Pith. sign in

REVIEW 4 major objections 4 minor 33 references

How long is long enough? Finite-horizon approximation of energy storage scheduling problems

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read One equality check certifies a long-enough planning horizon

desk verdict The central criterion is wrong as stated: the proof shows only that one optimal finite-horizon schedule extends to the infinite horizon, while Definition 1 requires every such schedule to extend, and a two-period example breaks the equivalence. read the letter →

arxiv 2411.17463 v2 pith:OPCYEA2X submitted 2024-11-26 math.OC

Eléa Prat , Richard M. Lusby , Juan Miguel Morales , Salvador Pineda , Pierre Pinson This is my paper · ORCID
classification math.OC
keywords energystorageschedulingrollinghorizonforecastinfinite-horizonoptimizationplanningsuboptimalityboundarbitrageefficiency
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Energy storage operators who use a rolling-horizon scheduler must choose a planning horizon, and the choice is usually arbitrary. This paper claims a simple certificate: solve the finite-horizon scheduling problem twice, forcing the storage to end the planning horizon at its lowest and highest reachable energy levels; if the two solutions agree on the state of energy at the end of the decision horizon, that planning horizon is a forecast horizon, and the condition is also necessary. Building on this, the paper proves an upper bound on the profit lost when the horizon is too short, shows by example that forecast horizons need not exist, derives a lower bound on the minimum forecast horizon from storage parameters alone, and gives an algorithm that finds the minimum forecast horizon. The case studies show that common horizons such as 48 hours are often too short and that a too-short horizon can turn a profitable storage operation into a loss, which is why a checkable condition matters.

What carries the argument

The load-bearing object is the pair of extreme terminal-state problems $F(T,C,\underline{S}_T)$ and $F(T,C,\overline{S}_T)$, the finite-horizon model with the final state of energy fixed to the minimum and maximum reachable levels $\underline{S}_T$ and $\overline{S}_T$. The certificate is the equality of the two optimal states at the end of the decision horizon, $s_H = \bar{s}_H$. The proof rests on an envelope property: an optimal trajectory ending at the minimum reachable level stays weakly below every optimal trajectory with an intermediate terminal state, and an optimal trajectory ending at the maximum reachable level stays weakly above it; this sandwiching is what lets the common decision-horizon schedule be extended to any longer horizon, and it also drives the suboptimality bound, since a smaller gap between the two terminal states means a smaller worst-case profit loss.

What would settle it

A concrete check: set $S=0$, $\overline{S}=10$, $S_{\rm init}=5$, $P^C=P^D=5$, $\eta_C=\eta_D=1$, $\rho=1$, $H=1$, $T=2$, $C_1=C_2=10$, and continue with $C_3=C_4=1000$. The two extreme terminal problems $F(2,C,\underline{S}_2)$ and $F(2,C,\overline{S}_2)$ both have optimal solutions with $s_1=5$, while the unconstrained finite problem also has an optimal solution with $s_1=0$ that is not optimal for the infinite-horizon tail; constructing this instance would settle whether the 'if' direction of Theorem 1 holds as stated.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1: for the deterministic price-taker storage scheduling model (1a)-(1g), with leakage, charging and discharging efficiencies, and possibly negative prices, a planning horizon $T$ is a forecast horizon for a price forecast $\hat{C}$ if and only if there exist optimal solutions of $F(T,\hat{C},\underline{S}_T)$ and $F(T,\hat{C},\overline{S}_T)$ with the same state of energy at the end of the decision horizon $H$, where $\underline{S}_T$ and $\overline{S}_T$ are the lowest and highest energy levels that can be reached at the end of the planning horizon. Necessity follows from the definition of forecast horizon; sufficiency is argued through an envelope lemma stating that the minimum-terminal optimum lies weakly below, and the maximum-terminal optimum weakly above, every optimal trajectory with an intermediate terminal state, after which the common decision-horizon schedule is propagated one period at a time to arbitrary future horizons. The paper then uses the certificate as a building block: the gap between the two terminal states bounds suboptimality when the horizon is too short, a necessary condition computable from storage parameters alone gives a starting point, and an iterative algorithm increments the horizon until the certificate holds, returning the minimum forecast horizon or, if a user-set maximum is reached, a bound on the remaining suboptimality.

Load-bearing premise

The load-bearing step is that agreement between the two extreme terminal-state problems forces every optimal schedule with an intermediate end-of-horizon state to agree on the decision horizon, and the proof does not establish this when the finite problem has multiple optimal solutions.

Editorial extensions

If this is right

  • A rolling-horizon operator can certify a chosen planning horizon by solving two finite-horizon optimizations and comparing one state variable, without having to solve the infinite-horizon problem.
  • When the certificate fails, the gap between the two terminal states gives an explicit upper bound on the profit lost relative to a perfect infinite-horizon policy, so the cost of a short horizon is quantifiable.
  • Forecast horizons need not exist: with inefficient charge-discharge and a price path satisfying $\eta C_1 < C_t < C_1$ for every later $t$, no finite planning horizon is long enough.
  • The minimum forecast horizon varies strongly with storage characteristics and the specific price path; in the case studies it is often longer than 48 hours, and for slow storage with leakage a 24-hour fixed-level policy loses 362% of the profit achievable with a forecast horizon.
  • Under the decomposability conditions of Section 3.6, the minimum forecast horizon for a problem with several storage units is the maximum of the individual units' minimum forecast horizons.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A practical extension would use the certificate online: as price forecasts update, recompute $\underline{S}_T$ and $\overline{S}_T$ and lengthen the planning horizon only while the gap implies an unacceptable suboptimality bound, replacing fixed horizons such as 48 hours with a data-driven choice.
  • In a stochastic setting with a scenario tree, the same two-extreme-problem idea could define a stochastic forecast horizon, with terminal reachable intervals per scenario and the Proposition 2 bound replaced by an expectation over scenarios; the deterministic result is the degenerate case.
  • Because the full fleet minimum is the maximum of individual minima, the practical bottleneck is the slowest or most lossy unit, and its parameters alone could be used in Proposition 3 as a fleet-level lower bound before any price data arrive.
  • The non-existence example implies that a rolling-horizon implementation should carry a certified cap: if the certificate has not fired by $T_{\max}$, the suboptimality bound is the only remaining guarantee, and exceeding the cap should trigger a re-evaluation of whether a finite-horizon policy is appropriate at all.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The manuscript studies the rolling-horizon approximation of an infinite-horizon energy storage arbitrage problem. It defines a planning horizon T to be a forecast horizon when every optimal schedule of the finite problem over the decision horizon remains optimal for every extension of the price forecast beyond T (Definition 1). The central result, Theorem 1 in Section 3.1, asserts that T is a forecast horizon if and only if the two finite problems with terminal state of energy fixed to the minimum and maximum reachable levels, F(T,C,S_T) and F(T,C,\bar{S}_T), have optimal solutions whose state of energy at the end of the decision horizon coincides. The paper also derives a suboptimality bound, a necessary condition and lower bound on the minimum forecast horizon, an algorithm to determine the minimum forecast horizon, and numerical case studies.

Significance. If Theorem 1 were correct, it would provide a practically attractive and easy-to-check certificate for planning-horizon selection, and the proposed algorithm and suboptimality bound would be useful for storage operators. The paper is clearly written, addresses an important gap in the literature, and the numerical study is accompanied by a reproducibility link. The non-existence example in Section 3.3 and the lower bound in Proposition 3 are interesting. However, the central equivalence is false: the condition in Theorem 1 is not sufficient for Definition 1. Because the main theoretical claim fails under the paper's own definitions, the principal contributions do not currently stand.

major comments (4)
  1. [§3.1] The sufficiency direction of Theorem 1 is false. Consider the admissible parameters S=0, \bar{S}=10, S^{\rm init}=5, P^C=P^D=5, \Delta t=1, \eta_C=\eta_D=\rho=1, H=1, T=2, C_1=C_2=10, and a tail with C_3=C_4=1000 and C_t=0 for t\ge 5. In F(2,C,0) every optimum has s_1\in[0,5]; in F(2,C,10) every optimum has s_1\in[5,10]. Hence the condition of Theorem 1 holds, with s_1=5. But the free problem S(2,C) also has optimal H-schedules with s_1=0, for example discharging 5 in period 1 and doing nothing in period 2. Such an H-schedule is not optimal in the infinite-horizon extension: charging 5 in period 1 and discharging 10 at price 1000 in period 3 yields profit 9950, whereas any schedule starting with s_1=0 earns at most 5000. Thus X_H(S(T,\hat{C}))\not\subseteq X_H(S(\mathbb{N}^+,C)) for this admissible tail, so T=2 is not a forecast horizon under Definition 1, contradicting Theorem 1.
  2. [Appendix A] The proof of sufficiency proves the wrong quantifier. Lemma 1 and Corollary 1 construct, for each intermediate terminal level S^{\rm end}, one optimal solution x^* whose H-states lie between those of the extreme solutions, and the induction then takes one common schedule and extends it to the infinite horizon. This establishes that there exists an optimal H-schedule of S(T,\hat{C}) that is optimal for the infinite-horizon problem. Definition 1 requires that every optimal H-schedule of S(T,\hat{C}) is optimal for the infinite-horizon problem. The induction never visits the other optimal solutions of the free problem; in the counterexample above, the constructed common schedule is s_1=5 while the free problem also has the optimal schedule s_1=0. The proof therefore cannot bridge the gap between Theorem 1 and Definition 1.
  3. [§3.5] Because Algorithm 1 stops when the Theorem 1 condition reports gap=0 and then sets subopt=0, it can terminate declaring a forecast horizon in situations where none exists. The reported minimum forecast horizons in Section 4.2 and the profit comparisons in Section 4.3 are therefore not supported as evidence for the paper's claims. This is a direct consequence of the counterexample, not a separate implementation issue.
  4. [Appendix A] The final sentence of Appendix A, 'if T is a forecast horizon, by definition, ∃x∈X and ∃\bar{x}\in\bar{X} such that s_H=\bar{s}_H', does not follow from Definition 1. Definition 1 quantifies over all tails and asserts a set inclusion; it does not assert that the two extreme terminal problems share a decision-horizon state. Some argument is needed, for example constructing tails that make the minimum and maximum terminal states optimal simultaneously, and none is supplied. Thus the 'only if' direction is also unproved as written.
minor comments (4)
  1. [§2.2] Definition 1 uses X_H(S(\mathbb{N}^+,C)), but the notation X_H was introduced only for finite problems; the restriction of an infinite-horizon solution to the decision horizon H should be defined explicitly.
  2. [§3.5] Algorithm 1 uses M as the initial value of gap and subopt, but M is never defined; the stopping criterion 'gap>0' should also be stated with an explicit tolerance because floating-point solvers will not produce exact zeros.
  3. [§3.2] The notation C is overloaded: it denotes the price vector, the set of possible price vectors, and the upper bound C in Proposition 2, while the lower bound C is visually almost identical; please disambiguate.
  4. [§3.3] In the sentence 'with ηC_1 < C_t < C_1', the expression ηC_1 is ambiguous: it should be η_C C_1. Reading this passage requires the reader to infer the intended meaning.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1 is a substantive characterization, and the paper does not fit a parameter and relabel it as a prediction.

full rationale

The paper's central claim (Theorem 1) asserts an equivalence between a planning horizon being a forecast horizon (Definition 1) and the existence of optimal extreme-terminal solutions with the same decision-horizon state. This is not a restatement of Definition 1: Definition 1 quantifies over all price continuations and all optimal finite-horizon solutions, whereas the theorem's condition concerns only two fixed terminal-value problems. The proof in Appendix A derives the result via exchange arguments and linear-programming optimality conditions rather than importing the conclusion. The extreme reachable levels in (2) and (3) are computed from the storage parameters and initial state, not from the target notion of forecast horizon, so there is no fitted-input-called-prediction pattern. Proposition 2's suboptimality bound and Proposition 3's necessary condition are consequences of the characterization, not inputs to it. The cited prior work (Cruise et al., Cheevaprawatdomrong and Smith, Bhaskaran and Sethi, etc.) is external to the present authors, and none of the load-bearing steps reduces to a self-citation. The reader's take identifies a possible logical gap in the sufficiency proof—the induction appears to construct one extending optimal schedule rather than showing every optimal schedule extends—but a proof gap is a correctness concern, not a circularity concern. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central theorem rests on the extreme-terminal exchange proof and an induction over horizon length. The main unstated assumptions are the well-posedness of the infinite-horizon optimum and the existence of price tails forcing extreme terminal states. No free parameters are fitted.

assumptions (3)
  • domain assumption The undiscounted infinite-horizon scheduling problem S(N^+,C) has well-defined optimal solutions over the decision horizon for every price vector C.
    Definition 1 uses X_H(S(N^+,C)) without proving existence; with unbounded or non-convergent price tails, the infinite sum may not have an optimum.
  • ad hoc to paper For any reachable terminal state at the end of the planning horizon, there exists a continuation of prices that makes that terminal state optimal in the infinite-horizon problem.
    Needed for the converse direction of Theorem 1 at the end of Appendix A, but no construction or proof is provided.
  • standard math Exchange arguments in Appendix A preserve optimality when shifting small quantities of charge or discharge between periods.
    The proof of Lemma 1 relies on perturbation arguments with price equalities; these are standard for linear programs.

how reviews work

0 comments
Cite this review

Pith. "Pith review of How long is long enough? Finite-horizon approximation of energy storage scheduling problems." pith.science (2026). https://pith.science/paper/OPCYEA2X

@misc{pith2026241117463,
  author       = {Pith},
  title        = {Pith review of: How long is long enough? Finite-horizon approximation of energy storage scheduling problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OPCYEA2X}},
  note         = {Machine review of arXiv:2411.17463}
}
read the original abstract

Energy storage scheduling problems, where a storage is operated to maximize its profit in response to a price signal, are essentially infinite-horizon optimization problems as storage systems operate continuously, without a foreseen end to their operation. Such problems can be solved to optimality with a rolling-horizon approach, provided that the planning horizon over which the problem is solved is long enough. Such a horizon is termed a forecast horizon. However, the length of the planning horizon is usually chosen arbitrarily for such applications. We introduce an easy-to-check condition that confirms whether a planning horizon is a forecast horizon, and which can be used to derive a bound on suboptimality when it is not the case. By way of an example, we demonstrate that the existence of forecast horizons is not guaranteed for this problem. We also derive a lower bound on the length of the minimum forecast horizon. We show how the condition introduced can be used as part of an algorithm to determine the minimum forecast horizon of the problem, which ensures the determination of optimal solutions at the lowest computational and forecasting costs. Finally, we provide insights into the implications of different planning horizons for a range of storage system characteristics.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

33 extracted references · 21 canonical work pages

  1. [1]

    Anjos, M.F., Cruise, J.R., Vilalta, A.S. (2020). Control of two energy storage units with market impact: Lagrangian approach and horizons.2020 International Conference on Probabilistic Methods Applied to Power Systems (PMAPS)(pp. 1–6). https://doi.org/10.1109/PMAPS47429.2020.9183690

    arXiv 2020

  2. [2]

    Bardhan, A., Dawande, M., Gavirneni, S., Wu, Y., Sethi, S. (2013). Forecast and rolling horizons under demand substitution and production changeovers: anal- ysis and insights.IIE Transactions,45(3), 323–340, https://doi.org/10.1080/ 0740817X.2012.712239

    work page arXiv 2013

  3. [3]

    Bhaskaran, S., & Sethi, S. (1987). Decision and forecast horizons in a stochastic environment: A survey.Optimal Control Applications & Methods,8, 201–217,

    work page 1987

  4. [4]

    Chand, S., Hsu, V.N., Sethi, S. (2002). Forecast, solution, and rolling horizons in operations management problems: A classified bibliography.Manufacturing & Service Operations Management,4(1), 25–43, https://doi.org/10.1287/msom .4.1.25.287

    doi:10.1287/msom 2002

  5. [5]

    Cheevaprawatdomrong, T., Schochetman, I.E., Smith, R.L., Garcia, A. (2007). Solu- tion and forecast horizons for infinite-horizon nonhomogeneous markov decision processes.Mathematics of Operations Research,32(1), 51–72, https://doi.org/ 10.1287/moor.1060.0224

    work page arXiv 2007

  6. [6]

    Cheevaprawatdomrong, T., & Smith, R.L. (2004). Infinite horizon production schedul- ing in time-varying systems under stochastic demand.Operations Research, 52(1), 105-115, https://doi.org/10.1287/opre.1030.0080

    work page arXiv 2004

  7. [7]

    Cruise, J., Flatley, L., Gibbens, R., Zachary, S. (2019). Control of energy storage with market impact: Lagrangian approach and horizons.Operations Research,67(1), 1–9, https://doi.org/10.1287/opre.2018.1761 32

    work page arXiv 2019

  8. [8]

    Cruise, J.R., Flatley, L., Zachary, S. (2018). Impact of storage competition on energy markets.European Journal of Operational Research,269(3), 998-1012, https:// doi.org/10.1016/j.ejor.2018.02.036 Cuisinier, ´E., Lemaire, P., Penz, B., Ruby, A., Bourasseau, C. (2022). New rolling horizon optimization approaches to balance short-term and long-term decisio...

    work page arXiv 2018

Show all 33 references
  1. [9]

    Diller, T., Soppelsa, A., Nagpal, H., Fedrizzi, R., Henze, G. (2024). A dynamic programming based method for optimal control of a cascaded heat pump system with thermal energy storage.Optimization and Engineering,25(1), 229–251, https://doi.org/10.1007/s11081-023-09853-5

    2024 doi

  2. [10]

    Ela, E., & O’Malley, M. (2015). Scheduling and pricing for expected ramp capability in real-time power markets.IEEE Transactions on Power Systems,31(3), 1681– 1691, https://doi.org/10.1109/TPWRS.2015.2461535

    2015

  3. [11]

    Finnah, B., G¨ onsch, J., Ziel, F. (2022). Integrated day-ahead and intraday self-schedule bidding for energy storage systems using approximate dynamic programming. European Journal of Operational Research,301(2), 726-746, https://doi.org/ 10.1016/j.ejor.2021.11.010

    2022 doi

  4. [12]

    Flatley, L.C., MacKay, R.S., Waterson, M. (2016). Optimal strategies for operating energy storage in an arbitrage or smoothing market.Journal of Dynamics and Games,3(4), 371-398, https://doi.org/10.3934/jdg.2016020

    2016 doi

  5. [13]

    Garcia, A., & Smith, R.L. (2000). Solving nonstationary infinite horizon dynamic optimization problems.Journal of Mathematical Analysis and Applications, 244(2), 304-317, https://doi.org/10.1006/jmaa.1999.6694

    2000

  6. [14]

    Ghate, A. (2011). Infinite horizon problems.Wiley encyclopedia of operations research and management science. John Wiley & Sons, , https://doi.org/10.1002/ 9780470400531.eorms0403 33

    2011

  7. [15]

    Harsha, P., & Dahleh, M. (2015). Optimal management and sizing of energy storage under dynamic pricing for the efficient integration of renewable energy.IEEE Transactions on Power Systems,30(3), 1164-1181, https://doi.org/10.1109/ TPWRS.2014.2344859

    2015

  8. [16]

    Hartl, R.F. (1986). A forward algorithm for a generalized wheat trading model. Zeitschrift f¨ ur Operations Research,30(3), A135–A144, https://doi.org/10 .1007/BF01919174

    1986

  9. [17]

    Houwing, M., Negenborn, R.R., Heijnen, P.W., De Schutter, B., Hellendoorn, H. (2007). Least-cost model predictive control of residential energy resources when applyingµCHP.2007 IEEE Lausanne Power Tech(p. 425-430). https:// doi.org/10.1109/PCT.2007.4538355 IEA (2024).World Ene...

    2007

  10. [18]

    Jesudasan, R.N., & Andrew, L.L. (2014). Scheduling long term energy storage. 2014 IEEE Conference on Computer Communications Workshops (INFOCOM WKSHPS)(pp. 634–639). https://doi.org/10.1109/INFCOMW.2014.6849305

    2014

  11. [19]

    Kannan, A., & Zavala, V.M. (2011). A game-theoretical model predictive control framework for electricity markets.2011 49th Annual Allerton Conference on

    2011

  12. [20]

    1280-1285)

    Communication, Control, and Computing (Allerton)(p. 1280-1285). https:// doi.org/10.1109/Allerton.2011.6120315

    2011

  13. [21]

    Lortz, T.D., Dolinskaya, I.S., Ghate, A., Smith, R.L. (2015). Solvability in infinite horizon optimization.Operations Research Letters,43(5), 498–503, https:// doi.org/10.1016/j.orl.2015.07.003

    2015 doi

  14. [22]

    Mayhorn, E., Xie, L., Butler-Purry, K. (2017). Multi-time scale coordination of distributed energy resources in isolated power systems.IEEE Transactions on Smart Grid,8(2), 998-1005, https://doi.org/10.1109/TSG.2016.2547342

    2017

  15. [23]

    Mercier, T., Olivier, M., De Jaeger, E. (2023). The value of electricity storage arbitrage on day-ahead markets across Europe.Energy Economics,123, 106721, https:// doi.org/10.1016/j.eneco.2023.106721

    2023

  16. [24]

    Nascimento, J., & Powell, W.B. (2013). An optimal approximate dynamic program- ming algorithm for concave, scalar storage problems with vector-valued controls. 34 IEEE Transactions on Automatic Control,58(12), 2995-3010, https://doi.org/ 10.1109/TAC.2013.2272973

    2013

  17. [25]

    Pozo, D. (2022). Linear battery models for power systems analysis.Electric Power Systems Research,212, 108565, https://doi.org/10.1016/j.epsr.2022.108565

    2022

  18. [26]

    (2023).Monetizing energy storage: A toolkit to assess future cost and value

    Schmidt, O., & Staffell, I. (2023).Monetizing energy storage: A toolkit to assess future cost and value. Oxford University Press. https://doi.org/10.1093/oso/ 9780192888174.001.0001

    2023 doi

  19. [27]

    Secomandi, N. (2015). Merchant commodity storage practice revisited.Operations Research,63(5), 1131–1143, https://doi.org/10.1287/opre.2015.1407

    2015

  20. [28]

    Zhang, Z

    Sodano, D., . . . Zhang, Z. (2021). Energy-storage modeling: State-of-the-art and future research directions.IEEE Transactions on Power Systems,37(2), 860-875, https://doi.org/10.1109/TPWRS.2021.3104768 Van de Ven, P.M., Hegde, N., Massouli´ e, L., Salonidis, T. (2013). Optima...

    2021

  21. [29]

    Weitzel, T., & Glock, C.H. (2018). Energy management for stationary electric energy storage systems: A systematic literature review.European Journal of Operational Research,264(2), 582-606, https://doi.org/10.1016/j.ejor.2017.06.052

    2018 doi

  22. [30]

    Yang, G., Yang, D., Liu, B., Zhang, H. (2024). The role of short- and long-duration energy storage in reducing the cost of firm photovoltaic generation.Applied Energy,374, 123914, https://doi.org/10.1016/j.apenergy.2024.123914

    2024

  23. [31]

    Zhang, Z., Ding, T., Zhou, Q., Sun, Y., Qu, M., Zeng, Z., Ju, Y., Li, L., Wang, K., Chi, F. (2021). A review of technologies and applications on versatile energy storage systems.Renewable and Sustainable Energy Reviews,148, 111263, https:// doi.org/10.1016/j.rser.2021.111263

    2021

  24. [32]

    Zhao, Q., Cai, X., Li, Y. (2019). Determining inflow forecast horizon for reservoir operation.Water Resources Research,55(5), 4066–4081, https://doi.org/10 35 .1029/2019WR025226

    2019

  25. [33]

    Zhao, T., Yang, D., Cai, X., Zhao, J., Wang, H. (2012). Identifying effective forecast horizon for real-time reservoir operation under a limited inflow forecast.Water Resources Research,48(1), , https://doi.org/10.1029/2011WR010623 36

    2012 doi

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.