## Quality quantifier of indirect measurements |

Optics Express, Vol. 20, Issue 5, pp. 5151-5167 (2012)

http://dx.doi.org/10.1364/OE.20.005151

Acrobat PDF (1059 KB)

### Abstract

A quality quantifier, referred to as *measurement quality quantifier* (MQQ), is proposed for indirect measurements. It satisfies the property that the MQQ of the data fusion of two or more independent measurements is the sum of the MQQs of the individual measurements and can also be determined in absolute terms for ill-posed problems. It is calculated from the covariance and Jacobian matrices of the observations, but the same result is also obtained using the covariance and averaging kernel matrices of the retrieved quantities. In the case of measurements of a continuous distribution a quantifier that provides the information distribution can be derived from the MQQ. The proposed quantifiers are herewith used for the quality assessment of atmospheric ozone measurements performed by IASI and MIPAS instruments.

© 2012 OSA

## 1. Introduction

2. S. Ceccherini, P. Raspollini, and B. Carli, “Optimal use of the information provided by indirect measurements of atmospheric vertical profiles,” Opt. Express **17**(7), 4944–4958 (2009). [CrossRef] [PubMed]

4. S. Ceccherini, U. Cortesi, S. Del Bianco, P. Raspollini, and B. Carli, “IASI-METOP and MIPAS-ENVISAT data fusion,” Atmos. Chem. Phys. **10**(10), 4689–4698 (2010). [CrossRef]

*additivity property*). In this paper, starting from these two basic properties we identify a parameter, that we call

*measurement quality quantifier*(MQQ), and evaluate its performances.

*Shannon information content*[1]. This quantifier is used in the framework of the optimal estimation and evaluates the information gain brought by the measurement with respect to the a priori information. Consequently the value of the Shannon information content is a relative quality quantifier depending on the covariance matrix (CM) of the a priori profile and it cannot be adapted to quantify the absolute information coming from the observations in case of ill-posed problems. Furthermore, as illustrated in [4

4. S. Ceccherini, U. Cortesi, S. Del Bianco, P. Raspollini, and B. Carli, “IASI-METOP and MIPAS-ENVISAT data fusion,” Atmos. Chem. Phys. **10**(10), 4689–4698 (2010). [CrossRef]

5. M. Carlotti and L. Magnani, “Two-dimensional sensitivity analysis of MIPAS observations,” Opt. Express **17**(7), 5340–5357 (2009). [CrossRef] [PubMed]

*information load*. It describes the information brought by the observations with respect to a set of target parameters and we will show that it is closely linked with the MQQ introduced in this paper.

*grid normalized MQQ*. Because of this property, the value of this quantifier coincides with that obtained for an infinitely small grid step. Consequently the grid normalized MQQ has the property of providing the measurement quality of the vertical profile when it is represented by a continuous function of altitude.

6. C. Clerbaux, A. Boynard, L. Clarisse, M. George, J. Hadji-Lazaro, H. Herbin, D. Hurtmans, M. Pommier, A. Razavi, S. Turquety, C. Wespes, and P.-F. Coheur, “Monitoring of atmospheric composition using the thermal infrared IASI/MetOp sounder,” Atmos. Chem. Phys. **9**(16), 6041–6054 (2009). [CrossRef]

7. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. Lopez-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller, and R. Zander, “MIPAS: an instrument for atmospheric and climate research,” Atmos. Chem. Phys. **8**(8), 2151–2188 (2008).

## 2. Quality quantifier for direct measurements

### 2.1 Direct measurement of a scalar quantity

*x*and look for a quantifier that properly describes the measurement quality. The standard deviation σ of the Gaussian probability distribution of the value of the measured quantity represents the error of the measurement and, therefore, as stated in the introduction, we look for a quality quantifier of the measurement that decreases when σ increases.

*x*. We indicate the results of the two measurements with

*y*

_{1}and

*y*

_{2}that are characterized by the errors σ

_{1}and σ

_{2}. On the basis of the additivity property stated in the introduction we require that the quality quantifier of the fusion of the two measurements is the sum of the quality quantifiers of the two original measurements. It is well known that the best estimation of

*x*from the two measurements is obtained determining the

*x*value that minimizes the chi-square function:

*measurement quality quantifier*(MQQ) and indicate with

*Q*.

### 2.2 Direct measurement of a vector quantity

**S**

*contains in the diagonal the MQQs of the components of*

_{x}**x**:

*n*independent measurements of

*n*different quantities and, on the basis of the additivity property, we define the MQQ associated with the measurement of

**x**as the sum of the MQQs of its components:where

*tr*(…) denotes the trace of the matrix.

**x**are dimensionally homogeneous.

## 3. Quality quantifier for indirect measurements

### 3.1 Indirect measurement of a scalar quantity

*y*we call the former

*the measurement of x*and indicate the latter as

*the observation y*.

*y*by means of an expansion of Eq. (7) at the first order:where

*k*is the derivative of

*F*(

*x*) with respect to

*x*calculated in

*x*:

*x*that minimizes

*x*and, therefore, satisfies the following equation:where

*k*

_{1}and

*k*

_{2}are the derivatives of

*F*

_{1}(

*x*) and

*F*

_{2}(

*x*) at

*y*and

_{1}*y*and from a first order expansion of this function we can calculate the error propagation from

_{2}*y*and

_{1}*y*into

_{2}*σ*:where

*y*and

_{1}*y*(neglecting the dependence of

_{2}*k*

_{1}and

*k*

_{2}on

*F*

_{1}(

*x*) and

*F*

_{2}(

*x*) around the minimum of

*g*and

_{1}*g*can be determined. Substituting these values of

_{2}*g*and

_{1}*g*in Eq. (14) we obtain the equation:

_{2}### 3.2 Indirect measurement of a vector quantity

*m*indirect measurements of a vector quantity

**x**made of

*n*components. We represent the

*m*observations with a vector

**y**characterized by the CM

**S**

*and the relationship between*

_{y}**x**and

**y**is expressed by a function

**F**(

**x**) from

**R**

*to*

^{n}**R**

*(including all the vectors made of ordered*

^{m}*n*-tuples and

*m*-tuples of real numbers, respectively):

**x**is given by the value

**x**are equal to zero we find that the value

**K**is the Jacobian matrix of

**F**(

**x**) (including the partial derivatives of

**F**(

**x**) with respect to the elements of

**x**) calculated in

*n*equations in

*n*unknown. If the

*n*equations are independent, Eq. (18) determines

**y**and we can calculate the error

**y**:where

**G**is the

*gain matrix*

*ij*-th element is

**S**

*of*

_{x}**y**(neglecting the dependence of

**K**on

**F**(

**x**) around the minimum of

**G**:

**S**

*.*

_{x}**y**:

*n*equations of Eq. (18) are independent of each other. If this assumption does not apply, it is not possible to calculate the inverse of

**U**

^{Tx^}have not been measured and do not contribute to the MQQ. Nevertheless the MQQ can still be determined accounting for the measured components. Therefore, the quantity defined in Eq. (25) provides the MQQ of indirect measurements regardless of whether the inverse problem is well-posed or not.

### 3.3 General considerations

*Shannon information content*[1], also use the inverse of the CM, but calculate instead the determinant of this matrix. In this case the generalization to ill-posed problems cannot be done because if some eigenvalues of the inverse of the CM are zero also the determinant is zero independently of the values of the others. Therefore, for ill-posed problems, the determinant of the inverse of the CM is not able to quantify the measurement quality on the basis of the errors of the measured components.

*n*-1 dimensions and the

*n*-dimensional volume is zero. On the other hand, the diagonal of the (

*n*-1)-hypercuboid is different from zero and its value depends on the lengths of the edges different from zero.

*Fisher information matrix*[8

8. R. A. Fisher, “The logic of inductive inference,” J.R. Stat. Soc. **98**(1), 39–54 (1935). [CrossRef]

*L*(

**x**) =

*P*(

**y**|

**x**) (that is the conditional probability distribution to obtain

**y**given

**x**, considering

*P*(

**y**|

**x**) as a function of

**x**) when we assume a Gaussian distribution for

*P*(

**y**|

**x**):that is generally appropriate for describing the noise associated with experimental data. Indeed, the Fisher information matrix

**F**relative to

*L*(

**x**) is defined as:and substituting

**S**

*.*

_{y}*F*is closely linked with the

_{ii}*information load*(defined as the square root of the diagonal elements of

**K**

^{T}**K**) introduced in [5

5. M. Carlotti and L. Magnani, “Two-dimensional sensitivity analysis of MIPAS observations,” Opt. Express **17**(7), 5340–5357 (2009). [CrossRef] [PubMed]

7. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. Lopez-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller, and R. Zander, “MIPAS: an instrument for atmospheric and climate research,” Atmos. Chem. Phys. **8**(8), 2151–2188 (2008).

**S**

*and has the additivity property.*

_{y}**y**

_{1}(of

*m*

_{1}elements) and

**y**

_{2}(of

*m*

_{2}elements) of the set of parameters represented by the vector

**x**. The measurements are characterized, respectively, by the Jacobian matrices

**K**

_{1}and

**K**

_{2}and by the CMs

**S**

_{y}_{1}and

**S**

_{y}_{2}. In order to calculate the MQQ components

*F*of the data fusion of the two measurements we consider the two measurements as a single measurement

_{ii}*m*

_{1}+

*m*

_{2}observations) with Jacobian matrix

**M**below the rows of the matrix (vector)

**L**. Consequently

*F*of the data fusion are the sum of the MQQ components (

_{ii}*F*

_{1})

*and (*

_{ii}*F*

_{2})

*of the two original measurements. This result can be extended to the data fusion of any number of measurements.*

_{ii}## 4. Relative quality quantifier

*F*we explicitly write the expression of these terms:

_{ii}*F*represents the sum of the quadratic variations of the observations, weighted with the CM

_{ii}**S**

_{y}, that correspond to a variation Δ

*x*divided by the square of Δ

_{i}*x*.

_{i}*x*, it can be useful to use a

_{i}*relative MQQ*instead of the absolute MQQ defined in subsection 3.2. The definition of a relative quality quantifier can be deduced from Eq. (32) substituting the square absolute variation (Δ

*x*)

_{i}^{2}with the square relative variation

*F*

_{ii}x_{i}^{2}we can define a relative MQQ as equal to:

## 5. The invariant of the retrieval problem

**x**is no longer equal to the inverse of the Fisher matrix of the observations. Therefore, the Fisher matrix and the MQQ, which are defined by quantities (

**K**and

**S**

*) characterizing the observations, can no longer be simply derived from the quantities characterizing the retrieval. In order to verify how the Fisher matrix can be calculated from the diagnostics that are generally distributed to the data user together with the retrieval products the mathematics of the constrained retrieval is briefly recalled.*

_{y}*c*(

**x**) that is the sum of the chi-square function plus a constraint function

*R*(

**x**):

*c*(

**x**) with respect to the components of

**x**equal to zero we find that the value

**x**that minimizes

*c*(

**x**) satisfies the following equation:

**y**and is characterized by the CM

**S**

*and by the averaging kernel matrix (AKM) that is defined as*

_{x}9. S. Ceccherini, B. Carli, E. Pascale, M. Prosperi, P. Raspollini, and B. M. Dinelli, “Comparison of measurements made with two different instruments of the same atmospheric vertical profile,” Appl. Opt. **42**(32), 6465–6473 (2003). [CrossRef] [PubMed]

10. S. Ceccherini and M. Ridolfi, “Technical Note: Variance-covariance matrix and averaging kernels for the Levenberg-Marquardt solution of the retrieval of atmospheric vertical profiles,” Atmos. Chem. Phys. **10**(6), 3131–3139 (2010). [CrossRef]

**y**and solve the obtained equation with respect to the gain matrix

**G**, which turns out to be equal to:where we have defined the matrix

**S**

_{x}does not coincide with that given in Eq. (22). Therefore, now it is not correct to estimate the MQQ of the measurement calculating the trace of the inverse of

**S**

_{x}because this quantity contains the information that we have added with the constraint. In order to obtain the correct expression for the MQQ we need to consider also the AKM.

**S**

_{x}expressed by Eq. (38) is invertible. In this case it is easy to verify that, because of Eqs. (38) and (39), the matrix

**S**

_{x}is not invertible. However, we can consider the generalized inverse [11] of

**S**

*that we indicate with*

_{x}**S**

_{x}^{#}and that can be calculated using the singular value decomposition. It is possible to demonstrate (see Appendix) that:

*R*(

**x**) and it provides the quantity from which we can calculate the MQQ:

**R**

^{−1}and the solution is interpreted as the weighted mean of the two measurements (the actual measurement and the constraint). This is for instance the case of the “optimal estimation method”. In this case the CM of

**K**and

**S**

_{y}) and that, independently of the adopted constraint, can also be determined from the quantities that characterize the measurements (through

**A**and

**S**

_{x}). The existence of this invariant proves that the retrieval, if properly made and fully characterized, is a process that does not destroy any information.

## 6. Information distribution

*x*of

_{i}**x**is associated an altitude interval Δ

*z*, a variation of the parameter

_{i}*x*determines a variation of the observations

_{i}**y**proportional to the altitude interval Δ

*z*, and the element

_{i}*K*of

_{ij}**K**is proportional to Δ

*z*. Consequently the diagonal elements

_{j}*F*are proportional to Δ

_{ii}*z*

_{i}^{2}and, as stated above, their values depend on the grid on which the vertical profile is represented. Also the MQQ (the sum of the

*F*) depends on the grid and in particular it approaches zero for very fine grids when Δ

_{ii}*z*tend to zero.

_{i}*z*tend to zero, and, therefore, this value represents the measurement quality referred to the vertical profile represented as a continuous function of altitude. Since

_{i}*F*are proportional to Δ

_{ii}*z*

_{i}^{2}we define the grid normalized MQQ components as:and call this quantity the

*information distribution*.

*f*values in the neighborhood of each altitude are independent of the grid. Accordingly we can define the

_{i}*grid normalized MQQ*as:that in the limit of

*f*(

*z*) and provides an overall assessment of the quality of a distribution measurement (independently of the selected retrieval grid).

*relative information distribution*as

*f*

_{i}x_{i}^{2}and the

*grid normalized relative MQQ*as:

## 7. Some applications

### 7.1 Quality of IASI and MIPAS ozone measurements and of their data fusion

4. S. Ceccherini, U. Cortesi, S. Del Bianco, P. Raspollini, and B. Carli, “IASI-METOP and MIPAS-ENVISAT data fusion,” Atmos. Chem. Phys. **10**(10), 4689–4698 (2010). [CrossRef]

**10**(10), 4689–4698 (2010). [CrossRef]

6. C. Clerbaux, A. Boynard, L. Clarisse, M. George, J. Hadji-Lazaro, H. Herbin, D. Hurtmans, M. Pommier, A. Razavi, S. Turquety, C. Wespes, and P.-F. Coheur, “Monitoring of atmospheric composition using the thermal infrared IASI/MetOp sounder,” Atmos. Chem. Phys. **9**(16), 6041–6054 (2009). [CrossRef]

^{−1}. The retrieval of the ozone vertical profile was performed using a version of the MARC (Millimetre-wave Atmospheric-Retrieval Code) retrieval code [12

12. B. Carli, G. Bazzini, E. Castelli, C. Cecchi-Pestellini, S. Del Bianco, B. M. Dinelli, M. Gai, L. Magnani, M. Ridolfi, and L. Santurri, “MARC: a code for the retrieval of atmospheric parameters from millimetre-wave limb measurements,” J. Quantum Spectrosc. Radiat. **105**(3), 476–491 (2007). [CrossRef]

13. L. Palchetti, G. Bianchini, B. Carli, U. Cortesi, and S. Del Bianco, “Measurement of the water vapour vertical profile and of the Earth’s outgoing far infrared flux,” Atmos. Chem. Phys. **8**(11), 2885–2894 (2008). [CrossRef]

14. G. Bianchini, B. Carli, U. Cortesi, S. Del Bianco, M. Gai, and L. Palchetti, “Test of far infrared atmospheric spectroscopy using wide-band balloon borne measurements of the upwelling radiance,” J. Quantum Spectrosc. Radiat. **109**(6), 1030–1042 (2008). [CrossRef]

7. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. Lopez-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller, and R. Zander, “MIPAS: an instrument for atmospheric and climate research,” Atmos. Chem. Phys. **8**(8), 2151–2188 (2008).

^{−1}. The retrieval of the ozone profile was performed using the ORM (Optimized Retrieval Model) [15

15. M. Ridolfi, B. Carli, M. Carlotti, T. von Clarmann, B. M. Dinelli, A. Dudhia, J. M. Flaud, M. Höpfner, P. E. Morris, P. Raspollini, G. Stiller, and R. J. Wells, “Optimized forward model and retrieval scheme for MIPAS near-real-time data processing,” Appl. Opt. **39**(8), 1323–1340 (2000). [CrossRef] [PubMed]

18. S. Ceccherini, C. Belotti, B. Carli, P. Raspollini, and M. Ridolfi, “Technical Note: Regularization performances with the error consistency method in the case of retrieved atmospheric profiles,” Atmos. Chem. Phys. **7**(5), 1435–1440 (2007). [CrossRef]

*F*

_{ii}x_{i}^{2}of the relative MQQ as a function of altitude for the IASI measurement, the MIPAS measurement and their data fusion. Since the two retrievals have been performed using the same retrieval grid (from 1 to 80 km of altitude at 1 km steps), the MQQ components provide an adequate parameter for the comparison. Figure 1 shows clearly that the MIPAS measurement contains information on the ozone profile between 15 and 60 km, while the IASI measurement contains information below 30 km with a minimum at 15 km. The data fusion contains information on the ozone profile between 1 and 60 km with a minimum around to 15 km where both instruments have a contribution to the relative MQQ that is very small. The oscillations in the line of the MIPAS measurement correspond to maxima of the information for tangent altitudes of the observations. The analysis of the MQQ components highlights the relative merits of the two measurements as well as the strength and the weakness of the product of their fusion.

*Q*is 50156 for the IASI measurement, 125225 for the MIPAS measurement and 175381 for the data fusion, confirming again the additivity property of the MQQ parameter.

_{r}**10**(10), 4689–4698 (2010). [CrossRef]

**10**(10), 4689–4698 (2010). [CrossRef]

### 7.2 Comparison of quality of MIPAS full and optimized resolution measurements

^{−1}. A limb sequence in the nominal observation mode was composed of 17 spectra that looked at different tangent altitudes from 6 to 68 km, with a step of 3 km in the troposphere and lower stratosphere and of up to 8 km in the high stratosphere. The measurements acquired between July 2002 and March 2004 are referred to as

*full resolution*(FR) measurements.

^{−1}. In the nominal observation mode adopted after January 2005, a MIPAS limb scan consists of 27 spectra that look at different tangent altitudes from 7 to 72 km with a step of 1.5 km in the troposphere and lower stratosphere and of up to 4.5 km in the high stratosphere. The measurements acquired since January 2005 are referred to as

*optimized resolution*(OR) measurements.

19. A. Dudhia, V. L. Jay, and C. D. Rodgers, “Microwindow selection for high-spectral-resolution sounders,” Appl. Opt. **41**(18), 3665–3673 (2002). [CrossRef] [PubMed]

20. S. Ceccherini, U. Cortesi, P. T. Verronen, and E. Kyrölä, “Technical Note: Continuity of MIPAS-ENVISAT operational ozone data quality from full- to reduced-spectral-resolution operation,” Atmos. Chem. Phys. **8**(8), 2201–2212 (2008). [CrossRef]

^{−1}for the FR orbit and 75845 km

^{−1}for the OR orbit, proving the improvement attained with the new measurement scenario.

## 8. Conclusion

## Appendix

**S**

_{x}is not invertible, we have:

**S**

_{x}

^{#}is the generalized inverse of

**S**

_{x}[11]. The generalized inverse of

**S**

_{x}can be calculated in the following way. Since

**S**

_{x}is a symmetric matrix we can find an orthogonal matrix

**U**for which

**Λ**is a diagonal matrix with some diagonal values equal to zero. The generalized inverse of

**S**

_{x}is the matrix

**Λ**

^{#}is the diagonal matrix whose diagonal elements are given by Λ

^{#}

*=1/Λ*

_{ii}*if Λ*

_{ii}**is different from zero and by Λ**

_{ii}^{#}

**=0 if Λ**

_{ii}**is equal to zero.**

_{ii}**S**

_{x}is a symmetric matrix, we obtain:

## Acknowledgments

## References and links

1. | C. D. Rodgers, |

2. | S. Ceccherini, P. Raspollini, and B. Carli, “Optimal use of the information provided by indirect measurements of atmospheric vertical profiles,” Opt. Express |

3. | S. Ceccherini, B. Carli, U. Cortesi, S. Del Bianco, and P. Raspollini, “Retrieval of the vertical column of an atmospheric constituent from data fusion of remote sensing measurements,” J. Quantum Spectrosc. Radiat. |

4. | S. Ceccherini, U. Cortesi, S. Del Bianco, P. Raspollini, and B. Carli, “IASI-METOP and MIPAS-ENVISAT data fusion,” Atmos. Chem. Phys. |

5. | M. Carlotti and L. Magnani, “Two-dimensional sensitivity analysis of MIPAS observations,” Opt. Express |

6. | C. Clerbaux, A. Boynard, L. Clarisse, M. George, J. Hadji-Lazaro, H. Herbin, D. Hurtmans, M. Pommier, A. Razavi, S. Turquety, C. Wespes, and P.-F. Coheur, “Monitoring of atmospheric composition using the thermal infrared IASI/MetOp sounder,” Atmos. Chem. Phys. |

7. | H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. Lopez-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller, and R. Zander, “MIPAS: an instrument for atmospheric and climate research,” Atmos. Chem. Phys. |

8. | R. A. Fisher, “The logic of inductive inference,” J.R. Stat. Soc. |

9. | S. Ceccherini, B. Carli, E. Pascale, M. Prosperi, P. Raspollini, and B. M. Dinelli, “Comparison of measurements made with two different instruments of the same atmospheric vertical profile,” Appl. Opt. |

10. | S. Ceccherini and M. Ridolfi, “Technical Note: Variance-covariance matrix and averaging kernels for the Levenberg-Marquardt solution of the retrieval of atmospheric vertical profiles,” Atmos. Chem. Phys. |

11. | R. E. Kalman, “Algebraic aspects of the generalized inverse of a rectangular matrix,” in |

12. | B. Carli, G. Bazzini, E. Castelli, C. Cecchi-Pestellini, S. Del Bianco, B. M. Dinelli, M. Gai, L. Magnani, M. Ridolfi, and L. Santurri, “MARC: a code for the retrieval of atmospheric parameters from millimetre-wave limb measurements,” J. Quantum Spectrosc. Radiat. |

13. | L. Palchetti, G. Bianchini, B. Carli, U. Cortesi, and S. Del Bianco, “Measurement of the water vapour vertical profile and of the Earth’s outgoing far infrared flux,” Atmos. Chem. Phys. |

14. | G. Bianchini, B. Carli, U. Cortesi, S. Del Bianco, M. Gai, and L. Palchetti, “Test of far infrared atmospheric spectroscopy using wide-band balloon borne measurements of the upwelling radiance,” J. Quantum Spectrosc. Radiat. |

15. | M. Ridolfi, B. Carli, M. Carlotti, T. von Clarmann, B. M. Dinelli, A. Dudhia, J. M. Flaud, M. Höpfner, P. E. Morris, P. Raspollini, G. Stiller, and R. J. Wells, “Optimized forward model and retrieval scheme for MIPAS near-real-time data processing,” Appl. Opt. |

16. | P. Raspollini, C. Belotti, A. Burgess, B. Carli, M. Carlotti, S. Ceccherini, B. M. Dinelli, A. Dudhia, J. M. Flaud, B. Funke, M. Hopfner, M. Lopez-Puertas, V. Payne, C. Piccolo, J. J. Remedios, M. Ridolfi, and R. Spang, “MIPAS level 2 operational analysis,” Atmos. Chem. Phys. |

17. | S. Ceccherini, “Analytical determination of the regularization parameter in the retrieval of atmospheric vertical profiles,” Opt. Lett. |

18. | S. Ceccherini, C. Belotti, B. Carli, P. Raspollini, and M. Ridolfi, “Technical Note: Regularization performances with the error consistency method in the case of retrieved atmospheric profiles,” Atmos. Chem. Phys. |

19. | A. Dudhia, V. L. Jay, and C. D. Rodgers, “Microwindow selection for high-spectral-resolution sounders,” Appl. Opt. |

20. | S. Ceccherini, U. Cortesi, P. T. Verronen, and E. Kyrölä, “Technical Note: Continuity of MIPAS-ENVISAT operational ozone data quality from full- to reduced-spectral-resolution operation,” Atmos. Chem. Phys. |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(010.1280) Atmospheric and oceanic optics : Atmospheric composition

(100.3190) Image processing : Inverse problems

(120.0280) Instrumentation, measurement, and metrology : Remote sensing and sensors

(110.3055) Imaging systems : Information theoretical analysis

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: November 23, 2011

Revised Manuscript: January 19, 2012

Manuscript Accepted: January 31, 2012

Published: February 16, 2012

**Citation**

Simone Ceccherini, Bruno Carli, and Piera Raspollini, "Quality quantifier of indirect measurements," Opt. Express **20**, 5151-5167 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-5-5151

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### References

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