Kling–Gupta Efficiency¶

A hydrological model-performance score combining correlation, relative variability and mean bias against observed flow.

Version: v1 · 2026-10-17 · History
Domain-specific #: 14208
Domain group: Natural Sciences
Origin domain: Geology & Earth Sciences
Subdomain: Rainfall Runoff Model Evaluation → Geology & Earth Sciences
Aliases: KGE

Core Idea¶

Kling–Gupta efficiency (KGE), in the original unweighted formulation considered here, is a score for a simulated hydrological flow series compared with observations at matching times. It combines how the series co-vary, how their spread compares, and how their means compare. The score is one minus the Euclidean distance of those three components from their ideal values of one. A perfect component triple gives KGE = 1; a lower score records departure from that ideal but does not, by itself, identify which component is responsible or declare a model adequate.^[1]^[2]

This entry concerns the original formulation with the simulated-to-observed standard-deviation and mean ratios. It can score a deliberately constructed series as well as output from a calibrated rainfall–runoff model; the score does not turn the former into a predictive model. Knoben and colleagues use a synthetic offset to expose a benchmark misconception, while Mathevet and colleagues use the same component structure in calibration and split-sample evaluation.^[1]^[2]

Structural Signature¶

Sig role-phrases: observed reference → time-aligned simulation → correlation r → relative spread α → relative mean β → unweighted distance/readout.

Observed reference. An observed discharge series supplies the reference timing, mean μobs and standard deviation σobs. Without a defined observed mean and spread, the two ratios in this bounded formulation cannot be calculated.
Aligned simulation. A simulated discharge value is paired with the observation at each evaluated time step. Unpaired series cannot yield the same correlation or component comparisons.
Correlation r. Pearson's linear correlation describes co-movement of the paired values. Matching means and spread alone can coexist with poor timing or dynamics.
Relative variability α. σsim/σobs compares simulated with observed standard deviation. Removing it conceals amplitude error that correlation and mean may not expose.
Relative mean β. μsim/μobs compares simulated with observed mean. An additive offset can leave correlation and spread unchanged while moving this ratio.
Unweighted distance and readout. KGE = 1 − √[(r−1)² + (α−1)² + (β−1)²]. The ideal is (r, α, β) = (1, 1, 1). Three diagnostics without this aggregation are informative, but are not the named scalar KGE.^[1]^[2]

Ordinary Pearson correlation is undefined when an input series is constant. Both ratios also need their observed denominators to be nonzero. A reported score for such an edge case therefore requires an explicit convention or a different handling rule, rather than silent use of the ordinary formula.^[1]

What It Is Not¶

Not Pearson correlation alone. r is one component; it does not check mean or relative spread, and is not KGE's three-dimensional distance.
Not Nash–Sutcliffe efficiency (NSE). NSE normalizes squared residuals against observed variation. Its zero point has a mean-flow benchmark interpretation that cannot be copied onto KGE.^[1]
Not an automatic good/bad certificate. Neither KGE = 0 nor a particular negative value supplies a universal adequacy threshold. A comparison needs a declared benchmark and modeling purpose.^[1]
Not every later KGE variant. Modified, weighted and nonparametric forms can change the components or their aggregation. Their formulas must be named separately rather than silently assigned to this original unweighted entry.^[1]
Not a forecast from a synthetic perturbation. A series constructed from observations can be scored, but its score does not establish independently predictive rainfall–runoff skill.^[1]

Scope of Application¶

The evidenced domain here is hydrological model-performance evaluation using paired observed and simulated discharge. The score is literal when the time support and units are commensurate, its component ratios are defined or their exceptional handling is declared, and the original unweighted formula is used. The sources do not establish use in crop-yield or other unrelated model domains.^[1]^[2]

Knoben, Freer and Woods construct a same-time offset from observed CAMELS daily discharge to show why the NSE and KGE scores can diverge under chosen thresholds. Mathevet and colleagues instead calibrate conceptual rainfall–runoff models over one period and evaluate them on another; their Hérault watershed case supplies a named fitted-model habitat. The first case tests the score's interpretation, and the second shows its use with model parameters and period-specific diagnostics.^[1]^[2]

Clarity¶

The direction of the ratios matters: α is simulated spread divided by observed spread, and β is simulated mean divided by observed mean. Their value one means agreement with the reference on that dimension; it does not mean a discharge of one unit. Correlation one records perfect linear co-movement, not agreement in the level of the flow. KGE = 1 requires all three component ideals together.^[1]^[2]

The word Efficiency can suggest the same zero benchmark as NSE or the input–output frontier described by other efficiency concepts. Here the numerical zero is merely a particular distance from (1,1,1). Knoben and colleagues show that a constant mean-flow predictor does not naturally land at zero on this scale. Model assessment therefore has two distinct questions: what score follows from the stated formula, and what benchmark makes that score useful for the decision at hand.^[1]

Manages Complexity¶

Three departures in a time-series comparison become one scalar, so a modeler can order candidate simulations under a single declared scoring rule. The compression is useful for calibration, but it discards the direction of each component error: different triples can lie at the same distance from (1,1,1). Reading r, α and β beside KGE restores the diagnosis that the scalar alone cannot supply.^[1]^[2]

There is also a time-aggregation choice. Mathevet and colleagues report full-period and annual-subperiod component views for their Hérault case. A small full-period bias can coexist with substantial yearly mean or variability bias, so a decision about seasonal behavior cannot rest on the full-period score alone.^[2]

Abstract Reasoning¶

Place the three component values at a point in r–α–β space. The ideal point is (1,1,1), and KGE subtracts its ordinary Euclidean distance from one. This makes the invariant clear: a change that moves the component point farther from ideal lowers the score, while two different error patterns can have the same distance. The scalar ranks distances, not error mechanisms.^[1]^[2]

The mean-flow thought experiment exposes a convention. If Qsim(t) equals the constant observed mean at every time, then α = 0 and β = 1; ordinary Pearson r is undefined because the simulation has zero spread. Knoben and colleagues explicitly assign r = 0 for this calculation, obtaining KGE = 1 − √2 ≈ −0.41. That value is conditional on the assignment; it is not an ordinary, universally defined KGE of every constant simulation, nor a universal adequacy cutoff.^[1]

Knowledge Transfer¶

The roles transfer between a constructed offset and a fitted hydrological model without transferring their evidential conclusions. In both, one first pairs observed and simulated flow, calculates correlation and the two simulated-to-observed ratios, then obtains the same distance score. A positive additive offset tests how mean bias can coexist with unchanged correlation and spread; split-sample GRX output tests a fitted model on periods distinct from calibration.^[1]^[2]

What does not transfer automatically is a threshold, physical explanation or measured skill. Knoben's illustrative KGE > 0.3 versus NSE > 0.5 rule belongs to its synthetic demonstration. Mathevet's component plots belong to particular watersheds and periods. Any new hydrological setting needs its own reference series, declared convention and evaluation purpose.^[1]^[2]

Examples¶

Canonical interpretation case: synthetic offset from CAMELS flow¶

In Knoben and colleagues' §3.1 Figure 1c, the observed daily discharge comes from CAMELS catchment 04124000. The paper constructs Qsim(t) = Qobs(t) + 0.45 mm/day at every step, reporting a mean bias of +39%. The constant offset leaves correlation and spread ratio at one for the nonconstant observed series; the mean ratio moves. The resulting KGE is 0.61, while NSE is −0.95. Under the paper's illustrative thresholds, KGE > 0.3 accepts the constructed series and NSE > 0.5 does not. This contrast demonstrates threshold nontransfer, not the predictive skill of a calibrated model.^[1]

Mapped back: Observed reference = catchment 04124000 daily flow; aligned simulation = its same-time +0.45 mm/day construction; correlation = one; relative spread = one; relative mean = the +39% bias expressed against observed mean; unweighted readout = KGE 0.61. The equalities for correlation and spread follow algebraically from a constant positive offset on a nonconstant series; the score and bias are reported by the source.^[1]

Applied model case: Hérault GRX split-sample evaluation¶

Mathevet and colleagues assess daily streamflow in 2050 watersheds with GRX and MRX rainfall–runoff models. They split each watershed's available record into two time halves, calibrate on one and evaluate on the other, then reverse the halves. Their §4.2 Figure 8 names Hérault station Y2102010: GRX is calibrated with KGE(Q), and component displays compare calibration and evaluation, full periods and individual years. The case shows why apparently small period-wide bias can coexist with marked yearly mean and variability departures. These are described from the methods, text and caption; no exact figure-pixel values are inferred.^[2]

Mapped back: Observed reference = gauged Hérault daily streamflow; aligned simulation = GRX daily output in each split; correlation = paired-flow R; relative spread = simulated/observed α; relative mean = simulated/observed βr; unweighted readout = Equation (24) KGE, used as an objective and evaluated with component and period diagnostics. Hérault is a documented named case, not a pattern claimed for every watershed.^[2]

Structural Tensions¶

T1 — One comparable score versus component-level diagnosis. A single KGE gives a compact objective for ranking simulations under one formula, but different combinations of correlation, variability and mean departure can yield the same scalar. Choosing only the aggregate simplifies comparison while hiding which error matters for a particular decision. Reporting components and yearly slices reveals more of that structure, but makes the evaluation less reducible to one ranking and requires a stated hydrological purpose. Knoben and colleagues warn against ad hoc metric interpretation; Mathevet and colleagues' Hérault analysis shows the period-scale cost in a named case.^[1]^[2] Diagnostic: Which component and evaluation timescale would change the current model decision if reported alongside KGE?

The conditional −0.41 mean-flow calculation and the absence of an inherent zero cutoff are interpretation boundaries, rather than separate opposed objectives. Neither turns a context-specific benchmark into a second built-in scale.^[1]

Structural–Framed Character¶

The score has a structural-leaning form inside a framed hydrological use. Its algebraic vocabulary—paired series, correlation, relative spread, relative mean and distance—can be written without naming a catchment. Yet its evaluated object in the inspected sources is observed versus simulated discharge, and adequacy depends on hydrological purpose and reference choice. A numerical score does not carry a moral judgment; its evaluative weight arises when a modeler declares what counts as adequate for a particular task.^[1]^[2]

The named formulation is associated with hydrological model evaluation, not with one institution's rules. The words Efficiency and benchmark travel poorly if they are read as NSE's zero point or an input–output frontier. Importing KGE to another series requires deliberately selecting a reference, aligning observations and simulations, defining the component ratios and defending the benchmark; merely recognizing that two things were compared is insufficient. The portable skeleton is a co-framed comparison of component departures, while this named score retains its hydrological evidence boundary.

Its character: a domain-specific performance measure with a precise structural formula and framed interpretation; the formula travels across the two documented hydrological workflows, but the inspected evidence does not establish universal model-quality thresholds or cross-domain validity.^[1]^[2]

Structural Core vs. Domain Accent¶

The core is the original unweighted computation: aligned observed and simulated series, r, α = σsim/σobs, β = μsim/μobs, and 1 − the Euclidean distance from (1,1,1). Removing a term or changing its weight changes the bounded named formulation. As a Similarity Measure, KGE maps an ordered pair under a fixed relevance frame to a degree of match. Its particular hydrological carrier and formula keep it distinct from other rules that do the same kind of mapping.^[1]^[2]

The accent is the catchment, the way the simulated series was made, the period split, the plotting style and any declared decision threshold. Knoben's constructed CAMELS offset and Mathevet's fitted GRX output vary on those dimensions while retaining the score. No demonstrated transfer here makes this particular hydrological performance measure a substrate-independent Prime. A more general comparison pattern can be tested for Prime reach separately, without equating that pattern with KGE.

This entry is a kind of Similarity Measure.

Similarity Measure is the broader rule that maps an ordered pair under a declared representation and relevance frame to a comparable degree of likeness. KGE specializes it to time-aligned observed and simulated flow and the original unweighted r–α–β score. Similarity Measure in turn is a Function Mapping and presupposes Comparison: KGE's fixed formula maps admissible pairs to scores, while its shared time and discharge frame makes their component differences intelligible. A separate performed Comparison is not needed to define an unapplied scoring rule. Measurement includes an instrument and calibration/uncertainty chain. A KGE calculation from already supplied series need not contain that whole chain, so the word Measure alone does not make it a Measurement instance.

Efficiency has a broader input–output or feasible-frontier identity that a KGE score need not instantiate. Pearson Correlation Coefficient supplies one component, not the three-component distance. Concordance Correlation Coefficient uses a different agreement algebra, and Goodness-of-Fit Test adds a null-calibrated inference absent from deterministic KGE scoring. These links distinguish structures without making their full identities necessary to KGE.

Relationships to Other Abstractions¶

Current abstraction Kling–Gupta Efficiency Domain-specific

Parents (1) — more general patterns this builds on

Kling–Gupta Efficiency is a kind of Similarity Measure Prime

Original unweighted KGE is a hydrology-specific Similarity Measure mapping aligned observed and simulated flow series to a comparable degree of match through a fixed three-component rule.

Hierarchy paths (2) — routes to 2 parentless roots

Kling–Gupta Efficiency → Similarity Measure → Function (Mapping)

Show alternative path (1)

Neighborhood in Abstraction Space¶

Kling–Gupta Efficiency sits in a sparse region of the domain-specific corpus (76^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2721 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-17

Not to Be Confused With¶

NSE: its squared-residual normalization gives zero a mean-flow reference, whereas KGE's zero is only a distance level. Their scores have no unique conversion.^[1]
The constant mean-flow convention: −0.41 follows after assigning zero to an otherwise undefined correlation. It is not an unconditional score or adequacy rule.^[1]
Modified, weighted or nonparametric KGE: these are variant formulas. This entry's exact component definitions and unweighted aggregation are the inclusion test.^[1]
Pearson correlation or concordance correlation: either can speak to agreement, but neither is this hydrological three-component score.
Knowledge graph embedding: it shares the abbreviation “KGE” in another field, not this metric's formula or evaluated carrier.
A universal model verdict: a higher score under one declared comparison can be useful, but the score alone neither selects a universally adequate threshold nor locates each error source.^[1]^[2]

References¶

[1] Wouter J. M. Knoben, Jim E. Freer, and Ross A. Woods (2019), “Technical note: Inherent benchmark or not? Comparing Nash–Sutcliffe and Kling–Gupta efficiency scores,” Hydrology and Earth System Sciences 23:4323–4331, DOI 10.5194/hess-23-4323-2019. Original open publisher article, §1 Equations (2)–(3), §2 Equations (4)–(6), §3.1 Figure 1 caption and accompanying text. Equations and text/caption were inspected; figure pixels were not independently evaluated. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s ↩t ↩u ↩v ↩w ↩x ↩y ↩z ↩27 ↩28

[2] Thibault Mathevet, Nicolas Le Moine, Vazken Andréassian, Hoshin Gupta, and Ludovic Oudin (2023 volume), “Multi-objective assessment of hydrological model performances using Nash–Sutcliffe and Kling–Gupta efficiencies on a worldwide large sample of watersheds,” Comptes Rendus Géoscience 355(S1):117–141, DOI 10.5802/crgeos.189. Original open publisher article, §3.3 Equation (24), §§3.1–3.4 model and split-sample procedure, §4.2 Hérault Y2102010 discussion and Figure 8 caption. The publisher lists online first 2023 and published online 2024; 2023 denotes the cited volume. Original HTML and version-of-record text were inspected; no exact figure-pixel values are claimed. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r