Evaluation of irrigation requirement for the design of an irrigation system using a probabilistic approach for the estimation of evapotranspiration and rainfall<sup/>

Melo, Verônica Gaspar Martins Leite; Frizzone, José Antônio; Camargo, Antonio Pires de; Melo, Leonardo Leite de

doi:10.5935/1806-6690.20230048

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Scientific Article • Rev. Ciênc. Agron. 54 • 2023 • https://doi.org/10.5935/1806-6690.20230048 copy

Evaluation of irrigation requirement for the design of an irrigation system using a probabilistic approach for the estimation of evapotranspiration and rainfall¹ 1 Biosystems Engineering Department, College of Agriculture “Luiz de Queiroz”, University of São Paulo, CEP 13418-900, Piracicaba-SP, Brazil

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

Reference evapotranspiration (ET₀) and rainfall are basic variables for estimating the net irrigation depth (NID). The objective of this study was to estimate the NID for designing irrigation systems in Piracicaba, SP, Brazil, using ET₀ and rainfall probability distributions. A 30-year ET₀ and rainfall dataset (1990-2019) was obtained from the ESALQ/USP weather station. The water balance between ET₀ and rainfall indicated July, August, and September as months of higher water deficit. Based on the first-order Markov chain, August presented the highest water deficit. Rainfall and ET₀ were estimated on 19 probability levels, and four probability distributions such as normal, log-normal, beta, and mixed gamma were evaluated. The analysis of historical August series using accumulated values in periods of five, ten, or 15 days is recommended for sizing irrigation designs in Piracicaba, SP, Brazil. The log-normal and mixed gamma probability distributions presented the best fit for ET₀ and rainfall data, respectively. To reach a crop coefficient K_c = 1 in Piracicaba, SP, Brazil in August, the irrigation system should be designed for an NID of 4.1 mm day^-1. The use of mean monthly rainfall and ET₀ values for designing irrigation systems underestimates the NID by a mean of 26.6% compared to estimates made at a probability of 75% at five-, ten-, and 15-day intervals because the mean rainfall values occurred with exceedance probabilities of < 36%, and mean ET₀ values occurred with non-exceedance probabilities of < 56%.

Keywords:
net irrigation depth; supplementary irrigation; probable rainfall; probable evapotranspiration

Distribution	Probability density function	Expected value	Variance
Normal	$f (x) = \frac{1}{σ \sqrt{2 π}} e^{- \frac{1}{2} {(\frac{x - μ}{α})}^{2}} - \infty < χ < \infty; - \infty < μ < \infty; σ > 0$	$E (χ) = μ$	$V (χ) = σ^{2}$
Log-normal	$f (x) = \frac{1}{χ ω \sqrt{2 π}} e^{- \frac{1}{2} {[\frac{\ln (χ) - θ}{ω}]}^{2}} 0 < χ < \infty$	$E (χ) = e^{θ + \frac{ω^{2}}{2}}$	$V (χ) = e^{2 θ + ω^{2}} (e^{ω^{2}} - 1)$
Beta	$f (χ) = \frac{Γ (α + β)}{Γ (α) Γ (β)} χ^{α - 1} {(1 - χ)}^{β - 1} 0 \leq χ \leq 1; α > 0; β > 0$	$E (χ) = \frac{α}{α + β}$	$V (χ) = \frac{α β}{{(α + β)}^{2} (α + β + 1)}$
Gama	$f (χ) = \frac{λ^{Γ} χ^{Γ - 1} e^{- λ χ}}{Γ (Γ)} χ > 0; λ > 0; Γ > 0$	$E (χ) = \frac{Γ}{λ}$	$V (χ) = \frac{Γ}{λ^{2}}$

Month	Period	ET0 (mm)	Rainfall (mm)	P(D)	P(W)	P(D/D)	P(W/D)	P(W/W)	P(D/W)
July	1-10	26.2	10.7	0.907	0.093	0.938	0.062	0.440	0.560
	11-20	28.0	12.3	0.913	0.087	0.948	0.052	0.375	0.625
	21-30	31.8	12.0	0.906	0.094	0.937	0.063	0.414	0.586
August	1-10	32.6	9.4	0.920	0.080	0.953	0.047	0.440	0.560
	11-20	36.2	7.5	0.930	0.070	0.961	0.039	0.556	0.444
	21-30	42.4	11.6	0.912	0.088	0.934	0.066	0.321	0.679
September	1-10	40.5	20.8	0.857	0.143	0.902	0.098	0.409	0.591
	11-20	42.2	21.7	0.857	0.143	0.885	0.115	0.325	0.675
	21-30	41.4	22.2	0.827	0.173	0.875	0.126	0.396	0.604

P(-)	Probable rainfall, mm
	Month	Fortnightly		Decennial			Quinquennial
	August	1	2	1	2	3	1	2	3	4	5	6
0.05	92.1	52.5	66.0	51.6	38.5	43.8	32.7	24.3	7.7	37.5	24.7	32.8
0.10	69.9	34.0	47.4	29.8	24.0	31.3	17.3	14.0	5.0	20.9	16.1	23.2
0.15	56.8	23.7	39.7	20.3	17.5	26.1	9.7	8.1	3.3	12.3	10.6	18.5
0.20	47.6	17.1	33.1	12.8	12.0	22.5	4.5	4.2	2.1	6.6	6.4	13.9
0.25	40.3	12.3	27.6	7.2	7.6	18.7	2.0	1.6	1.1	3.0	3.9	9.8
0.30	34.6	8.6	22.9	4.3	4.9	14.9	0.3	0.0	0.1	0.6	1.1	7.1
0.35	29.6	5.8	18.7	2.4	3.1	11.6	0.0	0.0	0.0	0.0	0.0	4.8
0.40	25.3	3.7	15.2	0.1	1.0	9.2	0.0	0.0	0.0	0.0	0.0	1.9
0.45	21.5	2.1	12.1	0.0	0.7	7.4	0.0	0.0	0.0	0.0	0.0	1.2
0.50	18.2	0.9	9.1	0.0	0.0	5.3	0.0	0.0	0.0	0.0	0.0	0.0
0.55	15.1	0.2	6.0	0.0	0.0	2.6	0.0	0.0	0.0	0.0	0.0	0.0
0.60	12.3	0.1	2.7	0.0	0.0	0.7	0.0	0.0	0.0	0.0	0.0	0.0
0.65	9.7	0.0	0.7	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
0.70	7.3	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
0.75	5.1	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
0.80	3.1	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
0.85	1.1	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
0.90	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
r	1.28	0.54	1.34	0.50	0.68	1.39	0.43	0.77	1.23	0.63	0.97	1.28
1/λ	27.80	33.26	20.86	40.03	23.72	14.42	34.12	17.04	3.61	29.16	13.81	12.94
$\bar{X}$	28.5	10.7	17.8	9.4	7.5	11.6	5.4	3.9	1.4	6.2	4.5	7.1
Sx	29.1	22.9	21.5	22.9	14.5	16.1	16.0	10.7	2.9	13.9	10.2	12.8
Max	105.7	105.7	72.5	95.0	95.0	59.4	68.8	52.2	12.3	51.2	49.6	59.4
Min	0	0	0	0	0	0	0	0	0	0	0	0
$P (X \geq \bar{X})$	0.36	0.27	0.28	0.23	0.26	0.35	0.18	0.22	0.23	0.22	0.23	0.30
D_sup	0.074	0.130	0.090	0.078	0.076	0.049	0.107	0.059	0.042	0.069	0.073	0.062
D_critical	0.276	0.309	0.301	0.349	0.349	0.309	0.391	0.432	0.432	0.410	0.361	0.410

P(-)	Reference evapotranspiration (ET₀), mm
	Month	Fortnightly		Decennial			Quinquennial
	August	1	2	1	2	3	1	2	3	4	5	6
0.05	107.0	46.6	53.0	30.5	32.2	36.1	13.9	15.2	16.2	14.1	14.0	19.0
0.10	109.2	48.8	55.1	32.4	33.9	37.8	14.9	16.2	16.9	15.7	15.2	20.3
0.15	110.8	50.6	56.4	33.4	34.8	38.8	15.5	16.7	17.4	16.4	15.9	20.9
0.20	112.1	51.7	57.2	33.8	35.3	39.5	15.9	17.1	17.7	16.7	16.3	21.4
0.25	113.1	52.7	57.9	34.2	35.7	40.0	16.2	17.3	17.9	16.9	16.6	21.7
0.30	114.0	53.5	58.5	34.6	36.0	40.5	16.4	17.7	18.1	17.2	17.0	22.0
0.35	114.8	54.3	59.2	35.0	36.4	41.1	16.7	17.8	18.4	17.5	17.5	22.4
0.40	115.7	55.1	59.9	35.6	36.9	41.6	17.0	18.2	18.6	17.9	18.0	22.8
0.45	116.5	55.9	60.7	36.1	37.4	42.2	17.4	18.5	18.8	18.3	18.5	23.2
0.50	117.4	56.8	61.4	36.7	37.9	42.7	17.7	18.8	19.1	18.7	19.0	23.6
0.55	118.3	57.7	62.1	37.2	38.3	43.3	18.1	19.1	19.3	19.0	19.4	24.0
0.60	119.2	58.5	62.7	37.7	38.8	43.8	18.4	19.5	19.6	19.3	19.8	24.3
0.65	120.1	59.3	63.4	38.1	39.2	44.3	18.7	19.8	19.8	19.6	20.2	24.7
0.70	121.0	60.1	64.1	38.6	39.6	44.8	19.1	20.1	20.0	20.0	20.6	25.1
0.75	122.0	61.1	64.9	39.1	40.1	45.4	19.4	20.5	20.3	20.5	21.2	25.5
0.80	123.1	62.7	65.9	39.9	40.7	46.1	19.9	21.0	20.7	21.0	21.9	26.1
0.85	124.5	65.3	67.1	40.9	41.4	47.1	20.5	21.6	21.0	21.3	22.7	26.8
0.90	126.6	70.0	68.5	42.1	42.4	48.3	21.3	22.3	21.5	21.6	23.4	27.6
0.95	128.9	78.1	70.1	43.4	43.7	50.0	22.4	22.9	22.3	21.8	23.9	28.5
$\bar{X}$	118.7	57.2	61.5	37.1	38.3	43.3	17.9	18.9	19.2	18.8	19.1	23.8
Sx	6.6	7.5	5.1	2.2	3.3	4.0	2.4	2.3	1.8	2.2	3.0	2.8
Max	129.8	78.1	69.6	26.2	44.0	50.4	22.6	22.4	22.9	21.9	23.4	28.3
Min	106.1	46.6	49.7	15.8	29.4	33.4	11.9	13.3	15.3	10.6	11.4	16.5
$P (X \geq \bar{X})$	0.56	0.53	0.50	0.54	0.55	0.55	0.52	0.51	0.51	0.51	0.51	0.52
D_sup	0.113	0.196	0.112	0.095	0.061	0.077	0.078	0.177	0.071	0.180	0.160	0.087
D_critical	0.248	0.248	0.248	0.248	0.248	0.248	0.248	0.248	0.248	0.248	0.248	0.248

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