Crop Yield Response to Water (Ky)
Enter yield response factor Ky, actual evapotranspiration ETa and maximum ETm to compute evapotranspiration deficit, relative yield loss and relative yield (FAO-33 Ky model), assessing drought impact on crops.
Input Data
Results
At a glance:Crop yield response to water describes how yield falls as available water (actual evapotranspiration) decreases — the core quantitative basis of irrigation management, answering 'how much yield do I lose per unit of water saved?'. Its authoritative framework is the yield response factor Ky from FAO Irrigation and Drainage Paper 33 (Doorenbos & Kassam, 1979). Start with two relative quantities. First, relative ET deficit: a fully watered, vigorous crop reaches a maximum (potential) evapotranspiration ETm; under water stress, actual ETa is lower, and (1 − ETa/ETm) is the relative water deficit — e.g. ETa/ETm = 0.8 means 80% of demand met, 20% deficit. ET is a good indicator because it reflects both transpiration (directly tied to photosynthesis and growth) and soil evaporation; deficit closes stomata, lowers transpiration and photosynthesis, and finally depresses yield. Second, relative yield Ya/Ym: actual yield Ya over maximum (fully irrigated) yield Ym. FAO-33 assumes these two relative quantities are roughly linear, with constant Ky: 1 − Ya/Ym = Ky × (1 − ETa/ETm). So relative yield loss = Ky × relative water deficit. Ky is the amplifier translating 'percent water deficit' into 'percent yield loss', i.e. crop sensitivity to water deficit. Ky = 1 means loss and deficit are 1:1 (20% less water → 20% less yield); Ky > 1 means very sensitive (20% less water → >20% loss, e.g. maize whole-season Ky ≈ 1.25), so such crops or critical stages should not be stressed; Ky < 1 means drought-tolerant (smaller loss per deficit). Ky varies by crop and growth stage — most crops are most sensitive (highest Ky) at flowering and yield formation (heading, grain-filling, podding) and less sensitive in vegetative or maturity stages, so precise irrigation protects the sensitive stages first. The practical form: Ya/Ym = 1 − Ky × (1 − ETa/ETm); relative loss = Ky × (1 − ETa/ETm). Example: crop whole-season Ky = 1.1, ETa = 400 mm, ETm = 500 mm → deficit = 1 − 400/500 = 0.2 (20%); relative loss = 1.1 × 0.2 = 0.22 (22%); relative yield = 1 − 0.22 = 0.78 (78%). Such a deficit lowers yield to 78% of full irrigation, a 22% loss. At full water (ETa = ETm) deficit, loss and yield loss are 0 and relative yield is 100%. Uses: (1) deficit-irrigation decisions — with limited water, use Ky to evaluate directing scarce water to sensitive stages or reducing supply overall, maximizing output per drop rather than yield; (2) irrigation scheduling & economic analysis — convert loss to revenue loss and compare with saved water cost and energy; (3) compare crop/variety drought tolerance and plan crop choice in water-scarce areas. Caveats: it is an empirical linear approximation, most accurate for moderate deficit; at extreme deficit (low ETa/ETm) real loss is nonlinear and the model underestimates; use the right Ky for crop and stage (whole-season vs stage Ky differ, and deficit at a critical stage hurts far more than the same total spread evenly); estimate ETm (ET₀ × Kc) and ETa (water balance or soil-moisture monitoring) correctly; the model covers only water-stress loss, not pests, nutrients, salinity or lodging.
Formula
FAO-33: 1 − Ya/Ym = Ky × (1 − ETa/ETm).
Relative water deficit = 1 − ETa/ETm; relative yield loss = Ky × deficit.
Relative yield Ya/Ym = 1 − Ky × (1 − ETa/ETm).
$$1 - \dfrac{Y_a}{Y_m} = K_y\left(1 - \dfrac{ET_a}{ET_m}\right)$$$$\dfrac{Y_a}{Y_m} = 1 - K_y\left(1 - \dfrac{ET_a}{ET_m}\right)$$How to Use
- Look up the yield response factor Ky for the crop (and growth stage) from FAO-33.
- Enter actual evapotranspiration ETa and maximum ETm (ETm ≈ ET₀ × Kc).
- The tool returns ET deficit, relative yield loss and relative yield for deficit-irrigation and economic decisions.
FAO-33 Ky yield–water response example
| Item | Value | Note |
|---|---|---|
| Yield response factor Ky | 1.1 | Whole-season example |
| Actual ETa | 400 mm | Water-deficit scenario |
| Maximum ETm | 500 mm | Full irrigation |
| ET deficit | 20% | = 1 − 400/500 |
| Relative yield loss | 22% | = 1.1 × 20% |
| Relative yield | 78% | = 1 − 22% |
Ky varies by crop and stage; sensitive stages (flowering/grain-filling) have the highest Ky and should be protected first.
Case Studies
Yield-loss cost of deficit irrigation
Crop Ky = 1.1; water restriction makes actual ETa = 400 mm, maximum ETm = 500 mm.
ET deficit = 1 − 400/500 = 20%; relative loss = 1.1 × 20% = 22%; relative yield = 78%.
Decision-makers compare saved water against the 22% revenue loss to judge whether this deficit level pays off.
Never stress the sensitive stage
Same 20% deficit, if it occurs at flowering/grain-filling, the stage Ky can reach 1.5.
Relative loss = 1.5 × 20% = 30%, far above the 22% whole-season average.
Shows scarce water should be prioritized to high-Ky critical stages to avoid major loss.
FAQ
How do I choose the yield response factor Ky?
Ky is looked up by crop and growth stage from FAO Irrigation and Drainage Paper 33. Whole-season Ky is mostly 0.8–1.3, e.g. maize ~1.25, wheat ~1.0, potato ~1.1, sunflower ~0.95. The same crop has different Ky across stages — most are highest (most sensitive) at flowering and yield formation (heading, grain-filling, podding) and lower in vegetative and maturity stages. For stage-by-stage analysis use the stage Ky rather than only the whole-season value for better accuracy.
What is the difference between Ky > 1 and Ky < 1?
Ky is the amplifier from 'relative water deficit' to 'relative yield loss'. Ky = 1 means loss and deficit are 1:1 (20% less water → 20% less yield). Ky > 1 (e.g. maize 1.25) means very sensitive to water deficit — 20% less water causes >20% loss, so such crops or critical stages should not be stressed. Ky < 1 means relatively drought-tolerant — smaller loss per the same deficit, better suited to deficit irrigation or unstable water supply.
How to estimate maximum ETm and actual ETa?
Maximum (potential) evapotranspiration ETm is usually ET₀ × crop coefficient Kc: ETm = ET₀ × Kc, the crop's demand at that stage with no water stress. Actual ETa reflects real water supply, estimated from soil water balance, soil-moisture sensors, eddy covariance or crop water-stress indicators; under deficit ETa < ETm. Use the same period (whole season or a stage) and consistent units (mm) for both.
What is deficit irrigation?
Deficit irrigation deliberately supplies less than the crop's maximum ET demand, accepting some yield loss to save water, expand irrigated area or raise water productivity (output per unit water). The Ky model is its decision basis: estimate the loss at a given supply level, then compare with saved water, energy and expanded-area benefits. In practice, scarce water is prioritized to high-Ky sensitive stages and reduced in low-Ky drought-tolerant stages, minimizing total loss and maximizing water-use efficiency.
What are the limitations of this model?
FAO-33 Ky is an empirical linear approximation, most accurate for moderate deficit; at extreme deficit (low ETa/ETm) real loss is nonlinearly amplified and the model underestimates. It covers only water-stress loss, not pests, nutrient shortage, salinity, lodging or other limiting factors. Use the correct Ky for crop and stage, and estimate ETm and ETa correctly. For detailed cumulative stage-deficit effects, use stage Ky multiplied period-by-period or a more complex crop model such as AquaCrop.
Related Tools
References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.