Simulation-Based Schemes to Determine Economical Irrigation Depths Considering Volumetric Water Price and Weather Forecasts

We used the WASH_2D model (Fujimaki et al. 2014) to determine irrigation depths for the proposed simulation schemes. The model simulates the two-dimensional movement of water, solute, and heat in soils using the finite-difference method. It separately calculates the evaporation using a bulk transfer equation (van Bavel and Hillel 1976) and the transpiration using a combination of the dual crop coefficient approach (Allen et al. 1998) and a macroscopic root water uptake model. It also includes a growth module to simulate the response of plant to irrigation. The software with the open-source code can be downloaded freely from the website of the Arid Land Research Center, Tottori University (Fujimaki et al. 2014).

To determine $W$, two steps were performed (Fig. 2). In Step 1, we ran the simulation to update the initial conditions using the information of the previous irrigation interval (48 h): irrigation records, meteorological data, and cumulative transpiration. In this step, $W$ was calculated for the refilling scheme to return updated values of ${\theta }_x$ to ${\theta }_{\mathrm{fc}}$. In Step 2, the optimization was performed for both the three-point and two-point schemes using information from the update from Step 1 in addition to weather forecast data to determine values of $W$ that maximize $I_n$ in each irrigation interval. Weather forecast data were used to predict liquid water inputs from rain, evaporation from soil surface, and $E{T}_0$.

The proposed schemes were evaluated using a field experiment located in the Arid Land Research Center of Tottori University in 2021. It was carried out as a randomized complete block design with four independent blocks and the following four treatments (Fig. 3):

A drip irrigation network was installed, and each replicate was irrigated with seven drip tubes spaced at 60 cm intervals along 10 m. The emitter spacings and discharge rate were 20 cm and $2\times {10}^{-3}{\mathrm{m}}^3{\mathrm{h}}^{-1}$, respectively. The soil was sand (99.1% sand); the hydraulic properties are shown in Fig. 4.

Actual weather data [air temperature (°C), rainfall (mm), wind speed ($\mathrm{m}{\mathrm{s}}^{-1}$), solar radiation ($\mathrm{MJ}{\mathrm{m}}^{-2}{\mathrm{h}}^{-1}$), and relative humidity (%)] were collected from a weather station installed in the field (Fig. 5). Weather forecast data were accessed and downloaded from the website of Yahoo! Japan (Yahoo Japan Corporation 2022). Measured weather data were used to compute ET over the growing season. The Yahoo! website provides numerical values for all parameters required to compute ${\mathrm{ET}}_0$ except solar radiation. Instead, it provides nonnumerical forecasts of sky cover such as “rain,” “cloudy,” or “clear.” Those descriptions were used to calculate the solar radiation in accordance with Fujimaki et al. (2014). The wind speed values provided by the Yahoo! website refer to a standard height of 10 m, so those data were converted to 2-m heights based on the approach described by Allen et al. (1998).

The feasibility of the WASH_2D model to simulate soil water flow was evaluated as follows:

The cuttings of sweet potato [Ipomoea batatas (L.), cv. Kintoki] were transplanted on June 10 at 40 cm intervals along the drip tubes. The parameter values of the stress response function [Eq. (13)] were determined to ${\psi }_{50}=-45\mathrm{cm}$, ${\psi }_{o50}=-\mathrm{1,378}\mathrm{cm}$, and $p=4.44$, from a pot experiment. The parameter values of the $k_{\mathrm{cb}}$ function [Eq. (16)] were obtained by fitting to those reported by Allen et al. (1998), assuming that ${\mathrm{ET}}_0$ during the initial, development, middle, and late stages was 2, 3, 4, and $3\mathrm{mm}{\mathrm{d}}^{-1}$, respectively (Fig. 7). We estimated the net income by setting $P_c=\$1/\mathrm{kg}\mathrm{DM}$ based on typical value of producer prices in 2020 (FAOSTAT 2021); $P_w$ was set to $0.0002/kg, similar to a case in Israel (Cornish et al. 2004); and $\epsilon$ was set to 0.003 (Siqinbatu et al. 2013).

Nitrogen-phosphorus-potassium (NPK) fertilizers were applied throughout the growing seasons in two forms: (1) a solid form in two types, $\mathrm{NPK}=8–8–8$ and $\mathrm{NPK}=14–14–14$, at total rates of 60, 35, and $55\mathrm{kg}{\mathrm{ha}}^{-1}\mathrm{N}$, P, and K, respectively; and (2) liquid fertilizer ($\mathrm{NPK}=12–5–8$) at total rates of 95, 15, and $45\mathrm{kg}{\mathrm{ha}}^{-1}\mathrm{N}$, P, and K, respectively (Laurie et al. 2012). The daily application rate of NPK liquid fertilizer was constant and equal among treatments. Plant samples were collected at 35, 53, 74, 97, 119, and 144 days after transplanting (DAT) to evaluate the plant growth in terms of leaf area index (LAI), biomass, and harvest index, defined as the ratio of dry weight (DW) of tubers (g) to the total plant dry weight (g) (Bourke 1985). Plants were harvested on November 1, 2021. Yield and its components—dry weight of fresh leaves ($\mathrm{g}{\mathrm{plant}}^{-1}$), dry weight of shriveled leaves ($\mathrm{g}{\mathrm{plant}}^{-1}$), fresh weight (FW) of tubers ($\mathrm{g}{\mathrm{plant}}^{-1}$), dry weight of dry tubers ($\mathrm{g}{\mathrm{plant}}^{-1}$), dry weight of stem ($\mathrm{g}{\mathrm{plant}}^{-1}$), and total biomass ($\mathrm{g}{\mathrm{plant}}^{-1}$)—were analyzed statistically using a randomized complete block design. The ANOVA test was conducted using SPSS version 26.0 software to estimate the significant differences among the four treatments.

The LAI, biomass, and harvest index as functions of time are shown in Fig. 8. The maximum mean values of LAI for Treatments A, B, C, and D were 2.9, 2.5, 2.6, and 2.2, respectively, which were measured at 119 DAT (October 7), after which values declined. The harvest index indicated similar values for Treatments A, B, C, D, i.e., 70%, 69%, 74%, and 73%, respectively. The results of LAI and harvest index agreed with those reported by Bourke (1985). The growth rate of biomass, which is the slope of Fig. 8(b), was stagnant from 74 DAT (August 23) to 97 DAT (September 15), with a mean value of $3.5\mathrm{g}{\mathrm{m}}^{-2}{\mathrm{day}}^{-1}$. This likely was due to lesser solar radiation and heavy rainfall events that may have caused deep nutrient leaching beyond the plant rootzone. After 97 DAT, the growth rate recovered and varied among the treatments, which may have been due to the effect of irrigation on nutrients uptake by plants. Treatment C tended to have less water applied than other treatments until August 6, after which irrigation gradually increased (Fig. 9). This may have led to less nutrient leaching in Treatment C than in other treatments, and partly offset negative effects of drought stress in the early stage. As a result, the fresh tuber yield was 19.7, 19.4, 19.7 and $18.8\mathrm{Mg}{\mathrm{ha}}^{-1}$ for Treatments A, B, C and D, respectively. The statistical analysis of yield components is presented in Table 1. There were no significant differences for whole components among the treatments ($F\mathrm{probability}>0.05$). Treatment A had a higher dry mass of leaves and LAI. Treatments A and C had slightly higher values of fresh tuber yield than the other treatments, probably due to higher nutrient uptake, which may have increased the photosynthetic rates, which enhanced the transport of carbohydrates from the leaves to the tubers (Abd El-Baky et al. 2010).

We focused on the cost of applied water to motivate farmers to save water. The main concern of farmers is not water productivity (yield per applied water), but net income (Fujimaki et al. 2014). Hence, the effectiveness of the proposed schemes in terms of net income is presented in Fig. 10. Treatment A achieved net income by 16%, 5% and 22% higher than that of Treatments B, C, and D, respectively. This was due to producing 13%, 4% and 17% higher biomass and applying 24%, 11%, and 27% less water than in Treatments B, C, and D, respectively. One of the major advantages of the three-point and two-point schemes is maximizing the farmers’ net income in each irrigation event, rather than the seasonal net income, as employed in other studies (e.g., Wang and Cai 2009). To confirm that simulated net income matched the observed value, the cumulative virtual net income for the three-point and two-point schemes is illustrated in Fig. 11(a). The total simulated net income achieved for the three-point and two-point schemes was $9,677 and $\$\mathrm{9,409}{\mathrm{ha}}^{-1}$, respectively, whereas the actual values were $9,983 and $\$\mathrm{8,368}{\mathrm{ha}}^{-1}$, respectively. The optimization gave almost the same gross net income as the actual values. The $k_i$ values decreased sharply from 6 to 1 as the crop grew [Fig. 11(b)].

The fluctuation of simulated ET was compared with the observed value measured with a weighing lysimeter (Fig. 12). The model tended to underestimate ET values, with a RMS error (RMSE) of $0.08\mathrm{mm}{\mathrm{h}}^{-1}$. The discrepancy may have been caused partly by disturbance of weight measurement by wind over $3\mathrm{m}{\mathrm{s}}^{-1}$ [Fig. 5(b)], as reported by Schrader et al. (2013) and Lorite et al. (2012), particularly because the level of lysimeter surface was slightly higher than the surrounding field, by 3–5 cm. This also may have caused the drip tubes to touch the surface of the lysimeter, affecting the weight measurement due to tension change caused by temperature fluctuation.

The measured and simulated $\theta$ for the period July 15–August 14 is illustrated in Fig. 13. We showed the response of $\theta$ to both irrigation and rainfall events in Treatment A for top and deep soil layers below the plant. The model accurately simulated $\theta$, with a RMSE of 0.015 and $0.005{\mathrm{cm}}^3{\mathrm{cm}}^{-3}$ compared with the measured values at 5 and 45 cm below an emitter, respectively. These results agreed with those reported by Fujimaki et al. (2014) and Abd El Baki et al. (2020) for Tottori sand and by Zhu et al. (2018) for loamy soil with the same bulk density as Tottori sand.

To check the accuracy of rain forecasts and WF data, which greatly affected the determination of $W$, we compared daily forecasted rainfall and daily forecasted ${\mathrm{ET}}_0$ with the daily observed values (Fig. 14). As WFs were downloaded in the morning of each irrigation day at 09:00 until the day end at 23:59, the term “daily” means from 9:00 to 24:00. Total measured and forecasted rainfall across the growing season were similar, 598 and 648 mm, respectively. In contrast, the forecasted amount of each rainfall event often deviated greatly from the measured amounts. The total number of measured and forecasted rainfall events were 39 and 52 over 120 days, respectively. Setting effective rainfall to $25\mathrm{mm}{\mathrm{d}}^{-1}$, the RMSE between measured and forecasted values was $5.6\mathrm{mm}{\mathrm{d}}^{-1}$, in which 17% and 15% of rain events exceeded this setting (25 mm d⁻¹), respectively. This is consistent with the findings of Shrestha et al. (2013), who found that the RMSE for daily forecasted rainfall ranged from 4.67 to $14.64\mathrm{mm}{\mathrm{day}}^{-1}$. Relative RMSE, which is the ratio of the average value to the RMSE, was 0.77. Those underestimations and overestimations of rain may have caused overestimation and underestimation of $W$, respectively. On the other hand, daily ${\mathrm{ET}}_0$ calculated using hourly forecasted values generally was overestimated, with a RMSE of $0.7\mathrm{mm}{\mathrm{d}}^{-1}$ compared with ${\mathrm{ET}}_0$ calculated using the measured values. Relative RMSE was 0.81. Those overestimations may have caused overestimations of $W$. This likely was due to overestimation of wind speed, with a RMSE of $0.9\mathrm{m}{\mathrm{s}}^{-1}$, and underestimation of relative humidity, with a relative RMSE of 1.17. These results agreed with the possible reasons reported by Lorite et al. (2015), who observed that high wind speed and close proximity to the sea caused larger errors in ${\mathrm{ET}}_0$ estimation, and by Xiong et al. (2016), who indicated that relative humidity is an important factor to estimate ${\mathrm{ET}}_0$ in humid regions. An interval longer than 2 days in this study is appropriate in finer-textured soils, but accuracy at longer forecast horizons would be even lower than that at 2 days. Gowing and Ejieji (2001) observed that using short-term weather forecasts (up to 5 days ahead) is useful to save both cost and irrigation depth, and thereby increases net income in wet seasons, whereas it led to improved water-use efficiency of limited water supply in dry seasons. Cai et al. (2011) reported that even an imperfect 1-week forecast still was valuable in terms of both profit gain and water saving. In parallel to the present study, Abd El Baki and Fujimaki (2021a) observed that using 1-day weather forecasts led to more-accurate estimates of ${\mathrm{ET}}_0$, with a RMSE $0.5\mathrm{mm}{\mathrm{d}}^{-1}$ for a sandy filed. We expect that the accuracy of numerical weather forecast will continue improving with the use of faster supercomputers using finer grids and improved submodels. Despite the current accuracy of WFs, they may still have benefits for determining irrigation depths.

The combination of numerical simulation, volumetric water pricing, and weather forecasts could be useful in making optimal irrigation decisions. In this regard, Fig. 15 shows an example of determining $W$ using the three-point and two-point schemes on July 15. Both gave similar recommended values of $W$, as 0.69 and 0.71 cm at a maximum value of $I_n$, respectively. We found that both schemes gave similar values of recommended $W$ only when irrigation was not performed in the previous event. In other cases, the two-point scheme often gave a higher $W$, partly due to inappropriate setting of $W_1$ (0.55 cm in Fig. 15), which is the main drawback of the scheme (Abd El Baki and Fujimaki 2021). Scheme B may give higher net income than other treatments only when the relationship between $W$ and cumulative transpiration can be described by two linear segments (trapezoidal). Our results indicate that the assumption invoked in Scheme B generally was inappropriate in the experimental condition. On the same day, the refilling scheme suggested refilling water into the root zone with $W=0.7\mathrm{cm}$ for the next 2 days. The drawback of the refilling scheme is that it neglects the near-future rainfall events [Fig. 16(a)]. For example, the refilling scheme suggested $W=1.09\mathrm{cm}$ to return ${\theta }_x$ to ${\theta }_{\mathrm{fc}}$ on September 17. Irrigation began at 09:00, and rainfall started at 17:00. Such a case occurred only once across the entire season. This may be another reason why Treatment C gave higher values of $I_n$ than Treatment B. The automated irrigation scheme also tended to neglect the near-forecast rainfall events [Fig. 16(b)]. On August 7, water was applied automatically with $W=0.64\mathrm{cm}$ at 13:00 when the trigger suction value approached $-40\mathrm{cm}$. Soon after that, 2.5 cm of rainfall occurred at 17:00. This overirrigation repeated six times, which may be a reason why Treatment D applied more water and leached nutrients beyond the plant rootzone than did the other schemes. Automated irrigation systems also are sensitive to the selection of optimum trigger suction value, which requires additional field studies (Migliaccio et al. 2010). If one tensiometer was set under a clogged emitter, the observed suction value could result in the application of more water. Thus, the proposed simulation schemes, as an innovative technology considering weather forecasts, would be more affordable to farmers to maximize their net income or, at the very least, provide promising trends for optimizing water use. The proposed schemes are a type of appropriate and/or intermediate technology (Schumacher 1999) that farmers can adopt easily after being trained using the most effective agricultural extension methods, such as demonstration, farmer-to-farmer, and household extension methods.