Thank you for your detailed response to reviewer comments. Can you please revise the manuscript as proposed in your response for a further review by editor and referees.

Thank you for your revised manuscript, which has now been reviewed by two reviewers. The first reviewer still has concerns regarding the mathematical implementation of the model. A particular concern relates to the estimated slope coefficients relating transfers of the logarithm of TP from a source node to a recipient node being near 1, with the resulting implications for transfers on the raw (un-logged) scale increasing by about 2.8. The reviewer also has concerns regarding the calculations of the Bayesian R-sq values and why they are so high and would like to see a clearer explanation of the K-fold cross-validation calculation. The reviewer likes the idea of using a state-space model for this application but has concerns regarding the implementation of the approach. I agree with both reviewers that the application of the SSM approach to the understanding of total phosphorus pollution in a large lake system is novel and worth pursuing. Could you please therefore take careful account of the constructive criticism offered in the referee 1 report and consider revising your manuscript to address the raised concerns?

Thank you for your reply to reviewer comments and manuscript revisions. I believe that the manuscript has been substantially improved and clarified during the review process and I hope that you found the reviewer comments helpful. I have a few remaining suggestions.

R1C7 In your reply to reviewer comment, you state that truncating the values of the latent state does not set a prior based on the data, rather it constrains the later state to plausible values. Does this point relate to your explanation of the high Bayesian R2 values (in response to R1C4), whereby ‘the model passes through the observed data and the daily observation in the river datasets are likely driving Bayesian R2 values higher with their constrained predictive intervals’? You also say that the Bayesian R2 may not be a fair metric to evaluate the model performance. Hence, is there a link between the Reviewer’s point that such constraints (or priors) should not be informed by observations to which the data is fitted and your explanation of the high Bayesian R2 values? Or perhaps I have misunderstood. However, if there is a link, could this be reflected in the discussion as a potential limitation and area for future improvement?

R1C8 I found the format suggested by the reviewer to explain the model clear and helpful. Can you please take another look at the wording proposed by the reviewer and consider adopting it, or parts of it, in the manuscript? I believe it makes the model structure clear and explicit, especially the explanation of the two models – data=observation and latent=process.

R1C9 I think this explanation is of interest and would be helpful to include in the manuscript for a better understanding of the physical processes operating on the lake.

R2C1and R2C2 I believe that your response to these reviewer questions may be of interest to the readers. Can you please consider including these points in the manuscript?

Thank you for your contribution to this Special Issue and for your constructive consideration of reviewer comments. Thank you also for making the model code publicly available - this will no doubt facilitate further scientific discussion and development. Congratulations on the completion of this extensive work.