The Complete Data Fusion (CDF) is a powerful method that combines multiple independent atmospheric measurements into a single, improved estimate of a vertical profile. By merging observations from different instruments or satellites, CDF produces fused products with reduced uncertainty and enhanced vertical resolution.
Concept and Motivation
When multiple independent measurements of the same atmospheric parameter are available at the same location, each carries unique information with different vertical sensitivities and error characteristics. Rather than choosing a single “best” measurement, CDF systematically fuses all measurements to construct an optimal combined product.
The method is founded on optimal estimation theory (Rodgers, 2000). We assume we have N independent, simultaneous measurements of an atmospheric vertical profile at a specific geolocation. Each measurement is retrieved using optimal estimation, producing:
- State vector \widehat{\mathbf{x}}_{i} (i = 1, 2, …, N) — the retrieved profile
- Averaging Kernel Matrix \mathbf{A}_{i} — characterizing the vertical sensitivity
- Noise Covariance Matrix \mathbf{S}_{ni} — quantifying measurement uncertainty
Each measurement provides an independent estimate of the true profile on its own vertical grid, but with limited vertical resolution. CDF fuses all N estimates into a single product with superior characteristics: lower total error, higher degrees of freedom, and improved information content.
The CDF Formulation
The fused profile \mathbf{x}_{f} is obtained by minimizing a cost function that balances fidelity to each input measurement against deviation from a priori information:
where:
- \mathbf{x} — the fused profile (to be optimized)
- \mathbf{x}_{a}, \mathbf{S}_{a} — a priori profile and its covariance, constraining the solution
- \mathbf{\alpha}_{i} — a transformed measurement vector (defined below)
- \mathbf{S}_{ni} — noise covariance matrix of measurement i
The transformed measurement vector is defined as:
where \mathbf{x}_{ai} is the a priori profile used in the retrieval of measurement i.
Why CDF Matters
CDF offers several advantages over using a single measurement:
- Lower uncertainty — Fusion reduces noise and smoothing errors by combining independent measurements
- Enhanced vertical resolution — The fused product often has more degrees of freedom than individual input products
- Better information content — Combining different instruments with different sensitivities captures features that no single instrument can resolve
- Multi-instrument synergy — Satellites with complementary vertical sensitivities can be fused to improve coverage across the entire atmosphere
Scope of This Section
This section of the website covers the theory and practice of the CDF algorithm:
- CDF Prerequisites — Requirements that input products must satisfy
- CDF Formulas — Mathematical formulations for different measurement configurations (perfect coincidence, with interpolation, with coincidence errors)
- Special Cases — Fusion with total columns and multi-parameter (multi-target) retrievals
- Extensions — Extended algorithms for non-overlapping grids and critical numerical implementation aspects
- Non-OE and incomplete products — Products retrieved without an a priori covariance (Tikhonov, Levenberg–Marquardt) or with incomplete files: what the CDF needs, how it is reconstructed, which checks replace the auto-consistency test
- CDF Tests — Validation tests to ensure correct application
- To Be Studied — Open research questions and future developments
Reference: The CDF method was first described in Ceccherini et al. (2015) and has since been extended and refined (Ceccherini et al., 2018, 2022). This website documents the current CDF(2022) formulation, which is recommended for operational use.