The manuscript is focused on numerical techniques to 'distribute' residuals of a water balance equation over the contributing terms, avoiding negative values. The manuscript seems focused on the numerical techniques, with limited efforts for a hydrological interpretation. It could be more convincing if authors bring a bit more on the 'explanation' side.

It is not immediately evident to this reviewer that negative values are a problem, especially for the soil storage term (TWSC). In fact one may expect this term to be symmetric around zero, and maybe the same holds for the errors?

One would assume, based on considerations of the various terms of the water balance that a comparison between the negative (or positive) residuals over time will help identifying which term may be primarily responsible: the soil term can dominate in the short term (e.g. days), but will be small for annual comparisons and may become negligible at decadal scale (except for the long-term desiccation discourse). Spatial patterns are also expected, as frontal rainfall patterns are much easier to represent correctly than thunderstorms (much of tropics and arid zone rainfall) -- indeed your later results (Fig. 6) seem to match this expectation. 

Maybe further reference can be made to the 'Budyko' literature that looks at an annual balance, while your current analysis takes a monthly perspective.

The abstract could become more attractive to readers if the time unit (monthly balance calculations) is made explicit, as results for daily or annual balance calculations will likely be different.

Details:

The Highlights should be understandable for a non-technical expert -- at the moment they are too full of jargon to attract readers.

Line 57 Indeed a closed budget gives some confidence in the underlying estimates, but not if the closure is obtained by 'fudging' the data, without 'understanding'. So I disagree that 'closing the budget' helps with 'understanding'.

Line 66. Before delving into the details it will be good for the reader to be reminded of the physical aspects of uncertainty in the various terms, as these are of different natures:

P precipitation input -- the typically are fairly reliable point data from rainfall gauge data, often with some need t gap fill missing data. The main uncertainty here is in the spatial distribution and representativeness of rainfall gauges, in relation to rainfall types (for frontal rains the spatial uncertainty is low, for local storms it can be high). The distribution of rainfall gauges is often determined in part by accessibility and convenience, and overall uncertainty of daily rainfall may be easily underestimated. More recent satellite based estimates of rainfall appear to perform well for frontal rains, but not in other rainfall types.  

ET Evapotranspiration equations have been fairly well calibrated, but there can be uncertainty over the advection term especially in small catchments. For larger areas energy balance equations may be sufficient.

Q monitoring of outflow can have low uncertainty if ;rating curves' are frequently calibrated. However, the delineation of the watershed (and area used for the calculations) can be off where groundwater flows don't necessarily follow surface catchment delineations and can be underestimated.

TWSC can become negligible if a multi-year balance is considered (verifying the P and Q estimates) but can dominate the balance at a daily time-scale. A major challenge is the depth over which TWSC is to be assessed, as changes in the topsoil can be more easily assessed than that deeper in the soil.

Line 98 Negative ET is possible under 'dew formation' conditions... (be it in only part of a daily temperature cycle)

Line 244 There can be 'bias' (systematic error, e.g. if groundwater flows mean that the basin is not closed and part of outflowing Q is missed; the area of the basin can also be incorrect), part 'measurement error'. As you focus on relatively large basins, the bias term may be relatively small, but for smaller watersheds the bias terms cannot be ignored. Standard techniques such as plotting cumulative Q vs cumulative P give indications, especially if nested Q data exist beyond outflow data.

Line 702-714, Figure 7 - would it make sense to compensate S Hemisphere data for a 6 month shift in seasons? Or even more flexibly to use a hydrological year concept with a standardized month for maximum P.

Greetings. The manuscript entitled “A novel method for 1 correcting water budget components and reducing their uncertainties by optimally distributing the imbalance residual without full closure” deals with the closure of water budget problems, and specifically with uncertainties therein. The structure and goals are clear, and the results offering is well-suited. This paper can for sure be published after some adjustments, listed below. I think that these itemized improvements would make the work more scientifically sound and robust. These considerations come from my expertise as a hydrogeologist, so they will pertain to this sphere of competency. Best regards.

• In equation 2, jkl should be written as subscripts, as well as 123, etc in eq. 3.
• We need a more detailed specification, both in the introduction and in the methodology (e.g., from line 64 on) of the TWSC term. It is not sufficient to describe the ground-underground part of the water cycle. Two major points should be at least touched: (i) a portion of the TWSC term is the water infiltrating to aquifer, but that is returned to major water bodies soon or later (see e.g. Levison et al., 2016); (ii) the major role in the aquifer ability to store or drain the portion of water infiltrating is played by local geology, precisely its spatial distribution and the eventual presence of fractures (again, Levison et al., 2016) or highly permeable conduits (Schiavo, 2023). I think these two major points should be supported leveraging the suggested references.
• Global precipitation models, as well as other kinds of climate products, need to be bias corrected to be employable, even if these issues and the opportunity of such procedures are still subject of scientific debate (e.g., Ehret et al., 2012). Are the employed data raw or bias corrected (if so, how)?
• The resolution of the spatial problem is crucial. It is well known that changes in resolution make employed variables (such as DEMs, as in Aziz et al., 2022) very different at the same location. How to tackle this point? Which resolution ‘advice’ for the reader? Any criterion? Otherwise, the errors will propagate in a sort of uncontrolled cascade.
• From line 289 on, I think that the noise to afflict the Kalman Filter with should be pointed out. Which is the observation noise covariance (and its quantification for the employed variables)?
• Is there any exit criterion (such as tolerances) to exit from equations 9 and 10?

Suggested References:

• Levison et al., 2016. Long-term trends in groundwater recharge and discharge in a fractured bedrock aquifer – past and future conditions. Canadian Water Resources Journal / Revue canadienne des ressources hydriques, Volume 41, 2016 - Issue 4: Special Issue: Groundwater – Surface Water Interactions in Canada. https://doi.org/10.1080/07011784.2015.1037795
• Schiavo, 2023. The role of different sources of uncertainty on the stochastic quantification of subsurface discharges in heterogeneous aquifers. J. Hydrol. 617 (4), 128930. DOI: 10.1016/j.jhydrol.2022.128930

Further reading references:

• Ehret, U., Zehe, E., Wulfmeyer, V., Warrach-Sagi, K., and Liebert, J.: HESS Opinions "Should we apply bias correction to global and regional climate model data?", Hydrol. Earth Syst. Sci., 16, 3391–3404, https://doi.org/10.5194/hess-16-3391-2012, 2012.
• Aziz, K.M.A., Rashwan, K.S. Comparison of different resolutions of six free online DEMs with GPS elevation data on a new 6th of October City, Egypt. Arab J Geosci15, 1585 (2022). https://doi.org/10.1007/s12517-022-10845-5