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data_assimilation_equations.html
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>Data Assimilation Mechanics</title>
<script>
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inlineMath: [['$', '$'], ['\\(', '\\)']],
displayMath: [['$$', '$$'], ['\\[', '\\]']]
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<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
<style>
body {
font-family: 'Segoe UI', Roboto, Helvetica, Arial, sans-serif;
background-color: #f4f7f6;
color: #333;
margin: 0;
padding: 40px;
display: flex;
justify-content: center;
}
.slide-container {
width: 100%;
max-width: 1400px;
background-color: #ffffff;
border-radius: 12px;
box-shadow: 0 10px 30px rgba(0,0,0,0.1);
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grid-template-columns: 1fr 1fr;
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font-size: 110%;
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.description {
font-size: 15px;
line-height: 1.6;
color: #555;
}
ul {
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padding-left: 20px;
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li {
margin-bottom: 8px;
}
strong {
color: #2c3e50;
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</style>
</head>
<body>
<div class="slide-container">
<div class="header">
<h1>The Mathematical Mechanics of Data Assimilation</h1>
<p>Bridging the gap between the Digital Twin forecast and Physical Twin reality</p>
</div>
<div class="grid">
<div class="panel full-width">
<h2>1. The Core DA Loop: Forecast & Update</h2>
<div class="description">
Data Assimilation is a continuous two-step process. First, we predict the future state using our physical models (or fast FNO surrogates). Second, we correct that prediction using real-time sensor observations.
</div>
<div style="display: flex; gap: 20px;">
<div style="flex: 1;">
<div class="equation-box">
<strong>Step 1: The Forward Forecast</strong><br>
$$ \mathbf{x}_{f, k} = \mathcal{M}(\mathbf{x}_{a, k-1}) + \mathbf{w}_{k} $$
</div>
<ul class="description">
<li><strong>$\mathcal{M}(\cdot)$</strong>: Non-Linear Forward Operator (Adamantine / FNO)</li>
<li><strong>$\mathbf{w}_{k}$</strong>: Process noise (model error)</li>
</ul>
</div>
<div style="flex: 1;">
<div class="equation-box">
<strong>Step 2: The State Update (Analysis)</strong><br>
$$ \mathbf{x}_a = \mathbf{x}_f + \mathbf{K} (\mathbf{y} - \mathbf{H} \mathbf{x}_f) $$
</div>
<ul class="description">
<li><strong>$\mathbf{K}$</strong>: Kalman Gain (Weighting Matrix)</li>
<li><strong>$(\mathbf{y} - \mathbf{H} \mathbf{x}_f)$</strong>: The "Innovation" (Sensor Data minus Expected Data)</li>
</ul>
</div>
</div>
</div>
<div class="panel warning">
<h2>2. The Error Covariance Problem ($\mathbf{P}_f$)</h2>
<div class="description">
To find the optimal weight ($\mathbf{K}$), we must track the uncertainty and spatial correlations of every variable in the mesh. The theoretical definition relies on the unknown "true" state:
</div>
<div class="equation-box">
$$ \mathbf{P}_f = \mathbb{E}[(\mathbf{x}_f - \mathbf{x}_{true})(\mathbf{x}_f - \mathbf{x}_{true})^T] $$
</div>
<div class="description">
For a thermal mesh with $N$ degrees of freedom, this creates a massive, computationally impossible $N \times N$ matrix:
</div>
<div class="equation-box">
$$ \mathbf{P}_f = \begin{bmatrix}
\sigma_{1}^2 & \dots & \text{cov}(1,N) \\
\vdots & \ddots & \vdots \\
\text{cov}(N,1) & \dots & \sigma_{N}^2
\end{bmatrix} \in \mathbb{R}^{N \times N} $$
</div>
</div>
<div class="panel math-heavy">
<h2>3. Deriving $\mathbf{K}$ and the EnKF Solution</h2>
<div class="description">
We find $\mathbf{K}$ by defining the Analysis Error Covariance ($\mathbf{P}_a$) and taking the derivative of its trace to minimize total system uncertainty:
</div>
<div class="equation-box">
$$ \frac{\partial \text{Tr}(\mathbf{P}_a)}{\partial \mathbf{K}} = 0 \implies \mathbf{K} = \mathbf{P}_f \mathbf{H}^T (\mathbf{H} \mathbf{P}_f \mathbf{H}^T + \mathbf{R})^{-1} $$
</div>
<div class="description" style="margin-top: 20px;">
<strong>The EnKF Solution:</strong> Because $\mathbf{x}_{true}$ is unknown and $\mathbf{P}_f$ is too large, the Ensemble Kalman Filter approximates it dynamically using the sample covariance of $N_{ens}$ parallel model runs:
</div>
<div class="equation-box" style="background-color: #e8f4f8; border: 1px solid #b3e0ff;">
$$ \mathbf{P}_f \approx \frac{1}{N_{ens}-1} \sum_{i=1}^{N_{ens}} (\mathbf{x}_{f}^{(i)} - \bar{\mathbf{x}}_f)(\mathbf{x}_{f}^{(i)} - \bar{\mathbf{x}}_f)^T $$
</div>
</div>
</div>
</div>
</body>
</html>
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