Dataset Documentation

[jejjohnson]

Documentation for use cases associated with the dataset in #4

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Use Cases

This is a major feature so it would be worth it to showcase a few examples for how this can be used for most problems. Here is an outline of some ways this can be used

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Coordinate-Based Interpolation

This is useful for Interpolation using Neural Fields (a.k.a. Implicit NNs, Coordinate-Based NNs). Some methods include NerFs or even GPs.

Model Formulation

$$ \mathbf{y} = \boldsymbol{f_\theta}(\mathbf{x}) + \epsilon $$

Input Variables:

$$\mathbf{x} \in \mathbb{R}^{D_x}$$

The $\mathbf{x}$ is a 3-dimension vector with of [lat,lon,time].

Output Variables:

$$\mathbf{y} \in \mathbb{R}^{D_y}$$

The output $\mathbf{y}$ is a 2-dimensional vector with of [ssh, sst].

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Full-Field Interpolation

This is useful where we want the full field to go into our model

Model Formulation:

$$ \begin{aligned} \mathbf{Z} &= \boldsymbol{F_e}(\mathbf{X}) \\ \hat{\mathbf{X}} &= \boldsymbol{F_d}(\mathbf{Z}) \end{aligned} $$

Input Variables:

$$\mathbf{X} \in \mathbb{R}^{H\times W\times T\times V}$$

The $\mathbf{X}$ is a multi-dimensional array of rank 4 of size[lat x lon x time x variables].

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Example Applications

• Interpolation: Implicit/Coordinate-Based methods, e.g. GPs, NerFs
  • x=[lat,lon,time]
  • f=[ssh,sst]
• Interpolation: Explicit/Coordinate-Based methods, e.g. CNN
  • f=u(lat x lon x time x var)
  • x=(lat,lon,time,mask)
• Inverse Problems: Operator Methods, e.g. 4DVar
  • f=u_sim(lat x lon x time x vars)
  • y=u_obs(lat x lon x time x vars)
  • x=(lat,lon,time,mask)
• Predictions Problems: Functional Methods, e.g. GPs, NerFs
  • f=[var1,var2,...]
  • c=[var1,var2,...]
  • x=[lat,lon,time]
• Predictions Problems: Operator Methods, e.g. FNO, MNO
  • f=u(lat x lon x time x var)
  • g=a(lat x lon x time x var)
  • x=(lat,lon,time,mask)