The Supervision of Dough Fermentation Using Image Analysis Complemented by a Continuous Discrete Extended Kalman Filter

Round 1

Reviewer 1 Report

See attached file. (If there is no attached file, something went wrong.)

Comments for author File: Comments.pdf

Author Response

First review

Review of \The supervision of dough fermentation using image analysis complemented by a continuous discrete extended Kalman filter"

November 20, 2020

This paper applies an extended Kalman filter to estimate state variables and one parameter for a model of dough fermentation. The focus is on the formation of CO2. The model is estimating three states (r, the radius of all bubbles, n, the amount of CO2 in a bubble, and CD, the CO2 concentration in non-gaseous dough) and one parameter, qCO2 , the specific CO2 production rate. The measurement is the volume of the dough obtained using an image acquisition system. The model is seemingly able to follow the measurements. The paper is in general easy to read, also for a reader not being an expert on dough fermentation. However, I have a few objections to how the parameter is estimated, and will therefore prefer to see some more results before recommending the paper for publication. Based on this, detailed in my comments below, my recommendation is that the paper goes through a major revision.

Suggestions of major changes

1. My major concern with this study is the estimation of the model parameter, qCO2 . Since this is an uncertain parameter, I do not understand why the initial uncertainty of the model parameter, i.e. PqCO2 ;qCO2 is set to zero. My interpretation is that this means that the parameter is actually known at time zero. Therefore I strongly suggest to rerun the experiment with a much higher initial value for PqCO2 ;qCO2 .

One choice could for instance be 0.75 * 10^-3 based on the assumption that the parameter is very uncertain, but at least positive. However, if you consider that even the order of magnitude of qCO2 is uncertain, you might consider to estimate log qCO2

instead. This will of course slightly complicate the equations, but since the assumption is that parameter is only influenced by the stochastic term zqCO2 it should be viable. Estimating the logarithm of the parameters is commonly done in geosciences in cases where the order of magnitude is uncertain (for examples see [Tar05]). The value of QqCO2 ;qCO2 must of course also be paid attention to. One might even consider to set this value to zero.

To me it seems strange that the uncertainty in an estimated parameter is increasing over time after you get information about it through measurements. (For the state variables that can of course happen because one think they can be influenced by unmodeled effects represented by the stochastic term ~z.)

Answer: (line 174) We set the value of PpCO2pCO2 to 7.5*10^-4 mol²*m⁶/kg²/s²

2. The measurement system is introduced at line 94-95. Then it is written like there are two measurements, the total dough volume and the radius of an average bubble inside of the fermentation dough (I suppose there is some error in the writing at the end of line 95). However, my impression is that there is only one measurement, the radius of an average bubble in fermentation dough. This should be clarified. It might also be made more clear how the average radius is calculated from the volume of the dough.

Answer: (line 140) We made the measurement more clear writing: “Therefore, as only measurement for the extended Kalman filter the radius of an average bubble is used in the following equation:…”

3. At line 142 two symbols are not coming out correctly.

Answer: Has been corrected.

4. What is X in Equation 9?

Answer: (line 126) We added: “X the yeast concentration”

5. For me the presentation will be better if you proceed to describe the measurement (line 159-169) before describing the preparation of the dough. I think the full description of the mathematical framework could be completed before the \Preparation of Dough" part, i.e. move line 159-176 before line 150-158.

Answer: (line 153-161) We moved the lines, as suggested.

6. Is the imaging evaluation software an in-house tool or available software? Is

r(ti) = r(0) +volume of dough(ti) ? volume of dought(t0) NB ?

Answer: The imaging evaluation software is developed by us.  
r(t) = (3*[V(t)-0.9*V(t=0)]/4/pi/NB)^(1/3) as described indirectly in M. Stanke, V. Zettel, S. Schütze, B. Hitzmann, Measurement and mathematical modeling of the relative volume of wheat dough during proofing, Journal of Food Engineering 131(2014)58-64. Usually the gas after kneading is 10 % of the total volume.

7. Line 180-181: Are the values of Qii and R obtained by doing experiments with the filter?

Answer: (line 168) We wrote in the text: A measurement error was assumed, which is more than half of the radius r(t=0); therefore the measurement noise variance R was fixed to 10^-9 m².

8. Table 1: Second parameter is misprinted.

Answer: (line 173) Has been corrected to

9. Table 1: I am not able to see where Xw (last line) is used elsewhere.

Answer: Line was deleted.

10. Table 2: Why is Prr(t = 0) = 0? Is there no initial uncertainty in r?

Answer: (line 174) We changed it to the value 10^-10 m².

11. Table 2: PqCO2 ;qCO2 : See discussion in item 1.

Answer: (line 174) We have changed all initial values (see Table 2).

12. Figure 2: I am only able to see one curve. It would be much better to show both the actual measurements and the estimate of the radius of the bubble obtained by the Kalman filter.

Answer: (line 194, now Figure 1) We increased the measurement symbols.

13. Figure 7: Are you running new experiments changing the values of X in Eq. 4 as well as the initial guess of the parameter qCO2? Please make it clear.

Answer: (line 258) Yes. Now we wrote: To prove the capability of the extended Kalman filter for the estimation of the specific CO₂ production rate two new runs of the extended Kalman filter was carried out with changing biomass concentration X.

Suggestions of minor changes

14. Chapter 2: In the description of the Kalman filter it seems that some conditions on time independence in ~z(t) and ~w(t), as well as independence between these two variables are left out. I think such assumptions should be added in the description. (I supposed it is mentioned in Gelb, otherwise you might consider to consult [Ste94].)

Answer: The whole chapter was removed as the reviewer 3 suggested. Just a reference is presented.

15. It seems like Q is introduced both on line 105 and 111.

Answer: The whole chapter was removed as the reviewer 3 suggested.

16. I do not think you should start new paragraphs at lines 103, 109, and 125.

Answer: The whole chapter was removed as the reviewer 3 suggested.

17. Line 111: \where" instead of \Where".

Answer: The whole chapter was removed as the reviewer 3 suggested.

18. I was a bit surprised while reading that only nitrogen is introduced into the dough while kneading. I assume that this is done in air, having a significant contribution of oxygen. For those well into dough fermentation this might be clear, but I will be happy to have short explanation (in paper or the response).

Answer: (line 115) We add in the paper: “(because we assumed just anaerobic consumption of the sugars)”. Aerobic consumption will change the amount of CO2 produced, which make everything more complicated – and most probably before dough fermentation all the oxygen is consumed already (in between kneading and fermentation there is a dough rest of several minutes).

19. Equation 9 will look better if the first and third bracket are better lined up with the second.

Answer: (line 123, now equation 1) We did as suggested.

20. Figure 3: Add legend for CO2 amount in one bubble (the red line).

Answer: (line 205, now Figure 2) We did as suggested.

21. Line 219-223: Please try to write it more concise.

Answer: (line 224) We improved the text. “The decrease might be a result of the limiting substrate concentration, because just damaged starch can be converted to fermentable sugars. The fermentation lasted more than 40 min, so most probably the cell number will increase due to growth. However, this is not considered in the model (X is constant). If the cell count increases in reality but not in the model, then the filter can compensate this fact by an increase in the specific CO₂ production rate, which might explain the moderate increase during the last 800 s.”

22. Figure 5 and 6: Consider to plot standard deviation (square root of variances) as those are easier to compare with the variables.

Answer: (line 224 and 256) We did as suggested.

23. Figure 5: The blue line is very thick? Why?

Answer: Because the values are changing very fast, whenever a filtering is performed the values fell down, and during simulation they are increasing.

24. Figure 6: Consider to use another color for the \Estimation error variance of CO2 in dough".

Answer: (line 256) We did as suggested.

25. Reference list: Since the numbers are not used, they can be skipped.

Answer: We removed all the numbers.

As a final comment, I would like to mention that if you can release the series of measurement this is a nice test case people can actually reproduce (or even try to improve on) with their own filter. That might be appreciated.

Answer: (line 360 ff) The measurements are given as supplementary materials.

References

[Ste94] Robert F. Stengel. Optimal control and estimation. Dover publications, inc., New York, 1994.

[Tar05] Albert Tarantola. Inverse Problem Theory and Methods for Model Parameter Estimation. SIAM, 2005.

Many thanks to the reviewer!!

Reviewer 2 Report

The paper concerns the monitoring of dough fermentation using Kalman filetring. The paper is well written, very pleasant to read and is very clear. The topic is very interesting.

However, in my opinion,, the presented Kalman filterer was not really validated through the presented experimental data. Indeed, the radius was the only measured data (or at least, the only one presented and analyzed). The results of the Kalman filter, presented in Figure 2, highlights the filtering of the measurement noise with the Kalman filter but do not illustrates the behavior of the estimation of the remaining state variables. Indeed, since R is less than Q11, the filter will use mainly the measurement instead of the dynamic model. The choice of the tuning parameters Q22, Q33 and Q44 are the key for a good estimation of the three remaining state variables. However, the authors do not explain how these parameters were determined, neither they compared the estimated data with the measured one.

Some comparison between measured and estimated data, other than measurement used in the Kalman filter, must be provided to highlight the Kalman filter efficiency.

Can authors add a figure with the predicted CO2 production provided by the model, and the one given by the Kalman filter ? It could be interesting to highlight the behavior the Kalman filter (if no additional experimental data are available). This figure can highlight the correction performed by the Kalman filter (but remains limited to ensure an accuracy of the Kalman filter for the estimation of CO2 production rate).

Minor remarks:

• Page 4, Line 120, R must be definite positive matrix.
• Page 4, Eq. 9, z_i must be non correlated noises, according to the presented Kalman Filter.
• Page 4, Line 142, Some variables appear as "@ "(or kind of symbol). The same in Table 1. Please check.
• Page 4, Line 148. H was introduced as the measurement matrix, but also as the Henry constant. Please change notation to avoid confusion.
• Page 5, Eq. 10. Please provide information about sampling time. Is the measurement equation evaluated at constant interval times?
• Figures 5 and 6. The estimation errors variances are interesting, but they do not really validate the Kalman filtering efficiency. The evolution of the error is also needed. 
• Figure 7. It is not clear how the yeast influence the model used in the Kalman filtering. Which parameter in the model is modified?

Author Response

Second Review

The paper concerns the monitoring of dough fermentation using Kalman filtering. The paper is well written, very pleasant to read and is very clear. The topic is very interesting.

However, in my opinion, the presented Kalman filterer was not really validated through the presented experimental data. Indeed, the radius was the only measured data (or at least, the only one presented and analyzed). The results of the Kalman filter, presented in Figure 2, highlights the filtering of the measurement noise with the Kalman filter but do not illustrates the behavior of the estimation of the remaining state variables. Indeed, since R is less than Q11, the filter will use mainly the measurement instead of the dynamic model. The choice of the tuning parameters Q22, Q33 and Q44 are the key for a good estimation of the three remaining state variables. However, the authors do not explain how these parameters were determined, neither they compared the estimated data with the measured one.

Answer: (line 168) We wrote in the text: A measurement error was assumed, which is more than half of the radius r(t=0); therefore the measurement noise variance R was fixed to 10^-9 m².

Some comparison between measured and estimated data, other than measurement used in the Kalman filter, must be provided to highlight the Kalman filter efficiency.

Answer: We did not measure anything else.

Can authors add a figure with the predicted CO2 production provided by the model, and the one given by the Kalman filter ? It could be interesting to highlight the behavior the Kalman filter (if no additional experimental data are available). This figure can highlight the correction performed by the Kalman filter (but remains limited to ensure an accuracy of the Kalman filter for the estimation of CO2 production rate).

Answer: We add purely simulated data in all the corresponding Figures (1-4).

Minor remarks:

• Page 4, Line 120, R must be definite positive matrix.

Answer: The text of that line was removed, as the reviewer 3 suggested.

• Page 4, Eq. 9, z_i must be non correlated noises, according to the presented Kalman Filter.

Answer: The text of that line was removed, as the reviewer 3 suggested.

• Page 4, Line 142, Some variables appear as "@ "(or kind of symbol). The same in Table 1. Please check.

Answer: We changed, as suggested.

• Page 4, Line 148. H was introduced as the measurement matrix, but also as the Henry constant. Please change notation to avoid confusion.

Answer: We removed the introduction of the measurement matrix, as the reviewer 3 suggested.

• Page 5, Eq. 10. Please provide information about sampling time. Is the measurement equation evaluated at constant interval times?

Answer: (line 142) We add: “Every 3 s a measurement is send to the extended Kalman filter and add the measurements in the Supplementary Materials.“

• Figures 5 and 6. The estimation errors variances are interesting, but they do not really validate the Kalman filtering efficiency. The evolution of the error is also needed. 

Answer: (Figures 5 and 6) Now the errors are presented instead of variances.

• Figure 7. It is not clear how the yeast influence the model used in the Kalman filtering. Which parameter in the model is modified?

Answer: (Figure 7) X is the yeast concentration (we add this information) which was modified.

Many thanks to the reviewer!!

Reviewer 3 Report

The work represents a traditional type of research. In an applied field of fermented baking goods preparation, the authors propose a mathematical model of a production process of dough fermentation. The model describes the evolution of state variables and connects these variables with available observations - data from a camera placed on a black baking pad (processed observations provide a "radius of an average bubble"). To the resulting dynamic observation system, the authors apply a typical analysis tool - Extended Kalman Filter (EKF). A real experiment is carried out using the model and the analysis tool. The obtained numerical results are interpreted from the point of the analyzed production process.

The main advantages of the work are a good review of the subject area, with a focus on "Kalman filter for fermentation processes" and work with real data in a real experimental setting.

Minor issues:

1. inaccurate wordings, e.g., "mathematical dough model" instead of a process model or "Kalman Filter" instead of the Extended Kalman Filter (this also connected to one of the major issues, see below);

2. undefined abbreviations (PID) and highly specialized terms (it is hard to understand what is "gari fermentation plant" even using the quoted source);

3. instead of "expectation value" it is better to use the mathematical notation; instead of using the "noise spectral matrix", it is better to properly define the noise z(t);

4. incorrect symbols instead of variables, e.g., in the notation of "viscosity";

5. indistinguishable plots in Figure 2.

Principal issues in the theoretical part

The fundamental problem is the misconception about the Extended Kalman Filter. This is evidenced by the omitting of the word "Extended" throughout the paper. This is not correct, because the studied model is nonlinear, the Kalman filter is not applicable and the guaranteed properties of the optimal filter estimate do not take place. Moreover, the Extended Kalman Filter is suboptimal, it is obtained as a result of linearization, and hence it has no guaranteed properties. Moreover, specialists are well aware of examples of inadequate work of EKF, the divergence of its estimates. More specific errors in the text:

1. In equation (1) authors use the derivative of a random process. The derivative is not defined, the process z(t) cannot be arbitrary, and what is said below about z(t) is not enough for equation (1) to be mathematically correct.

2. Relation (2) requires the existence of density (pdf), which is not necessary to determine the quality of the estimate. Also, there is nothing stated about the process \hat{x} except that it is "estimated process variables".

3. Formula (3) is incorrect. More precisely, it is valid only in the linear case, when \hat{x} is a Kalman filter estimate given the linear observations with Gaussian white noise. In this case, the Jacobian F(t) does not depend on x.

4. Fig. 1 brings no additional information and leaves open the question of what to do with the estimate in the interval between upcoming discrete observations.

5. Equations (6) indeed describe an EKF estimate, but they are misinterpreted. In particular, (6) does not determine the "estimation error variance", which means there is no point in minimizing it, i.e. writing down (7).

Thus, the entire contents of section 2 up to the reference Gelb, 1999 is, unfortunately, incorrect. It is advisable to remove it entirely since EKF is a widely known instrument and concentrate on the application, which is a subject matter of the paper.

Principal issues in the application part

Taking into account the particular features of EKF, which limit its usability, the sufficiency of the performed analysis of the numerical data obtained as a result of the experiment raises doubts. This doubt bases on the absence of any justification for choosing the parameters of disturbances and observation noises. In (9), the variance (scale) of disturbances for the “radius of all bubbles” does not depend on this radius. That is, the same absolute deviations from the mean take place for bubbles of any radius. Is there any physical justification for this assumption? A similar situation is with observation noises for the "radius of an average bubble". Usually in such models, the error is characterized by deviation from the mean, but in the proposed model, the errors are the same for any possible radius. 

Finally, there is no analysis of the model experiment, when the state x and observations y are simulated strictly within the framework of the observation system model. This could allow one to confirm the adequacy of the model to real data and to analyze the estimation accuracy. Here, the model and filter are applied only to the data of the experiment, so the adequacy of the results is impossible to assess.

Сonclusion

The paper shows real potential, but unfortunately, can not be recommended for publication in the present form. The revision should include: 

• rewriting/removing of the section with the mathematical formulation of the filtration problem;
• simulation experiments for the proposed mathematical model;
• study of the dependence of the results on the size and variable nature of disturbances and observation noises.

Author Response

Third review

The work represents a traditional type of research. In an applied field of fermented baking goods preparation, the authors propose a mathematical model of a production process of dough fermentation. The model describes the evolution of state variables and connects these variables with available observations - data from a camera placed on a black baking pad (processed observations provide a "radius of an average bubble"). To the resulting dynamic observation system, the authors apply a typical analysis tool - Extended Kalman Filter (EKF). A real experiment is carried out using the model and the analysis tool. The obtained numerical results are interpreted from the point of the analyzed production process.

The main advantages of the work are a good review of the subject area, with a focus on "Kalman filter for fermentation processes" and work with real data in a real experimental setting.

Minor issues:

1. inaccurate wordings, e.g., "mathematical dough model" instead of a process model or "Kalman Filter" instead of the Extended Kalman Filter (this also connected to one of the major issues, see below);

Answer: The wording has been corrected.

2. undefined abbreviations (PID) and highly specialized terms (it is hard to understand what is "gari fermentation plant" even using the quoted source);

Answer: (line 52 and 91) The abbreviation as well the term gari is explained.  

3. instead of "expectation value" it is better to use the mathematical notation; instead of using the "noise spectral matrix", it is better to properly define the noise z(t);

Answer: We removed the part as suggested.

4. incorrect symbols instead of variables, e.g., in the notation of "viscosity";

Answer: we corrected the symbol.

5. indistinguishable plots in Figure 2.

Answer: We improved Figure 1 (old Figure 2).

Principal issues in the theoretical part

The fundamental problem is the misconception about the Extended Kalman Filter. This is evidenced by the omitting of the word "Extended" throughout the paper. This is not correct, because the studied model is nonlinear, the Kalman filter is not applicable and the guaranteed properties of the optimal filter estimate do not take place. Moreover, the Extended Kalman Filter is suboptimal, it is obtained as a result of linearization, and hence it has no guaranteed properties. Moreover, specialists are well aware of examples of inadequate work of EKF, the divergence of its estimates.

Answer: We now use throughout the manuscript “extended Kalman filter”.

More specific errors in the text:

1. In equation (1) authors use the derivative of a random process. The derivative is not defined, the process z(t) cannot be arbitrary, and what is said below about z(t) is not enough for equation (1) to be mathematically correct.

Answer: We removed that part as suggested.

2. Relation (2) requires the existence of density (pdf), which is not necessary to determine the quality of the estimate. Also, there is nothing stated about the process \hat{x} except that it is "estimated process variables".

Answer: We removed that part as suggested.

3. Formula (3) is incorrect. More precisely, it is valid only in the linear case, when \hat{x} is a Kalman filter estimate given the linear observations with Gaussian white noise. In this case, the Jacobian F(t) does not depend on x.

Answer: We removed that part as suggested.

4. Fig. 1 brings no additional information and leaves open the question of what to do with the estimate in the interval between upcoming discrete observations.

Answer: We removed that part as suggested.

5. Equations (6) indeed describe an EKF estimate, but they are misinterpreted. In particular, (6) does not determine the "estimation error variance", which means there is no point in minimizing it, i.e. writing down (7).

Answer: We removed that part as suggested.

Thus, the entire contents of section 2 up to the reference Gelb, 1999 is, unfortunately, incorrect. It is advisable to remove it entirely since EKF is a widely known instrument and concentrate on the application, which is a subject matter of the paper.

Answer: We removed that part as suggested.

Principal issues in the application part

Taking into account the particular features of EKF, which limit its usability, the sufficiency of the performed analysis of the numerical data obtained as a result of the experiment raises doubts. This doubt bases on the absence of any justification for choosing the parameters of disturbances and observation noises. In (9), the variance (scale) of disturbances for the “radius of all bubbles” does not depend on this radius. That is, the same absolute deviations from the mean take place for bubbles of any radius. Is there any physical justification for this assumption? A similar situation is with observation noises for the "radius of an average bubble". Usually in such models, the error is characterized by deviation from the mean, but in the proposed model, the errors are the same for any possible radius.

Answer: (line 168) We wrote in the text: A measurement error was assumed, which is more than half of the radius r(t=0); therefore the measurement noise variance R was fixed to 10^-9 m².

Finally, there is no analysis of the model experiment, when the state x and observations y are simulated strictly within the framework of the observation system model. This could allow one to confirm the adequacy of the model to real data and to analyze the estimation accuracy. Here, the model and filter are applied only to the data of the experiment, so the adequacy of the results is impossible to assess.

Answer: (Figure 1-4) We presented the simulation of the model and compared it with the estimation of the extended Kalman filter.

 Сonclusion

The paper shows real potential, but unfortunately, can not be recommended for publication in the present form. The revision should include: 

• rewriting/removing of the section with the mathematical formulation of the filtration problem;

Answer: We removed that part as suggested.

• simulation experiments for the proposed mathematical model;

Answer: (Figure 1-4) We present simulation experiments of the mathematical model.

• study of the dependence of the results on the size and variable nature of disturbances and observation noises.

Answer: (Figure 1-4) We compared the simulated and estimated values to show the influence of the extended Kalman filter.

Many thanks to the reviewer!!

Round 2

Reviewer 1 Report

See sttached file.

Comments for author File: Comments.pdf

Author Response

Review of The supervision of dough fermentation using image analysis complemented by a continuous discrete extended Kalman Filter" (revised version)

December 3, 2020

The paper has improved significantly through revision and might now qualify for publication. I have still some comments that authors might like to take into account for further improvement.

1. Line 13-14: \By estimation a fixed number..." This sentence is not clear to me. Do you mean by \By assuming a fixed number ...."?

Answer: (line 13) yes, we changed.

2. In Equation 1 a quantity r0 is used but not properly defined. I am guessing r0 = r(0).

If so it might be worthwhile to state it.

Answer (line 120): We wrote: “r₀ the radius at t=0 s“.

3. Equation 1: The unknown parameter qCO2 is only appearing through the product qCO2X where X is appearing to be another uncertain quantity. This might be discussed more. It becomes very clear that it is the product that determines the behavior of the system, not only qCO2 , by looking at Figure 7. At about 100s the parameter is set to zero. After that one can see clear trends in the estimates. While the curves are increasing the curves using highest values for X (green) moves to the top, lowest values (blue) to the bottom, and the opposite while the curves are decreasing. Did you do any reflections on that?

Answer: Yes in a paper (Stanke, M.; Zettel, V.; Schütze, S.; Hitzmann, B.; Measurement and mathematical modeling of the relative volume of wheat dough during proofing, Journal of Food Engineering 131(2014)58-64) we used different biomass X (2 % and 4 %) to verify the stated equations. So we would like to use the same model and not combine both quantities.

4. Consider to add some more information in the Figure captions. In Fig. 1 the measured data is not mentioned in the caption, the expression \Kalman estimation" could be made more concise etc. Moreover, for Fig. 1-4 I assume the blue curves are all made by doing a simulation with fixed parameter and initial condition.

Answer: (Figures 1 - 4) We add “using initial values and fixed parameters”. In Figure 1 the legend is as follows: Bubble radii calculated by the imaging system (measured data) compared to the ones estimated by the EKF as well as simulated values, using initial values and fixed parameters.

5. Line 189-190: You might state that the simulated values are a free run based on the provided initial conditions and the initial guess of the parameter.

Answer: (lines 189-190) We added “based on the provided initial conditions and the initial guess of the parameters.”

6. Line 193-196: I am not able to digest the meaning here, and I think it might be written more concisely.

Answer: (lines 192 ff) We improved it to: “Especially two observations can be link directly to the measurements of the radius, which are short before 100 s and after 645 s. The measurements of the radius decreases before 100 s therefore the estimates of the amount of CO₂ decreased as well. After 645 s the measurements of the radius kept constant, and therefore the EKF reduced the amount of CO₂ correspondingly.”

7. Line 238-241: Try to write it more concisely.

Answer: (lines 245 ff) We improved it to: “As can be seen in Figure 6 the estimation error of the amount of CO₂ sqrt(P_nn) in the bubble with an exception at the beginning increases slowly. The estimation error of values for CO₂ in non-gas phase sqrt(P_CDCD) jumped at the beginning to 16 mol/m³ and after a while decrease almost exponentially to roughly 2 mol/m³. The reason, why they went down might be due to the fact that the maximal CO₂ concentration in non-gas phase is reached.”

8. Line 242-249: I am still puzzled by the fact that the estimation uncertainty is so large in specific production rate. However, it seems that you might still be able to construct a warning system that dough is fermenting properly. However, an alarm would be going at about 100 seconds since the CO2 drops to zero at that point. Is this due to temporal failure of the measurement system?

Answer: Yes, at the beginning there seems to be higher measurement errors, which might be due to temporal failure of the measurement system.

9. Figure 1: A final comment. It seems that some dynamic of the system is missing by comparing the free run (blue curve) of the model with the measurements as they have slightly different forms.

Answer: The different forms result from the influence of the measurements.

I really appreciate the release of the measurement data, this opens up for other groups to investigate both the mathematical modeling and the estimation.

Thank you for your comments!

Reviewer 2 Report

The paper was significantly improved. Thank you for taking into account reviewers' suggestions. 

I have some minor remarks:

• I think it is a good idea to remove the equations of the EKF. Could authors provide a reference to Kalman filtering theory ? Even if it is well known, it could be interesting for readers who are looking for more information about continuous-discrete  EKF.
• In Page 4, authors say that the process noise covariance matrix was determined by simulation and by performing experiments. I think they should explain a little more how it was determined. It is OK to be a trial/error method. I think it should be more precise in the paper. 
• In figure 6, the error on CD is increasing. Could authors comment this behaviour ? I think that it could be linked to results of Figure 2. I mean, In figure 2, estimated data is below simulated one, but with higher variance of the error as seen in Figure 6. Could you check this please ?

I think that additional experimental data for all state variable could be a very nice result to assess the Kalman filter efficiency. But I understand that it is a difficult (or impossible ?) task. The paper is still very interesting without those data.  

Author Response

The paper was significantly improved. Thank you for taking into account reviewers' suggestions. 

I have some minor remarks:

• I think it is a good idea to remove the equations of the EKF. Could authors provide a reference to Kalman filtering theory ? Even if it is well known, it could be interesting for readers who are looking for more information about continuous-discrete  EKF.

Answer: (lines 103 and 324) We add the following book as reference for the Kalman filter theory: Grewal, M.S.; Andrews, A.P., Kalman Filtering: Theory and Practice using MATLAB, Second Edition, John Wiley & Sons, New York, 2001

• In Page 4, authors say that the process noise covariance matrix was determined by simulation and by performing experiments. I think they should explain a little more how it was determined. It is OK to be a trial/error method. I think it should be more precise in the paper. 

Answer: (line 158) We add: Here the values were changed in a way, so that a balance was obtained between the influence of the measurements and the model during filtering.

• In figure 6, the error on CD is increasing. Could authors comment this behaviour ? I think that it could be linked to results of Figure 2. I mean, In figure 2, estimated data is below simulated one, but with higher variance of the error as seen in Figure 6. Could you check this please ?

Answer: The error of CD sqrt(P_CDCD) (the initial value of PCDCD=250 mol²/m6, sqrt(PCDCD)=15,8 mol/m³) is decreasing and then increasing at the beginning, because the initial value might be wrong, but shortly later (around 200 s) it is decreasing, because the maximal CO₂ concentration in non-gas phase is reaching. The increasing in sqrt(Pnn) might reflect the lager difference between the estimated and simulated data in Figure 2 compared to Figure 1.

I think that additional experimental data for all state variable could be a very nice result to assess the Kalman filter efficiency. But I understand that it is a difficult (or impossible ?) task. The paper is still very interesting without those data.  

Answer: Unfortunately, it is impossible to get additional experimental data.

Thank you for your comments!

Reviewer 3 Report

The authors made improvements, but a misconception about the Extended Kalman Filter is still an issue.

1. there still are places where Kalman filter is used without "Extended", plus in most of the cases, it is "extended" with lowercase. The capital letter makes sense here because EKF does not simply extend the classical filter, it has its own name since it is a separate nonlinear filter. I propose to introduce a widely used abbreviation EKF in the very beginning and use it throughout the paper.
2. It should be taken into account that EKF is a suboptimal filter, and hence EKF estimate has no guaranteed properties. This means that the differential equation (6) from the original version of the paper does not yield an "estimation error variance" and hence all the consequent analysis of its square root is unfortunately useless. Perhaps the authors are misled by some sources, which indicate that equation (6) gives some approximation for the estimation error variance, but it does not. If there is no way to characterize the accuracy of an "approximation", it can't be called an approximation in the first place. Consequently, figs 5 and 6 bring no sense.
3. The new line on the figs (called "simulated") has to be defined. But it seems that again it is a result of a misunderstanding. The correspondent remark in the review meant that a series of numerical simulations must be performed. In each of these experiments, the system dynamics is simulated using equations (1) given various random noise samples; observations are determined using equations (2); and, finally, EKF estimates are calculated. The difference in these experiments is that the ground-truth values of estimated variables are available. So based on this simulated data it is possible to evaluate biases and variances of the EKF estimate.

Author Response

The authors made improvements, but a misconception about the Extended Kalman Filter is still an issue.

1. there still are places where Kalman filter is used without "Extended", plus in most of the cases, it is "extended" with lowercase. The capital letter makes sense here because EKF does not simply extend the classical filter, it has its own name since it is a separate nonlinear filter. I propose to introduce a widely used abbreviation EKF in the very beginning and use it throughout the paper.

Answer: We used “Extended” (lines 12, 22, 78, 94) and EKF were possible. In the Introduction different Kalman filters are mentioned. In M&M it is present just once (because the references discuss different Kalman filters), in Results and Conclusion the word “Kalman” does not appear.

1. It should be taken into account that EKF is a suboptimal filter, and hence EKF estimate has no guaranteed properties. This means that the differential equation (6) from the original version of the paper does not yield an "estimation error variance" and hence all the consequent analysis of its square root is unfortunately useless. Perhaps the authors are misled by some sources, which indicate that equation (6) gives some approximation for the estimation error variance, but it does not. If there is no way to characterize the accuracy of an "approximation", it can't be called an approximation in the first place. Consequently, figs 5 and 6 bring no sense.

Answer: (lines 233 ff) To present Figure 5 and 6 was suggested by another reviewer. So we include the following sentence: However one should keep in mind that the EKF is a suboptimal filter and therefore these values must be considered with care and they might not represent the correct errors.

1. The new line on the figs (called "simulated") has to be defined. But it seems that again it is a result of a misunderstanding. The correspondent remark in the review meant that a series of numerical simulations must be performed. In each of these experiments, the system dynamics is simulated using equations (1) given various random noise samples; observations are determined using equations (2); and, finally, EKF estimates are calculated. The difference in these experiments is that the ground-truth values of estimated variables are available. So based on this simulated data it is possible to evaluate biases and variances of the EKF estimate.

Answer: (line 169, Figures 1 - 4) We stated more clear what “simulated” meant: “using just the initial conditions and the fixed parameters”.

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