel times from the sea floor to the sea surface, which were used to create vertical profiles of density, salinity, and temperature. These pressure records combined with estimated horizontal density gradients were used to calculate geostrophic water velocities. This array was located to cover both east and west sides of the LC between the West Florida Slope and the Mississippi Fan and was also centered over the zone where LC eddies typically separate from the LC. The horizontal separation between moorings was around 50–80 km and between the PIES sensors was around 40–50 km. These recorded data were used to construct the measurement-based water velocity matrix used in this study. More details on the creation of the velocity matrix can be found in [31]. The horizontal resolution varies between 30 and 50 km, and only the first 500 m, corresponding to 26 vertical layers, was used in this study. The time resolution for the velocity data was 12 h, which corresponds to 1810 data frames for each u and v velocity component. The final matrix dimensions were of 1810 × 26 × 29 × 36 for each component.

3.2. Transform Model

The deep learning network of choice for our Transform Model is the U-shaped Convolution Neural Network (CNN), also known as U-Net [45]. The typical use of convolutional networks is in classification tasks, where the output to an image is a single class label. However, in many visual tasks, especially in biomedical image processing, the desired output should include localization, i.e., a class label is supposed to be assigned to each pixel. The architecture of this network consists of two parts. The first part is the analysis path called the encoder, which is similar to the CNN architecture that is widely used for image feature extraction. The second part is the synthesis path called the decoder, where the compact features are expanded to the input dimension. In the image case, the feature set, or code, is decoded back to a (segmented) image of the same dimension as the input image. The name of U-Net is based on the U structure of the network. This type of network has been widely used in medical image segmentation applications [46,47], for land segmentation studies in remote sensing images [48], for cloud and shadow segmentation [49], for modeling and prediction of coastal weather events in the Netherlands [50], and for coastal wetland classification [51], among many other applications.

For our application, the U-Net was modified at the output layers to make it suitable for a regression problem instead of the original structure that was designed for image segmentation. The problem can be seen as a time series transformation through a regression analysis. Hence, the last layers after the convolution layers have been replaced by regression layers, instead of a Softmax and a segmentation layer in the original structure. The regression layer is a type of neural network layer used for predicting continuous values, as opposed to classification or segmentation. It is typically used in machine learning tasks where the goal is to predict a numerical value or a set of numerical values. The output value can be a scalar, a vector, or a matrix, depending on the specific task and architecture of the neural network. Generally, the choice of layer architecture is determined by the particular activity and data that are being utilized, and there is no solution that is universally applicable to all situations. Because the U-Net will learn both the spatial and temporal features of the training data, it is called hereafter the Spatio-Temporal U-Net (STU-Net). The final model structure consists of four stages of encoders, as shown in Figure 2. The input is made of a four-dimensional input layer, which is organized as $x,y,u/v,t$. Here, $x,y$ are the coordinates of the sample location, $u/v$ are the zonal or meridional velocities, respectively, and t denotes the time of the measurements. The output of the model is the transformed velocity field time series, thus the same variables as the input. In the vertical direction, a Transform Model is created for each depth layer.

The Transform Model is the outcome of the training phase, where both the numerical model and the observations are fed to the STU-Net. The relationship between the measurements and the numerical model is represented by the neural network structure, including its hyper parameters and weights. However, in order to apply bias corrections both forward and backward in time, a Transform Model is created for each temporal direction. During the training phase, the errors between the two fields are used to adjust the weights of the Transform Model until it converges.

For the experiments conducted in this study, the hyperparameters were set by trial and error to be as follows: InitialLearnRate = $5\times {10}^{-4}$, MiniBatchSize = 4, MaxEpochs = 300, LearnRateDropFactor = 0.1. The loss function utilized for the regression output is the Root Mean Squared Error (RMSE), and an Adam optimizer was used in the training phase. We arbitrarily set the limit of the overall loss to 0.1 and Mini-batch RMSE to 0.5, for which only 120 epochs were necessary. Traditionally, the model is left to reach a plateau that determines the loss value. However, in our case, the congruence between the transformed field of the numerical model and the observations (Dynloop) was considered adequate with a loss value set at 0.1. At the end of the training stage, the obtained regression layers constitute the Transform Model that will be used for bias corrections. This model was implemented with the Matlab Deep Learning Toolbox (R2021b). The same loss value and hyperparameters were used to create the transformer of each numerical model.

3.3. Experimental Strategy and Input Data Formatting

In this section, the strategy to evaluate our Transform Model concept is presented. First, the short-term transformation of the HYCOM and MITgcm-GoM velocity fields by the in-situ observations is assessed. The training is conducted with the first 855 days and the transformation results are evaluated with the remaining 50 days of measurements. The purpose of this experiment is to assess the transformative potential of limited observation periods and the efficacy on assimilated and non-assimilated numerical model fields, respectively. Second, for the long-term transformation capability of the concept, longer-term time series are required. In this case, HYCOM was used as virtual observations to transform the MITgcm-GoM velocity fields. The training took place over a period of three years and the testing period was either the preceding or following year of the training period. The relatively long training period over the testing period warrants that a large diversity of relationships are learned by the model, which prevents overfitting [52]. For the short-term Transform Model, the period March 2009–June 2011 was used for training and validation, and July–August 2011 was used for testing. For the long-term Transform Model, the years 2009–2011 (2012–2010) were used for training and validation, and 2012 (2009) was used for testing in the forward (backward) experiment. Although a longer training period could have been selected, the three years was found to be sufficient to enable acceptable bias corrections up to one year before or after the training period, as shown in the results section. In the remainder of this study, we refer to a bias-corrected field as a transformed field—the product of the Transform Model.

Both model fields and observations were interpolated on the same spatial and temporal grids. Therefore, the time interval was set to daily, i.e., the coarsest temporal resolution of the three time series. Because the spatial resolution of the three datasets differed over the geographical area of the measurements, $29\times 36$ for the Dynloop data, $64\times 88$ for HYCOM and $52\times 70$ for the MITgcm-GoM simulations, all three data sets were interpolated to a $64\times 64$ grid using a bicubic interpolation [53]. Vertically, the same depth layers as in the Dynloop dataset were used to reinterpolate the other two model fields, which resulted in 26 layers of 20 m intervals from the surface to 500 m (maximum depth used in this study). The interpolated fields were checked for spatial consistency with the original fields, and increasing the temporal resolution to twice daily (original temporal resolution of the Dynloop dataset) did not change the performance of the bias correction.

3.4. Bias Correction Evaluation Metrics

To evaluate the bias correction of the numerical model field by the Transform Model, we applied metrics that are commonly used in the field of physical oceanography to compare the output of different numerical models, as done, for example, in [29,54]. The time series of the spatial Correlation Coefficients (CCs) and Root Mean Squared Errors (RMSEs) between the “observed” and predicted fields were calculated. The observed field is either the in-situ observations for the short-term transformations or HYCOM—virtual observations—for the long-term transformations. The degree of transformation, also called correction gain, can be estimated and was calculated as follows:

$$Gain={\displaystyle \frac{|M_{transformed}-M_{model}|}{M_{model}}}\times 100$$

(1)

where $M_{model}={\displaystyle \frac{{\sum }_{n=1}^{T}RMS{E}_1\left(n\right)}{T}}$, $M_{transformed}={\displaystyle \frac{{\sum }_{n=1}^{T}RMS{E}_2\left(n\right)}{T}}$, T is the length of the time series, $RMS{E}_1$ is the RMSE between the model and the observed data, and the $RMS{E}_2$ is the RMSE between the transformed field and observed data. A high (low) gain signifies that the transformed model field errors are low (high).

Taylor diagrams [55] were used as an assessment method that both quantifies various aspects of model performance and visually summarizes these aspects within compact diagrams. The Taylor diagrams used herein are applied to the comparison of the two-dimensional fields. They capture the CC, the Root Mean Square Difference (RMSD), the so-called RMSE, and the standard deviation between the models and the in-situ observations, which are the reference fields.

In order to capture whether the dynamical field is properly resolved in the corrected model fields, we applied an EOF decomposition, which is a form of principal component analysis, for spatio-temporal data [56]. The data are decomposed on orthogonal spatial modes, whose net response as a function of time accounts for the combined variance in all of the modes. The first modes usually account for most of the variance in the LC system [29] and are used to demonstrate that the corrected fields contain the same modes as the observations. The EOF decomposition is applied to the 50-day testing period of the short-term bias correction and the first 120 days of the testing period in the long-term bias correction experiment.

4. Results

4.1. Comparison between Observed and Modeled Fields

To evaluate how different the three concurrent field times series are, we calculated the Taylor diagrams of the spatially averaged velocity magnitudes for each of them at four depths, i.e., 0, 20, 100, and 500 m, respectively (Figure 3). Because the MITgcm-GoM is a free-run, the model velocity exhibits the lowest correlation coefficient (CC) and largest RMSE and standard deviation at all depths. While the assimilated HYCOM model exhibits a lower standard deviation than the MITgcm-GoM, the RMSE values and difference between the two models are similar at 0, 20 and 100 m but not at 500 m (Figure 3d). A large RMSE signifies that the amplitude of the variation in the velocity components is overestimated. At 500 m, both models’ RMSE is an order of magnitude lower than at shallower depths, which is explained by the weaker velocities at that depth. HYCOM’s CC is lower at 500 m than at shallower depths, suggesting that the data assimilation at 500 m might be less efficient than in shallower waters.

4.2. Evaluation of the Bias Correction by the Short-Term Transform Model

Surface observations are, in general, more commonly available than subsurface observations, such as three-dimensional arrays. Therefore, predicting subsurface dynamics has remained more challenging than at the ocean surface because of the lack of continuous sampling at the same location. Vertical projections of the surface corrections in data assimilation models are still reliant on the methods described in [38,57]. The transformation of three-dimensional fields is, therefore, critical to full water column prediction. For this evaluation, two types of experiments were conducted. In the first one, the transformation was only applied to the two-dimensional surface velocity, and in the second experiment, the transformation was applied to a three-dimensional tensor of the velocity field. The efficacy of the transformation was evaluated by calculating the RMSE between the transform model solution and the reference, which are the observations in this case, over the last 50 days of the observational period not used in the training phase. The same evaluation was performed for the original numerical model velocity field.

4.2.1. Two-Dimensional Field Bias Correction

The results of the RMSE for both the original numerical model and the transformed model fields are shown in Figure 4 for both HYCOM and MITgcm-GoM. Figure 4a shows a significant reduction in the HYCOM model velocity RMSE, both in the mean and in the variability. The RMSE of the transformed HYCOM field remained almost constant throughout the 50-day period, unlike the original HYCOM field, with a slight increase toward the end of the period. Figure 4b shows that the transformation of the MITgcm-GoM fields was as efficient as for the HYCOM fields in terms of reduction in the RMSE but the RMSE of the transformed fields increased over time. HYCOM fields’ RMSE, although quite variable, did not exhibit an increasing trend over time as the MITgcm-GoM fields did. This difference is most likely due to the fact that data were assimilated in the HYCOM simulation and not in the MITgcm-GoM model. The alignment of the dynamical features between the observations and the numerical model is shown by the increased CC of the transformed model for both HYCOM and MITgcm-GOM (Figure 4c,d). However, for the latter, the CC of the transformed model exhibits larger variations than the CC of the transformed HYCOM due to the larger deviation of the MITgcm-GOM dynamics from the observations.

The transformation gain is shown in

Table 1. The 50-day average transformation gain (percentage) of the velocity corrections at different depths by applying the Transform Model to HYCOM and MITgcm velocity fields.

Dataset	Zonal	Zonal	Zonal	Meridional	Meridional	Meridional
Depth	0 m	100 m	500 m	0 m	100 m	500 m
HYCOM	$78.85\%$	$76.23\%$	$71.78\%$	$74.80\%$	$69.16\%$	$71.99\%$
MITgcm	$54.88\%$	$43.43\%$	$35.53\%$	$48.35\%$	$21.29\%$	$42.91\%$

. It was higher for the HYCOM model than for the MITgcm-GoM field. However, the non-assimilated model transformation exhibited an RMSE < 0.4 m.s⁻¹ for up to 30 days, which is less than the RMSE of the assimilated model fields (Figure 4b). This result suggests that the correction of the flow field with in-situ observations by the Transform Model can be as effective as, if not better than, the data assimilation method currently used in the HYCOM operational forecast. This result is significant because the transformation was made outside of the observation period. In terms of the spatial structure of the flow field variability, the transformation is also able to carry through the spatial modes contained in the observed field, as shown in Figure 5. The efficacy of the transformation is stronger for the HYCOM than for the MITgcm-GoM field, which also shows that there is room for improvement in the data assimilation process, whether in the NCODA system or by applying our Transform Model.

4.2.2. Three-Dimensional Field Bias Correction

For the three-dimensional velocity field transformation, a Transform Model was created for each depth level between the numerical model and the observations, and the transformation was applied to the last 50 days not included in the training.

The results in terms of the RMSE and correlation coefficient (CC) for each depth level for the transformed HYCOM model over the 50-day period are shown in Figure 6. As seen in the two-dimensional case, the transformation efficacy is significant. The RMSE is reduced by a factor of 2.5 from the surface down to 200 m and by half at the 500 m level (Figure 6a,b). The HYCOM field’s RMSE decreases with depth whereas the gain of the transformation only changes by a factor of 9% (4%) for the zonal (meridional) velocity between 0 and 500 m (Table 1). While the RMSE of the HYCOM zonal velocity field is higher than the one of the meridional velocity field, when transformed, the resulting RMSEs of each component are nearly identical. The CC of the zonal field is much less than the CC of the meridional field across all levels (Figure 6c,d). The difference persists in the transformed fields but is strongly reduced. In addition, the CC of the transformed fields is larger than the CC of the original fields and varies less vertically as well. The difference between zonal and meridional velocity CC is not surprising since the current meridional velocities dominate the flow pattern of the LC.

For the MITgcm-GoM three-dimensional transformed velocity fields, the RMSE is reduced by a factor of 1.5 at the surface to about 1 at 200 m and by 1.5 at the 500 m level (Figure 7a,b). For the CC, the MITgcm-GoM original model shows a very small correlation with the observations. The transformed fields, however, show an increase in the meridional velocity by a factor of sixteen (Figure 7c,d), which is relatively constant over depth, in contrast with the CC of the transformed zonal velocity, which is not as uniformly improved; although, the correction can reach a factor of fifteen near 100 and 350 m. The 50-day gain is, therefore, less for the MITgcm-GoM overall, as shown by Table 1, but the correction factor is much higher for the latter than for HYCOM. The RMSE of the transformed MITgcm-GoM fields (red lines in Figure 7a,b) is similar to the HYCOM’s original RMSE (blue lines in Figure 6a,b). From a correlation standpoint, there is a similar improvement, in particular for the meridional velocity (red lines in Figure 7d vs. blue lines in Figure 6d) and to a lesser degree for the zonal velocity. This shows that our method is at least as efficient at correcting the free-run model with just the observed velocity field as the NCODA is with all the different data sources used for HYCOM. The latter, on the other hand, requires less correction by the Transform Model, but the improvements by the Transform Model can still be significant, especially near the surface.

While the statistical metrics such as RMSE and CC may indicate a good agreement between the quantities compared, the spatial features that comprise the system may look dissimilar. Therefore, we show here the efficacy of the transformation in the three-dimensional structures of the transformed field on day 50 after the end of the observation period (Figure 8 and Figure 9). For the HYCOM model, the vertical and horizontal flow features in the observations are present in the transformed model field, showing similarities in magnitude and location in space. The presence of these features and their timing is critical to the development of the instabilities that lead to the growth of the frontal cyclones near the surface and then in the deep waters and ultimately to the separation of a LC eddy [43,44,58,59]. The three-dimensional transformation confirms the results seen in the two-dimensional transformation. The Transform Model is capable of extending the correction of observations in HYCOM at least 50 days past the last measurement. For the MITgcm-GoM model, the flow field difference is quite significant, as shown by Figure 9. The transformed flow field is shifted toward the path of the observed flow, both horizontally and vertically, although the magnitude of the transformed flow is reduced from the observations.

4.3. Evaluation of the Bias Correction by the Long-Term Transform Model

For this evaluation, HYCOM is used for virtual observations to transform the MITgcm-GoM surface velocity field. In the forward transformation experiment, the transformation was applied for up to 365 days after the virtual observations ended. As seen in the short-term transformation, the RMSE of the transformed field is reduced by up to a factor of four and exhibits less temporal variability than the MITgcm-GoM RMSE (Figure 10a,b). The RMSE of the transformed field varies over time along with the evolution of the LC system, whose dynamics can affect the efficacy of the transformation. Indeed between days 120 and 160, the LC evolution between the HYCOM and the MITgcm-GoM significantly diverged. While the eddy shed and reattached in the HYCOM flow, the LC remained in its extended position in the MITgcm-GoM model. This event affected the capacity of the Transform Model to correct the divergent dynamics, which increased the RMSE of the transformed model. Nonetheless, the transformation is able to strongly reduce the MITcgm-GoM velocity field RMSE even 300 days after the virtual observation ended, which reveals the long-lasting potential of the transformation process. In terms of velocity field variability, the CC shown in Figure 10c confirms the skill of the transformer at improving the flow field features, especially when the HYCOM and MITgcm-GoM diverge the most. After 300 days, the CC of the transformed field is less than the one of the MITgcm-GoM although the RMSE of the transformed field is much smaller.

The backward transformation also shows a decrease in the RMSE of the transformed MITgcm-GoM velocity fields for most parts except during periods when the original numerical model RMSE is also low. In this case, both MITgcm-GoM and HYCOM circulation features exhibit dynamical similarities. In addition, the transformed RMSE is less variable than the original one. The CC of the transformed field is also improved over the CC of the MITgcm-GoM field and is similar to the one of the MITgcm-GoM field when the latter improves (days 100–150). As in the forward transformation, the transformed field CC becomes less than the MITgcm-GoM field CC toward the end of the time window. The error field variance is, however, strongly reduced by the transformation.

Although the transformation could lead to unrealistic features after several months, the EOF analysis of the velocity field reveals that the modal structure of the virtual observations is present in the transformed field, both in the forward and backward transformations (Figure 11). The transformation is also able to convey the timeliness of the separation of the LC eddy (Figure 12) in the forward transformation, which is paramount to the generation of reliable and useful predictions of the LC. In the backward direction, the Transform Model is capable of readjusting the LC state in the MITgcm-GoM to closely follow the state of the LC in the virtual observations (Figure 13).

5. Conclusions

In this study, we have presented a new method based on deep learning that enables the extension of the bias correction of a numerical ocean model velocity field up to one year after the observations ended. We have focused on ocean observations, but we believe that our approach could be applied to any biogeochemical or physical bias correction exercise, as long as the observations and model output are concurrently available.

Our approach is based on the re-purpose of an existing deep learning model, called U-Net, designed specifically for image segmentation analysis in the biomedical field. We have modified the network in such a way that its original segmentation layer is replaced by a regression layer. That layer is used to create a Transform Model that retains the temporal and spatial evolution of the differences between the model and observations to produce a correction in the form of regression weights that evolves spatially and temporally with the model both forward and backward in time, beyond the observation period. The results shown in this study also reveal the efficacy of the Transform Model at correcting the three-dimensional flow field, even on existing reanalyses, such as the HYCOM data used herein. Finally, with the increased availability of surface current data from high-frequency (HF) radars, a Transform Model could be easily created for any numerical model with just a few years of concurrent model data and observations. Such a model, which can be updated daily with new data, would provide useful corrections to the model’s surface flow field that would be transferred vertically to other variables, either by the numerical model itself or through a similar type of AI technique as the one described here.

Overall, our U-Net application has shown promising results in the numerical ocean model’s bias correction of the velocity field. Most bias corrections are designed for SST or SSH, and even sea surface salinity, but none for the velocity field. The reason for this is the lack of systematic in-situ current measurements. Despite the increase in collection efforts and duration because of the climate study needs, ocean full water column measurement arrays remain scarce and very expensive to operate. However, another measurement array in the GoM, funded by the National Academies of Science, Engineering and Medicine under the Understanding Gulf Ocean Systems initiative (UGOS), was deployed from June 2019 to May 2021 and expanded to the north, west, and south beyond the DynLoop array used in this study. Although not used in this study, this new GoM array could be used to train another STU-Net transformer to conduct bias correction of the velocity in numerical simulations of the LC system before, during and after that period. Other longer-term arrays have been deployed or are currently active as part of the international program such as the Global Tropical Moored Array [60]. Such arrays, because of their duration, offer opportunities for the application of transformers like ours that can virtually expand their longevity. There is, nonetheless, an effort to provide real-time surface current information in various areas along the coastline, which is happening in various regions around the world. However, the footprint is limited to about 200 km from shore [61]. As long-term deployments become more available and measurements more reliable, our Transformer method could be used to correct regional numerical current predictions that exhibit significant challenges in coastal areas [62].