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Ask what the score rewards before ranking the model
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Full chapter index · Independent practice · Hydrology subject guide
Nash–Sutcliffe efficiency, usually abbreviated NSE, compares a model’s squared prediction errors with the squared deviations of the observations from their own mean. An NSE of 0.94 can coexist with substantial errors in low flows or total volume. The score becomes useful when a learner can calculate it, identify its benchmark, and explain which practical questions it leaves unanswered.
This guide works through original four-interval river-flow datasets, compares two models whose rankings differ by purpose, and tests changes in units, evaluation windows and timing. The datasets are fictional teaching examples. The article explains a model-evaluation tool; it does not certify a real flood forecast, drought assessment or engineering design.
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Full chapter index
Understand the question · Chapters 1–2
Build and calculate · Chapters 3–4
Change the assumptions · Chapters 5–6
Check the evidence · Chapters 7–8
CHAPTER 1 OF 12 · Understand the question
1. A model score begins with a prediction target
Before calculating anything, state what is being predicted. A hydrological model may estimate instantaneous discharge, hourly mean discharge, daily mean discharge, a flood peak, a low-flow threshold or an accumulated volume. These targets are related, but accuracy for one does not automatically establish accuracy for the others.
Define matched observations oi and predictions pi at the same timestamps and with the same units. A modelled hourly peak cannot be paired casually with an observed daily average. Even an excellent prediction of the underlying hydrograph can appear inaccurate if its timestamps are shifted or its intervals represent different quantities.
Now define the evaluation window. A wet season, a dry spell and a multi-year record may have very different means and variability. NSE uses properties of the observations in the selected window, so the window is part of the score’s meaning. Report its dates, sampling interval, missing-data treatment and sample size.
Finally, state the decision. If the concern is keeping discharge above an ecological threshold, examine low-flow behaviour. If it is warning time for a flood, examine peak timing and threshold crossings. NSE can contribute to either evaluation, but it does not choose the relevant objective on the user’s behalf.
This order matters because a model ranking has no universal interpretation detached from its task. A single number can summarise one discrepancy efficiently while concealing another. The aim is not to reject summary scores, but to understand exactly what information each score preserves and what supporting checks the decision needs.
Let the observed mean be ō = (1/n)Σoi. The sum of squared model errors is SSE = Σ(pi − oi)². The reference sum is SST = Σ(oi − ō)². NSE is defined as 1 − SSE/SST, provided SST is positive.
The denominator is the squared-error total obtained by predicting the same observed mean at every evaluated time. NSE therefore compares the model’s squared errors with a constant observed-mean benchmark for that record. The benchmark is useful for interpretation, although the full evaluation-period mean would not necessarily be available to a real-time forecaster in advance.
If the model is perfect on every matched observation, SSE = 0 and NSE = 1. If the model has the same squared-error total as the observed-mean benchmark, NSE = 0. If its errors exceed the benchmark’s, NSE is negative. There is no finite universal lower bound because squared model errors can grow arbitrarily large.
An NSE of 0.8 means that SSE is 20% of SST on the evaluated record. It does not mean that 80% of individual predictions are correct, that the flow is within 20% at each timestamp, or that the model has an 80% probability of being valid.
The formula is a deterministic calculation once the paired data are specified. Uncertainty enters through observations, estimated model parameters, the selected record and future conditions. A precisely computed score is not the same as a precise estimate of future predictive performance.
Use observed equal-interval mean discharges o = [1, 2, 3, 10] m³/s. Their sum is 16 and their mean is 4. The deviations from the mean are −3, −2, −1 and 6. Squaring and adding gives SST = 9 + 4 + 1 + 36 = 50 (m³/s)².
Model A predicts [0, 1, 2, 10] m³/s. Its errors pi − oi are −1, −1, −1 and 0. SSE = 1 + 1 + 1 + 0 = 3. NSE = 1 − 3/50 = 0.94. The arithmetic is short enough to audit by hand.
The zero prediction in the first interval is deliberate. It shows that an excellent overall score can coexist with a 100% relative error at one low-flow observation. The score is high because those absolute errors contribute only three units of squared error against a denominator of fifty.
For an independent check, the root mean squared error is RMSE = √(SSE/n) = √(3/4) ≈ 0.8660 m³/s. This is an error magnitude in the original discharge units, unlike NSE. Reporting both helps explain the scale of the discrepancy, although neither alone describes every interval.
The observed-mean benchmark predicts [4, 4, 4, 4] and has SSE = 50, confirming NSE = 0. A model predicting the observations exactly has SSE = 0 and NSE = 1. Checking these two endpoints is a useful way to detect reversed subtraction, an incorrect denominator or a spreadsheet that accidentally calculates variance over the predictions instead of the observations.
| Interval | Observed | Model A | Model B |
|---|---|---|---|
| 1 | 1 | 0 | 1 |
| 2 | 2 | 1 | 2 |
| 3 | 3 | 2 | 3 |
| 4 | 10 | 10 | 8 |
CHAPTER 4 OF 12 · Build and calculate
4. A higher NSE need not give the better low-flow model
Model B predicts [1, 2, 3, 8] m³/s for the same record. Its errors are 0, 0, 0 and −2, so SSE = 4. NSE = 1 − 4/50 = 0.92. On NSE alone, Model A ranks above Model B.
But Model B reproduces all three low-flow observations exactly, while Model A underpredicts each by 1 m³/s. If the question is whether the model captures the low-flow conditions in these first three intervals, B performs better in this example. If the aim is squared-error performance over the full record, A performs better by construction.
This is not a contradiction. The criteria weight different discrepancies. Squaring an error of 2 gives 4, which exceeds the three squared errors of 1 in Model A. NSE expresses that choice through its numerator. A practical decision may require different or additional information.
For example, suppose a flow below 0.5 m³/s triggers an investigation. None of the observed values crosses that threshold. Model A predicts a crossing in the first interval, while Model B does not. This threshold comparison is different from the NSE ranking and should be evaluated directly rather than assumed from the overall score.
Do not turn this into a new universal rule that B is always the better model. The fictional example has only four intervals and one chosen threshold. It demonstrates that a model can be preferable under one explicit purpose and less accurate under another. A defensible evaluation states those purposes before looking for a single winner.
CHAPTER 5 OF 12 · Change the assumptions
5. Volume bias and correlation reveal other dimensions
Because the example values are equal-interval mean discharges, their sums are proportional to total volume. Model A sums to 13 versus an observed 16. Its relative volume bias, defined here as 100 × (predicted volume − observed volume)/observed volume, is −18.75%. Model B sums to 14, giving −12.5%.
If each interval lasts one day, the observed total volume is 16 × 86,400 = 1,382,400 m³. A predicts 1,123,200 m³ and B predicts 1,209,600 m³. Both underestimate total volume, and B has the smaller volume deficit despite its lower NSE.
The equal interval and interval-mean qualifications matter. A sum of instantaneous discharge samples is not itself an exact volume. Unequal intervals require appropriate time weights or integration. Reporting a volume bias without checking those details can introduce an error before model quality is even discussed.
Now define Model C as each observation plus 3: [4, 5, 6, 13]. Its correlation with the observations is exactly one because it is a perfect positive linear shift. Yet every prediction is 3 m³/s too high. SSE = 4 × 9 = 36, NSE = 1 − 36/50 = 0.28, and the relative volume bias is 12/16 = +75%.
Correlation describes co-variation, not agreement on the identity line. The C example makes that distinction visible. NSE, volume bias, correlation and interval-specific errors describe different aspects of performance. Reporting several complementary diagnostics is useful when each is tied to a stated question rather than collected as an unexplained list of impressive numbers.
Separate mean bias, variability and correlation algebraically
For nonconstant observations and predictions, define α = σp/σo, r as their correlation, and b = (μp − μo)/σo. Use the same population-divisor convention for both standard deviations over the paired record. Expanding the mean squared error gives MSE = (μp − μo)² + σp² + σo² − 2rσpσo.
Dividing by σo² and substituting into NSE produces the exact identity NSE = 2αr − α² − b². Equivalently, NSE = r² − (α − r)² − b². The identity shows that correlation, variability mismatch and mean bias all contribute to the result.
Use a new model p = 0.5o + 2 for the original observations. It predicts [2.5, 3, 3.5, 7]. Its mean is 4, so b = 0; its standard deviation is half the observed standard deviation, so α = 0.5; and r = 1. The identity gives NSE = 2(0.5)(1) − 0.5² = 0.75. Perfect correlation and correct mean do not prevent the attenuated variability from reducing NSE.
For p = 2o, α = 2, r = 1 and the mean difference is 4. Observed population variance is SST/n = 50/4 = 12.5, so b² = 16/12.5 = 1.28. NSE = 4 − 4 − 1.28 = −1.28. Direct calculation gives SSE = Σoi² = 114 and 1 − 114/50 = −1.28, providing an independent check.
This decomposition is descriptive, not a causal diagnosis. A mean bias may arise from rainfall error, evaporation, a rating relation or a parameter choice. The algebra identifies the form of disagreement; further evidence is needed to explain its physical source.
| Model | SSE | NSE | Relative volume bias |
|---|---|---|---|
| A | 3 | 0.94 | −18.75% |
| B | 4 | 0.92 | −12.50% |
| C = observed + 3 | 36 | 0.28 | +75.00% |
CHAPTER 6 OF 12 · Change the assumptions
6. Change units and windows to test the meaning
Convert every observed and predicted discharge from m³/s to litres per second by multiplying by 1,000. Both SSE and SST multiply by 1,000², so their ratio is unchanged. NSE remains 0.94 for A and 0.92 for B. RMSE, by contrast, changes units and numerical scale as expected.
Adding the same constant to both series also leaves NSE unchanged, as does multiplying both by the same nonzero scale factor. These algebraic properties do not mean every transformed discharge series is physically equivalent. They show how the metric behaves under a common affine change to the paired data.
Now restrict evaluation to the first three observations [1, 2, 3]. The new observed mean is 2 and SST = 1 + 0 + 1 = 2. Model A’s SSE is still 3 over those three intervals, so NSE = 1 − 3/2 = −0.5. Model B is exact on those intervals, so its NSE is 1.
The change from A’s full-record 0.94 to low-flow-window −0.5 arises from both the selected errors and the new denominator. The model itself did not change. The score now answers a different window-specific comparison.
This is why NSE values from different catchments or periods should not be ranked as though they shared one identical difficulty scale. A record with large observed variability can supply a large denominator. Inspect the underlying errors, benchmark and flow regime alongside the summary. A higher score can reflect a different evaluation problem as well as a better model.
Averaging scores is different from pooling errors
Suppose a first evaluation record has SST = 100 and SSE = 10, giving NSE = 0.9. A second record has SST = 10 and SSE = 10, giving NSE = 0. Their unweighted average score is 0.45. If the purpose is to give both records equal weight as separate tasks, that average has a clear interpretation.
Combining the two error totals and the two within-record reference totals instead gives 1 − 20/110 ≈ 0.81818. This weights the records according to their reference variation, so the first record has much greater influence. It answers a different question from the equal-record average.
Even that combined ratio is not necessarily the ordinary NSE of the raw concatenated observations. A concatenated score uses one grand observed mean; if the record means differ, its denominator includes between-record differences as well as within-record variation. State the aggregation rule before interpreting a regional or multi-catchment score.
This distinction matters when one record is long or highly variable and another is short or comparatively stable. An aggregate number can improve while a particular catchment remains poorly predicted. Preserve individual results where local performance is relevant to the decision.
CHAPTER 7 OF 12 · Check the evidence
7. Handle constant observations, zeros and transformations explicitly
If all observed values equal 5, the observed mean is 5 and SST = 0. Ordinary NSE is undefined because its denominator is zero. Even if the model also predicts five everywhere, the expression contains 0/0. Report the undefined condition and use a suitable alternative error description.
A nearly constant record can also make NSE unstable in interpretation. When SST is very small, modest absolute errors produce a large ratio and a strongly negative score. That may be mathematically correct while RMSE and the actual deviations provide a more intuitive account of the error’s practical size.
Raw-flow NSE can include zero observations if SST remains positive. However, a log-transformed analysis cannot simply take the logarithm of zero. Adding an offset changes the transformation and potentially the ranking. State the offset, its units and the reason for using it rather than treating it as an invisible numerical repair.
Calculating NSE after a square-root or log transformation changes the evaluated discrepancies. It is no longer the same raw-discharge score. Such transformations may be useful for a stated purpose, but compare models under the same transformation and explain which parts of the flow range receive greater emphasis.
Never silently drop difficult observations merely to make the score computable or larger. Missing-data rules should be established consistently for both series. Document exclusions and inspect whether missingness is concentrated during floods, low flows or instrument failures, because that concentration can change the apparent performance of the model.
CHAPTER 8 OF 12 · Check the evidence
8. Peak timing and aggregation can change the conclusion
Consider observed flow [1, 1, 10, 1] and a model predicting [1, 10, 1, 1]. Both have the same sum and the same peak magnitude. The model peak occurs one interval early. Its errors are 0, 9, −9 and 0, so SSE = 162.
The observed mean is 3.25. SST = 3(2.25²) + 6.75² = 60.75. NSE = 1 − 162/60.75 = −1.6667 approximately. The model conserves total volume in this equal-interval-mean example but performs poorly on the timed sequence.
If all four intervals are aggregated into one overall mean, both series have the same mean. The timing error disappears from that one aggregated value, and ordinary NSE is undefined for a single observation because its variance denominator is zero. Reporting the matched aggregate means would not establish that the original hydrograph was predicted well.
Less extreme aggregation can likewise smooth peaks or obscure shorter timing errors. Evaluate at a temporal resolution relevant to the decision, and make sure model and observations represent the same intervals. A daily score may not answer an hourly warning question.
Do not shift predictions after inspecting the evaluation record and then present the improved score as untouched forecast skill. If timing adjustment is a fitted component, it belongs in calibration and must be tested on independent data. Otherwise information from the answers has entered the evaluation procedure.
CHAPTER 9 OF 12 · Practise and continue
9. Separate calibration from an independent test
Calibration chooses parameters using observed data. If NSE is used as an optimisation objective, a high calibration score partly reflects a successful search against those observations. It does not by itself demonstrate performance on an unseen season or catchment.
Set aside an evaluation period and document how it differs from the calibration period. Hydrological records have temporal structure, so randomly mixing neighbouring timestamps between training and test sets can create overly similar conditions across the split. The appropriate design depends on the intended transfer question.
Use benchmarks that match operational information. The observed evaluation-period mean is NSE’s mathematical reference, but an actual forecast may need comparison with persistence, a seasonal climatology or another implementable baseline. State which benchmark is being used for which claim.
Inspect observed hydrographs and residuals as well as the score. Look for timing offsets, persistent bias, missed low flows, changing seasonal behaviour and implausible values. Observations also have uncertainty, particularly where discharge is inferred through a rating relation. A model may be scored against an uncertain estimate rather than a perfect truth series.
For a student, the minimum complete report contains the paired-data definition, period, observed mean, SSE, SST, NSE, one complementary diagnostic and a limitation. For a real model assessment, the same structure extends to uncertainty, independent testing and the decision consequences of the remaining errors.
Build a concise model comparison that a reader can audit
For the A and B example, a useful report would state that both models were assessed against four equal-interval mean flows, with no missing values and the same observed-mean reference. It would give A’s NSE of 0.94 and B’s 0.92, then report the volume deficits of 18.75% and 12.5% and identify the three low-flow errors in A.
That report supports a narrow conclusion: A has the smaller squared-error total on this record, while B better reproduces the supplied low flows and total volume. It does not support a general claim that either model is ready for operational forecasting. The short invented record has no independent validation or evidence about other events.
In a real assessment, append information about the calibration split, uncertainty in observed discharge and the target consequences of errors. For a flood-warning task, a missed threshold or late peak may deserve a direct event-based measure. For an allocation study, cumulative bias and low-flow persistence may matter more than one full-record score.
An audit can therefore remain brief without becoming superficial. The reader needs enough information to reproduce the main calculation and understand its limits. A long list of metrics without definitions, windows and purposes is harder to interpret than a small number of well-chosen checks.
CHAPTER 10 OF 12 · Practise and continue
10. Independent practice and full arithmetic checks
A spreadsheet check before interpreting the result
Use one row per matched interval and separate columns for observation, prediction, error, squared error and squared deviation of the observation from the fixed observed mean. The mean reference must stay fixed across the rows. Accidentally using a moving mean or subtracting each observation from itself destroys the intended denominator.
Check that both SSE and SST use exactly the same paired rows. If a model prediction is missing, a documented paired-data rule should determine whether that timestamp is excluded from both sums. Calculating SSE on three rows and SST on four creates a score with an inconsistent reference, even if each individual formula looks plausible.
Retain an explicit unit label in the original columns. If observations are in m³/s but predictions are in litres per second, the error calculation is invalid until units are aligned. The metric’s invariance to a common conversion does not protect against converting only one series.
For a hand-check, choose the row with the largest residual and trace its contribution to SSE. Then inspect the observation furthest from the mean and its contribution to SST. In the original example, the 10 m³/s observation contributes 36 of the denominator’s 50 units. That one fact helps explain why a record containing a pronounced peak can have a very different reference scale from a quiet low-flow window.
Keep the original series alongside the score. A single formula cell cannot show whether the residuals represent a persistent offset, a shifted peak or scattered errors. The arithmetic and the hydrograph are complementary evidence, and a clear report lets the reader inspect both.
Practice A: Observations are [2, 4, 6] and predictions [3, 4, 5]. Find the mean, SST, SSE and NSE. Check: the mean is 4; SST = 4 + 0 + 4 = 8; SSE = 1 + 0 + 1 = 2; NSE = 0.75. A score of 0.75 means the model’s squared-error total is one quarter of the observed-mean benchmark’s total.
Practice B: Use predictions [4, 4, 4] for A. Check: SSE = 8 and NSE = 0. The constant prediction equals the observed mean. It does not reproduce the changing sequence even though its mean and total match the observations.
Practice C: Use predictions [8, 10, 12] for A. Check: each error is 6, SSE = 108 and NSE = 1 − 108/8 = −12.5. Correlation is still exactly one. The negative score has no lower bound at −1.
Practice D: All observations are [3, 3, 3] and predictions [2, 3, 4]. Check: SST = 0, so ordinary NSE is undefined. SSE = 2 and RMSE = √(2/3) ≈ 0.8165 in the discharge unit remain calculable.
Practice E: A model has NSE = 0.6 on a record with SST = 200. Find SSE. Check: 1 − SSE/200 = 0.6 gives SSE = 80. If another record has SST = 20 with the same SSE = 80, its NSE is −3. This compares metric behaviour, not equal real-world modelling difficulty or an automatically transferable ranking.
Record enough significant figures to reproduce the comparison, then round the reported score consistently. Extra decimal places do not repair a mismatched evaluation design or remove uncertainty in the observations.
Is NSE the same as R²?
Not generally. NSE compares squared errors with the observed-mean benchmark using the predictions as supplied. A squared correlation measures association and can remain one under a biased linear shift. Under particular fitted regression conditions some quantities coincide, but that special case should not be assumed for hydrological simulations.
Does NSE = 0.94 mean errors are only 6%?
It means SSE/SST = 0.06 for the specified record. It is not a mean percentage error or a guarantee about each timestamp. In Model A, a low-flow prediction is zero despite the high overall NSE.
Is a negative value automatically useless?
It means the model’s squared-error total exceeds the observed-mean benchmark on that record. That is a clear limitation under that criterion. The model might still contain useful information for another purpose, but that claim requires direct evidence rather than dismissing the negative score.
Is there one universal pass mark?
A threshold should be justified for the application, data quality, temporal resolution and consequences of error. A convenient label does not replace checking the flow regime and the actual missed events. Two models with the same NSE can have very different residual patterns.
Can an NSE value be compared directly with a Kling–Gupta score?
The metrics have different constructions and benchmark behaviour. Equal numerical values do not necessarily imply equal performance, and the zero benchmark interpretation of NSE should not be copied automatically to another score. Use the definitions and the intended task.
CHAPTER 12 OF 12 · Practise and continue
12. Sources, evidence checks and the next reading route
For primary research on the interpretation of efficiency benchmarks, see Knoben, Freer and Woods’s Technical note: Inherent benchmark or not? Comparing Nash–Sutcliffe and Kling–Gupta efficiency scores. It supports the distinction between metric definitions and warns against transferring benchmark interpretations without examining their construction.
For a further research perspective, see Le Duc and Sawada’s A signal-processing-based interpretation of the Nash–Sutcliffe efficiency. The fictional examples here use the conventional paired-data definition and are intended to make the arithmetic and decision limits inspectable.
Before relying on a reported score, ask for the evaluation period, interval, paired-data rule, observation quality, calibration separation, benchmark, missing-data handling and transformation. Then inspect at least one hydrograph and the errors most relevant to the decision. A ranking becomes meaningful when those conditions are visible.
Continue with rainfall time steps to investigate why temporal resolution can change a model response, and paired catchments to distinguish prediction performance from attribution. Return to the Hydrology subject guide for connected study.