# Sprinkler Drip Infiltration Quality Prediction for Moisture Space Distribution Using RSAE-NPSO

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{w}) at 224.8 KPa, irrigation duration time (I

_{d}) at 2.68 h, flow discharge amount (F

_{q}) at 1682.5 L/h, solar radiation (S

_{r}) at 17.2 MJ/m

^{2}, average wind speed (A

_{w}) at 1.18 m/s, average air temperature (A

_{t}) at 22.8 °C, and average air relative humidity (A

_{h}) at 72.8%, as well as the key variables of the irrigation environment, including the soil bulk density (S

_{b}) at 1.68 g/cm

^{3}, soil porosity (S

_{p}) at 68.7%, organic carbon ratio (O

_{c}) at 63.5%, solute transportation coefficient (S

_{t}) at 4.86 × 10

^{−6}, evapotranspiration rate (E

_{v}) at 33.8 mm/h, soil saturated hydraulic conductivity rate (S

_{s}) at 4.82 cm/s, soil salinity concentration (S

_{c}) at 0.46%, saturated water content (S

_{w}) at 0.36%, and wind direction W

_{d}in the north–northwest direction (error tolerance = ±5%, the same as follows), an optimal data set of SDIQ indices can be ensured, as shown by the exponential entropy of the soil infiltration pressure (ESIP) at 566.58, probability of moisture diffusivity (PMD) at 96.258, probabilistic density of infiltration effectiveness (PDIE) at 98.224, modulus of surface radial runoff (MSRR) at 411.25, infiltration gradient vector (IGV) at [422.5,654.12], and normalized infiltration probabilistic coefficient (NIPC) at 95.442. The quality inspection of the SDIQ prediction process shows that a high agreement between the predicted and actual measured SDIQ indices is achieved. RSAE-NPSO has extraordinary predictive capability and enables much better performance than the other prediction methods in terms of accuracy, stability, and efficiency. This novel prediction method can be used to ensure the infiltration uniformity of the moisture space distribution in sprinkler drip irrigation. It facilitates productive SDIQ management for precision soil irrigation and agricultural crop production.

## 1. Introduction

## 2. Methods and Materials

#### 2.1. SDIQ Indices of Moisture Space Distribution

_{o}, the maximum infiltration rate Z

_{o}, the exponential entropy of soil infiltration pressure N

_{f}

^{k}(ESIP), and the soil volume subjected to infiltration effectiveness ΔV, which is expressed as follows:

_{o}determines the durability of the moisture concentration, which ensures a uniform distribution of water and infiltration effectiveness over the total area of the irrigated soil field. The drip irrigation determining the infiltration quality can be calibrated using the maximum orthogonal infiltration rate, σ. This is comparable to the depth of the infiltrated soil level, so that the infiltration quality determines the influence of the moisture content and soil infiltration on the survival probability. The mathematical equation determining the SDIQ of infield soil is demonstrated as follows [22,23]:

_{xψ}, could be defined as the product of the elementary water/soil mass ratio measured in the targeted soil specimen, Δψ(N

_{f}

^{k})·r

_{bx}, and the volume determined by the monitored field area, Δx. The drip infiltration depth at which the maximum infiltration rate occurs, Z

_{xψ}, is as follows:

_{f}, and the drip infiltration quality index, P, can be determined. The coefficients c

_{0}and c

_{1}characterize the moisture regression of the infiltration quality of infield soil, providing a clear description of P depending on N

_{f}, which is denoted as the probability of moisture diffusivity (PMD) [28,29,30]:

_{f}is the random variable of the infiltration depth based on its normal distribution, and the drip infiltration quality, P, is the mathematical expectation of N

_{f}, which is expressed as P = (N

_{fx}, N

_{fy})

^{T}. Here, Σ is the covariance matrix of N

_{f}[31]; therefore, the computational function of the normal distribution covariance matrix is expressed as:

_{fx}= N

_{fy}= P, σ(N

_{fx}, N

_{fx}) = σ(N

_{fy}, N

_{fy}) = σ

_{2}

^{2}, σ(N

_{fx}, N

_{fy}) = σ(N

_{fy}, N

_{fx}) = 0. To simplify the predictive analysis, a fast Fourier transformation (mathematical) is applied to determine the spectrum of the probabilistic density of the drip infiltration depth, which can be updated with the probabilistic frequency of the controlled parameter sampling. This yields the following mathematical expression of infiltration quality [32,33]:

_{d}(N

_{f}

^{k}) denotes the probabilistic density of infiltration effectiveness (PDIE) and s(N

_{fx}N

_{fy}) denotes the moisture infiltration gradient vector (IGV). Thus, the modulus of surface radial runoff (MSRR) can be given as [34,35]:

_{f}

^{k}. For any k ∈ K, there is a closed interval [N

_{f_lower}

^{k}, N

_{f_upper}

^{k}] that satisfies N

_{f_lower}

^{k}< N

_{f}

^{k}< N

_{f_upper}

^{k}. N

_{f_lower}

^{k}and N

_{f_upper}

^{k}stand for the envelopes of the probability family of infiltration effectiveness. The drip infiltration exponents of a soil field at time t can be characterized by [36]:

_{f}

^{k};θ) stands for the nonlinear drift term of the overall infiltration trend, μ is the drift coefficient, θ is a vector with unknown parameters, δ is a constant moisture diffusion coefficient, and B

_{H}(N

_{f}

^{k}) is f

_{Bm}. A generalized drip infiltration model describing the moisture space distribution can be obtained by replacing μ and δ by the time functions φ(x) and ω(x), respectively. Thereafter, the interference term B

_{H}(N

_{f}

^{k}) is replaced by ψ(x), meaning Equation (9) can be rewritten as [37,38]:

_{f}

^{t}) is a time-dependent function used to describe the overall trend of the probability density of the moisture content diffusivity, equivalent to the normalized infiltration probabilistic coefficient N

_{fx}

^{i}

_{qua}(NIPC), and which can be described by the linear, power rate, and exponential drift functions [37,38]. Based on this arrangement, the mathematical properties of SDIQ can be described.

#### 2.2. Working Mechanism of Regularized Sparse Autoencoder

_{d}[(N

_{f}

_{x}

^{k},N

_{f}

_{y}

^{k});μ,σ

_{2}] [40,41].

_{RSAE}denotes the weight matrix of RSAE. Here, σ

_{r}(z) is a rectified linear unit employed as an activation function in the RSAE–decoder, which guarantees more efficient training during network prediction than other traditional functions. Thus, the RSAE–decoder reconstructs $\left\{{\stackrel{\u2322}{I}}_{d}\left[({N}_{fx}{}^{k},{N}_{fx}{}^{k});\mu ,{\sigma}^{2}\right]\right\}$ using a mapping function g

_{RSAE}:

_{RSAE}

_{1}or W

_{RSAE}

_{2}denotes the weight matrix of the RSAE, ensuring that its computational performance is maintained in a highly efficient state. Here, λ

_{xkyk}denotes the regular coefficient of the RSAE corresponding to the meshed grid coordinated by (N

_{fx}

^{k}, N

_{fy}

^{k}

_{)}. As W

_{RSAE}

_{1}and W

_{RSAE}

_{2}are replaced by W and W

^{T}, the gradient of J

_{RSAE}with respect to W is calculated as follows:

^{T}

_{σr}(W∙(N

_{f}

_{x}

^{k},N

_{fy}

^{k})) − (N

_{f}

_{x}

^{k},N

_{fy}

^{k}), sgn denotes the sign function of σ

_{r}(W∙(N

_{f}

_{x}

^{k}, N

_{fy}

^{k})), σ

_{r}′ denotes the derivative function of the rectified linear unit, σ

_{r}, and (N

_{f}

_{x}

^{k}, N

_{fy}

^{k}) denotes the matrix form of (N

_{f}

_{x}

^{k}) and (N

_{fy}

^{k}). The reiterative update process of W

^{(i+1,j+1)}can be written as:

_{RSAE}

_{1}and W

_{RSAE}

_{2}, respectively; H

^{(i,j)}denotes the inverse of the Hessian matrix [42,43], and η denotes the step size of the SDIQ update process. The RSAE is trained an adequate number of times, allowing the drip infiltration properties of the soil field to be accurately calculated and inputted into the NPSO algorithm for the highly efficient calculation of SDIQ indices, making NPSO an universal and reliable prediction network for infiltration qualities [44,45].

#### 2.3. Working Mechanism of NPSO Incorporated with RSAE

## 3. Results and Discussion

#### 3.1. Experiment Preparation

^{2}under calm conditions. Environmental parameters such as the average air temperature, average relative humidity, and solar radiation were collected every 10 min. Moreover, the local air temperature was measured using an HW-F7 thermometer (LD Products Inc. Long Beach City, CA, USA), the air relative humidity was determined using Testo 610 hygrometers (Testo Pty Ltd., Croydon South, Victoria, Australia), the solar radiation value obtained from an LI-1500 irradiance radiometer (HUATEC., Beijing, China), and the wind speed was measured using RK100-02 RS485 wind speed measurement units (RIKA Sensor Inc., Changsha, China). The measurements of soil moisture evaporation were carried out using 40 sets of microlysimeters, which were equally distributed throughout whole field area. A moderate flow pressure of 320 kPa was applied to ensure the infiltrated soil depths were measured easily. A pair of sprinkler heads with a vertical height of 80 cm were placed at the linear interval of 2.5 m to provide a uniformly distributed soil wetting pattern and to guarantee a wetting diameter overlap of 55–60% at all times.

#### 3.2. Experimental Data Measuring

_{w}/KPa), irrigation duration time (I

_{d}/h), flow discharge amount (F

_{q}/L/h), solar radiation (S

_{r}/MJ/m

^{2}), average wind speed (A

_{w}/m/s), average air temperature (A

_{t}/°C), average air relative humidity (A

_{h}/%), soil bulk density (S

_{b}/g/cm

^{3}), soil porosity (S

_{p}/%), organic carbon ratio (O

_{c}/%), solute transportation coefficient (S

_{t}/×10

^{−6}), evapotranspiration rate (E

_{v}/mm/h), soil saturated hydraulic conductivity rate (S

_{s}/cm/s), soil salinity concentration (S

_{c}/%), saturated water content (S

_{w}/%), and wind direction (W

_{d}) [49]. Because the parametric data in Table 3 conformed to the stochastic normal distribution of experimental variables, their equalized intervals were applied to cover 97% of all possible values, with the lowest value denoted as “1” and the highest as “10”. Based on these equalized intervals, other parametric grouping stages were achieved sequentially. Therefore, the value ranges can be defined to ensure the high repeatability of the testing conditions, providing accurate data ranges and facilitating subsequent quantified computation in RSAE-NPSO. Table 4, Table 5, Table 6, Table 7 and Table 8 list the representative parametric sets of sprinkler drip irrigation data for loam soil (A), sandy soil (B), chernozem soil (C), saline–alkali soil (D), and clay soil (E), based on orthogonal test arrangements. The moisture content on each soil layer was obtained from 100 × 10 = 1000 grid positions, covering 94–96% of the overall irrigated zone.

_{w}and I

_{d}were determined using the BF1000-3EB-X full-bridge moisture content gauges buried on each soil subsurface layer. F

_{q}, S

_{r}, and A

_{w}were obtained by using a DN6-DN300 metering system and calibrated before each test. A

_{t}and A

_{h}were monitored using a Coriolis mass flow meter and transmitted online via signal acquisition using a quadratic encoder. Concurrently, S

_{b}, S

_{p}, O

_{c}, S

_{t}, E

_{v}, S

_{s}, S

_{c}, S

_{w}, and W

_{d}were determined by employing the prearranged parametric data [50]. It is worth noting that a computer-controlled data acquisition system was applied to collect the irrigation data [51,52,53,54].

#### 3.3. Intelligent Prediction of SDIQ Indices

_{w}, A

_{t}, and F

_{q}should be focused on to determine S

_{b}, S

_{p}, O

_{c}, S

_{t}, E

_{v}, S

_{s}, S

_{c}, S

_{w}, and W

_{d}, because they have a remarkable influence on ESIP, PDIE, and PMD, presenting intensified accuracy improvements of up to 33–35% of NIPC after one complete epoch of RSAE-NPSO prediction. The irrigation parameters of A

_{w}and A

_{t}cause 16–18% reductions in the stability of PDIE and IGV, owing to increased information entropy or measurement chaos. The change tendencies of A

_{t}, A

_{h}, and S

_{b}indicate that IGV and MSRR decrease by 15–18%, depending on F

_{q}, S

_{r}, A

_{w}, and their accompanying variation. In contrast, PDIE and IGV were found to be most sensitive to the correlation mechanism of P

_{w}, I

_{d}, F

_{q}, S

_{r}, A

_{w}, and A

_{t}, owing to the high drip infiltration rate. PMD and NIPC are sensitive to S

_{p}, O

_{c}, S

_{t}, E

_{v}, S

_{s}, S

_{c}, and S

_{w}, owing to the corresponding variation in drip infiltration conditions. It can also be found that when a set of irrigation parameters such as P

_{w}= 224.8 KPa, I

_{d}= 2.68 h, F

_{q}= 1682.5 L/h, S

_{r}= 17.2 MJ/m

^{2}, A

_{w}= 1.18 m/s, A

_{t}= 22.8 °C, and A

_{h}= 72.8%; and key variables of the irrigation environment, including S

_{b}= 1.68 g/cm

^{3}, S

_{p}= 68.7%, O

_{c}= 63.5%, S

_{t}= 4.86 × 10

^{−6}, E

_{v}= 33.8 mm/h, S

_{s}= 4.82 cm/s, S

_{c}= 0.46%, S

_{w}= 0.36%, and W

_{d}(north–northwest; error tolerance = ±5%, the same as follows) were prepared, an optimal data set of ESIP, PDIE, IGV, MSRR, PMD, and NIPC could be obtained for the purpose of improving the performance quality of the sprinkler drip irrigation approach [55].

#### 3.4. Significant Analysis Using F-Ratio Tests

_{j}levels for the jth index, where each value level is tested c times; n denotes the total number of orthogonal irrigation tests; the obtained values for SDIQ verification cases are marked as x

_{1}, x

_{2}, …, x

_{n}, and K

_{1}, K

_{2}, …, K

_{mj}denote the sum of levels of the jth index with respect to the testing time [56]:

_{e}, Df

_{T}, Q, F

_{j}, and P, whereby ** represents a highly significant influence, * denotes a significant influence, and O indicates no influence. Thus, the predictive quality levels of SDIQ indices were calibrated in a clear mathematical manner.

_{e}, Df

_{T}, Q, F

_{j}, and P sourced from the variance F-ratio tests showed the slightly lower heritability values than those corresponding to the original SDIQ index data. The normality and homogeneity variability studies in this research were likely proposed to satisfy the SDIQ index prediction conditions.

_{w}, I

_{d}, and F

_{q}on IGV and MSRR are remarkable, while S

_{r}, A

_{w}, A

_{t}, and A

_{h}have larger impacts on PMD and NIPC than on other SDIQ indices. Simultaneously, the influence of P

_{w}, S

_{p}, O

_{c}, and S

_{c}on ESIP, IGV, and MSRR cannot be ignored. Further analysis shows that A

_{h}, S

_{b}, and S

_{p}have the most influence on PDIE and IGV, whereas P

_{w}, I

_{d}, A

_{t}, and S

_{p}are crucial in terms of the infiltration quality deviations. After identifying the mathematical effects of these parameters, the optimized results of the RSAE-NPSO system can be anticipated by controlling the corresponding factors. A comparison of IGV, MSRR, PMD, and NIPC values for the given soil samples with and without using the optimized irrigation parameters demonstrated the remarkable improvements achieved by the RSAE-NPSO prediction approach.

#### 3.5. Calibration Coefficients of Prediction Error

_{findexij}and $\overline{Nf}{}_{indexij}$ are the actual and averaged SDIQ indices, and [N

_{findexij}]

_{measurement}and [N

_{findexij}]

_{prediction}denote the data limits of actual measured and predicted infiltration quality indices, respectively.

_{t}and S

_{r}are highly focused. RSAE-NPSO has strong processing capability for data uncertainty in the determination of MSRR and NIPC values. It uses the conditional probability density to express the correlation mechanism among various influential factors, which makes the learning and reasoning of SDIQ indices under different limited, incomplete, and uncertain conditions possible. Even when the experimental SDIQ data are not rich, they can be quantified using the expert estimations and statistical corrections or by utilizing the expectation maximization (EM). Therefore, RSAE-NPSO combines adaptive prediction and extension reasoning to ensure the reliability and stability of SDIQ indices.

_{h}, S

_{b}, S

_{p}, and O

_{c}when the PMD and NIPC are considered. This is particularly obvious in the irrigation cases when saline–alkali soil and clay soil are considered. The MAPE provides a critical criterion for A

_{w}, A

_{t}, A

_{h}, S

_{b}, and S

_{p}; thus, the IGV effectively assesses the multi-source information expression and data fusion of the drip infiltration properties. This shows that it is suitable for the heterogeneous infiltration knowledge representation and reasoning under uncertain irrigation conditions. ESIP and PDIE provide reliable mathematical models for uncertainty knowledge representation and logical reasoning in the case of SDIQ determination. The predictive reasoning principle of RSAE-NPSO is essentially a set of probability calculations considering the influences of mean SDIQ percentage errors.

_{d}, S

_{p}, and W

_{d}are considered. Its value variation causes a corresponding fluctuation in the predictive precision of PMD and NIPC when S

_{r}, A

_{w}, and A

_{t}are concerned. RSAE-NPSO has causal and probabilistic semantics, which facilitate the combination of prior knowledge and a probability distribution of infiltrated moisture in the experimental cases involving chernozem and clay soil. Furthermore, it effectively avoids overfitting and ensures the robustness of the prediction of PDIE and IGV, considering the fluctuations in A

_{h}, S

_{b}, S

_{p}, O

_{c}, S

_{t}, and E

_{v}values.

_{r}, A

_{w}, A

_{t}, A

_{h}, S

_{b}, S

_{p}, and O

_{c}are considered, and it is remarkably affected by the water mass redistribution also. The quantitative knowledge regarding moisture infiltration includes edge and conditional probabilities, which should be fully considered in the computation of MSRR and PMD. Because the quantitative relationships (probabilities) are mainly derived from the statistical calculations, the professional literature, and expert experience, the RSAE-NPSO system has a conditional independence. It only needs to consider the finite variables associated with the S

_{b}, S

_{p}, O

_{c}, S

_{t}, E

_{v}, S

_{s}, S

_{c}, S

_{w}, and W

_{d}variables, making the SDIQ determination approach a feasible solution in many complex infiltration prediction cases.

_{c}, S

_{t}, and E

_{v}on sandy and chernozem soils. This highly depends on the value ranges of MSRR, PMD, and NIPC when S

_{b}, S

_{p}, O

_{c}, S

_{t}, and W

_{d}are observed closely. This refers to the structural relationship with the prediction network and expresses the mathematical correlations between PDIE, MSRR, and PMD. The qualitative relationships between all SDIQ indices are mainly derived from expert experience, the professional literature, and statistical learning. Using this coefficient, RSAE-NPSO can be used to deal with different prediction errors.

_{w}, S

_{b}, S

_{p}, E

_{v}, and S

_{s}, and that its operational mechanism is remarkably affected by the accuracy of the P

_{w}, O

_{c}, S

_{t}, E

_{v}

_{,}S

_{s}, S

_{c}, and S

_{r}values, particularly in the irrigation of clay and saline–alkali soil. RSAE-NPSO itself is a calculation system that visualizes the probabilistic representation and reasoning computation processes to determine the mathematical and conditional correlations between various neuron variables. IGV, MSRR, and NIPC describe the interrelationships among the drip infiltration data via graphic representations of relative exponent error, with clear semantics, making this approach easy to understand. The mathematical description of Rob makes it easy to assess the consistency and integrity of SDIQ indices and to demonstrate the RSAE-NPSO module in the investigation of sprinkler drip infiltration.

_{w}at 224.8 KPa, I

_{d}at 2.68 h, F

_{q}at 1682.5 L/h, S

_{r}at 17.2 MJ/m

^{2}, A

_{w}at 1.18 m/s, A

_{t}at 22.8 °C, and A

_{h}at 72.8%, as well as key irrigation environmental variables of S

_{b}at 1.68 g/cm

^{3}, S

_{p}at 68.7%, O

_{c}at 63.5%, S

_{t}at 4.86 × 10

^{−6}, E

_{v}at 33.8 mm/h, S

_{s}at 4.82 cm/s, S

_{c}at 0.46%, S

_{w}at 0.36%, and W

_{d}as north–northwest, the optimal SDIQ indices of ESIP at 566.58, PMD at 96.258, PDIE at 98.224, MSRR at 411.25, IGV at [422.5,654.12], and NIPC at 95.442 could be achieved. This novel prediction method contributes greatly to performance improvements in soil infiltration using sprinkler drip irrigation and agricultural crop production in practice.

## 4. Conclusions

_{w}, I

_{d}, F

_{q}, S

_{r}, A

_{w}, A

_{t}, A

_{h}) and key variables of the irrigation environment (S

_{b}, S

_{p}, O

_{c}, S

_{t}, E

_{v}, S

_{s}, S

_{c}, S

_{w}, W

_{d}). This research makes the following theoretical and technological contributions. It presents the accurate prediction of SDIQ indices for moisture space distribution analyses. Based on the comparison of these predicted indices with actual measurement data, we show that the average relative error between them is within ±5%, which clearly verifies the accuracy and correctness of the SDIQ estimation process. A novel RSAE-NPSO system is designed and subsequently its working mechanism and constructive effects on the SDIQ predictive computation are examined. A complete set of SDIQ indices are covered, including ESIP, PDIE, IGV, MSRR, PMD, and NIPC. A quality inspection is performed to assess the predicted SDIQ results from innovative perspectives, meaning the prediction accuracy and calculation stability of the RSAE-NPSO system are confirmed. This research compares the SDIQ indices obtained from the RSAE-NPSO system with those of other typical prediction methods under identical test conditions, confirming the extraordinary accuracy, generalizability, and logical reliability of this newly proposed prediction method. Therefore, the irrigation efficiency and infiltration quality of soil fields can be monitored instantaneously and planned precisely, while the SDIQ prediction also facilitates remarkable improvements in agricultural crop production under complex working conditions.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**The kernel density figure describing the drip infiltration on the example soil surface. Here, the key distributive area with high SDIQ values is identified by white circles.

**Figure 3.**Data illustration of drip infiltration considering the moisture space distribution labeled as sampled layer A.

**Figure 4.**The loam soil specimen and its corresponding SDIQ index values obtained from the sampled layer A.

**Figure 5.**The loam soil specimen and its corresponding SDIQ index values obtained from the sampled layer B.

**Figure 6.**The loam soil specimen and its corresponding SDIQ index values obtained from the sampled layer C.

**Figure 7.**The loam soil specimen and its corresponding SDIQ index values obtained from the sampled layer D.

**Figure 8.**The loam soil specimen and its corresponding SDIQ index values obtained from the sampled layer E.

**Figure 9.**Demonstration of the actual and predicted ESIP values of chernozem soil on tests A_1 to E_10, implemented on layers A–E, respectively.

**Figure 10.**Demonstration of the actual and predicted PMD values of chernozem soil on tests A_1 to E_10, implemented on layers A–E, respectively.

**Figure 11.**Demonstration of the actual and predicted PDIE values of chernozem soil on tests A_1 to E_10, implemented on layers A–E, respectively.

**Figure 12.**Demonstration of the actual and predicted MSRR values of chernozem soils on tests A_1 to E_10, implemented on layers A–E, respectively.

**Figure 13.**Demonstration of the actual and predicted IGV values of chernozem soils on tests A_1 to E_10, implemented on layers A–E, respectively.

**Figure 14.**Demonstration of the actual and predicted NIPC values of chernozem soils on tests A_1 to E_10, implemented on layers A–E, respectively.

**Figure 15.**Accuracy comparison between the actual measured and predicted SDIQ indices considering soil types.

Condition | Initial Positions | Initial Velocities | pBest_{i}^{k} | Level |
---|---|---|---|---|

Initial particle | $\{\begin{array}{c}{N}_{fx}{}_{i=1}^{k=1}={\left[\mathsf{\Delta}\alpha {N}_{fx},\mathsf{\Delta}{N}_{fx},\mathsf{\Delta}{N}_{fy},\mathsf{\Delta}{N}_{fz}\right]}^{T}\\ {N}_{fx}{}_{i=2}^{k=1}=rand\xb7{\left[\mathsf{\Delta}\alpha {N}_{fx},\mathsf{\Delta}{N}_{fx},\mathsf{\Delta}{N}_{fy},\mathsf{\Delta}{N}_{fz}\right]}^{T}\\ \vdots \\ {N}_{fx}{}_{i={n}_{p}}^{k=1}=rand\xb7{\left[\mathsf{\Delta}\alpha {N}_{fx},\mathsf{\Delta}{N}_{fx},\mathsf{\Delta}{N}_{fy},\mathsf{\Delta}{N}_{fz}\right]}^{T}\end{array}$ | $\{\begin{array}{c}{v}_{i=1}^{k=1}={\left[{v}_{\mathsf{\Delta}\alpha fx},{v}_{\mathsf{\Delta}fx},{v}_{\mathsf{\Delta}fy},{v}_{\mathsf{\Delta}fz}\right]}^{T}\\ {v}_{i=2}^{k=1}=rand\xb7{\left[{v}_{\mathsf{\Delta}\alpha fx},{v}_{\mathsf{\Delta}fx},{v}_{\mathsf{\Delta}fy},{v}_{\mathsf{\Delta}fz}\right]}^{T}\\ \vdots \\ {v}_{i={n}_{p}}^{k=1}=rand\xb7{\left[{v}_{\mathsf{\Delta}\alpha fx},{v}_{\mathsf{\Delta}fx},{v}_{\mathsf{\Delta}fy},{v}_{\mathsf{\Delta}fz}\right]}^{T}\end{array}$ | $\begin{array}{l}pBes{t}_{3,i}^{k}=\\ \mathrm{min}(pBes{t}_{i-1}^{k},pBes{t}_{i}^{k},pBes{t}_{3,i+1}^{k})\end{array}$ | Level 1 (Measurement Level) |

f_{objki}(X_{i}) | $\begin{array}{c}{N}_{fx}{}_{i=1}^{k}\Rightarrow {f}_{obj}^{k,i=1}\Rightarrow \{\begin{array}{c}pBes{t}_{i=1}^{k}=\mathrm{min}({f}_{obj}^{1,i=1},{f}_{obj}^{2,i=1},{f}_{obj}^{3,i=1},\dots ,{f}_{obj}^{k,i=1})\\ pBes{t}_{3,i=1}^{k}=\mathrm{min}(pBes{t}_{i=np}^{k},pBes{t}_{i=1}^{k},pBes{t}_{i=2}^{k})\end{array}\\ {N}_{fx}{}_{i=2}^{k}\Rightarrow {f}_{obj}^{k,i=2}\Rightarrow \{\begin{array}{c}pBes{t}_{i=2}^{k}=\mathrm{min}({f}_{obj}^{1,i=2},{f}_{obj}^{2,i=2},{f}_{obj}^{3,i=2},\dots ,{f}_{obj}^{k,i=2})\\ pBes{t}_{3,i=2}^{k}=\mathrm{min}(pBes{t}_{i=1}^{k},pBes{t}_{i=2}^{k},pBes{t}_{i=3}^{k})\end{array}\\ {N}_{fx}{}_{i=3}^{k}\Rightarrow {f}_{obj}^{k,i=3}\Rightarrow \{\begin{array}{c}pBes{t}_{i=3}^{k}=\mathrm{min}({f}_{obj}^{1,i=3},{f}_{obj}^{2,i=3},{f}_{obj}^{3,i=3},\dots ,{f}_{obj}^{k,i=3})\\ pBes{t}_{3,i=2}^{k}=\mathrm{min}(pBes{t}_{i=2}^{k},pBes{t}_{i=3}^{k},pBes{t}_{i=4}^{k})\end{array}\\ \begin{array}{c}\vdots \\ {N}_{fx}{}_{i=np}^{k}\Rightarrow {f}_{obj}^{k,i=np}\Rightarrow \{\begin{array}{c}pBes{t}_{i=np}^{k}=\mathrm{min}({f}_{obj}^{1,i=np},{f}_{obj}^{2,i=np},{f}_{obj}^{3,i=np},\dots ,{f}_{obj}^{k,i=np})\\ pBes{t}_{3,i=np}^{k}=\mathrm{min}(pBes{t}_{i=np-1}^{k},pBes{t}_{i=np}^{k},pBes{t}_{i=np+1}^{k})\end{array}\end{array}\end{array}$ | $\begin{array}{l}{v}_{i}^{k+1}=\omega {v}_{i}^{k}+{c}_{1}{r}_{1}(pbes{t}_{i}-{x}_{i}^{k})\\ +{c}_{2}{r}_{2}(gbes{t}_{i}-{x}_{i}^{k})\end{array}$${N}_{fx}{}_{i}^{k+1}={N}_{fx}{}_{i}^{k}+{v}_{i}^{k+1}$ ${N}_{fxi}^{k}={\left(\mathsf{\Delta}\alpha {N}_{fxi}^{k},\mathsf{\Delta}{N}_{fxi}^{k},\mathsf{\Delta}{N}_{fyi}^{k},\mathsf{\Delta}{N}_{fzi}^{k}\right)}^{T}$ | Level 2 (Calculation Level) | |

Objective function (Quasi-Monte Carlo simulation) | ${f}_{obj}=\mathrm{min}\left(\begin{array}{l}\begin{array}{c}\underset{\_}{{\epsilon}_{1}{({f}_{\gamma o}({N}_{fx}{{}^{k}}_{i})-{\gamma}_{o})}^{2}+{\epsilon}_{2}{({f}_{\phi}({N}_{fx}{{}^{k}}_{i})-\phi )}^{2}+{\epsilon}_{3}{({f}_{r}({N}_{fx}{{}^{k}}_{i})-{r}_{c})}^{2}}\\ \Downarrow \\ \begin{array}{cc}Parametric& Fitness\end{array}\end{array}+\\ \begin{array}{c}\underset{\_}{{\epsilon}_{4}{\displaystyle \sum _{jj=1}^{{n}_{core}}{f}_{dist}(jj)+{\epsilon}_{5}{\displaystyle \sum _{jj={n}_{core}+1}^{{n}_{core}}{f}_{dist}(jj)}}}\\ \Downarrow \\ \begin{array}{cc}\mathrm{Pr}ofile& Fitness\end{array}\end{array}+\begin{array}{c}\underset{\_}{u({N}_{fx}{{}^{k}}_{i})}\\ \Downarrow \\ \begin{array}{cc}Constra\mathrm{int}& Condition\end{array}\end{array}\end{array}\right)$ Above, ε _{1}, ε_{2}, ε_{3}, ε_{4}, and ε_{5} are the weight coefficients meeting different irrigation accuracy requirement of γ_{0}, r_{c}, is rake profile, and φ is rear profile. | Level 3 (Prediction level) | ||

Penalty function: | $\overrightarrow{{S}_{j+1}{}^{i}}=\overrightarrow{{S}_{j}{}^{i}}+{c}_{1}{g}_{1}({S}_{lb}{}^{i}-{S}_{j}{}^{i})+{c}_{2}{g}_{2}({S}_{gb}{}^{i}-{S}_{j}{}^{i});{x}_{j}{}^{i+1}={x}_{j}{}^{i}+{S}_{j}{}^{i}$; ${S}_{i}{}^{k+1}=\omega \xb7{S}_{i}{}^{k+1}+{\lambda}_{1}\xb7rand\xb7(pBes{t}_{i}^{k}-{N}_{fx}{}_{i}^{k})+{\lambda}_{2}\xb7rand\xb7(gBes{t}_{}^{k}-{N}_{fx}{}_{i}^{k});{N}_{fx}{{}_{i}}^{k+1}={N}_{fx}{{}_{i}}^{k+1}+{S}_{i}{}^{k}$ N _{fx}^{i} is the inertia weight factor. c_{1} and c_{2} denote the local learning factor and the global learning factor, respectively. r_{1} and r_{2} are the random numbers in the range of (0, 1). v^{k}_{i} and x^{k}_{i} denote the position and velocity vector of particle i at the k^{th} iteration, respectively. |

_{i}

^{k}is the velocity of particle N

_{fx}

^{i}after kth iterations; N

_{fxi}

^{k}is the position of particle N

_{fx}

^{i}after kth iterations; λ

_{1}and λ

_{2}are the accelerating constants; rand is a random value in the range of [0, 1]; pBest

_{i}

^{k}is the best position of particle N

_{fx}

^{i}after kth iterations; gBest

^{k}denotes the best positions of all particles after kth iterations; ω is an inertia weight in the range of (0.8, 1.2).

Soil Type | The Physical and Chemical Properties (Error Tolerance = ±5%) | ||||||
---|---|---|---|---|---|---|---|

pH Value | Electrical Conductivity (ECe) (dS/m) | Average Volumetric Moisture Content (%) | Wilting Point (%) | Organic Content (g/kg) | Nitrogen Content (mg/kg) | Mean Bulk Density (g/cm^{3}) | |

loam | 6.624 | 0.143 | 40.44 | 28.93 | 22.601 | 1.68 | 1.354 |

sandy | 6.672 | 0.152 | 42.43 | 29.94 | 24.113 | 1.72 | 1.441 |

Chernozem | 6.782 | 0.182 | 41.46 | 30.43 | 23.841 | 1.58 | 1.389 |

Saline–alkali | 6.651 | 0.167 | 39.77 | 31.21 | 22.467 | 1.64 | 1.522 |

Clay | 6.722 | 0.149 | 38.98 | 29.98 | 23.903 | 1.66 | 1.55 |

Factor Level | P_{w}/KPa | I_{d}/h | F_{q}/L/h | S_{r}/MJ/m ^{2} | A_{w}/m/s | A_{t}/°C | A_{h}/% | S_{b}/g/cm ^{3} | S_{p}/% | O_{c}/% | S_{t}/×10 ^{−6} | E_{v}/mm/h | S_{s}/cm/s | S_{c}/% | S_{w}/% | W_{d} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 210 | 2.1 | 1100 | 11 | 0.1 | 10 | 60 | 1.0 | 30 | 40 | 2.0 | 20 | 1.0 | 0.23 | 30 | Northeast |

2 | 220 | 2.2 | 1200 | 12 | 0.3 | 12 | 62 | 1.1 | 35 | 43 | 2.6 | 22 | 1.5 | 0.26 | 35 | Southeast |

3 | 230 | 2.3 | 1300 | 13 | 0.5 | 14 | 64 | 1.2 | 40 | 46 | 3.2 | 24 | 2.0 | 0.29 | 40 | West |

4 | 240 | 2.4 | 1400 | 14 | 0.7 | 16 | 66 | 1.3 | 45 | 49 | 3.8 | 26 | 2.5 | 0.32 | 45 | Northwest |

5 | 250 | 2.5 | 1500 | 15 | 0.9 | 18 | 68 | 1.4 | 50 | 52 | 4.4 | 28 | 3.0 | 0.35 | 50 | North |

6 | 260 | 2.6 | 1600 | 16 | 1.1 | 20 | 70 | 1.5 | 55 | 55 | 5.0 | 30 | 3.5 | 0.38 | 55 | South |

7 | 270 | 2.7 | 1700 | 17 | 1.3 | 22 | 72 | 1.6 | 60 | 58 | 5.6 | 32 | 4.0 | 0.44 | 60 | Southwest |

8 | 280 | 2.8 | 1800 | 18 | 1.5 | 24 | 74 | 1.7 | 65 | 61 | 6.2 | 34 | 4.5 | 0.47 | 65 | East |

9 | 290 | 2.9 | 1900 | 19 | 1.7 | 26 | 76 | 1.8 | 70 | 64 | 6.8 | 36 | 5.0 | 0.50 | 70 | North/Northwest |

10 | 300 | 3.0 | 2000 | 20 | 1.9 | 28 | 78 | 1.9 | 75 | 67 | 7.4 | 38 | 5.5 | 0.53 | 75 | Southwest/West |

Test | P_{w} | I_{d} | F_{q} | S_{r} | A_{w} | A_{t} | A_{h} | S_{b} | S_{p} | O_{c} | S_{t} | E_{v} | S_{s} | S_{c} | S_{w} | W_{d} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A_1 | 5 | 4 | 8 | 8 | 6 | 6 | 8 | 9 | 8 | 9 | 8 | 9 | 8 | 8 | 8 | 8 |

A_2 | 3 | 2 | 2 | 5 | 2 | 6 | 7 | 8 | 9 | 6 | 9 | 6 | 8 | 9 | 9 | 5 |

A_3 | 2 | 8 | 5 | 2 | 3 | 3 | 5 | 9 | 5 | 5 | 6 | 8 | 5 | 5 | 5 | 4 |

A_4 | 1 | 6 | 6 | 3 | 5 | 2 | 4 | 6 | 2 | 2 | 5 | 5 | 4 | 2 | 1 | 8 |

A_5 | 8 | 2 | 5 | 6 | 4 | 5 | 2 | 3 | 1 | 10 | 8 | 2 | 2 | 3 | 2 | 9 |

Test | P_{w} | I_{d} | F_{q} | S_{r} | A_{w} | A_{t} | A_{h} | S_{b} | S_{p} | O_{c} | S_{t} | E_{v} | S_{s} | S_{c} | S_{w} | W_{d} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

B_1 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 9 | 8 | 8 | 8 | 10 | 8 | 8 | 8 | 8 |

B_2 | 5 | 9 | 5 | 5 | 5 | 6 | 5 | 6 | 5 | 8 | 5 | 8 | 5 | 6 | 9 | 9 |

B_3 | 8 | 5 | 2 | 2 | 6 | 2 | 6 | 5 | 9 | 5 | 9 | 5 | 2 | 5 | 5 | 5 |

B_4 | 7 | 8 | 6 | 6 | 4 | 5 | 1 | 8 | 5 | 7 | 6 | 6 | 6 | 2 | 1 | 4 |

B_5 | 4 | 5 | 9 | 2 | 2 | 3 | 2 | 9 | 4 | 7 | 2 | 1 | 9 | 4 | 6 | 6 |

Test | P_{w} | I_{d} | F_{q} | S_{r} | A_{w} | A_{t} | A_{h} | S_{b} | S_{p} | O_{c} | S_{t} | E_{v} | S_{s} | S_{c} | S_{w} | W_{d} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

C_1 | 5 | 8 | 8 | 8 | 8 | 9 | 10 | 8 | 8 | 8 | 8 | 8 | 9 | 8 | 8 | 8 |

C_2 | 5 | 5 | 7 | 9 | 4 | 5 | 5 | 9 | 9 | 5 | 9 | 5 | 9 | 5 | 9 | 6 |

C_3 | 6 | 6 | 5 | 5 | 7 | 8 | 6 | 5 | 5 | 6 | 5 | 4 | 5 | 6 | 5 | 5 |

C_4 | 2 | 2 | 4 | 6 | 5 | 2 | 8 | 6 | 6 | 2 | 6 | 7 | 8 | 2 | 6 | 4 |

C_5 | 5 | 4 | 2 | 2 | 2 | 4 | 2 | 2 | 6 | 4 | 2 | 2 | 4 | 4 | 2 | 7 |

**Table 7.**Orthogonal test parametric arrangement for sprinkler drip irrigation in saline–alkali soil.

Test | P_{w} | I_{d} | F_{q} | S_{r} | A_{w} | A_{t} | A_{h} | S_{b} | S_{p} | O_{c} | S_{t} | E_{v} | S_{s} | S_{c} | S_{w} | W_{d} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

D_1 | 8 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 2 | 8 | 9 | 8 | 8 | 8 | 8 |

D_2 | 8 | 6 | 9 | 5 | 9 | 9 | 6 | 2 | 9 | 5 | 2 | 6 | 5 | 5 | 9 | 5 |

D_3 | 9 | 9 | 5 | 4 | 8 | 6 | 5 | 5 | 5 | 1 | 5 | 9 | 6 | 6 | 5 | 6 |

D_4 | 5 | 5 | 6 | 2 | 5 | 5 | 9 | 6 | 6 | 4 | 1 | 8 | 2 | 2 | 6 | 3 |

D_5 | 6 | 8 | 2 | 1 | 7 | 3 | 2 | 2 | 3 | 10 | 4 | 5 | 3 | 3 | 3 | 2 |

Test | P_{w} | I_{d} | F_{q} | S_{r} | A_{w} | A_{t} | A_{h} | S_{b} | S_{p} | O_{c} | S_{t} | E_{v} | S_{s} | S_{c} | S_{w} | W_{d} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

E_1 | 8 | 9 | 8 | 9 | 8 | 8 | 8 | 1 | 8 | 8 | 1 | 9 | 8 | 8 | 8 | 8 |

E_2 | 5 | 6 | 5 | 9 | 7 | 5 | 9 | 10 | 8 | 6 | 2 | 6 | 9 | 9 | 9 | 9 |

E_3 | 6 | 9 | 6 | 5 | 4 | 4 | 5 | 8 | 6 | 9 | 1 | 8 | 5 | 5 | 5 | 5 |

E_4 | 2 | 5 | 3 | 6 | 5 | 7 | 6 | 9 | 5 | 2 | 4 | 5 | 6 | 6 | 6 | 6 |

E_5 | 1 | 3 | 2 | 3 | 2 | 2 | 3 | 9 | 8 | 5 | 5 | 3 | 3 | 4 | 3 | 3 |

Index | Value | SS_{e} | Df_{T} | Q | F_{j} | p | Significance |
---|---|---|---|---|---|---|---|

ESIP | 566.58(±5%) | 78.25/69.25/32.47/85.24/17.45 | 8 | 14.77/16.25/8.97 /14.77/9.31 | 13.25/15.42/16.65 /17.82/9.98 | <0.0002 | **/*/**/**/O |

PMD | 96.258(±5%) | 9.22/28.55/10.26/ 9.98/6.47 | 8 | 6.47/10.05/8.98/14.51/18.47 | 20.01/18.77/13.24/14.42/13.95 | <0.0154 | O/**/O/*/** |

PDIE | 98.224(±5%) | 11.47/9.24/28.78/ 18.42/46.52 | 8 | 13.25/8.98/13.25/6.98/14.77 | 17.74/16.21/17.75/16.64/10.22 | <0.0523 | **/O/*/***/ |

MSRR | 411.25(±5%) | 15.45/28.65/36.54/9.47/26.35 | 8 | 10.33/18.74/13.34/12.28/14.75 | 11.47/18.82/14.46/18.65/17.74 | <0.0315 | **/O/***//** |

IGV | [422.5,654.12] (±5%) | 18.57/19.58/22.64/18.75/26.54 | 8 | 9.99/7.14/8.02 /10.25/11.47 | 19.25/10.22/16.32/18.85/17.77 | <0.0005 | **/**/O/*/* |

NIPC | 95.442(±5%) | 6.51/17.52/28.22/ 9.25/18.11 | 8 | 12.25/6.62/3.32 /14.25/8.87 | 10.25/14.47/13.25/6.65/18.87 | <0.0010 | O/**/**/*/** |

Test | ESIP (×10) | PMD | PDIE | MSRR (×10) | NIPC | IGV (×10) | MAE | MAPE | RMSE | Err | Corr | Rob |
---|---|---|---|---|---|---|---|---|---|---|---|---|

set A_1 | 33.2 | 50.1 | 89.2 | 66.3 | 65.2 | 12.3 | 0.5589 | 96.52 | 88.25 | 75.23 | 0.5563 | 69.32 |

set B_2 | 55.2 | 45.3 | 43.2 | 25.8 | 52.4 | 16.3 | 0.6985 | 54.82 | 82.56 | 82.56 | 0.8526 | 85.26 |

set C_3 | 42.6 | 26.1 | 50.6 | 75.2 | 65.2 | 49.5 | 0.8625 | 81.54 | 96.25 | 77.49 | 0.5952 | 44.72 |

set D_4 | 63.2 | 26.3 | 26.3 | 44.7 | 45.6 | 72.5 | 0.6954 | 56.32 | 86.25 | 85.64 | 0.8256 | 66.25 |

set E_5 | 72.5 | 56.3 | 72.5 | 48.5 | 47.6 | 79.6 | 0.8825 | 48.26 | 78.25 | 63.25 | 0.6332 | 47.26 |

**Table 11.**Performance comparison of SDIQ index prediction results using RSAE-NPSO and other typical methods.

Calibration Coefficients | Typical Prediction Approaches | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

RSAE- NPSO | LSSVM | MLR | MLR-BR | MLR-BR-SOM | NPSO- SOM | PTP | CTP | GR | SMSLP | ||

Network training | Computation Accuracy (%) | 97.58 | 93.22 | 91.52 | 95.44 | 93.15 | 93.25 | 96.24 | 94.15 | 93.55 | 94.26 |

Standard deviation (%) | 0.296 | 0.335 | 0.542 | 0.685 | 0.665 | 0.725 | 0.824 | 0.645 | 0.558 | 0.625 | |

Network testing | Computation Accuracy (%) | 97.42 | 93.22 | 91.45 | 96.23 | 93.65 | 96.55 | 95.87 | 94.56 | 93.66 | 95.26 |

Standard deviation (%) | 0.256 | 0.336 | 0.542 | 0.863 | 0.553 | 0.642 | 0.635 | 0.475 | 0.635 | 0.558 | |

Average Computation Storage (kb) | 1856.2 | 1554.5 | 1963.2 | 1556.2 | 1725.6 | 1663.2 | 1845.2 | 1965.2 | 1753.2 | 1695.2 | |

Computation Time (s) | 1.88 | 2.36 | 2.54 | 3.65 | 4.26 | 1.95 | 1.89 | 2.03 | 2.56 | 2.95 | |

Standard error of prediction (%) | 4.15 | 5.36 | 5.89 | 6.32 | 5.78 | 5.68 | 6.12 | 6.05 | 5.65 | 6.34 | |

Confidence interval of 94% | Upper error limit (%) | 5.14 | 6.32 | 6.25 | 5.58 | 5.89 | 5.75 | 5.65 | 5.48 | 6.32 | 6.01 |

Lower error limit (%) | 3.25 | 4.26 | 5.21 | 4.15 | 3.65 | 3.98 | 3.66 | 4.03 | 3.58 | 3.95 |

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## Share and Cite

**MDPI and ACS Style**

Liang, Z.; Zou, T.; Zhang, Y.; Xiao, J.; Liu, X. Sprinkler Drip Infiltration Quality Prediction for Moisture Space Distribution Using RSAE-NPSO. *Agriculture* **2022**, *12*, 691.
https://doi.org/10.3390/agriculture12050691

**AMA Style**

Liang Z, Zou T, Zhang Y, Xiao J, Liu X. Sprinkler Drip Infiltration Quality Prediction for Moisture Space Distribution Using RSAE-NPSO. *Agriculture*. 2022; 12(5):691.
https://doi.org/10.3390/agriculture12050691

**Chicago/Turabian Style**

Liang, Zhongwei, Tao Zou, Yupeng Zhang, Jinrui Xiao, and Xiaochu Liu. 2022. "Sprinkler Drip Infiltration Quality Prediction for Moisture Space Distribution Using RSAE-NPSO" *Agriculture* 12, no. 5: 691.
https://doi.org/10.3390/agriculture12050691