This page collects the key references for the Complete Data Fusion algorithm, organized by topic and annotated to guide the reader. The focus is on the algorithm itself — its mathematical foundations, successive improvements, and applications to real atmospheric data. References specific to individual input datasets are documented in the respective Tested Datasets pages.

1. Mathematical and Methodological Foundations

The CDF algorithm is built on optimal estimation theory. The following two references provide the mathematical background that underpins the entire approach.

Rodgers, C.D. — Inverse Methods for Atmospheric Sounding: Theory and Practice, Vol. 2 of Series on Atmospheric, Oceanic and Planetary Physics, World Scientific, Singapore, 2000.

The standard reference for optimal estimation in atmospheric remote sensing. Introduces the concepts of state vector, averaging kernel, error covariance matrices, degrees of freedom for signal, and the general inversion framework on which CDF inputs are defined. Essential background for understanding CDF prerequisites.

Kalman, R.E. — A New Approach to Linear Filtering and Prediction Problems, J. Basic Eng.-T. ASME, 82, 35–45, 1960. DOI: 10.1115/1.3662552

The Kalman filter paper. Provides the optimal linear estimation framework that is closely related to the CDF formulation: the CDF solution can be seen as an extension of Kalman-style merging to the atmospheric profile setting, using averaging kernels rather than state-transition matrices.

2. The CDF Algorithm — Original Formulation and Improved Formula

These two papers define the two formulations of the CDF algorithm. They should be read in sequence: the first introduces the concept and the original solution; the second replaces it with a numerically superior formula.

Ceccherini, S., Carli, B., and Raspollini, P. — Equivalence of data fusion and simultaneous retrieval, Opt. Express, 23, 8476–8488, 2015. DOI: 10.1364/OE.23.008476

This is the founding paper of the CDF algorithm — CDF(2015). It derives the formula for the optimal fusion of N independent atmospheric retrievals, and proves that the result is mathematically equivalent to a simultaneous retrieval of all measurements. The paper establishes the theoretical basis of the approach and the structure of the fused state vector, averaging kernel, and error covariance matrices.

Ceccherini, S., Zoppetti, N., and Carli, B. — An improved formula for the complete data fusion, Atmos. Meas. Tech., 15, 7039–7048, 2022. DOI: 10.5194/amt-15-7039-2022

This is the recommended CDF formulation — CDF(2022). The key improvement is the replacement of the noise covariance matrix Sni (which is often singular and requires pseudo-inversion) with the total error covariance matrix Si = Sni + Ssi, which is always invertible. This eliminates the ambiguity in choosing SVD thresholds and greatly improves numerical robustness. The paper also derives the extended formulas for configurations with interpolation and coincidence errors (corresponding to Configurations B and C in this website), superseding the corresponding CDF(2015) formulas from Ceccherini et al. (2018).

3. Extensions, Refinements, and Simulation Studies

These papers extend the CDF algorithm to handle progressively more complex scenarios — interpolation and coincidence errors, fusion of total columns with profiles, multi-target retrievals, feasibility studies on realistic satellite configurations, and two-dimensional tomographic fields — and validate each extension on simulated data. The section closes with a methodological note on the correct use of the algorithm.

Ceccherini, S., Carli, B., Tirelli, C., Zoppetti, N., Del Bianco, S., Cortesi, U., Kujanpää, J., and Dragani, R. — Importance of interpolation and coincidence errors in data fusion, Atmos. Meas. Tech., 11, 1009–1017, 2018. DOI: 10.5194/amt-11-1009-2018

Derives the CDF(2015) formulas for the general case where measurements are retrieved on different vertical grids (interpolation error) and/or are not perfectly coincident in space and time (coincidence error). Demonstrates through simulated experiments that ignoring these error sources leads to overoptimistic fused products, and shows how to correctly account for them. The CDF(2022) counterpart of these results is given in Ceccherini et al. (2022).

Tirelli, C., Ceccherini, S., Zoppetti, N., Del Bianco, S., Gai, M., Barbara, F., Cortesi, U., Kujanpää, J., Huan, Y., and Dragani, R. — Data fusion analysis of Sentinel-4 and Sentinel-5 simulated ozone data, J. Atmos. Ocean. Tech., 37, 573–587, 2020. DOI: 10.1175/JTECH-D-19-0063.1

Introduces and applies the extension of CDF to the fusion of total-column measurements with vertical profile products. Derives the formulas for handling column-type inputs (where the averaging kernel reduces to a row vector and covariances to scalars) and validates the approach using simulated Sentinel-4 and Sentinel-5 ozone data. Demonstrates the expected improvement in information content when combining a geostationary total column sounder with a LEO profiler.

Tirelli, C., Ceccherini, S., Zoppetti, N., Del Bianco, S., and Cortesi, U. — Generalization of the complete data fusion to multi-target retrieval of atmospheric parameters and application to FORUM and IASI-NG simulated measurements, J. Quant. Spectrosc. Radiat. Transfer, 276, 107925, 2021. DOI: 10.1016/j.jqsrt.2021.107925

Extends the CDF(2015) formulation to multi-target retrievals (MTRs), where the state vector of each input product contains multiple atmospheric species (e.g., temperature and several trace gases simultaneously). This generalization is necessary for modern operational products from IASI and future missions such as FORUM and IASI-NG, which routinely retrieve multiple parameters in a single fit. Validates the MTR-CDF on simulated measurements and quantifies the information gain from combining FORUM and IASI-NG.

Zoppetti, N., Ceccherini, S., Carli, B., Del Bianco, S., Gai, M., Tirelli, C., Barbara, F., Dragani, R., Arola, A., Kujanpää, J., van Peet, J.C.A., van der A, R., and Cortesi, U. — Application of the Complete Data Fusion algorithm to the ozone profiles measured by geostationary and low-Earth-orbit satellites: a feasibility study, Atmos. Meas. Tech., 14, 2041–2053, 2021. DOI: 10.5194/amt-14-2041-2021

A feasibility study that applies CDF to simulated ozone profiles representative of three satellite instruments — GOME-2 (Metop, LEO), IASI (Metop, LEO), and SEVIRI (MSG, geostationary). Demonstrates the practical workflow from input data selection and completeness checks, through auto-consistency testing, to the production and characterization of fused profiles. Discusses the improvement in degrees of freedom for signal achieved by combining nadir LEO and geostationary observations.

Tirelli, C., Ceccherini, S., Del Bianco, S., Funke, B., Höpfner, M., Cortesi, U., and Raspollini, P. — Extension of the Complete Data Fusion algorithm to tomographic retrieval products, Atmos. Meas. Tech., 18, 5619–5636, 2025. DOI: 10.5194/amt-18-5619-2025

Extends CDF from one-dimensional vertical profiles to two-dimensional atmospheric fields, enabling the fusion of nadir and limb measurements with their intrinsically different spatial geometries. Validated on simulated ozone data from IASI-NG (nadir) and CAIRT (limb), demonstrating gains in degrees of freedom, Shannon information content, and horizontal spatial resolution. Opens the way to combining the vertical resolution of limb sounders with the horizontal coverage of nadir instruments.

Ceccherini, S. — Comment on “Synergetic use of IASI profile and TROPOMI total-column level 2 methane retrieval products” by Schneider et al. (2022), Atmos. Meas. Tech., 15, 4407–4410, 2022. DOI: 10.5194/amt-15-4407-2022

A methodological comment that identifies an incorrect application of the data fusion concept in the literature. Points out that the approach in Schneider et al. does not constitute a proper data fusion in the CDF sense, because the required mathematical equivalence between data fusion and simultaneous retrieval is not preserved. Useful reading for understanding what CDF is not, and for avoiding analogous errors in other satellite-synergy studies.

5. Applications to Real Atmospheric Data

The following paper documents the first production and independent validation of an extended CDF ozone dataset derived from real satellite observations.

Guidetti, L., Brattich, E., Ceccherini, S., Hegglin, M.I., Raspollini, P., Tirelli, C., Zoppetti, N., and Cortesi, U. — Development and validation of a new ozone dataset using Complete Data Fusion of MIPAS and IASI observations: a step towards understanding stratospheric ozone intrusions, Atmos. Meas. Tech., 19, 167–184, 2026. DOI: 10.5194/amt-19-167-2026

The first extended CDF dataset produced and validated on real data. Applies CDF to fuse MIPAS (Envisat, limb) and IASI (Metop, nadir) ozone profiles over a multi-year period (2008–2011), producing a new ozone dataset that is then validated against independent reference measurements. Demonstrates that the fused product achieves better vertical resolution and reduced uncertainty compared to either input dataset alone. Also illustrates the scientific use case: the improved dataset allows the detection and characterization of stratospheric ozone intrusions that are not resolved by either instrument individually.