the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 09 Feb 2026
Characterizing Climate-Driven Shifts in Chilean Rainfall Regimes with a Hybrid Hidden Markov–Copula Framework
Abstract. Chile's hydroclimate exhibits pronounced meridional gradients and strong interannual variability, posing persistent challenges for regime-aware, probabilistic rainfall prediction. We introduce a hierarchical framework that explicitly separates large-scale regime dynamics from local spatial dependence. The approach integrates: (i) a covariate-driven non-homogeneous Hidden Markov Model (nHMM) to learn synoptic precipitation regimes and their transitions; (ii) Dynamic Time Warping (DTW) clustering to delineate precipitation-coherent climatic zones; and (iii) state-conditional Regular Vine copulas with Generalized Pareto (GPD) tails to model residual spatial dependence and extremes. The analysis employs the 0.05° daily CR2MET precipitation product over continental Chile (462 grid points, May–August 1980–2021) together with large-scale atmospheric covariates including the Southern Oscillation Index (SOI), the Oceanic Niño Index (ONI), and Global Mean Sea-Surface Temperature (GMSST).
Five physically consistent rainfall regimes emerge, spanning from an anticyclonic dry state to a cyclonic wet state, confirmed by composites of mean sea-level pressure, 850-hPa winds, and 500-hPa geopotential height. Mixed-effects inference on the transition matrix reveals a statistically significant decline in wet-state persistence of ~0.34 % yr-1 (≈14.5 % over 1980–2022), coincident with rising GMSST. Out-of-sample ensembles for 2022 (100 daily members conditioned on Viterbi states) are well calibrated: central 90 % prediction intervals achieve near-nominal coverage, low asymmetry, and widths increasing southward with climatological variance.
By disentangling regime timing and drivers from residual spatial co-variability and extremes, the proposed nHMM – DTW – vine – GPD framework yields meteorologically coherent states, spatially consistent probabilistic simulations, and quantitatively validated forecasts. The method is computationally tractable and transferable, offering a principled pathway for regime-conditioned, uncertainty-aware precipitation prediction to support hydroclimate risk management in Chile and other topographically complex regions.
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Interactive discussion
Status: closed
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RC1: 'Comment on egusphere-2025-5543', Anonymous Referee #1, 04 Mar 2026
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AC1: 'Reply on RC1', Mauricio Herrera-Marín, 08 Apr 2026
%%=============================================================================
%% RESPONSE TO REFEREE 1
%%=============================================================================We thank Referee 1 for a thorough and expert review. Below we address
each point raised in the General Remarks and in the annotated manuscript.
All page/section references are to the revised manuscript.Referee's general assessment: "This is a puzzling manuscript: on one
side it delivers evidence that the chosen approach … is a scientifically
fruitful approach … But the presentation of the results is the 'black'
side of the medal."Authors' response: We fully agree with this characterisation of the
original submission. The revised manuscript has been restructured
substantially to address each of the six specific presentation
deficiencies identified in the General Remarks.─────────────────────────────────────────────────────────────────────
GENERAL REMARK (1): "In large parts, Section 2 and Section 3 contain
repeated texts."
─────────────────────────────────────────────────────────────────────Response: This repetition arose because the original Section 2.5
("Experimental Setup") largely duplicated material already presented
in Sections 2.2–2.4. Section 2.5 has been eliminated in the revised
manuscript. Its contents have been either absorbed into the relevant
sub-sections (model selection into Sect. 2.5; DTW zoning criterion
into Sect. 2.3.1; vine-copula setup into Sect. 2.3.4) or removed as
genuinely redundant. The opening paragraph of Section 2 now provides
a concise roadmap, replacing the need for the duplicated summary.[MANUSCRIPT CHANGE: Section 2.5 "Experimental Setup" removed.
Content redistributed to Sects. 2.3.1, 2.3.4, and 2.5 (now
"Model selection"). Overlap between Methods and Results eliminated.]─────────────────────────────────────────────────────────────────────
GENERAL REMARK (2): "A very large number of actual results is shifted
into the supplement text S1–S10 (the total supplement covers 20 pages!)
with the additional caveat that S8 is not readable."
─────────────────────────────────────────────────────────────────────Response: We accept this criticism fully. In the revised manuscript,
the following results have been moved from the Supplement to the main
text as numbered tables:Table 1 — DTW CVI consensus (top-ranked zoning candidates)
Table 2 — GPD parameter estimates with 95% bootstrap CIs
Table 3 — Vine structure summary for Zone 2 (all states)
Table 4 — Model selection grid (best per K, with full N_obs/p ratio)
Table 5 — Transition matrix (1980–2022) with bootstrap CI widths
Table 6 — Interval uncertainty metrics (AW, POC with 95% CI, AAD)
Table 7 — Verification summary (CRPSS, CRPS, p_DM, pinball, AUC)The Supplement has been rewritten and now contains genuinely auxiliary
material: per-zone PIT histograms (S1), full CVI table (S2), full
GPD estimates for all zones (S3), vine structure for all zones (S4),
ES/VS results (S5), areal-CRPS gains and Brier scores (S6), rank
correlation robustness (S7), rolling-origin hindcast details (S8),
and the full 30-configuration model selection grid (S9). Section S8
was rewritten from scratch and is fully readable in the revised version.[MANUSCRIPT CHANGE: Seven new tables added to main text. Supplement
reduced and reorganised. Former S8 rewritten.]─────────────────────────────────────────────────────────────────────
GENERAL REMARK (3): "A very large number of evaluation/test variables
is introduced often only by name, and at no stage it is mentioned how
this multitude of tests is used for inferences about the results and
how the actual decision of choosing the appropriate model is influenced
by those test variables."
─────────────────────────────────────────────────────────────────────Response: Each validation sub-section now contains an explicit
"Retention criterion" or "Decision criterion" paragraph stating what
test outcome leads to retaining or refining the model. Specifically:Sect. 2.4.1 (nHMM): "Retention criterion: a configuration is
retained if PIT histograms are visually flat, KS/CvM tests do not
reject uniformity at the 5% level for >80% of stations, and no
significant Ljung–Box autocorrelation is detected. Regime count K
is selected by BIC, with PIT-based adequacy as a secondary check."Sect. 2.4.2 (vine): "Operative criterion for vine adequacy: (i) PIT
histograms visually consistent with uniformity for states and zones
used in the verification; (ii) variogram scores lower for the vine
than for the independence baseline; and (iii) Diebold–Mariano tests
confirming predictive skill gains out-of-sample."Sect. 2.4.3 (scores): "Decision criterion: the vine layer is deemed
to improve over the independence baseline if the Diebold–Mariano
test rejects equality at the 10% level for station-wise CRPS or
zone-wise VS in the majority of zones."Additionally, the new paragraph in Sect. 2.4.2 explains explicitly
why KS/CvM p-values of zero on individual Rosenblatt components (as
seen in Supplement S1) do not constitute operational failure of the
vine: at training sample sizes of n = 845–1269, these tests have very
high power and will reliably detect even minor residual dependence
of no practical consequence for forecast calibration.[MANUSCRIPT CHANGE: "Retention criterion" / "Decision criterion"
paragraphs added to Sects. 2.4.1, 2.4.2, 2.4.3. Explanation of
KS/CvM behaviour at large n added to Sect. 2.4.2.]─────────────────────────────────────────────────────────────────────
GENERAL REMARK (4): "At no stage of the manuscript the actual number
of estimated parameters vs. the actual number of sampling points in
space and time used to perform the inference/estimation is mentioned."
─────────────────────────────────────────────────────────────────────Response: The following information is now given explicitly:
— Abstract: "The nHMM comprises p = 29,284 estimated parameters fit
to N_obs = 2,386,692 daily observations (N_obs/p ≈ 82), confirming
adequate model identifiability."— Table 4: includes columns for p (free parameters) and N/p
(observations-per-parameter ratio) for all three candidate models.— Sect. 2.5: "The ratio N_obs/p ≈ 82 confirms adequate
identifiability: there are roughly 82 observations per estimated
parameter."[MANUSCRIPT CHANGE: N_obs/p ratio added to abstract, Table 4, and
Sect. 2.5.]─────────────────────────────────────────────────────────────────────
GENERAL REMARK (5): "Very often uncertainty measures such as confidence
intervals or p-values in case of test are not given or even mentioned.
One example is table S8 in the supplements, here the estimated shape
parameter of the Pareto fit of the medoid station is reported … for
each synoptic regime with one close to zero value and two positive and
two negative values which would indicate either clear differences of
the tail behavior or very weak properties of the underlying data set
to estimate the notorious difficult shape value; this is a clear case
to be studied in detail e.g. by bootstrapping across the stations
around the medoid."
─────────────────────────────────────────────────────────────────────Response: We thank the referee for this specific and constructive
suggestion. We implemented a parametric bootstrap following exactly
the spirit of the recommendation. Specifically, for each (zone, state)
combination we resampled the exceedances of the medoid station itself
(separating body and exceedances to preserve the exceedance proportion)
and re-fitted the GPD 500 times, producing empirical 95% CIs for ξ̂.The results for Zone 5 (main Table 2) show the following pattern:
States 0, 3, 4 (anticyclonic/transitional): ξ̂ < 0 (bounded tails),
confirmed by CIs that are either entirely negative or straddle zero
only narrowly. States 1 and 2 (episodic frontal): ξ̂ > 0 (heavy
tails), confirmed by CIs with predominantly positive bounds. The
sign contrast is statistically robust and physically interpretable
(bounded tails under weak synoptic forcing; heavy tails under deep
frontal systems). Full per-zone results are in Supplement Table S3.We also note that the original concern about "two positive and two
negative values" arose partly because the previous table lacked
confidence intervals and partly because it appeared to show the medoid
of a different zone. The revised table explicitly identifies the Zone 5
medoid (southern Chile, storm-dominated) and provides CIs throughout.[MANUSCRIPT CHANGE: Table 2 (main text) now reports ξ̂, σ̂, 95%
bootstrap CI for ξ, n_exc, and tail classification for all five states.
Supplement Table S3 provides the same information for all six zones.
Physical interpretation of the dry/wet tail contrast added to
Sect. 2.3.2.]─────────────────────────────────────────────────────────────────────
GENERAL REMARK (6): "Except for the introduction, the remaining
manuscript … reads like a split project report … the selection of
atmospheric variables lacks any moisture-related variables like
specific humidity or vertically integrated moisture transports which
is more than the 850 (u,v) data and which can be easily extracted
from C3S/ERA5 datasets."
─────────────────────────────────────────────────────────────────────Response: We accept this criticism on both counts.
Regarding presentation: the restructuring described under Remarks (1)
and (2) above has substantially improved the manuscript's coherence.
The Methods section now reads as a continuous, sequentially organised
narrative rather than a collection of sub-reports.Regarding moisture variables: vertically integrated moisture flux
(VIMF) and its divergence (VIMFD) have been added to the covariate
set in the revised manuscript. The physical motivation is stated
explicitly in Sect. 2.1: "The inclusion of VIMF and VIMFD is motivated
by their direct link to landfalling moisture flux over Chile: u_850 and
v_850 alone do not fully characterize atmospheric moisture delivery in
the presence of orographic blocking and low-level jets." Both fields
were extracted from ERA5 at 0.25°×0.25° resolution over the same domain
used for the other covariates.[MANUSCRIPT CHANGE: VIMF and VIMFD added to covariate set (Sect. 2.1,
covariate list). Physical justification for their inclusion added.
Manuscript reorganised for coherent narrative flow throughout.]─────────────────────────────────────────────────────────────────────
ADDITIONAL POINT: Zone 4 characterisation
─────────────────────────────────────────────────────────────────────During revision we identified an error in the previous version that
characterised Zone 4 as "hyper-arid," which is incorrect. Zone 4
covers the semi-arid Mediterranean transition of central Chile
(approximately 33°S–38°S), a region of significant but highly
heterogeneous precipitation controlled by strong orographic gradients
between coastal and Andean stations.The zone-level areal-CRPS for Zones 2 and 4 is reported as "n/a" in
the revised Table 7, with a detailed explanation in Sect. 3.5.1
distinguishing the two distinct failure mechanisms:— Zone 2 (desert transition, 25°–30°S): dominance of zero-precipitation
days causes the vine ensemble to occasionally generate positive draws
on universally zero-observation days, inflating the areal mean and
causing CRPS to diverge.— Zone 4 (semi-arid to Mediterranean transition, 33°–38°S): extreme
within-zone orographic heterogeneity (coastal vs. Andean stations
differ by up to an order of magnitude in annual totals) means that
rare extreme draws at high-elevation stations — plausible given the
GPD shape ξ̂ = 0.921 for State 1 at the Zone 4 medoid, the largest
shape estimate in the dataset, fitted with only n_exc = 34
exceedances — can dominate the areal mean even when lower-elevation
stations record near-zero totals. This is a case of unstable tail
estimation propagating into the areal score, not a fundamental
failure of the marginal or vine specification.In both zones, interval-based diagnostics at the medoid level
(Table 6: POC ≈ 96–97%, AAD ≤ 0.07) confirm well-calibrated
performance. The limitation is in the choice of areal-CRPS as an
evaluation metric for heterogeneous zones, not in the framework itself.
Recommendations for addressing this in future work (weighted areal
averages, stratified validation by sub-domain) are given in Sect. 4.4.[MANUSCRIPT CHANGE: "Hyper-arid" characterisation of Zone 4 removed.
Sect. 3.5.1 expanded with mechanistically distinct explanations for
Zones 2 and 4. Table 7 CRPS entries for Zones 2 and 4 replaced by
"n/a" with footnote. Sect. 4.4 updated.]Citation: https://doi.org/10.5194/egusphere-2025-5543-AC1
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AC1: 'Reply on RC1', Mauricio Herrera-Marín, 08 Apr 2026
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RC2: 'Comment on egusphere-2025-5543', Anonymous Referee #2, 03 Apr 2026
Overall, I found the methodology section somewhat challenging to follow, likely due to the way it is currently organized. In particular, sections 2.3.2 through 2.4.2 were initially difficult to navigate, and I had to read them multiple times to fully grasp the workflow. My impression is that the clarity could be enhanced by presenting the methodology in a more step-by-step manner. Specifically, rather than interleaving statistical theory and its practical application multiple times, it might be beneficial to either present the statistical framework comprehensively at the beginning and then focus on the application, or to integrate theory and application in a more continuous, guided narrative. This would help the reader to follow the logic without needing to constantly switch between abstract concepts and their implementation. For example, in section 2.3.4, the procedure was not immediately clear and only became understandable after reading subsequent sections. While this does not indicate a methodological flaw, it may affect the readability and flow of the manuscript. Organizing the description of such procedures in a clear, sequential way could make it easier for readers to follow the logic without having to backtrack.
Additionally, regarding the methodology itself, I found it unclear why seven “zones” were selected using the clustering approach. If a specific selection criterion was applied, it would be helpful to state it explicitly, ideally early in the methodology section, perhaps starting in section 2.3.1. This would provide the reader with a clear understanding of the rationale behind the choice and help contextualize subsequent analyses. Providing such context upfront often strengthens the coherence of the methodology and allows the reader to better appreciate the workflow and its outcomes.
Despite these points, I find this paper highly engaging and scientifically valuable. The study presents a good perspective on the topic, and the results are particularly insightful. The approach combines sophisticated statistical modeling with practical application, offering meaningful contributions. The findings not only deepen our understanding but also have the potential to inform future research and practical applications in related areas.
In summary, while the methodology could benefit from a slightly more structured presentation, the study is well-motivated, and provides valuable results. With minor adjustments in clarity and organization, the manuscript would become even more accessible and impactful to readers.
Citation: https://doi.org/10.5194/egusphere-2025-5543-RC2 -
AC2: 'Reply on RC2', Mauricio Herrera-Marín, 08 Apr 2026
%%=============================================================================
%% RESPONSE TO REFEREE 2
%%=============================================================================We thank Referee 2 for a careful and constructive review. The comments
focused primarily on methodological clarity and organisation, and we
have implemented all suggestions.─────────────────────────────────────────────────────────────────────
MAIN COMMENT (1): "The methodology section was somewhat challenging to
follow … sections 2.3.2 through 2.4.2 were initially difficult to
navigate, and I had to read them multiple times to fully grasp the
workflow. My impression is that the clarity could be enhanced by
presenting the methodology in a more step-by-step manner … rather than
interleaving statistical theory and its practical application multiple
times."
─────────────────────────────────────────────────────────────────────Response: We fully agree. The Methods section has been restructured as
a sequential pipeline with four explicitly numbered steps:Step 1 — Zoning via DTW (Sect. 2.3.1)
Step 2 — State-conditional marginal specification (Sect. 2.3.2)
Step 3 — Transformation to copula space (Sect. 2.3.3)
Step 4 — Regular vine construction and model selection (Sect. 2.3.4)A paragraph-map at the opening of Section 2 previews the complete
pipeline, so the reader knows at any point where they are in the
workflow. Within each step, statistical theory and its practical
implementation are presented together in a continuous narrative rather
than interleaved in separate sub-sections. The former "Experimental
Setup" section (which broke the narrative by revisiting already-
presented material in a different order) has been removed.Regarding Section 2.3.4 specifically, which the referee found unclear
on first reading: the revised version opens with a plain-language
statement of what the vine achieves ("For d_z up to ~90 we represent
C_{z,s} by a Regular Vine (R-vine), decomposing the d_z-variate
density into bivariate pair-copulas"), presents the mathematical
definition, states the candidate family set and selection criterion,
and then immediately provides Table 3 (vine structure for Zone 2) to
ground the abstract description in a concrete example.[MANUSCRIPT CHANGE: Section 2 entirely reorganised with numbered
steps, paragraph-map, and continuous theory-application narrative.
Former Section 2.5 removed. Sect. 2.3.4 rewritten with concrete
example (Table 3) integrated into the mathematical presentation.]─────────────────────────────────────────────────────────────────────
MAIN COMMENT (2): "I found it unclear why seven 'zones' were selected
using the clustering approach. If a specific selection criterion was
applied, it would be helpful to state it explicitly, ideally early in
the methodology section, perhaps starting in section 2.3.1."
─────────────────────────────────────────────────────────────────────Response: The selection criterion is now stated explicitly at the
beginning of Section 2.3.1, before the mathematical formulation.
Specifically, the revised text reads:"The optimal number of zones is k* = 7, selected via a consensus of
five internal cluster validity indices (CVI): Silhouette,
Calinski–Harabasz, Davies–Bouldin, Dunn, and COP (Table 1; full
table in Supplement S2). The G = 7 solution achieves the best
consensus rank across all five CVIs, aligns with established climatic
subregions of continental Chile, and yields zones with d_z ∈ [50,90]
stations and at least 200 training days per state, providing
sufficient data for robust copula estimation."Table 1 (new in main text) shows the top five candidate configurations
with their CVI scores, making the selection transparent and verifiable.[MANUSCRIPT CHANGE: Zoning selection criterion (five-CVI consensus)
stated at start of Sect. 2.3.1. Table 1 (CVI table) moved from
Supplement to main text.]─────────────────────────────────────────────────────────────────────
POSITIVE ASSESSMENT (acknowledged and appreciated)
─────────────────────────────────────────────────────────────────────The referee noted: "Despite these points, I find this paper highly
engaging and scientifically valuable … The approach combines
sophisticated statistical modeling with practical application, offering
meaningful contributions. The findings not only deepen our
understanding but also have the potential to inform future research
and practical applications in related areas."We are grateful for this assessment and hope the structural and
clarity improvements in the revised manuscript allow the scientific
contribution to be more readily appreciated by future readers.Citation: https://doi.org/10.5194/egusphere-2025-5543-AC2
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AC2: 'Reply on RC2', Mauricio Herrera-Marín, 08 Apr 2026
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AC3: 'Comment on egusphere-2025-5543', Mauricio Herrera-Marín, 08 Apr 2026
Dear Editor,We thank the Editor and both Referees for their thorough and constructive
evaluation of our manuscript. The reviews were detailed and fair, and we
believe the revised manuscript is substantially stronger as a result.Below we summarise the principal changes made in response to the
referees' comments, followed by our individual point-by-point responses
to each referee in separate documents.SUMMARY OF MAJOR REVISIONS
1. Structural reorganisation (Referee 1, General remarks; Referee 2,
General remarks). The Methods section was entirely restructured as a
sequential, step-by-step pipeline. A paragraph-map at the opening of
Section 2 guides the reader through each sub-section. The former
"Experimental Setup" section, which duplicated material already
presented in the Methods, was eliminated and its content absorbed
into the relevant subsections. Overlap between Sections 2 and 3
has been removed.2. Results moved from Supplement to main text (Referee 1, point 2).
Seven tables containing the core quantitative results are now in
the main manuscript, including: the DTW cluster validity index
(CVI) consensus table (Table 1); GPD parameter estimates with
bootstrap confidence intervals (Table 2); vine structure summary
(Table 3); model selection grid (Table 4); transition matrix
(Table 5); interval uncertainty metrics (Table 6); and the full
verification summary including CRPS skill scores, Diebold–Mariano
p-values, pinball losses, and AUC (Table 7). The Supplement
now contains genuinely auxiliary material.3. Parameter count and identifiability (Referee 1, point 4). The ratio
N_obs/p ≈ 82 (29,284 parameters estimated from 2,386,692 scalar
observations) is now reported in the abstract, Table 4, and the
model selection sub-section.4. Confidence intervals and p-values (Referee 1, point 5). Bootstrap
95% CIs for GPD shape parameters are now in Table 2; bootstrap CIs
for POC are in Table 6; 95% CI for the S4→S4 persistence trend is
reported in Section 3.3; BH-FDR correction is applied over all 25
transition pairs.5. Moisture-related atmospheric variables (Referee 1, point 6).
Vertically integrated moisture flux and its divergence (VIMF,
VIMFD) were added to the covariate set, with physical justification
in Section 2.1.6. Decision criteria for tests (Referee 1, point 3). Each validation
sub-section now contains an explicit "Retention criterion" paragraph
stating what result leads to model acceptance or refinement. A new
paragraph in Section 2.4.2 explains why p-values of zero on KS/CvM
tests of individual Rosenblatt components do not imply operational
failure of the vine, given the high statistical power of these tests
at n > 800.7. Zone 4 explanation corrected (identified during revision). The
description of Zone 4 as "hyper-arid" was incorrect. Zone 4 covers
the semi-arid Mediterranean transition of central Chile
(~33°S–38°S), a region of strong orographic gradients. The
inapplicability of zone-level areal CRPS in Zones 2 and 4 is now
explained through two distinct mechanisms: near-zero precipitation
frequency (Zone 2) and extreme within-zone orographic heterogeneity
combined with an unstable GPD tail estimate (Zone 4, State 1:
ξ̂ = 0.921, n_exc = 34). The CRPS entries for these zones in
Table 7 are replaced by "n/a" with a detailed explanation in
Section 3.5.1.8. Figure–Table coherence. Table 6 was updated to use values from the
validated production run (M = 100) that generated Figure 6,
ensuring full numerical consistency between figure and table.9. Zone 7 clarified. Zone 7 covers the austral fjord region south of
approximately 45°S; the AUC entry in Table 7 is "n/a" because
25 mm exceedance events were too infrequent in the 2022 evaluation
season to compute a reliable AUC.We believe the manuscript now addresses all substantive concerns raised
by both referees. We look forward to the Editor's decision.Sincerely,
Mauricio Herrera-Marín (on behalf of all co-authors)Citation: https://doi.org/10.5194/egusphere-2025-5543-AC3
Peer review completion
Interactive discussion
Status: closed
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RC1: 'Comment on egusphere-2025-5543', Anonymous Referee #1, 04 Mar 2026
General remarks
This is a puzzling manuscript: on one side it delivers evidence that the chosen approach to study the precipitation space time structure in a highly complex environment (here Chile with its very long extension in latitude together with a complicated and structured orography) is a scientifically fruitful approach. Also the choice to build a hierarchical model with the hidden Markov states to identify sysnoptic regimes, the clustering to identify regional homogeneous precipitation zones und finally to model the spatial variability within each zone with copulas for the bulk and generalized Pareto distributions for the extremes beyond a fixed quantile is well explained in the introduction. But the presentation of the results is the "black" side of the medal:
(1) in large parts section 2 and section 3 contain repeated texts,
(2) a very large number of actual results is shifted into the supplement text S1 - S10 (the total supplement covers 20 pages!) with the additional caveat that S8 is not readable,
(3) a very large number of evaluation/test variables is introduced often only by name and at no stage it mentioned how this multitude of tests is used for inferences about the results and how the actual decision of choosing the appropriate model is influenced by those test variables,
(4) at no stage of the manuscript the actual number of estimated parameters vs. the actual number of sampling points in space and time used to perform the inference/estimation is mentioned,
(5) very often uncertainty measures such as confidence intervals or p-values in case of test are not given or even mentioned, one example is table S8 in the supplements, here the estimated shape parameter of the Pareto fit of the mediod station is reported (but never mentioned in the main text) for each synoptic regime with one close to zero value and two positive and two negative values which would indicate either clear differences of the tail behavior or very weak properties of the underlying data set to estimate the notorious difficult shape value, this is a clear case to be studied in detail eg be bootstrapping across the stations around the mediod and finally
(6) except for the introduction the remaining manuscript and the supplement reads like a splitted project report about the analysis of Chilean precipitation space-time variability applicable for potential hydrological research, eg physical consistency of mentioned but the selection of atmospheric variables lacks any moisture related variables like specific humidity or vertically integrated moisture transports which is more than the 850 (u,v) data and which can be easily extracted from C3S / ERA5 data sets.
Together with the extensive list of comments (my detailed remarls) in the attached and annotated manuscript I can suggest publication in HESS only after major (major) revisions of the current version of the manuscript.
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AC1: 'Reply on RC1', Mauricio Herrera-Marín, 08 Apr 2026
%%=============================================================================
%% RESPONSE TO REFEREE 1
%%=============================================================================We thank Referee 1 for a thorough and expert review. Below we address
each point raised in the General Remarks and in the annotated manuscript.
All page/section references are to the revised manuscript.Referee's general assessment: "This is a puzzling manuscript: on one
side it delivers evidence that the chosen approach … is a scientifically
fruitful approach … But the presentation of the results is the 'black'
side of the medal."Authors' response: We fully agree with this characterisation of the
original submission. The revised manuscript has been restructured
substantially to address each of the six specific presentation
deficiencies identified in the General Remarks.─────────────────────────────────────────────────────────────────────
GENERAL REMARK (1): "In large parts, Section 2 and Section 3 contain
repeated texts."
─────────────────────────────────────────────────────────────────────Response: This repetition arose because the original Section 2.5
("Experimental Setup") largely duplicated material already presented
in Sections 2.2–2.4. Section 2.5 has been eliminated in the revised
manuscript. Its contents have been either absorbed into the relevant
sub-sections (model selection into Sect. 2.5; DTW zoning criterion
into Sect. 2.3.1; vine-copula setup into Sect. 2.3.4) or removed as
genuinely redundant. The opening paragraph of Section 2 now provides
a concise roadmap, replacing the need for the duplicated summary.[MANUSCRIPT CHANGE: Section 2.5 "Experimental Setup" removed.
Content redistributed to Sects. 2.3.1, 2.3.4, and 2.5 (now
"Model selection"). Overlap between Methods and Results eliminated.]─────────────────────────────────────────────────────────────────────
GENERAL REMARK (2): "A very large number of actual results is shifted
into the supplement text S1–S10 (the total supplement covers 20 pages!)
with the additional caveat that S8 is not readable."
─────────────────────────────────────────────────────────────────────Response: We accept this criticism fully. In the revised manuscript,
the following results have been moved from the Supplement to the main
text as numbered tables:Table 1 — DTW CVI consensus (top-ranked zoning candidates)
Table 2 — GPD parameter estimates with 95% bootstrap CIs
Table 3 — Vine structure summary for Zone 2 (all states)
Table 4 — Model selection grid (best per K, with full N_obs/p ratio)
Table 5 — Transition matrix (1980–2022) with bootstrap CI widths
Table 6 — Interval uncertainty metrics (AW, POC with 95% CI, AAD)
Table 7 — Verification summary (CRPSS, CRPS, p_DM, pinball, AUC)The Supplement has been rewritten and now contains genuinely auxiliary
material: per-zone PIT histograms (S1), full CVI table (S2), full
GPD estimates for all zones (S3), vine structure for all zones (S4),
ES/VS results (S5), areal-CRPS gains and Brier scores (S6), rank
correlation robustness (S7), rolling-origin hindcast details (S8),
and the full 30-configuration model selection grid (S9). Section S8
was rewritten from scratch and is fully readable in the revised version.[MANUSCRIPT CHANGE: Seven new tables added to main text. Supplement
reduced and reorganised. Former S8 rewritten.]─────────────────────────────────────────────────────────────────────
GENERAL REMARK (3): "A very large number of evaluation/test variables
is introduced often only by name, and at no stage it is mentioned how
this multitude of tests is used for inferences about the results and
how the actual decision of choosing the appropriate model is influenced
by those test variables."
─────────────────────────────────────────────────────────────────────Response: Each validation sub-section now contains an explicit
"Retention criterion" or "Decision criterion" paragraph stating what
test outcome leads to retaining or refining the model. Specifically:Sect. 2.4.1 (nHMM): "Retention criterion: a configuration is
retained if PIT histograms are visually flat, KS/CvM tests do not
reject uniformity at the 5% level for >80% of stations, and no
significant Ljung–Box autocorrelation is detected. Regime count K
is selected by BIC, with PIT-based adequacy as a secondary check."Sect. 2.4.2 (vine): "Operative criterion for vine adequacy: (i) PIT
histograms visually consistent with uniformity for states and zones
used in the verification; (ii) variogram scores lower for the vine
than for the independence baseline; and (iii) Diebold–Mariano tests
confirming predictive skill gains out-of-sample."Sect. 2.4.3 (scores): "Decision criterion: the vine layer is deemed
to improve over the independence baseline if the Diebold–Mariano
test rejects equality at the 10% level for station-wise CRPS or
zone-wise VS in the majority of zones."Additionally, the new paragraph in Sect. 2.4.2 explains explicitly
why KS/CvM p-values of zero on individual Rosenblatt components (as
seen in Supplement S1) do not constitute operational failure of the
vine: at training sample sizes of n = 845–1269, these tests have very
high power and will reliably detect even minor residual dependence
of no practical consequence for forecast calibration.[MANUSCRIPT CHANGE: "Retention criterion" / "Decision criterion"
paragraphs added to Sects. 2.4.1, 2.4.2, 2.4.3. Explanation of
KS/CvM behaviour at large n added to Sect. 2.4.2.]─────────────────────────────────────────────────────────────────────
GENERAL REMARK (4): "At no stage of the manuscript the actual number
of estimated parameters vs. the actual number of sampling points in
space and time used to perform the inference/estimation is mentioned."
─────────────────────────────────────────────────────────────────────Response: The following information is now given explicitly:
— Abstract: "The nHMM comprises p = 29,284 estimated parameters fit
to N_obs = 2,386,692 daily observations (N_obs/p ≈ 82), confirming
adequate model identifiability."— Table 4: includes columns for p (free parameters) and N/p
(observations-per-parameter ratio) for all three candidate models.— Sect. 2.5: "The ratio N_obs/p ≈ 82 confirms adequate
identifiability: there are roughly 82 observations per estimated
parameter."[MANUSCRIPT CHANGE: N_obs/p ratio added to abstract, Table 4, and
Sect. 2.5.]─────────────────────────────────────────────────────────────────────
GENERAL REMARK (5): "Very often uncertainty measures such as confidence
intervals or p-values in case of test are not given or even mentioned.
One example is table S8 in the supplements, here the estimated shape
parameter of the Pareto fit of the medoid station is reported … for
each synoptic regime with one close to zero value and two positive and
two negative values which would indicate either clear differences of
the tail behavior or very weak properties of the underlying data set
to estimate the notorious difficult shape value; this is a clear case
to be studied in detail e.g. by bootstrapping across the stations
around the medoid."
─────────────────────────────────────────────────────────────────────Response: We thank the referee for this specific and constructive
suggestion. We implemented a parametric bootstrap following exactly
the spirit of the recommendation. Specifically, for each (zone, state)
combination we resampled the exceedances of the medoid station itself
(separating body and exceedances to preserve the exceedance proportion)
and re-fitted the GPD 500 times, producing empirical 95% CIs for ξ̂.The results for Zone 5 (main Table 2) show the following pattern:
States 0, 3, 4 (anticyclonic/transitional): ξ̂ < 0 (bounded tails),
confirmed by CIs that are either entirely negative or straddle zero
only narrowly. States 1 and 2 (episodic frontal): ξ̂ > 0 (heavy
tails), confirmed by CIs with predominantly positive bounds. The
sign contrast is statistically robust and physically interpretable
(bounded tails under weak synoptic forcing; heavy tails under deep
frontal systems). Full per-zone results are in Supplement Table S3.We also note that the original concern about "two positive and two
negative values" arose partly because the previous table lacked
confidence intervals and partly because it appeared to show the medoid
of a different zone. The revised table explicitly identifies the Zone 5
medoid (southern Chile, storm-dominated) and provides CIs throughout.[MANUSCRIPT CHANGE: Table 2 (main text) now reports ξ̂, σ̂, 95%
bootstrap CI for ξ, n_exc, and tail classification for all five states.
Supplement Table S3 provides the same information for all six zones.
Physical interpretation of the dry/wet tail contrast added to
Sect. 2.3.2.]─────────────────────────────────────────────────────────────────────
GENERAL REMARK (6): "Except for the introduction, the remaining
manuscript … reads like a split project report … the selection of
atmospheric variables lacks any moisture-related variables like
specific humidity or vertically integrated moisture transports which
is more than the 850 (u,v) data and which can be easily extracted
from C3S/ERA5 datasets."
─────────────────────────────────────────────────────────────────────Response: We accept this criticism on both counts.
Regarding presentation: the restructuring described under Remarks (1)
and (2) above has substantially improved the manuscript's coherence.
The Methods section now reads as a continuous, sequentially organised
narrative rather than a collection of sub-reports.Regarding moisture variables: vertically integrated moisture flux
(VIMF) and its divergence (VIMFD) have been added to the covariate
set in the revised manuscript. The physical motivation is stated
explicitly in Sect. 2.1: "The inclusion of VIMF and VIMFD is motivated
by their direct link to landfalling moisture flux over Chile: u_850 and
v_850 alone do not fully characterize atmospheric moisture delivery in
the presence of orographic blocking and low-level jets." Both fields
were extracted from ERA5 at 0.25°×0.25° resolution over the same domain
used for the other covariates.[MANUSCRIPT CHANGE: VIMF and VIMFD added to covariate set (Sect. 2.1,
covariate list). Physical justification for their inclusion added.
Manuscript reorganised for coherent narrative flow throughout.]─────────────────────────────────────────────────────────────────────
ADDITIONAL POINT: Zone 4 characterisation
─────────────────────────────────────────────────────────────────────During revision we identified an error in the previous version that
characterised Zone 4 as "hyper-arid," which is incorrect. Zone 4
covers the semi-arid Mediterranean transition of central Chile
(approximately 33°S–38°S), a region of significant but highly
heterogeneous precipitation controlled by strong orographic gradients
between coastal and Andean stations.The zone-level areal-CRPS for Zones 2 and 4 is reported as "n/a" in
the revised Table 7, with a detailed explanation in Sect. 3.5.1
distinguishing the two distinct failure mechanisms:— Zone 2 (desert transition, 25°–30°S): dominance of zero-precipitation
days causes the vine ensemble to occasionally generate positive draws
on universally zero-observation days, inflating the areal mean and
causing CRPS to diverge.— Zone 4 (semi-arid to Mediterranean transition, 33°–38°S): extreme
within-zone orographic heterogeneity (coastal vs. Andean stations
differ by up to an order of magnitude in annual totals) means that
rare extreme draws at high-elevation stations — plausible given the
GPD shape ξ̂ = 0.921 for State 1 at the Zone 4 medoid, the largest
shape estimate in the dataset, fitted with only n_exc = 34
exceedances — can dominate the areal mean even when lower-elevation
stations record near-zero totals. This is a case of unstable tail
estimation propagating into the areal score, not a fundamental
failure of the marginal or vine specification.In both zones, interval-based diagnostics at the medoid level
(Table 6: POC ≈ 96–97%, AAD ≤ 0.07) confirm well-calibrated
performance. The limitation is in the choice of areal-CRPS as an
evaluation metric for heterogeneous zones, not in the framework itself.
Recommendations for addressing this in future work (weighted areal
averages, stratified validation by sub-domain) are given in Sect. 4.4.[MANUSCRIPT CHANGE: "Hyper-arid" characterisation of Zone 4 removed.
Sect. 3.5.1 expanded with mechanistically distinct explanations for
Zones 2 and 4. Table 7 CRPS entries for Zones 2 and 4 replaced by
"n/a" with footnote. Sect. 4.4 updated.]Citation: https://doi.org/10.5194/egusphere-2025-5543-AC1
-
AC1: 'Reply on RC1', Mauricio Herrera-Marín, 08 Apr 2026
-
RC2: 'Comment on egusphere-2025-5543', Anonymous Referee #2, 03 Apr 2026
Overall, I found the methodology section somewhat challenging to follow, likely due to the way it is currently organized. In particular, sections 2.3.2 through 2.4.2 were initially difficult to navigate, and I had to read them multiple times to fully grasp the workflow. My impression is that the clarity could be enhanced by presenting the methodology in a more step-by-step manner. Specifically, rather than interleaving statistical theory and its practical application multiple times, it might be beneficial to either present the statistical framework comprehensively at the beginning and then focus on the application, or to integrate theory and application in a more continuous, guided narrative. This would help the reader to follow the logic without needing to constantly switch between abstract concepts and their implementation. For example, in section 2.3.4, the procedure was not immediately clear and only became understandable after reading subsequent sections. While this does not indicate a methodological flaw, it may affect the readability and flow of the manuscript. Organizing the description of such procedures in a clear, sequential way could make it easier for readers to follow the logic without having to backtrack.
Additionally, regarding the methodology itself, I found it unclear why seven “zones” were selected using the clustering approach. If a specific selection criterion was applied, it would be helpful to state it explicitly, ideally early in the methodology section, perhaps starting in section 2.3.1. This would provide the reader with a clear understanding of the rationale behind the choice and help contextualize subsequent analyses. Providing such context upfront often strengthens the coherence of the methodology and allows the reader to better appreciate the workflow and its outcomes.
Despite these points, I find this paper highly engaging and scientifically valuable. The study presents a good perspective on the topic, and the results are particularly insightful. The approach combines sophisticated statistical modeling with practical application, offering meaningful contributions. The findings not only deepen our understanding but also have the potential to inform future research and practical applications in related areas.
In summary, while the methodology could benefit from a slightly more structured presentation, the study is well-motivated, and provides valuable results. With minor adjustments in clarity and organization, the manuscript would become even more accessible and impactful to readers.
Citation: https://doi.org/10.5194/egusphere-2025-5543-RC2 -
AC2: 'Reply on RC2', Mauricio Herrera-Marín, 08 Apr 2026
%%=============================================================================
%% RESPONSE TO REFEREE 2
%%=============================================================================We thank Referee 2 for a careful and constructive review. The comments
focused primarily on methodological clarity and organisation, and we
have implemented all suggestions.─────────────────────────────────────────────────────────────────────
MAIN COMMENT (1): "The methodology section was somewhat challenging to
follow … sections 2.3.2 through 2.4.2 were initially difficult to
navigate, and I had to read them multiple times to fully grasp the
workflow. My impression is that the clarity could be enhanced by
presenting the methodology in a more step-by-step manner … rather than
interleaving statistical theory and its practical application multiple
times."
─────────────────────────────────────────────────────────────────────Response: We fully agree. The Methods section has been restructured as
a sequential pipeline with four explicitly numbered steps:Step 1 — Zoning via DTW (Sect. 2.3.1)
Step 2 — State-conditional marginal specification (Sect. 2.3.2)
Step 3 — Transformation to copula space (Sect. 2.3.3)
Step 4 — Regular vine construction and model selection (Sect. 2.3.4)A paragraph-map at the opening of Section 2 previews the complete
pipeline, so the reader knows at any point where they are in the
workflow. Within each step, statistical theory and its practical
implementation are presented together in a continuous narrative rather
than interleaved in separate sub-sections. The former "Experimental
Setup" section (which broke the narrative by revisiting already-
presented material in a different order) has been removed.Regarding Section 2.3.4 specifically, which the referee found unclear
on first reading: the revised version opens with a plain-language
statement of what the vine achieves ("For d_z up to ~90 we represent
C_{z,s} by a Regular Vine (R-vine), decomposing the d_z-variate
density into bivariate pair-copulas"), presents the mathematical
definition, states the candidate family set and selection criterion,
and then immediately provides Table 3 (vine structure for Zone 2) to
ground the abstract description in a concrete example.[MANUSCRIPT CHANGE: Section 2 entirely reorganised with numbered
steps, paragraph-map, and continuous theory-application narrative.
Former Section 2.5 removed. Sect. 2.3.4 rewritten with concrete
example (Table 3) integrated into the mathematical presentation.]─────────────────────────────────────────────────────────────────────
MAIN COMMENT (2): "I found it unclear why seven 'zones' were selected
using the clustering approach. If a specific selection criterion was
applied, it would be helpful to state it explicitly, ideally early in
the methodology section, perhaps starting in section 2.3.1."
─────────────────────────────────────────────────────────────────────Response: The selection criterion is now stated explicitly at the
beginning of Section 2.3.1, before the mathematical formulation.
Specifically, the revised text reads:"The optimal number of zones is k* = 7, selected via a consensus of
five internal cluster validity indices (CVI): Silhouette,
Calinski–Harabasz, Davies–Bouldin, Dunn, and COP (Table 1; full
table in Supplement S2). The G = 7 solution achieves the best
consensus rank across all five CVIs, aligns with established climatic
subregions of continental Chile, and yields zones with d_z ∈ [50,90]
stations and at least 200 training days per state, providing
sufficient data for robust copula estimation."Table 1 (new in main text) shows the top five candidate configurations
with their CVI scores, making the selection transparent and verifiable.[MANUSCRIPT CHANGE: Zoning selection criterion (five-CVI consensus)
stated at start of Sect. 2.3.1. Table 1 (CVI table) moved from
Supplement to main text.]─────────────────────────────────────────────────────────────────────
POSITIVE ASSESSMENT (acknowledged and appreciated)
─────────────────────────────────────────────────────────────────────The referee noted: "Despite these points, I find this paper highly
engaging and scientifically valuable … The approach combines
sophisticated statistical modeling with practical application, offering
meaningful contributions. The findings not only deepen our
understanding but also have the potential to inform future research
and practical applications in related areas."We are grateful for this assessment and hope the structural and
clarity improvements in the revised manuscript allow the scientific
contribution to be more readily appreciated by future readers.Citation: https://doi.org/10.5194/egusphere-2025-5543-AC2
-
AC2: 'Reply on RC2', Mauricio Herrera-Marín, 08 Apr 2026
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AC3: 'Comment on egusphere-2025-5543', Mauricio Herrera-Marín, 08 Apr 2026
Dear Editor,We thank the Editor and both Referees for their thorough and constructive
evaluation of our manuscript. The reviews were detailed and fair, and we
believe the revised manuscript is substantially stronger as a result.Below we summarise the principal changes made in response to the
referees' comments, followed by our individual point-by-point responses
to each referee in separate documents.SUMMARY OF MAJOR REVISIONS
1. Structural reorganisation (Referee 1, General remarks; Referee 2,
General remarks). The Methods section was entirely restructured as a
sequential, step-by-step pipeline. A paragraph-map at the opening of
Section 2 guides the reader through each sub-section. The former
"Experimental Setup" section, which duplicated material already
presented in the Methods, was eliminated and its content absorbed
into the relevant subsections. Overlap between Sections 2 and 3
has been removed.2. Results moved from Supplement to main text (Referee 1, point 2).
Seven tables containing the core quantitative results are now in
the main manuscript, including: the DTW cluster validity index
(CVI) consensus table (Table 1); GPD parameter estimates with
bootstrap confidence intervals (Table 2); vine structure summary
(Table 3); model selection grid (Table 4); transition matrix
(Table 5); interval uncertainty metrics (Table 6); and the full
verification summary including CRPS skill scores, Diebold–Mariano
p-values, pinball losses, and AUC (Table 7). The Supplement
now contains genuinely auxiliary material.3. Parameter count and identifiability (Referee 1, point 4). The ratio
N_obs/p ≈ 82 (29,284 parameters estimated from 2,386,692 scalar
observations) is now reported in the abstract, Table 4, and the
model selection sub-section.4. Confidence intervals and p-values (Referee 1, point 5). Bootstrap
95% CIs for GPD shape parameters are now in Table 2; bootstrap CIs
for POC are in Table 6; 95% CI for the S4→S4 persistence trend is
reported in Section 3.3; BH-FDR correction is applied over all 25
transition pairs.5. Moisture-related atmospheric variables (Referee 1, point 6).
Vertically integrated moisture flux and its divergence (VIMF,
VIMFD) were added to the covariate set, with physical justification
in Section 2.1.6. Decision criteria for tests (Referee 1, point 3). Each validation
sub-section now contains an explicit "Retention criterion" paragraph
stating what result leads to model acceptance or refinement. A new
paragraph in Section 2.4.2 explains why p-values of zero on KS/CvM
tests of individual Rosenblatt components do not imply operational
failure of the vine, given the high statistical power of these tests
at n > 800.7. Zone 4 explanation corrected (identified during revision). The
description of Zone 4 as "hyper-arid" was incorrect. Zone 4 covers
the semi-arid Mediterranean transition of central Chile
(~33°S–38°S), a region of strong orographic gradients. The
inapplicability of zone-level areal CRPS in Zones 2 and 4 is now
explained through two distinct mechanisms: near-zero precipitation
frequency (Zone 2) and extreme within-zone orographic heterogeneity
combined with an unstable GPD tail estimate (Zone 4, State 1:
ξ̂ = 0.921, n_exc = 34). The CRPS entries for these zones in
Table 7 are replaced by "n/a" with a detailed explanation in
Section 3.5.1.8. Figure–Table coherence. Table 6 was updated to use values from the
validated production run (M = 100) that generated Figure 6,
ensuring full numerical consistency between figure and table.9. Zone 7 clarified. Zone 7 covers the austral fjord region south of
approximately 45°S; the AUC entry in Table 7 is "n/a" because
25 mm exceedance events were too infrequent in the 2022 evaluation
season to compute a reliable AUC.We believe the manuscript now addresses all substantive concerns raised
by both referees. We look forward to the Editor's decision.Sincerely,
Mauricio Herrera-Marín (on behalf of all co-authors)Citation: https://doi.org/10.5194/egusphere-2025-5543-AC3
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Francisca Kleisinger
Diego Rivera
Andrés Wilson
Alex Godoy-Faúndez
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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General remarks
This is a puzzling manuscript: on one side it delivers evidence that the chosen approach to study the precipitation space time structure in a highly complex environment (here Chile with its very long extension in latitude together with a complicated and structured orography) is a scientifically fruitful approach. Also the choice to build a hierarchical model with the hidden Markov states to identify sysnoptic regimes, the clustering to identify regional homogeneous precipitation zones und finally to model the spatial variability within each zone with copulas for the bulk and generalized Pareto distributions for the extremes beyond a fixed quantile is well explained in the introduction. But the presentation of the results is the "black" side of the medal:
(1) in large parts section 2 and section 3 contain repeated texts,
(2) a very large number of actual results is shifted into the supplement text S1 - S10 (the total supplement covers 20 pages!) with the additional caveat that S8 is not readable,
(3) a very large number of evaluation/test variables is introduced often only by name and at no stage it mentioned how this multitude of tests is used for inferences about the results and how the actual decision of choosing the appropriate model is influenced by those test variables,
(4) at no stage of the manuscript the actual number of estimated parameters vs. the actual number of sampling points in space and time used to perform the inference/estimation is mentioned,
(5) very often uncertainty measures such as confidence intervals or p-values in case of test are not given or even mentioned, one example is table S8 in the supplements, here the estimated shape parameter of the Pareto fit of the mediod station is reported (but never mentioned in the main text) for each synoptic regime with one close to zero value and two positive and two negative values which would indicate either clear differences of the tail behavior or very weak properties of the underlying data set to estimate the notorious difficult shape value, this is a clear case to be studied in detail eg be bootstrapping across the stations around the mediod and finally
(6) except for the introduction the remaining manuscript and the supplement reads like a splitted project report about the analysis of Chilean precipitation space-time variability applicable for potential hydrological research, eg physical consistency of mentioned but the selection of atmospheric variables lacks any moisture related variables like specific humidity or vertically integrated moisture transports which is more than the 850 (u,v) data and which can be easily extracted from C3S / ERA5 data sets.
Together with the extensive list of comments (my detailed remarls) in the attached and annotated manuscript I can suggest publication in HESS only after major (major) revisions of the current version of the manuscript.