A geostatistical spatially varying coefficient model for mean annual runoff that incorporates processbased simulations and short records
 ^{1}Norwegian University of Science and Technology, NTNU, Høgskoleringen 1, 7491 Trondheim, Norway
 ^{2}Norwegian Computing Center, NR, Gaustadalléen 23A, 0373 Oslo, Norway
 ^{3}The Norwegian Water Resources and Energy Directorate, NVE, Middelthuns gate 29, 0368 Oslo, Norway
 ^{1}Norwegian University of Science and Technology, NTNU, Høgskoleringen 1, 7491 Trondheim, Norway
 ^{2}Norwegian Computing Center, NR, Gaustadalléen 23A, 0373 Oslo, Norway
 ^{3}The Norwegian Water Resources and Energy Directorate, NVE, Middelthuns gate 29, 0368 Oslo, Norway
Abstract. We present a Bayesian geostatistical model for mean annual runoff that incorporates simulations from a processbased hydrological model by treating the simulations as a covariate in the statistical model. The regression coefficient of the covariate is modeled as a spatial field such that the relationship between the covariate (simulations from a hydrological model) and the response variable (observed mean annual runoff) is allowed to vary within the study area. Hence, it is a spatially varying coefficient. A preprocessing step for including short records in the modeling is also suggested and we obtain a model that can exploit several data sources by using state of the art statistical methods.
The geostatistical model is evaluated by predicting mean annual runoff for 1981–2010 for 127 catchments in Norway based on observations from 411 catchments. Simulations from the processbased HBV model on a 1 km × 1 km grid are used as input. We found that on average the proposed approach outperformed a purely processbased approach (HBV) when predicting runoff for ungauged and partially gauged catchments: The reduction in RMSE compared to the HBV model was 20 % for ungauged catchments and 58 % for partially gauged catchments, where the latter is due to the preprocessing step. For ungauged catchments the proposed framework also outperformed a purely geostatistical method with a 10 % reduction in RMSE compared to the geostatistical method. For partially gauged catchments however, purely geostatistical methods performed equally well or slightly better than the proposed combination approach. It is not surprising that purely geostatistical methods perform well in areas where we have data. In general, we expect the proposed approach to outperform geostatistics in areas where the data availability is low to moderate.
Thea Roksvåg et al.
Status: closed

RC1: 'Comment on hess20218', Anonymous Referee #1, 24 Jun 2021
The authors present a Bayesian geostatistical model for mean annual runoff that incorporates simulations from a processbased hydrological model (HBV model) by treating the simulations as a covariate in the statistical model.
The article describes an interesting application of a geostatistical model. The scientific significance is fair/good and I think that this work can be considered as an extension of a previous work (https://doi.org/10.5194/hess2441092020). The methodology is sound and results are discussed in an appropriate and balanced way. However, some criticisms emerge. The major problem is that is article is more similar to a report than a scientific article. Some parts of the article are not clearly presented (here I am referring to the first part of the article, the second part  discussion ad conclusions  are well structured). The document is too long and some parts are repeated several times in the text. The revision has been quite difficult due to the high number of pages and due to the fact that the authors repeated some information. Moreover, some key information is mentioned only in the second half of the article. I suggest performing a major revision of the text before publication.
Rows 1415. I suggest removing from the abstract the sentence “It is not surprising that purely geostatistical methods perform well in areas where we have data”. It is already expected and does not provide additional information to the reader. Moreover, it makes the article less robust.
Row 17 and following. The Introduction is too long and the description of the work performed in this study is split into several parts. I suggest removing some sentence (as rows 2122, “The temporal variability of runoff can also be used to study runoff’s sensitivity to climate change”) that do not provide additional information related to this work. I also suggest to summarize the work and the main novelties at the end of the paragraph.
Row 35. TopKriging is used to model only discharge and not other referenced data, as mentioned. Please correct.
Row 55. The correct and most used name is “kriging with external drift” and not “external drift kriging”.
Row 58. “Was estimated with a …”.
Row 59. Please explain what "g(.)" is.
Row 81. Please explain here what GRF is, considering that it is the first time that you mention this acronym.
Row 99. Please explain what INLA and SPDE are.
Row 121. You used a ratio equal to 0.2. Why? Please provide a reference or a motivation.
Rows 124125. Please check these rows, I am not sure that the grammar is correct.
Description of Figure 1. Please use “UTM” instead of “utm”.
Row 133. Please mention the spatial interpolation that you used.
Rows 164 – 165. I suggest removing “However, most of the 141 calibration catchments probably coincide with the 127 fully gauged catchments in Figure 1a”. If you are not sure about it, it is not advisable to mention it.
Rows 178179. I suggest to remove them.
Row 206. Is the square bracket correct? Or is it a typo?
Row 209. I suggest to remove “will”.
Row 224. Please insert the equation in a new line, by assigning number (3).
Row 226. Please explain what SPDE is.
Rows 231233. I suggest to remove them.
Row 235. I suggest to shorten the sentence in “Kriging is used to …”.
Row 236. "x" is an estimated variable: please insert the “hat”.
Row 246. Please add the brackets at the reference.
Row 249. Please correct with “According to Viglione et al. (2013) and Blöschl et al. (2013) …”.
Rows 279 – 284. I suggest to remove them.
Rows 307 – 310. You already mentioned this before. I suggest to remove this part.
Row 313. Why is “areas” in italics?
Row 329. Is the requirement of having positive runoff satisfied? If not, please provide additional information about how you managed this issue. If I am not wrong, you provide additional information about this only in row 713. I suggest to anticipate this statement.
Row 340. Please rewrite, removing “bear in mind”.
Row 352. Please remove or rewrite “because Norway is a diverse country when it comes to runoff generation”. For me, it is not clear what you meant.
Row 372. "Credibility interval" or "confidence interval"?
Row 409. I think you were referring to “confidence interval” and not to “credible interval”.
Equation 14. I suggest to insert it as three separate equations. Please change the blue font to black font.
Rows 464 – 467. Please remove them or move them before. It is not advisable to mention the goals of the study after 18 pages of text. You already provided some information about the goals in several parts of Section 1. I suggest to merge everything, providing a more detailed and structured statement.
Rows 470 – 471. Please remove “These are observations from 127 fully gauged catchments from 19812010 and 284 partially gauged catchments from 19652010”. You already mention it.
Row 491 and following rows. The bold font is not necessary.
Rows 561 – 563. I suggest to remove them.
Figure 6. Please remove the “[1]” above the first colorbar. Please insert the axis (with coordinates).
Figure 7. I suggest to use “UTM” instead of “utm”.
Row 637. Is “do” necessary?
Row 671. Please remove the second brackets after x(y).
Row 702. Please remove the empty row.
AC1: 'Reply on RC1', Thea Roksvåg, 12 Jul 2021
We would like to thank Anonymous referee #1 for the constructive review.
Referee 1’s main concern is connected to the structure of the paper and he/she has several suggestions on how to improve it. The suggestions are concrete and good, and we will revise the paper accordingly. More specifically, repeated information will be removed and we will shorten and sharpen the introduction. We will also edit/clarify the sentences that are mentioned by the referee.As referee 1’s comments mainly concern structure and grammar, we don’t have any additional comments. We will follow up on each of the comments in a revision.
Kind regards,
Thea Roksvåg and coauthors.

AC1: 'Reply on RC1', Thea Roksvåg, 12 Jul 2021

RC2: 'Comment on hess20218', Anonymous Referee #2, 24 Jun 2021
This paper proposes a method for the spatial prediction of annual runoff that uses processbased model outputs (model HBV), fully gauged catchment data and partially gauged catchment data. The latter are preprocessed using a method published by the same coauthors (https://doi.org/10.5194/hess2441092020) in HESS and it is optional if all catchment are fully gauged.
This contribution is a significant contribution since the proposed approach allows to provide improved maps of annual runoff at the national scale, by taking into account fully gauged and and partially gauged catchments.
The proposed method is hierarchical in a Bayesian framework; it includes a Gaussian observation process, a spatial latent model and hyperpriors. The latent random field is defined on a fine 1km x 1km grid. It is a Spatially Varying Coefficient regression model to the HBV outputs that involves two GRFs characterized with stationary Matern covariance functions. The statistical analysis is performed using INLA/SPDE.
The areal nature of the data is taken into account by relating the catchment data to the sum of the latent model over the catchment.
The approach is validated using a carefully designed kfold crossvalidation. The approach shows superior performances as compared to HBV only and to TopKriging.
General comment:
===============
The statistical analysis is stateofthe art. It uses the most modern tools for analyzing complex spatial data in a hierarchical Bayesian framework. The experimental setup and the validation study are very carefully designed. The results are well discussed.Overall, the paper is well written, maybe on the long side though considering that lots of material is common with https://doi.org/10.5194/hess2441092020. I spotted some rare typos and a few awkward sentences (see below). All illustrations are relevant. I believe that this paper will be a nice methodological followup to https://doi.org/10.5194/hess2441092020. I recommend publication after a minor revision that takes into account the specific comments below
Specific comments:
==================
1. A potential weakness of the method, which has been mentioned by the authors, is that the model does not prevent negative runoff predictions in some (unlikely situations). This is due to the Gaussian likelihood and Gaussian GRF. The authors mention that logrunoff could be used instead, but then linearity of Eq. (6) is lost, which is an impediment. Another way of preventing negative predictions would be to use logGaussian likelihood and logGaussian random fielsd for x(u) and α(u) in (4). This would be a marginal change, since INLA/SPDE allows for loggaussian likelihood and LG random fields at almost no cost. As a result, predictions for x and α would always be positive. I wonder how this would work. Ideally, I'd like the authors to try this option, but I'd be happy if they only discuss this possibility.2. The GRFs x(u) and α(u) are independent. This assumption is never clearly stated and it is not discussed. Is this a reasonable assumption? Is this an assumption you could check or validate? How useful/difficult would it be to relax this assumption?
3. The GRFs x(u) and α(u) are assumed to be stationary. Are you able to check that this assumption is supported by the data?
4. To my knowledge, the product of a an exponential variogram with a fractal variogram is not a valid variogram. However, the product of an exponential covariance function with a fractal variogram might be a valid variogram. Please doublecheck and provide references if necessary.
5. Regarding the results: is it really desirable to get a correlation of 1 between measures and predictions? I would relate this to the fact that the coverage is 83%, which shows that the SVC is overconfident in the UG setting. Please comment.
Typos, etc.
===========
41: counties > countries
51: there exist work > there exist works
206: remove the square bracket ]
222: Further is $\sigma^2$ > Further, $\sigma^2$ is
229: advice > advise
235 to 242: Kriging yields "predictions" not "estimations". Please change "estim*" to "predict*" everywhere. BLUE should become BLUP (lines 236, 240, 241)
248: subcatcment > subcatchment
266: in 3 > in Figure 3
335: remove the dot in $s_i \sigma^2_y$, since it is not used in most cases later. (Also line 359 and line 406)
358: are the scales > the scales are
383: I very much doubt that Norway is only 40 km wide (or less). Please double check
430 Further is the variable x > Further, the variable x is
449: I did not understand the sentence. I think it needs to be rephrased.
555: "... is defined as the probability that 90% of the observed values ..." should be "... is defined as the proportion of the observed values ..."
584: In order to respect the order (first α, then x) I suggest to change the order between short ranged and long ranged. Hence, write "the spatial fields have picked up long range and short range processes, respectively".
667: "only $x(u)$ or $\alpha(u)$ > "only either $x(u)$ or "$\alpha(u)$"
669: "indicates that it for many study area might ..." > "indicates that for many study area it might"
672: "only x(u) or only α(u)" > one random field only.
710: does not makes > does not make
734: that is > that it is
In several places, there are sentences beginning with "Mark". This is unusual. I recommend using "Remark" or "Notice". AC2: 'Reply on RC2', Thea Roksvåg, 12 Jul 2021
Status: closed

RC1: 'Comment on hess20218', Anonymous Referee #1, 24 Jun 2021
The authors present a Bayesian geostatistical model for mean annual runoff that incorporates simulations from a processbased hydrological model (HBV model) by treating the simulations as a covariate in the statistical model.
The article describes an interesting application of a geostatistical model. The scientific significance is fair/good and I think that this work can be considered as an extension of a previous work (https://doi.org/10.5194/hess2441092020). The methodology is sound and results are discussed in an appropriate and balanced way. However, some criticisms emerge. The major problem is that is article is more similar to a report than a scientific article. Some parts of the article are not clearly presented (here I am referring to the first part of the article, the second part  discussion ad conclusions  are well structured). The document is too long and some parts are repeated several times in the text. The revision has been quite difficult due to the high number of pages and due to the fact that the authors repeated some information. Moreover, some key information is mentioned only in the second half of the article. I suggest performing a major revision of the text before publication.
Rows 1415. I suggest removing from the abstract the sentence “It is not surprising that purely geostatistical methods perform well in areas where we have data”. It is already expected and does not provide additional information to the reader. Moreover, it makes the article less robust.
Row 17 and following. The Introduction is too long and the description of the work performed in this study is split into several parts. I suggest removing some sentence (as rows 2122, “The temporal variability of runoff can also be used to study runoff’s sensitivity to climate change”) that do not provide additional information related to this work. I also suggest to summarize the work and the main novelties at the end of the paragraph.
Row 35. TopKriging is used to model only discharge and not other referenced data, as mentioned. Please correct.
Row 55. The correct and most used name is “kriging with external drift” and not “external drift kriging”.
Row 58. “Was estimated with a …”.
Row 59. Please explain what "g(.)" is.
Row 81. Please explain here what GRF is, considering that it is the first time that you mention this acronym.
Row 99. Please explain what INLA and SPDE are.
Row 121. You used a ratio equal to 0.2. Why? Please provide a reference or a motivation.
Rows 124125. Please check these rows, I am not sure that the grammar is correct.
Description of Figure 1. Please use “UTM” instead of “utm”.
Row 133. Please mention the spatial interpolation that you used.
Rows 164 – 165. I suggest removing “However, most of the 141 calibration catchments probably coincide with the 127 fully gauged catchments in Figure 1a”. If you are not sure about it, it is not advisable to mention it.
Rows 178179. I suggest to remove them.
Row 206. Is the square bracket correct? Or is it a typo?
Row 209. I suggest to remove “will”.
Row 224. Please insert the equation in a new line, by assigning number (3).
Row 226. Please explain what SPDE is.
Rows 231233. I suggest to remove them.
Row 235. I suggest to shorten the sentence in “Kriging is used to …”.
Row 236. "x" is an estimated variable: please insert the “hat”.
Row 246. Please add the brackets at the reference.
Row 249. Please correct with “According to Viglione et al. (2013) and Blöschl et al. (2013) …”.
Rows 279 – 284. I suggest to remove them.
Rows 307 – 310. You already mentioned this before. I suggest to remove this part.
Row 313. Why is “areas” in italics?
Row 329. Is the requirement of having positive runoff satisfied? If not, please provide additional information about how you managed this issue. If I am not wrong, you provide additional information about this only in row 713. I suggest to anticipate this statement.
Row 340. Please rewrite, removing “bear in mind”.
Row 352. Please remove or rewrite “because Norway is a diverse country when it comes to runoff generation”. For me, it is not clear what you meant.
Row 372. "Credibility interval" or "confidence interval"?
Row 409. I think you were referring to “confidence interval” and not to “credible interval”.
Equation 14. I suggest to insert it as three separate equations. Please change the blue font to black font.
Rows 464 – 467. Please remove them or move them before. It is not advisable to mention the goals of the study after 18 pages of text. You already provided some information about the goals in several parts of Section 1. I suggest to merge everything, providing a more detailed and structured statement.
Rows 470 – 471. Please remove “These are observations from 127 fully gauged catchments from 19812010 and 284 partially gauged catchments from 19652010”. You already mention it.
Row 491 and following rows. The bold font is not necessary.
Rows 561 – 563. I suggest to remove them.
Figure 6. Please remove the “[1]” above the first colorbar. Please insert the axis (with coordinates).
Figure 7. I suggest to use “UTM” instead of “utm”.
Row 637. Is “do” necessary?
Row 671. Please remove the second brackets after x(y).
Row 702. Please remove the empty row.
AC1: 'Reply on RC1', Thea Roksvåg, 12 Jul 2021
We would like to thank Anonymous referee #1 for the constructive review.
Referee 1’s main concern is connected to the structure of the paper and he/she has several suggestions on how to improve it. The suggestions are concrete and good, and we will revise the paper accordingly. More specifically, repeated information will be removed and we will shorten and sharpen the introduction. We will also edit/clarify the sentences that are mentioned by the referee.As referee 1’s comments mainly concern structure and grammar, we don’t have any additional comments. We will follow up on each of the comments in a revision.
Kind regards,
Thea Roksvåg and coauthors.

AC1: 'Reply on RC1', Thea Roksvåg, 12 Jul 2021

RC2: 'Comment on hess20218', Anonymous Referee #2, 24 Jun 2021
This paper proposes a method for the spatial prediction of annual runoff that uses processbased model outputs (model HBV), fully gauged catchment data and partially gauged catchment data. The latter are preprocessed using a method published by the same coauthors (https://doi.org/10.5194/hess2441092020) in HESS and it is optional if all catchment are fully gauged.
This contribution is a significant contribution since the proposed approach allows to provide improved maps of annual runoff at the national scale, by taking into account fully gauged and and partially gauged catchments.
The proposed method is hierarchical in a Bayesian framework; it includes a Gaussian observation process, a spatial latent model and hyperpriors. The latent random field is defined on a fine 1km x 1km grid. It is a Spatially Varying Coefficient regression model to the HBV outputs that involves two GRFs characterized with stationary Matern covariance functions. The statistical analysis is performed using INLA/SPDE.
The areal nature of the data is taken into account by relating the catchment data to the sum of the latent model over the catchment.
The approach is validated using a carefully designed kfold crossvalidation. The approach shows superior performances as compared to HBV only and to TopKriging.
General comment:
===============
The statistical analysis is stateofthe art. It uses the most modern tools for analyzing complex spatial data in a hierarchical Bayesian framework. The experimental setup and the validation study are very carefully designed. The results are well discussed.Overall, the paper is well written, maybe on the long side though considering that lots of material is common with https://doi.org/10.5194/hess2441092020. I spotted some rare typos and a few awkward sentences (see below). All illustrations are relevant. I believe that this paper will be a nice methodological followup to https://doi.org/10.5194/hess2441092020. I recommend publication after a minor revision that takes into account the specific comments below
Specific comments:
==================
1. A potential weakness of the method, which has been mentioned by the authors, is that the model does not prevent negative runoff predictions in some (unlikely situations). This is due to the Gaussian likelihood and Gaussian GRF. The authors mention that logrunoff could be used instead, but then linearity of Eq. (6) is lost, which is an impediment. Another way of preventing negative predictions would be to use logGaussian likelihood and logGaussian random fielsd for x(u) and α(u) in (4). This would be a marginal change, since INLA/SPDE allows for loggaussian likelihood and LG random fields at almost no cost. As a result, predictions for x and α would always be positive. I wonder how this would work. Ideally, I'd like the authors to try this option, but I'd be happy if they only discuss this possibility.2. The GRFs x(u) and α(u) are independent. This assumption is never clearly stated and it is not discussed. Is this a reasonable assumption? Is this an assumption you could check or validate? How useful/difficult would it be to relax this assumption?
3. The GRFs x(u) and α(u) are assumed to be stationary. Are you able to check that this assumption is supported by the data?
4. To my knowledge, the product of a an exponential variogram with a fractal variogram is not a valid variogram. However, the product of an exponential covariance function with a fractal variogram might be a valid variogram. Please doublecheck and provide references if necessary.
5. Regarding the results: is it really desirable to get a correlation of 1 between measures and predictions? I would relate this to the fact that the coverage is 83%, which shows that the SVC is overconfident in the UG setting. Please comment.
Typos, etc.
===========
41: counties > countries
51: there exist work > there exist works
206: remove the square bracket ]
222: Further is $\sigma^2$ > Further, $\sigma^2$ is
229: advice > advise
235 to 242: Kriging yields "predictions" not "estimations". Please change "estim*" to "predict*" everywhere. BLUE should become BLUP (lines 236, 240, 241)
248: subcatcment > subcatchment
266: in 3 > in Figure 3
335: remove the dot in $s_i \sigma^2_y$, since it is not used in most cases later. (Also line 359 and line 406)
358: are the scales > the scales are
383: I very much doubt that Norway is only 40 km wide (or less). Please double check
430 Further is the variable x > Further, the variable x is
449: I did not understand the sentence. I think it needs to be rephrased.
555: "... is defined as the probability that 90% of the observed values ..." should be "... is defined as the proportion of the observed values ..."
584: In order to respect the order (first α, then x) I suggest to change the order between short ranged and long ranged. Hence, write "the spatial fields have picked up long range and short range processes, respectively".
667: "only $x(u)$ or $\alpha(u)$ > "only either $x(u)$ or "$\alpha(u)$"
669: "indicates that it for many study area might ..." > "indicates that for many study area it might"
672: "only x(u) or only α(u)" > one random field only.
710: does not makes > does not make
734: that is > that it is
In several places, there are sentences beginning with "Mark". This is unusual. I recommend using "Remark" or "Notice". AC2: 'Reply on RC2', Thea Roksvåg, 12 Jul 2021
Thea Roksvåg et al.
Thea Roksvåg et al.
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