# On the Accuracy and Consistency of Quintuple Collocation Analysis of In Situ, Scatterometer, and NWP Winds

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Data

## 3. Methods

#### 3.1. Multiple Collocation Formalism

#### 3.2. Linearization of the Covariance Equations

#### 3.3. Iterative Solution

^{2}s

^{−2}, and collocations are excluded whenever

#### 3.4. Representativeness Errors

^{2}s

^{−2}for the ASCATs.

#### 3.5. Precision Estimate

## 4. Results and Discussion

#### 4.1. Number of Solvable Models

#### 4.2. Calibration Coefficients and Error Standard Deviations

#### 4.3. Statistical Accuracy and Model Spread

^{2}s

^{−2}for $u$ and about 18 m

^{2}s

^{−2}for $v$. Its accuracy is about 0.2 m

^{2}s

^{−2}for both $u$ and $v$, and the same accuracy can be expected for the other term. If the terms were not correlated, the accuracy in the error variance would be about 0.4 m

^{2}s

^{−2}, but the correlations keep the accuracy down by two orders of magnitude to the values shown in Figure 3. Nevertheless, the observation error variances are obtained as the difference between two large numbers and are, therefore, sensitive to statistical noise. The same applies to the observation error covariances.

#### 4.4. Error Covariances

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Definition and Solution of a Particular Model

## Appendix B. Relation between Average of Model Solutions and Least-Squares Solution

## References

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**Figure 1.**Difference between the spatial variance of ASCAT-A, ASCAT-B, and ScatSat and that of ECMWF, $\Delta V\left(s\right)$, as a function of $s$ for the zonal wind component $u$ (

**a**) and for the meridional wind component $v$ (

**b**).

**Figure 2.**Quintuple collocation results as a function of the scale at which the representativeness errors are calculated for the calibration biases for $u$ (

**a**), the calibration biases for $v$ (

**b**), the calibration scalings for $u$ (

**c**), the calibration scalings for $v$ (

**d**), the error variances for $u$ (

**e**), and the error variances for $v$ (

**f**), all averaged over all models.

**Figure 3.**Quintuple collocation results for the accuracy (black curves) and the model spread (gray curves) as a function of the scale at which the representativeness errors are evaluated for the calibration biases for $u$ (

**a**), the calibration biases for $v$ (

**b**), the calibration scalings for $u$ (

**c**), the calibration scalings for $v$ (

**d**), the error variances for $u$ (

**e**), and the error variances for $v$ (

**f**).

**Figure 4.**Accuracy of the model average (black curves) and the least-squares solution (gray curves) for the observation error variances in $u$ (

**a**) and those in $v$ (

**b**).

**Figure 5.**Error covariances and their accuracies from the model average for $u$ (

**a**), the model average for $v$ (

**b**), the least-squares solution for $u$ (

**c**), and the least-squares solution for $v$ (

**d**).

**Table 1.**Number of observing systems, number of equations, number of models, and number of solvable and unsolvable models.

Observing Systems | Off-Diagonal Equations | Models | Solvable | Unsolvable |
---|---|---|---|---|

3 | 3 | 1 | 1 | 0 |

4 | 6 | 15 | 12 | 3 |

5 | 10 | 252 | 162 | 90 |

6 | 15 | 5005 | 2530 | 2475 |

7 | 21 | 116,280 | 45,615 | 70,665 |

8 | 28 | 3,108,105 | 937,440 | 2,170,665 |

9 | 36 | 94,143,280 | 21,685,132 | 72,458,148 |

Observing System | ${\mathit{\sigma}}_{\mathit{u}}$ | $\mathbf{s}\mathbf{t}\mathbf{d}\left({\mathit{\sigma}}_{\mathit{u}}\right)$ | ${\mathit{\sigma}}_{\mathit{v}}$ | $\mathbf{s}\mathbf{t}\mathbf{d}\left({\mathit{\sigma}}_{\mathit{v}}\right)$ |
---|---|---|---|---|

(m/s) | (m/s) | (m/s) | (m/s) | |

buoys | 0.914 | 0.017 | 1.063 | 0.020 |

ASCAT-A | 0.372 | 0.022 | 0.505 | 0.029 |

ASCAT-B | 0.390 | 0.025 | 0.444 | 0.020 |

ScatSat | 0.683 | 0.018 | 0.594 | 0.021 |

ECMWF | 0.845 | 0.017 | 1.006 | 0.021 |

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**MDPI and ACS Style**

Vogelzang, J.; Stoffelen, A.
On the Accuracy and Consistency of Quintuple Collocation Analysis of In Situ, Scatterometer, and NWP Winds. *Remote Sens.* **2022**, *14*, 4552.
https://doi.org/10.3390/rs14184552

**AMA Style**

Vogelzang J, Stoffelen A.
On the Accuracy and Consistency of Quintuple Collocation Analysis of In Situ, Scatterometer, and NWP Winds. *Remote Sensing*. 2022; 14(18):4552.
https://doi.org/10.3390/rs14184552

**Chicago/Turabian Style**

Vogelzang, Jur, and Ad Stoffelen.
2022. "On the Accuracy and Consistency of Quintuple Collocation Analysis of In Situ, Scatterometer, and NWP Winds" *Remote Sensing* 14, no. 18: 4552.
https://doi.org/10.3390/rs14184552