Increasing the measurement precision of optical instrumentation using Kalman-type filters

1. A method for increasing the measurement precision of an optical instrument, the method comprising:

determining, based on optical data and environmental data, a measured value of an optical property measured by the optical instrument, wherein:

the optical instrument comprises:

an optical path having two reflective surfaces and a transmission medium therebetween, the two reflective surfaces separated by a distance, the transmission medium having an index of refraction, and

a sensor configured to measure an environmental parameter that influences a magnitude of one or both of the distance and the index of refraction,

the optical data is generated by the optical instrument in response to an optical signal traversing the optical path, and

the environmental data is generated by the sensor measuring the environmental parameter;

determining a predicted value of the optical property based on a model representing time evolution of the optical instrument; and

by operation of one or more processors, calculating an effective value of the optical property based on:

the measured value,

the predicted value, and

a Kalman gain based on respective uncertainties in the measured and predicted values, the Kalman gain defining a relative weighting of the measured and predicted values in the effective value.

2. The method of claim 1 ,

wherein the Kalman gain is biased towards the measured value when the uncertainty in the measured value is less than the uncertainty in the predicted value; and

wherein the Kalman gain is biased towards the predicted value when the uncertainty in the predicted value is less than the uncertainty in the measured value.

3. The method of claim 1 ,

wherein the model comprises:

state variables comprising a first state variable representing the optical property and a second state variable representing the environmental parameter, and

a state vector comprising respective state values for the state variables;

wherein the time evolution occurs from a previous period to a current period; and

wherein the method comprises:

determining the Kalman gain based on a measurement noise matrix, a process noise matrix, and a covariance matrix, wherein:

the measurement noise matrix comprises values representing an uncertainty in the optical and environmental data,

the process noise matrix comprises values representing an uncertainty in the model, and

the covariance matrix comprises values representing an uncertainty in the state values.

4. The method of claim 3 , comprising:

repeating, over multiple iterations of respective periods, the operations of determining the measured value, determining the predicted value, determining the Kalman gain, and calculating the effective value; and

wherein the values of the measurement noise matrix, the process noise matrix, the covariance matrix, or any combination thereof, are updated for each iteration.

5. The method of claim 1 ,

wherein the model comprises:

state variables comprising a first state variable representing the optical property and a second state variable representing the environmental parameter,

a state vector comprising respective state values for the state variables, and

a state evolution function defining a change in the state values from a first set of state values associated with a previous period to a second set of state values associated with a current period; and

wherein the time evolution occurs from the previous period to the current period; and

wherein determining the predicted value comprises:

applying the state evolution function to the first set of state values to generate the second set of state values, a value of the second set of state values for the first state variable being the predicted value.

6. The method of claim 5 , wherein the state evolution function comprises a plurality of sigma points and respective weighting factors.

7. The method of claim 5 ,

wherein determining the measured value comprises:

obtaining measurement values for respective measurement variables of a measurement vector, wherein:

the measurement variables comprise a first measurement variable representing the optical property and a second measurement variable representing the environmental parameter, and

the measurement value obtained for the first measurement variable is the measured value; and

wherein calculating the effective value comprises:

calculating residual values of a residual vector based on a difference between the measurement values and the second set of state values, and

determining a third set of state values for the state vector based on the second set of state values, the Kalman gain, and the residual values, the third set of state values comprising the effective value.

8. The method of claim 7 ,

wherein the measurement variables define a measurement domain for the measurement vector and the state variables define a state domain for the state vector; and

wherein calculating the residual values comprises:

applying a measurement function to the second set of state values to generate a converted second set of state values, the measurement function defining a change in the state values upon conversion from the state domain to the measurement domain; and

subtracting the converted second set of state values from the measurement values to calculate the residual values of the residual vector.

9. The method of claim 1 , wherein the environmental parameter comprises a temperature of the transmission medium, a pressure of the transmission medium, a humidity of the transmission medium, or a concentration of carbon dioxide in the transmission medium.

10. The method of claim 1 , wherein the environmental parameter comprises a temperature of the transmission medium or a length of a spacer separating the two reflective surfaces.

11. A system comprising:

an optical instrument, configured to measure an optical property and comprising:

an optical path having two reflective surfaces and a transmission medium therebetween, the two reflective surfaces separated by a distance, the transmission medium having an index of refraction, and

a sensor configured to measure an environmental parameter that influences a magnitude of one or both of the distance and the index of refraction;

a control system comprising one or more processors and memory storing instructions that are configured to perform operations when executed by the one or more processors, the operations comprising:

determining, based on optical data and environmental data, a measured value of the optical property, wherein:

the optical data is generated by the optical instrument in response to an optical signal traversing the optical path, and

the environmental data is generated by the sensor measuring the environmental parameter;

determining a predicted value of the optical property based on a model representing time evolution of the optical instrument; and

calculating an effective value of the optical property based on:

the measured value,

the predicted value, and

a Kalman gain based on respective uncertainties in the measured and predicted values, the Kalman gain defining a relative weighting of the measured and predicted values in the effective value.

12. The system of claim 11 ,

wherein the Kalman gain is biased towards the measured value when the uncertainty in the measured value is less than the uncertainty in the predicted value; and

wherein the Kalman gain is biased towards the predicted value when the uncertainty in the predicted value is less than the uncertainty in the measured value.

13. The system of claim 11 ,

wherein the model comprises:

state variables comprising a first state variable representing the optical property and a second state variable representing the environmental parameter, and

a state vector comprising respective state values for the state variables;

wherein the time evolution occurs from a previous period to a current period; and

wherein the operations comprise:

determining the Kalman gain based on a measurement noise matrix, a process noise matrix, and a covariance matrix, wherein:

the measurement noise matrix comprises values representing an uncertainty in the optical and environmental data,

the process noise matrix comprises values representing an uncertainty in the model, and

the covariance matrix comprises values representing an uncertainty in the state values.

14. The system of claim 13 , wherein the operations comprise:

repeating, over multiple iterations of respective periods, the operations of determining the measured value, determining the predicted value, determining the Kalman gain, and calculating the effective value; and

wherein the values of the measurement noise matrix, the process noise matrix, the covariance matrix, or any combination thereof, are updated for each iteration.

15. The system of claim 11 ,

wherein the model comprises:

state variables comprising a first state variable representing the optical property and a second state variable representing the environmental parameter,

a state vector comprising respective state values for the state variables, and

a state evolution function defining a change in the state values from a first set of state values associated with a previous period to a second set of state values associated with a current period; and

wherein the time evolution occurs from the previous period to the current period; and

wherein determining the predicted value comprises:

applying the state evolution function to the first set of state values to generate the second set of state values, a value of the second set of state values for the first state variable being the predicted value.

16. The system of claim 15 , wherein the state evolution function comprises a plurality of sigma points and respective weighting factors.

17. The system of claim 15 ,

wherein determining the measured value comprises:

obtaining measurement values for respective measurement variables of a measurement vector, wherein:

the measurement variables comprise a first measurement variable representing the optical property and a second measurement variable representing the environmental parameter, and

the measurement value obtained for the first measurement variable is the measured value; and

wherein calculating the effective value comprises:

calculating residual values of a residual vector based on a difference between the measurement values and the second set of state values, and

determining a third set of state values for the state vector based on the second set of state values, the Kalman gain, and the residual values, the third set of state values comprising the effective value.

18. The system of claim 17 ,

wherein the measurement variables define a measurement domain for the measurement vector and the state variables define a state domain for the state vector; and

wherein calculating the residual values comprises:

applying a measurement function to the second set of state values to generate a converted second set of state values, the measurement function defining a change in the state values upon conversion from the state domain to the measurement domain; and

subtracting the converted second set of state values from the measurement values to calculate the residual values of the residual vector.

19. The system of claim 11 , wherein the environmental parameter comprises a temperature of the transmission medium, a pressure of the transmission medium, a humidity of the transmission medium, or a concentration of carbon dioxide in the transmission medium.

20. The system of claim 11 , wherein the environmental parameter comprises a temperature of the transmission medium or a length of a spacer separating the two reflective surfaces.

21. A non-transitory computer-readable medium comprising instructions that are operable, when executed by one or more processors, to perform operations comprising:

determining, based on optical data and environmental data, a measured value of an optical property measured by an optical instrument, wherein:

the optical instrument comprises:

an optical path having two reflective surfaces and a transmission medium therebetween, and

a sensor configured to measure an environmental parameter of the transmission medium between the two reflective surfaces,

the optical data is generated by the optical instrument in response to an optical signal traversing the optical path, and

the environmental data is generated by the sensor measuring the environmental parameter;

determining a predicted value of the optical property based on a model representing time evolution of the optical instrument; and

calculating an effective value of the optical property based on:

the measured value,

the predicted value, and

a Kalman gain based on respective uncertainties in the measured and predicted values, the Kalman gain defining a relative weighting of the measured and predicted values in the effective value.

22. The non-transitory computer-readable medium of claim 21 ,

wherein the Kalman gain is biased towards the measured value when the uncertainty in the measured value is less than the uncertainty in the predicted value; and

wherein the Kalman gain is biased towards the predicted value when the uncertainty in the predicted value is less than the uncertainty in the measured value.

23. The non-transitory computer-readable medium of claim 21 ,

wherein the model comprises:

state variables comprising a first state variable representing the optical property and a second state variable representing the environmental parameter, and

a state vector comprising respective state values for the state variables;

wherein the time evolution occurs from a previous period to a current period; and

wherein the operations comprise:

determining the Kalman gain based on a measurement noise matrix, a process noise matrix, and a covariance matrix, wherein:

the measurement noise matrix comprises values representing an uncertainty in the optical and environmental data,

the process noise matrix comprises values representing an uncertainty in the model, and

the covariance matrix comprises values representing an uncertainty in the state values.

24. The non-transitory computer-readable medium of claim 23 , wherein the operations comprise:

repeating, over multiple iterations of respective periods, the operations of determining the measured value, determining the predicted value, determining the Kalman gain, and calculating the effective value; and

wherein the values of the measurement noise matrix, the process noise matrix, the covariance matrix, or any combination thereof, are updated for each iteration.

25. The non-transitory computer-readable medium of claim 21 ,

wherein the model comprises:

state variables comprising a first state variable representing the optical property and a second state variable representing the environmental parameter,

a state vector comprising respective state values for the state variables, and

a state evolution function defining a change in the state values from a first set of state values associated with a previous period to a second set of state values associated with a current period; and

wherein the time evolution occurs from the previous period to the current period; and

wherein determining the predicted value comprises:

applying the state evolution function to the first set of state values to generate the second set of state values, a value of the second set of state values for the first state variable being the predicted value.

26. The non-transitory computer-readable medium of claim 25 , wherein the state evolution function comprises a plurality of sigma points and respective weighting factors.

27. The non-transitory computer-readable medium of claim 25 ,

wherein determining the measured value comprises:

obtaining measurement values for respective measurement variables of a measurement vector, wherein:

the measurement variables comprise a first measurement variable representing the optical property and a second measurement variable representing the environmental parameter, and

the measurement value obtained for the first measurement variable is the measured value; and

wherein calculating the effective value comprises:

calculating residual values of a residual vector based on a difference between the measurement values and the second set of state values, and

determining a third set of state values for the state vector based on the second set of state values, the Kalman gain, and the residual values, the third set of state values comprising the effective value.

28. The non-transitory computer-readable medium of claim 27 ,

wherein the measurement variables define a measurement domain for the measurement vector and the state variables define a state domain for the state vector; and

wherein calculating the residual values comprises:

applying a measurement function to the second set of state values to generate a converted second set of state values, the measurement function defining a change in the state values upon conversion from the state domain to the measurement domain; and

subtracting the converted second set of state values from the measurement values to calculate the residual values of the residual vector.

29. The non-transitory computer-readable medium of claim 21 , wherein the environmental parameter comprises a temperature of the transmission medium, a pressure of the transmission medium, a humidity of the transmission medium, or a concentration of carbon dioxide in the transmission medium.

30. The non-transitory computer-readable medium of claim 21 , wherein the environmental parameter comprises a temperature of the transmission medium or a length of a spacer separating the two reflective surfaces.