First of all, we express our gratitude to the referee for the valuable comments and feedback. We made efforts to incorporate all of their suggestions, and we believe that the manuscript has been enhanced as a result.

Below, we provide a detailed response to the reviewers' comments, with our responses in bold font.

1. The introduction would benefit from a more detailed and systematic literature review of the development of multivariate bias correction methods. Discussing the advantages and disadvantages of existing methods would provide valuable context and help highlight the contribution of this study.

We agree that we have provided a rather rapid overview of existing Multivariate Bias Correction (MBC) methods. However, it is not possible in this paper to provide a detailed description of all methods in this paper, which is not a review paper. We have thus decided to expand the paragraph presenting the MBC methods (lines 58-69) in the original manuscript, which will now read:

“To adjust biases in multivariate dependencies, some multivariate bias correction (MBC) methods have been recently developed in the literature. The objective of MBC is fundamentally the same as that of univariate BC: transforming climate model simulations so that selected statistical features match those of a reference data set over the calibration period. The difference with univariate BCs lies in the fact that statistical features are not only univariate, but also multivariate such as inter-variable correlations or spatial copula structure. MBC methods can be grouped into three categories based on how they adjust climate simulations, (Vrac (2018); Robin et al. (2019); François et al. (2020): 1) most methods belong to the "marginal/dependence'' category that gathers MBCs adjusting separately univariate distributions and dependence properties. For precipitation outputs of RCMs, Bárdossy and Pegram (2012) proposed a correction of the spatial structure using matrix recorrelation or sequential correlation based on copulas. For correcting spatiotemporal biases for multiple variables, Cannon (2018) proposed an image processing technique for multivariate and spatial bias correction. Sharma (2019) and Guo et al. (2019) proposed a modification of the ranks (obtained from the empirical quantiles) based on resampling for the former and shuffling for the latter. In the same spirit, but based on CDF-t for the univariate correction instead a simple quantile mapping, Vrac (2018) and Vrac and Thao (2020) proposed a "Rank Resampling for Distributions and Dependencies" bias correction (R2D2), which will be detailed later. In principle, these approaches are not limited in terms of the number of variables, grid points in space, and the time scale considered. Nahar et al. (2018) proposed an independent component analysis approach to correct biases at multiple locations conjointly, but this approach must be applied independently to each climate variable. Nguyen et al. (2019) considered a bias correction approach for time series using the spectrum in the frequency domain, applicable to multivariate time series or to several grid cells. François et al. (2021) proposed a Cycle-GAN, cycle-consistent adversarial network for the adjustment of spatial dependence structures of climate model projections. 2) The “successive conditional” correction approach, that adjusts the different simulated variables successively and conditionally on the previously adjusted ones (e.g., Piani and Haerter, 2012; Dekens et al., 2017). 3) The “all-in-one”' category that consists of MBCs simultaneously adjusting univariate and multivariate properties of climate simulations. Pan et al. (2021) proposed to learn an adversarial neural network for spatially coherent corrections of daily precipitations, while Robin et al. (2019) use optimal transport theory for multivariate and spatially coherent bias corrections."

2. It would be helpful if the subtitles in Figure 1 provided more detailed information about each panel. Additionally, labeling the subfigures with identifiers like (a), (b), (c) or numbers like 1, 2, 3 would improve readability and make it easier to reference specific parts of the figure in the text.

Done, thanks.

3. Line 180 Including a simple formula to show how the Soil Water Content (SWC) is calculated would enhance the reader's understanding of the methodology and the variables involved.

We added the following sentence describing how SWC is computed (now line 198) “by computing SWCt = SWCt-1 + (prt - k.ET0t )/ d, where d is the depth of the soil and k is a constant depending on the considered crop”.

4. Line 187: Please discuss the impact of the simplicity of the model or assumptions mentioned here. Elaborating on how this simplicity might affect the results or interpretations would strengthen the credibility of the study.

Multiplying the type of soils with possibly different depth and/or different soil water capacities would have significantly increased the experiment without significantly changing our results. We ran sensitivity tests in some settings. Even though some differences were visible in the SWC values, the conclusions in terms of bias correction methods remained unchanged (not shown). Actually, this result also holds when considering different wheat varieties for example. In order to keep the experimental design at a manageable size, we decided to run the full study for a single soil type and a single variety per crop. We thus added the following sentence: “Early sensitivity tests have shown that when changing these parameters, even though some differences were visible in the SWC values, the conclusions in terms of bias correction methods that will be reported later remained unchanged (not shown). In order to keep the experimental design at a manageable size, we decided to run the full study for a single soil type. This choice also avoids problems of soil heterogeneity in each region and allows us to catch as much climate effect as possible.”

5. Line 216: Under climate change, inter-variable correlations might change over time. How does the methodology account for potential changes in inter-variable correlations in future climate scenarios? Addressing this point would clarify the robustness of the bias correction methods when applied under changing climatic 

With dOTC, a transport map between the multivariate density on the model and that on the reference on the calibration period is estimated. This transport map is then used to correct the model outputs in the projection period. The underlying hypothesis is that this transport map remains identical in the future, but it does not assume that the inter-variable correlations do not change in the future. Indeed, dOTC accounts for the changes in the multivariate distribution (and thus in the inter-variable correlations) and incorporates them into the corrections (Robin et al., 2019). Regarding R2D2, the marginal distributions can change in the future while the copula is considered stationary in time, and identical to that in the reference. This might seem a strong assumption, but Vrac et al. (2023) showed that this was a conservative hypothesis preventing climate models from exhibiting over-varying inter-variable correlations, not corresponding to actual (historical) evolutions (Vrac et al., 2022). In conclusion, assuming a stationary copula corresponding to reference data is a safe choice. 

We added the following text at the end of the presentation of the R2D2 method (line 239).

“While this may appear to be a strong assumption, Vrac et al. (2023) showed that it serves as a conservative hypothesis. This prevents climate models from exhibiting over-varying inter-variable correlations that do not  correspond to actual (historical) evolutions (Vrac et al., 2022). Assuming a stationary copula corresponding to reference data is thus a safe choice.”

Additional references:

Vrac, M., Thao, S., and Yiou, P.: Should multivariate bias corrections of climate simulations account for changes of rank correlation over time?, Journal of Geophysical Research: Atmospheres, 127, e2022JD036 562, 2022.

Vrac, M., Thao, S., and Yiou, P. (2023) : Changes in temperature–precipitation correlations over Europe: are climate models reliable?, Climate Dynamics, 60, 2713–2733, 2023.