Neural Fields for Fast and Scalable Interpolation ofGeophysical Ocean Variables

J. Emmanuel Johnson
Univ. Grenoble Alpes, CNRS UMR IGE, Grenoble, Francejohnsonj@univ-grenoble-alpes.fr
Redouane Lguensat
Institut Pierre Simon Laplace, IRD, Sorbonne Université, Paris, France
Ronan Fablet
IMT Atlantique, CNRS UMR Lab-STICC, Brest, France
Emmanuel Cosme
Univ. Grenoble Alpes, CNRS UMR IGE, Grenoble, France
Julien Le Sommer
Univ. Grenoble Alpes, CNRS UMR IGE, Grenoble, France

Abstract
Optimal Interpolation (OI) is a widely used, highly trusted algorithm for interpolation and reconstruction problems in geosciences. With the influx of moresatellite missions, we have access to more and more observations and it is becoming more pertinent to take advantage of these observations in applications suchas forecasting and reanalysis. With the increase of the volume of available data,scalability remains an issue for standard OI and it prevents many practitioners from
effectively and efficiently taking advantage of these large sums of data to learn
the model hyperparameters. In this work, we leverage recent advances in Neural
Fields (NerFs) as an alternative to the OI framework where we show how they canbe easily applied to standard reconstruction problems in physical oceanography.
We illustrate the relevance of NerFs for gap-filling of sparse measurements of sea
surface height (SSH) via satellite altimetry and demonstrate how NerFs are scalable
with comparable results to the standard OI. We find that NerFs are a practical setof methods that can be readily applied to geoscience interpolation problems andwe anticipate a wider adoption in the future.

1

Introduction

Maps of sea surface height (SSH) provide a crucial source of information for the oceanographic
community. They are used in a number of applications including research involving mesoscale ocean
dynamics [1, 2] and their role in global climate change [3, 4]. A good characterization could even leadto a better quantification of the transport of biogeochemical components [4]. Our understanding of
SSH characteristics has gotten better not only through our improved understanding of the underlying
physics, but also due to the fact that we have more available observations with the introduction ofsatellite altimetry [5, 6]. However, we only ever have partial observations of SSH through satellite
Machine Learning and the Physical Sciences workshop, NeurIPS 2022.

altimetry, so the community has put a lot of research effort into methods that can help interpolate themissing observations for use in subsequent SSH mapping products [7, 8].
There have been a number of different algorithmic developments that have had success in sea
surface height reconstruction [9, 10]. In practice, the most notable of algorithms is the DUACS
framework [11] which is a carefully handcrafted solution based on Optimal Interpolation (OI).
Developed in 1997, this framework is capable of producing global daily maps of an effective resolutionof 100-200km [12, 8]. This single method is solely responsible for producing many Copernicus
products1 such as global maps of sea surface height and sea surface temperature [7, 13, 14, 15].
However, a major downside of the OI framework is that it is computationally expensive with avery high memory consumption even at a GPU scale. This makes it very impractical for training
and inference at very large scale especially with the onset of more and more available observation
data [16].
In this work, we implement an alternative solution to address the computational limitations of OI:
neural fields (NerF) [17]. NerFs are a family of coordinate-based, neural network algorithms which
map a set of spatial-temporal coordinates, xϕ , to a scalar or vector, y, quantity of interest. Thesemethod are parametric versions of the standard OI formulation where the parameters are learned from
data. This is only possible due to the copious amounts of data we now have given the introduction of
more altimetry satellites [16, 12]. We demonstrate that NerFs are a viable family of methods that
can interpolate the SSH field given partial observations while also being orders of magnitudes fasterand memory efficient than the standard OI solution. All of the preprocessing steps, machine learning
modules and training regimes used in this submission is available on Github2 .

2Methods

2.1

Data

In this work, we are looking at sea surface height as our observed variable of interest. The observation
data is recovered from the NADIR track satellite altimetry. In particular, we parsed all data over the
Gulf stream area during the time period of 01-01-2017 until 31-12-2017. We get an aggregated total of
7 altimetry tracks from the SARAL/Altika, Jason 2, Jason 3, Sentinel 3A, Cryosat-2, and Haiyang-2A
satellite altimeters which is roughly ∼1.8 million observations in total. The Cryosat-2 altimeter data
was left out as an independent validation set. We extract the spatial coordinates (latitude, longitude)
and the time coordinates (seconds) as the inputs, xϕ , to our model and the corresponding SSH values
as our outputs, yobs . All data is available on the data challenge that has been setup on Github3 [18].
2.2

Baselines

As mentioned in the introduction, there are many methods in the literature that have been used for
SSH reconstruction [19, 10, 9, 20, 21, 22]. However, the most prominent method that is foundational
for the Copernicus products is the DUACS algorithm [7]. The DUACS algorithm is an OI-based
scheme which tunes the kernel hyperparameters offline in a physically consistent manner with external
altimetry validation datasets. While the standard OI algorithm is an excellent and trusted method forinterpolation, this method suffers from scaling issues (training and inference) which can limit its use
in practice. The training has a time complexity of O(N 3 ) and memory complexity of O(N 2 ) whichmakes this method infeasible without the appropriate hardware [23].
To alleviate this issue with scalability, users apply this algorithm locally by only conditioning on theobservations available in the local region of interest and interpolating the remaining observations
within that region. The algorithm is run along in a patch-like manner until covering the entire globe.
This mosaic of many local patches results in a global map of interpolated SSH values. The local
application of the OI method has several advantages: 1) It allows the OI framework to be scaled tolarger datasets because it reduces the kernel matrix inversion to only the observations within the local
region and 2) it removes spurious long-range correlations that are physically inconsistent [24].
1

DOI:10.48670/moi-00147
https://github.com/jejjohnson/ml4ssh
3
DOI:10.5281/zenodo.5511905
2

2

In this work, we apply the standard OI method with preconfigured hyperparameters via the local
patch-based method described above to serve as the naive baseline which closely mimics the DUACS
algorithm. In addition to our naive patch-based OI implementation, we also include the publicly
available SSH maps, the solution to the DUACS method, as another baseline method.
2.3

Neural Fields

Neural Fields (NerFs) are implicit models that take coordinate-based inputs (e.g. spatial-temporal)
and output a scalar value or vector field for a variable of interest (e.g. sea surface height). This familyof neural networks have been very successful in many computer vision applications including image
regression and 3D rendering [17] however, their adoption in Earth sciences is still at a very earlystage, especially in ocean science applications [25].
Standard neural networks have problems modeling high frequency signals so there were many crucial
modifications that were introduced to make them successful. The first attempt included a Fourierfeatures transformation followed by any standard neural network, i.e. the Fourier Feature Network(FFN) [26]. This method was successfully able to capture the high frequency signals in 2D images
however the gradients were noisy or nonsensical [27]. Another proposed approach is the SIREN
method [27] which uses the sine function as an activation function within a fully connected neuralnetwork. This sinusoidal activation has been explored before [28] but the recent ingenuity is the
special initialization for the weights and biases enabling the network converge much faster and avoid
local minima. This method also results in calculable and relatively smooth first and second order
gradients which are more representative of physical systems. This is a non-exhaustive list and there
are many extensions to this including the modulated version [29] as well as Multiplicative Filter
Networks (MFN) [30]. In this work, we chose the SIREN method for its simplicity and smooth
gradients but a full comparison with other NerF variants for regression problems in physical systems
would be an excellent follow-up study.

3

Experiments

3.1

Data Preparation

NerFs. We used a coordinate transformation of the spatial data from spherical coordinates (lat,lon)
to Cartesian coordinates (x,y,z). After we aggregated the altimetry datasets, we had ∼1.6 million
data points for training and ∼179K data points for validation. A final evaluation is done over anentire spatial-temporal grid that is equivalent to DUACS product. The grid resolution is 0.2 degrees
spatially and 1 day temporally which results in ∼1 million data points for the final evaluation. The
statistical metrics are computed comparing this grid product to the left out validation altimetry dataset
and we plot maps for each algorithm for a visual comparison.
3.2

Hyperparameters

OI. The OI method was trained via the local patch-based version that was outlined in the previous
section. A Radial Basis Function (RBF) kernel was used with a length scale per dimension: A lengthscale of 7 days for the temporal dimension and a length scale of 1 degree (∼100 km) for the latitude
and longitude length scale. The noise likelihood was set to 0.05.
NerF. The SIREN method was trained using the standard Adam optimizer with a base learning rate of
1e-4. We used a cosine annealing scheduler with a linear warm-up. We used 100 epochs for the linear
warm-up with a learning rate between 0 and 1e-4 followed by 2,000 epochs for the subsequent cosine
annealing with a learning rate between 1e-4 and 0. We used Weights and Biases [31] to perform
hyperparameters sweeps on the batch size, the number of layers and the width of layers. We used
the standard mean squared error as the loss function and all training was done with one Nvidia Tesla
V100 GPU.

4

Results

Statistics. A snapshot of predictions, the norm of the gradients and the Laplacian can be seen in
figure 1. Each of the SSH maps are very similar as SIREN method can also capture some fine scale3

Table 1: Metrics
Algorithm

Normalized RMSE

OI
DUACs
SIREN

0.85 ± 0.09
0.88 ± 0.07
0.88 ± 0.08

Effective Resolution

CPU

GPU

140 km152 km136 km1 hr
N/A
30 secs

40 mins
N/A
5 secsstructures like the OI methods. This is even more evident in the Laplacian maps which result in more
non-smooth structures for the SIREN method. These higher order gradients are important for otherquantities of interest like the kinetic energy and vorticity and more evaluation is needed to check howeach compares method compares in respective energy levels in the spectral domain. In table 1, we
see the metrics computed with the independent validation dataset; i.e. the left out Cryosat-2 altimetry
track that was unseen by any model. The normalized root mean squared error (NRMSE) and effective
resolution (calculated from the signal-to-noise ratio) is comparable between all methods. For moredetails on the evaluation metrics, see section 3d in [32]. Note that this is simply a local region andmore evaluation is needed in other regions and at a global scale.
Timings. In table 1, we see that the SIREN method is the most scalable in terms of inference4 . The
naive OI framework on ∼ 1 million data points results in 1 hr on a CPU and 40 minutes on a GPU.
However, the NerF algorithm predicts the same amount of data in 30 seconds on a CPU and 5 seconds
on a GPU. This results in ×120 speedup and ×480 speedup respectively. This is expected due tothe time complexity of matrix inversions, O(N 3 ), versus matrix multiplications, O(N ). In addition,
the OI method requires one to condition on the observations which results in an additional time andmemory footprint that inhibits learning with more observational data.

5

Conclusions/Future Work

Discussion. In this work, we introduced Neural Fields as a viable interpolation method for sea surface
height. We demonstrated that we were able to match the results of the standard Optimal Interpolation
algorithm by training on observations. Not only did we get results that were statistically comparable
to the standard DUACS, we also achieve inference speeds and memory constraints that are viable
even with a modest computational budget. Neural networks are often only good when we have alarge amount of training data available. However, this work has made a case for NerFs coupled withaltimetry data as a candidate algorithm for interpolation at scale.
Future Work. Aside from the more detailed ablation studies related to noise and sampling density,
there is much to be done to make these models more appealing and more trustworthy for the broader
community. They lack immediate interpretability because they are fully parameterized machinesfrom data. Techniques such as Explainable AI [33] or Physics Informed Neural Nets [34] could be ofuse to understand and constrain the NN predictions. Quantifying the predictive uncertainty is anothermain future direction, in fact, there are extensions that have been explored in the literature [35, 36,37]. Lastly, we observed that the bulk of the computational limit is training these NNs. Under acomputational budget, faster training strategies [38] and better hardware would be necessary to makethese methods scale to global datasets with billions of data points.

Ethical Considerations
The spatial and temporal resolution of the altimetry datasets used in this study is too coarse to affectthe privacy of any individual and we only consider resolutions of ∼ 100 − 200km. In addition, alldatasets used were publically available via the Copernicus Marine Service. However, we believe
our method would be useful in higher resolution scenarios (∼ 1 − 50km) which would be touching
upon the territory of privacy in certain regions of the globe. The dataset itself is unlikely to enable
activities that are harmful for the environment or individuals. However, if we consider that one could
4

Note that we did not showcase the training times for each method because our OI is unoptimized and the
DUACs methods is closed-source. However, the SIREN method is a coordinate-based method which can take along time to train. Our model took an average of 6 hours to train not including the hyper-parameter sweeps. We
recommend multi-scale training strategies and multi-node GPU architectures to get drastic speedups [25].

4Observations

NAIVE OI

DUACS

SIREN

Figure 1: Snapshots of the daily aggregated altimetry observations (top row), the SSH field (2nd row),
the norm of the gradient of SSH field (3rd row) and the Laplacian of the SSH field (4th row) over the
Gulf-stream at time step 20-01-2017. Each column showcases each of the methods presented in thiswork: 1) Naive implementation of OI, 2) the OI in production (DUACS), and 3) the SIREN NerF.
The last row showcases the spectrum for each of the variables: bottom row (left) is SSH, bottom row(middle) is the Kinetic energy and bottom row (left) is the Vorticity.

5further develop these models for interpolating finer resolutions, one could imagine some downstream
potential adverse affects on the environmental applications like overfishing or security concerns. In
addition, these models are not limited to geophysical variables so one could easily apply these to landvariables or human-made objects. So one should always take care when working with high spatial
and temporal resolution maps.

Broader Impact
The theme of interpolation is present in many applied communities with different names, e.g. kriging
in ecology/hydrology, Optimal interpolation in oceanography, and Gaussian processes in statistics.
We hope that this work bridges this gap between the communities and we invite other works to try tohighlight concrete ways that machine learning and classic physics have commonalities.
In the oceanography community in particular, we especially hope to see more adoption of machinelearning methods for interpolation. DUACS is ultimately a closed-system so the wider scientificcommunity does not have access to the algorithm. Our OI baseline hopefully unveils some of the finer
details of the method. However, in general, the standard OI methods used in the applied community
cannot keep up with the massive influxes of observations we receive. So this work is a first step indemonstrating that neural networks (in particular NerFs) are a viable, simpler, and scalable alternative.

Acknowledgments and Disclosure of Funding
This work was supported by ANR (French National Research Agency, project number ANR-17CE01-0009-01) and CNES through the SWOT Science Team program. This work was also supported
by LEFE program (LEFE MANU project IA-OAC), CNES (OSTST DUACS-HR and SWOT STDIEGO) and ANR Projects Melody (ANR-19-CE46-0011) and OceaniX (ANR-19-CHIA-0016). It
benefited from HPC and GPU resources from GENCI-IDRIS (Grant 2021-101030). We would like tothank Weights & Biases for their support and for providing us with an academic license to use theirexperiment tracking tools for our work.

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