ML4EO - TOC Apps - Research Notebook

Application Track¶

Overview¶

Extreme Values Modeling

The first application track we have are looking at extreme values. This allows us to slowly introduce some core concepts that will be useful in the later application tracks.

\begin{aligned} D_s &= \left[ \text{Altitude, Longitude, Latitude}\right] \\ \Omega &= \left[ \text{Unstructured Weather Stations}\right] \\ \Delta t &= \left\{ \text{Daily, Hourly} \right\} \\ D_y &= \left[ \text{Temperature, Precipitation, Wind Speed}\right] \end{aligned}

(1)

Sea Surface Mapping

The second application track is working with sea surface mapping of variables like temperature, height, and ocean colour. In this setting, we will deal with how to transform between different geometries like the alongtrack satellite observations and the regular gridded applications. These measurements feature a lot of missing values, so we will learn how to deal with this. The bulk of this application will involve learning-to-learn, whereby we find various ways to learn prior models

\begin{aligned} D_s &= \left[ \text{Depth, Longitude, Latitude}\right] \\ \Omega &= \left[ \text{Irregular Along Track, Regular Gridded}\right] \\ \Delta t &= \left\{ \text{Daily, Hourly} \right\} \\ D_y &= \left[ \text{Temperature, Height, Salinity, Colour}\right] \end{aligned}

(2)

Instrument-2-Instrument Translation

In this third application track, we will investigate how we can apply a method to translate one type of satellite to another type of satellite. Here, we have to do all tasks including interpolation, X-casting, and variable transformation. The interpolation component is due to the non-consistent geometry amongst the satellites. The X-casting component comes from the We also have to deal with a

\begin{aligned} D_s &= \left[ \text{Height, Longitude, Latitude}\right] \\ \Omega &= \left[ \text{Irregular Polar-Orbiting Satellite, Regular Geostationary Satellite}\right] \\ \Delta t &= \left\{ \text{Days, 15 Minutes} \right\} \\ D_y &= \left[ \text{Spectral Signature}\right] \end{aligned}

(3)

Break Down¶

Here, we have a breakdown of how we will structure the application section. We will analyse the data which will be used to learn. Then, we will do whatever interpolation or discretization necessary to transform it to the form we want. Then, we will do some parameter estimation to learn a representation of the system. Lastly, we will do state estimation using whichever form is

Data Acquisition - GPs, Simulations, L1 Data, L2 Data, L3 Data
Interpolation & Discretization - Histograms, Nearest Neighbors, Gaussian Processes
Learning Priors - Gaussian Processes, L2 Data, L3 Data, Simulations
Parameter Estimation - Representation Learning, Dynamical Systems
State Estimation - 3DVar, Strong-Constrained 4D Var, Weak-Constrained 4DVar, Deep Equillibrium

¶

Walk-Through¶

For each of these sections

Download EO Data¶

Examples: Sea Surface Height/Temperature/Salinity, Satellite Data, Weather Stations

\begin{aligned} \text{LS Data}: && && &\left[\text{GP Priors}, \text{Simulations} \right] \\ \text{L1 Data}: && && &\left[\text{Weather Stations}, \text{ARGO Floats}, \text{Satellite}\right] \\ \text{L2 Data}: && && &\left[\text{Interpolated Data} \right] \\ \text{L3 Data}: && && &\left[\text{Reanalysis} \right] \\ \text{L4 Data}: && && &\left[\text{Reanalysis} \right] \\ \end{aligned}

(4)

Discretizations¶

Examples: Histogram, Nearest Neighbours, Radius Neighbours, Kernel Density Estimation

p(y_{1:T},\boldsymbol{\theta}) = \prod_{n=1}^{N_T}p(y_t|\theta)p(\theta|s_n, t_n)

(5)

Functional Regression¶

Examples: Nearest Neighbors, Gaussian Processes, Neural Fields

p(y_{1:N},s_{1:N},t_{1:N},\boldsymbol{\theta}) = p(\theta)\prod_{n=1}^{N_T}p(y_n|s_n, t_n,\theta)

(6)

Param. Est. - GP Priors¶

Examples: RBF, Matern

\begin{aligned} \text{Joint}: && && p(f, \alpha, z, \theta) &= p(f|z)p(z|\alpha,\theta)p(\theta) \\ \text{Posterior}: && && q(\alpha, z, f, \theta) &= q(\alpha) \prod_{n=1}^N q(z|f,\alpha) \\ \end{aligned}

(7)

State Estimation

q(y_{1:T},z_{1:T}^*,\alpha^*,\theta) = q(\theta^*)q()

(8)

Param. Est. - Simulations¶

Param. Est. - Spatial Field Priors¶

\begin{aligned} \text{Joint}: && && p(y,z,\theta) &= p(\theta)\prod_{n=1}^Np(y_n|z_n)p(z_n|\theta) \\ \text{Posterior}: && && q(z,y,\theta) &= q(\theta)\prod_{n=1}^N q(z|y,\theta) \end{aligned}

(9)

State Estimation¶

q(z,y,\theta) = q(z|\theta,\phi^*)q(\theta|\phi^*)

(10)

Equilibrium Models¶

z^* = f(z^*, y, x, \theta)

(11)

Param. Est. - Strong-Constrained Dynamical Priors¶

\begin{aligned} \text{Joint}: && && p(y_{1:T},z_{0:T},\theta) &= p(\theta)p(z_0) \prod_{n=1}^N p(y_t|z_t,\theta) \\ \text{Variational}: && && q(z_{0:T},y_{1:T},\theta) &= q(\theta) q(z_0) \prod_{n=1}^N q(z_t|y_t,\theta) \end{aligned}

(12)

State Estimation¶

Param. Est. - Weak-Constrained Dynamical Prior¶

\begin{aligned} \text{Joint}: && && p(y_{1:T},z_{0:T},\theta) &= p(\theta)p(z_0) \prod_{n=1}^N p(y_t|z_t)p(z_t|z_{t-1},\theta) \\ \text{Variational}: && && q(z_{0:T},y_{1:T},\theta) &= q(\theta) q(z_0) \prod_{n=1}^N q(z_t|z_{t-1},y_t) q(z_t|z_{t-1},\theta) \end{aligned}

(13)

State Estimation¶

q(y_{1:T},z_{1:T}^*,\theta^*) = q(\theta^*)q(z_0^*) \prod_{t=1}^Tq(z_t|z_{t-1},y_t)

(14)

We download some

Applications¶

Extreme Values. Here, we learn how we can estimate the parameters of a model for extreme values.

Ocean Surface Height. We will learn how to create mapping products for Sea Surface Height (SSH) and Sea Surface Temperature (SST).

Instrument-2-Instrument. We will learn how to map one set of spatio-spectral channels to a different spatio-spectral channels.

Application I - Extremes¶

In this application, we are interested in modeling temperatures, and in particular, temperature extremes.

Datasets. We will use some standard datasets that are found within the AEMET database.

\begin{aligned} \mathcal{D} &= \left\{(t_n, \mathbf{s}_n), \mathbf{x}_n, \mathbf{y}_n \right\}_{n=1}^{N}, && && \mathbf{y}_n\in\mathbb{R}^{D_y} && \mathbf{x}_n\in\mathbb{R}^{D_y} && \mathbf{s}_n\in\mathbb{R}^{D_s} && t_n\in\mathbb{R}^{+} \end{aligned}

(15)

Covariante, $x_n$
- Global Mean Surface Temperature Anomaly (GMSTA)
Measurement, $y_n$ :
- Daily Mean Temperature (TMax)
- Daily Maximum Temperature (TMean)

Extreme Events.

\mathcal{D} = \left\{y_n\right\}_{n=1}^N

(16)

We will use some standard techniques to extract the extreme events from the time series.

Block Maximum (BM)
Peak-Over-Threshold (POT)
Temporal Point Process (TPP)

Data Likelihood.

y \sim p(y|z)

(17)

We need to decide which likelihood we want to use. For the mean data, we can assume that it follows a Gaussian distribution or a long-tailed T-Student distribution. However, for the maximum values, we need to assume an EVD like the GEVD, GPD or TPP.