Predicting Tart Cherry Stem Water Potential Using UAV Multispectral Imagery and Environmental Data via Symbolic Regression

Highlights

What are the main findings?

• Stem water potential (Ψ_stem) in tart cherry orchards can be accurately estimated using UAV-based multispectral imagery combined with meteorological and soil moisture data through symbolic regression, achieving R² values up to 0.80 and RMSE as low as 0.08 MPa.
• The Red Chromatic Coordinate (RCC) index, derived from RGB bands, was the most informative spectral predictor of, particularly during the pre-harvest period.

What are the implications of the main findings?

• The proposed symbolic regression models provide interpretable, transferable equations that overcome the black-box limitations of common machine learning approaches for Ψ_stem estimation.
• In this study, symbolic regression allowed for interpretable numerical expressions that responded to available sources of information, while still maximizing the pattern contribution of selected inputs (aerial, meteorological, soil moisture) for description of Ψ_stem.

Abstract

Tart cherry is an important fruit crop in Utah, where irrigation is essential due to arid conditions. Precision irrigation requires reliable indicators of plant water status, and stem water potential (Ψ_stem), is among the most sensitive though labor-intensive and spatially limited. This study develops Ψ_stem estimation models using high-resolution multispectral Unmanned Aerial Vehicle (UAV) imagery combined with meteorological and soil moisture data, applying Symbolic Regression (SR). Results show a stronger correlation between optical bands and Ψ_stem during the pre-harvest period. Among 85 vegetation indices, the Red Chromatic Coordinate (RCC) index performed best (R² = 0.67). Six equations were generated for different data-availability scenarios and validated using a leave-one-tree-out (modified k-fold) approach, resulting in Ψ_stem estimates with R² values ranging from 0.67 to 0.80 and root mean square errors (RMSE) ranging from 0.11 to 0.08 MPa. Notably, SR was able to produce interpretable equations that enhance model transparency and transferability. Model robustness was further confirmed using an independent dataset from a different location. To our knowledge, this is the first application of SR for Ψ_stem estimation, offering a scalable and interpretable tool to support irrigation management in tart cherry orchards.

1. Introduction

Tart cherries represent a significant specialty crop in Utah. In 2022, tart cherry farms covered an area of 1503 ha, of which 1230 ha were in bearing age, producing 10,200 Mg of fruit [1]. The Cherry Industry Administrative Board [2] reported a production of 19,800 Mg in 2024, accounting for 16.6% of national output. Most orchards are located in Utah County, a temperate semi-arid region where irrigation is essential for achieving successful yields. An increasing population and limited farmland require growers to adopt irrigation practices that ensure both environmental and economic sustainability. In this context, irrigators must make effective decisions regarding irrigation systems, strategies, and scheduling methods [3]. Growing pressures on water resources further highlight the need for precise and efficient irrigation scheduling in tart cherry production.

Irrigation management in the Western United States varies widely depending on crop type and irrigation method [4]. Most tart cherry orchards utilize micro-sprinklers installed on fixed lateral lines, many of which were designed decades ago. In Northern Utah, ref. [5] reported that small farms often apply excess irrigation that contributes to deep percolation losses, particularly early in the season. These inefficiencies are influenced by how irrigation water is allocated. Ref. [6] noted that irrigation scheduling in Utah County often relies on “in turn” water deliveries, where growers irrigate when water becomes available rather than according to crop water demand. While crop coefficients for sweet cherries east of the Cascades Mountains have been established [7], no equivalent estimates exist for tart cherries. In regions with low summer precipitation and high vapor pressure deficit (VPD), sweet cherry trees may require between 750 and 1000 mm of irrigation during the growing season [8], suggesting that tart cherry orchards in similar climates may have comparable demands.

Stem water potential (Ψ_stem) and leaf water potential (Ψ_leaf) are widely used to quantify plant water stress. Water potential (Ψ_p) reflects the energy state of water within the Soil–Plant-Atmosphere-Continuum (SPAC), and in field conditions, the standard method for measuring is the Scholander Pressure Chamber [9]. When a leaf is enclosed in reflective foil for approximately 30 min or more, transpiration is assumed to cease, and the leaf equilibrates with the stem xylem, providing an estimate of Ψ_stem [10]. Among plant-based indicators, midday Ψ_stem has consistently been shown to be a reliable and sensitive metric of water stress in fruit trees, outperforming soil, predawn, and midday Ψ_leaf [11]. As such, Ψ_stem is considered a robust plant-based measurement for guiding deficit irrigation strategies.

Despite its value, routine Ψ_stem measurement faces practical limitations. The pressure chamber method is labor-intensive, time-consuming, and requires skilled personnel, contributing to its lack of widespread adoption outside of research [12]. Furthermore, Ψ_stem provides point-based measurements that do not capture the substantial spatial heterogeneity within orchards. Variations in soil properties, irrigation uniformity, root distribution, and microclimatic conditions can produce heterogeneous patterns of tree water status that are not represented by sparse sampling. These constraints limit the operational use of Ψ_stem and highlight the need for scalable approaches capable of estimating Ψ_stem across space and time.

Measurements of soil moisture and weather parameters are also important for understanding SPAC dynamics. The relationship between Ψ_stem and VPD can vary by phenological stage [13] and species [14,15]. Soil Water Content (SWC) further dictates the amount of water available for transpiration. To schedule irrigation, both Ψ_stem and the SWC must be monitored to know when to trigger irrigation and how much water is needed [16]. In fruit trees, Ψ_stem has been estimated as a function of SWC and atmospheric conditions such as air temperature and VPD [17], and similar relationships have been demonstrated in sweet cherries using soil matric potential and VPD [18].

Remote sensing from satellites and UAVs has recently shown potential for estimating Ψ_stem in fruit crops [19,20,21,22]. Thermal imagery can be used to assess spatial patterns of water stress relative to well-watered controls [8], and individual spectral bands have demonstrated good agreement with Ψ_stem [23,24,25]. Vegetation indices (VIs) show varying performance depending on crop type, with moderate correlations reported [26,27]. For example, ref. [28] found that the Photochemical Reflectance Index (PRI) correlated strongly with Ψ_stem in peaches and almonds, while a different index was most effective in apricots.

Most studies predicting Ψ_stem using remote sensing have relied heavily on machine learning models such as Random Forest, Support Vector Regression, and Artificial Neural Networks [21,29,30,31,32,33,34,35]. Although effective, these models function as “black boxes,” offering limited interpretability and restricting transferability across datasets and production environments. Symbolic regression provides an alternative by deriving analytical equations directly from data using genetic programming. Unlike traditional black-box approaches, SR generates symbolic expressions that are interpretable and often reflect underlying physical relationships [36]. As ref. [37] note, the absence of explicit internal logic in how black-box models mathematically map input variables to predictions limits their utility for decision-support applications. No published studies have evaluated SR for Ψ_stem estimation, leaving an important gap, particularly for tart cherry, where opportunities for implementing regulated deficit irrigation depend on improved plant water status monitoring.

The present study investigates the spatio-temporal relationships among Ψ_stem, meteorological variables, soil moisture, and high-resolution thermal and multispectral UAV reflectance with the objective of developing non-complex, equation-based models for Ψ_stem prediction in tart cherry trees. A wide range of VIs was evaluated to identify the most informative predictors, and SR was applied to derive interpretable and transferable equations. Models were formulated for multiple data-availability scenarios (remote sensing only, remote sensing plus weather variables, and remote sensing plus weather and soil moisture data) and validated using a leave-one-tree-out approach and using an external independent dataset. To our knowledge, this is the first application of SR for Ψ_stem estimation using combined remote sensing, meteorological, and soil moisture data.

2. Materials and Methods

2.1. Location of the Experimental Site

The study was conducted in a 1.21 ha block of ‘Montmorency’ tart cherry (Prunus cerasus L.) at the Kaysville Research Farm, part of the Utah Agricultural Experiment Station (41.0202N, −111.9301W, 1314 m elevation). The orchard was established in 2010 with a 3.6 m × 6 m tree spacing. According to the USDA soil survey, the site is characterized by uniform Kidman soil with fine sandy loam texture, well-drained conditions, and moderately rapid permeability. The region has a semi-arid climate, with hot, dry summers and an average precipitation of 558.3 mm. From May through October, reference evapotranspiration (ET₀) typically exceeds precipitation. Irrigation is applied via a micro-sprinkler system using Nelson R10 emitters (Nelson Irrigation Corporation, Walla Walla, WA, USA), spaced at 8 m × 6 m and delivering approximately 25.4 mm over 10 h irrigation periods. Meteorological variables, including hourly air temperature (T_a), relative humidity (RH), solar radiation, ET₀, and precipitation, were recorded by a Utah Climate Center weather station located within 200 m of the orchard (https://climate.usu.edu/), accessed on 3 March 2026. Vapor pressure deficit (VPD) was computed from T_a and RH [38]. Soil water content was monitored using seven sensor nodes, each equipped with four Teros 12 sensors connected to ZL6 dataloggers (METER Group Inc., Pullman, WA, USA), installed at depths of 15, 45, 60, and 76 cm.

A second orchard located in Santaquin, UT (39.9841N, −111.8129W), was utilized for external validation of the proposed models (Appendix A). This commercial orchard covers approximately 12 ha and was planted in 2011 using the same scion-rootstock combination. Rows are oriented north–south with a tree spacing of 4.3 m within rows and 5.5 m between rows. According to the web soil survey four types of soil are present in this site: Lakewin gravelly fine sandy loam (LaC; 1% to 6% slopes), Lakewin gravelly fine sandy loam (LaD; 6% to 15% slopes), Welby silt loam (WeB; 1% to 3% slopes), and Welby silt loam (WeC; 3% to 5% slopes). The climate is also semi-arid, with average annual precipitation of 442.9 mm. Irrigation is applied using a micro-sprinkler system equipped with Nelson R10 emitters, spaced similarly to the experimental orchard. Meteorological variables were recorded by a weather station located within 1 km of the orchard. Soil water content was measured using five sensor nodes, each equipped with four TDR-315 N (Acclima, Meridian, ID, USA), installed at depths of 15, 45, 60, and 76 cm.

2.2. Ψ_stem Sampling and Measurement Protocols

Fourteen trees were selected for Ψ_stem sampling. The locations of the sampled trees, soil moisture sensors, and meteorological station at the Kaysville site are shown in Figure 1.

A Scholander pressure chamber (Model 1505D, PMS Instruments, Albany, OR, USA), equipped with a digital gauge, was used for the Ψ_stem measurements. Three shaded leaves inside the canopy, close to the trunk or a main scaffold branch, were selected on each sampled tree. The leaves were enclosed in laminated foil bags at least one hour before measurements. A clean cut was made at the petiole, and leaves were immediately inserted into the pressure chamber. Nitrogen gas was applied until the water was extruded by the xylem (endpoint), and the pressure was recorded. Measurements were taken close to solar noon (12:00 to 14:00) to capture peak water-stress conditions. A total of 24 Ψ_stem data collection campaigns were conducted. In 2023, six Ψ_stem measurement campaigns were conducted from 18 June to 9 September, aiming for three measurements pre- and post-harvest. In 2024, data collection campaigns increased to 19, running from 15 May (three days after leaf-out) to 24 September, to capture finer-scale Ψ_stem dynamics.

2.3. Soil Apparent Electric Conductivity Mapping

Soil apparent electrical conductivity (EC_a) was measured to characterize within-orchard soil variability. Measurements were collected using an EM38 probe (Geonics, Mississauga, ON, Canada), which operates at 14.5 kHz with an effective sensing depth of approximately 1.5 m. The probe provides quad-phase (conductivity) and in-phase (magnetic susceptibility) readings in mS/m, with a conductivity noise level of 0.5 mS/m.

The EM38 probe was mounted on a wooden sled and towed across the field using an all-terrain vehicle (ATV) at 11–13 km/h. A Trimble Ag-114 GNSS receiver (Trimble Inc., Westminster, CO, USA), logged positional data at 1 s intervals [39]. The field setup is shown in Figure 2. EC_a point measurements were interpolated using ordinary Kriging in ArcGIS Pro v.2.9.5, (Esri, Redlands, CA, USA) and semivariogram model parameters were optimized during fitting.

2.4. UAV Multispectral and Thermal Data Acquisition and Calibration

High-resolution multispectral and thermal imagery was collected using a DJI Matrice 300 RTK (DJI Technology, Shenzhen, China) equipped with an Altum-PT multispectral and thermal camera (AgEagle Aerial Systems, Wichita, KS, USA). A total of 25 flights were conducted. Flights followed a 2D grid-based automatic mission with 85% frontal and side overlaps at 6 m s⁻¹ and 120 m altitude, resulting in a ground sampling distance of 2.5 cm. All flights were conducted within one hour of solar noon to minimize shadow effects.

The Altum-PT camera captures imagery in five multispectral bands: blue (475 nm, 32 nm bandwidth), green (560 nm, 27 nm), red (668 nm, 14 nm), red-edge (717 nm, 12 nm), and near-infrared (NIR) (842 nm, 57 nm). The 12.2 MP panchromatic sensor (634 nm, 463 nm) was used for pan-sharpening, improving spatial resolution to 2.5 cm per pixel. Thermal imagery was acquired through the integrated Long Wave Infrared (LWIR) sensor (11 μm, 6 μm) present in the Altum-PT camera, with a ground sampling distance of 30 cm. Figure 3 shows the UAV platform with the Altum-PT payload, Aeropoints ground control points (GCPs), and thermal calibration instruments.

Radiometric calibration was performed using a Micasense Calibrated Reflectance Panel (CRP2) placed flat on the ground and photographed before and after each flight. Geometric calibration used six Propeller Aeropoints ground control points (GCPs) positioned at the orchard corners and center. Post-processing resulted in a horizontal accuracy of 10 mm and a vertical accuracy of 20 mm.

Thermal calibration followed [40]. Three 3 m × 3 m tarps with different emissivity were placed on level ground, and their surface temperatures were measured using an Apogee SI-111 infrared radiometer (Apogee Instruments Inc., Logan, UT, USA) mounted 2 m above ground level at a 35° viewing angle. These measurements served as reference data for correcting the LWIR output of the Altum-PT. Temperature values were extracted from the thermal images using a 0.5 m × 0.5 m polygon, which resulted in a total of 400 pixels.

2.5. Orthomosaic Generation Processing

Orthomosaics were generated using Agisoft Metashape professional v.2.0.3 (Agisoft, LLC, St. Petersburg, Russia). Pan-sharpening and radiometric conversion were applied according to the manufacturer’s instructions. Reflectance (ρ) for each spectral band (λ) was computed using Equation (1):

$${\rho }_{\lambda }={(DN}_{\lambda }\left({\displaystyle \frac{{DN}_{634}}{0.2{DN}_{475}+0.2 {DN}_{560}+0.2{DN}_{668}+0.2{DN}_{717}+0.2{DN}_{942}}}\right))/32768,$$

(1)

where DN_λ is the band-specific digital number, DN₆₃₄ is the panchromatic DN, and 32,768 corresponds to the range of a 16-bit image.

Thermal digital numbers were converted to temperature using:

$$\mathrm{L}\mathrm{W}\mathrm{I}\mathrm{R}\left(\mathrm{°}\mathrm{C}\right)={\displaystyle \frac{{DN}_{LWIR}}{100}}-273.15,$$

(2)

To correct the LWIR temperature estimates, the average pixel values from the center of each calibration tarp were plotted against IRT measurements. The resulting calibration equation was:

$${LWIR }_{corrected}\left(\mathrm{°}\mathrm{C}\right)=0.5974LWIR\left(\mathrm{°}\mathrm{C}\right)+9.9875,$$

(3)

The goodness of fit containing the slope and coefficient obtained from the linear regression between pixel and infrared radiometers values, using the data from all 25 flights, is shown in Figure 4.

2.6. Canopy Segmentation, Extraction of Spectral Information, and Calculation of VIs

A two-stage segmentation workflow was developed to isolate pure canopy pixels from each orthomosaic. A dense point cloud was generated and classified using Metashape’s automatic ground-point detection. Parameters included a cell size of 50 m, a maximum angle threshold of 20°, and a maximum distance threshold of 1 m. A Digital Surface Model (DSM) was produced from unclassified points, while a Digital Terrain Model (DTM) was generated from the ground-classified points. The Canopy Height Model (CHM) was then calculated as the difference between DMS and DTM. Pixels with CHM values ≥ 1 m were retained to generate the first binary mask.

Because tart cherry trees are trained in an open-vase structure with sparse central canopies, a second refinement step was applied to remove non-foliage and radiometrically extreme pixels within the canopy using a binary NDVI mask. Tree-level NDVI zonal statistics (minimum, mean, and maximum) were examined to characterize NDVI distributions. Mean NDVI values were tightly clustered around the dominant canopy reflectance range (~0.88), while lower and upper extremes were associated with artifacts (Appendix B). Based on the histogram of these tree-level statistics, NDVI values ≤ 0.78 and ≥0.92 were identified as distribution tails and used to define the bounds of the binary mask applied to each spectral band. Figure 5 shows the workflow and results of the canopy segmentation approach.

Mean reflectance values for each sampled tree were extracted using a rectangular grid. Histograms of pixel values within each polygon showed no meaningful differences between mean and median values (Appendix C), validating the use of the mean. A Python 3.12.7 script was used to automate the extraction for each flight date. Eighty-five vegetation indices were computed from the multispectral imagery following the “Awesome Spectral Indices” catalog (https://github.com/awesome-spectral-indices/awesome-spectral-indices; accessed on 2 June 2024). Indices were ranked by their coefficient of determination (R²) with Ψ_stem. The relationships were computed using the full dataset prior to model training. Although this approach may introduce a potential risk of information leakage, model performance was subsequently evaluated using an independent validation dataset collected from a different orchard.

A correlation matrix was computed for the top 20 indices, and indices that were highly correlated with each other (R² > 0.95) were removed, and the top four independent indices were selected as model predictors. The canopy-air temperature difference (T_c − T_a) was computed using the corrected LWIR imagery.

2.7. Ψ_stem Model Formulation and Performance Evaluation

To estimate Ψ_stem, symbolic regression (SR) models were developed using the PySR library [41]. pySR applies genetic programming principles to identify simple, human-interpretable mathematical expressions through a Python interface to a Julia-based optimization engine. The algorithm initializes multiple populations of randomly generated candidate equations and iteratively evolves them through mutation and crossover operations. After each evolutionary cycle, candidate equations are evaluated using a loss function, and candidates are retained based on their loss and complexity. Additional implementation details on the PySR framework are available in the project repository (https://github.com/MilesCranmer/PySR; accessed on 4 September 2025). The statistical significance of coefficients in the final SR-derived equations was assessed using ordinary least regression (OLS) standard errors.

The algorithm was configured with eight populations, each containing 50 candidate equations. A total of 500 cycles were performed per iteration, with 40,000,000 evolutionary iterations. The operator set included addition, subtraction, multiplication, division, power, square root, and exponential functions.

Predictor variables were initially screened based on their R² and statistical significance (p-value) in relation to Ψ_stem. The dataset was divided into pre-harvest (n = 141) and post-harvest (n = 141). Within each subset, observations were randomly split into a training set (75%) and a validation set (25%). The training set was used to identify symbolic regression model structures and estimate coefficients, while the validation set was used for internal performance assessment. Model complexity was quantified independently by assigning weighted points to the number of variables (2 points each), arithmetic operations (1 point), and exponentiation terms (2 points).

Because Ψ_stem observations were collected repeatedly from multiple trees across sampling dates, a Leave-One-Tree-Out (LOTO) cross-validation procedure was implemented to evaluate the robustness and generalizability of SR equations. In this process, which is a modified version of k-fold cross-validation, during each iteration, all observations from one tree were excluded. The functional form of each SR-derived equation remained fixed, and only the coefficients were re-fitted using data from the remaining trees. Predictions were then generated for the excluded tree. This procedure was repeated until each tree had served once as a validation unit. Differences in predictive performance between equations were evaluated using a paired t-test on LOTO RMSE values.

In addition to SR-derived equations, a Generalized Additive Model (GAM) was fitted as a flexible, non-parametric benchmark. The GAM included all predictors used across models and was implemented using the mgcv package in R. Smooth functions were estimated automatically for each predictor, allowing non-linear relationships to emerge without manual specification of interaction terms. A null model predicting the mean Ψ_stem from the training trees was also included as a baseline comparator.

To evaluate temporal robustness, SR-derived functional forms identified from the primary training dataset were kept fixed, and coefficients were re-fitted using data from 2024. The recalibrated models were then applied to predict Ψ_stem in 2023, providing an additional evaluation of interannual transferability.

Model generalization and transferability were further evaluated using an independent external orchard dataset (Santaquin) that was not used during feature selection, model structure identification, or coefficient calibration. The SR equations and their original coefficients, identified from the primary orchard dataset (Kaysville), were applied directly to the external dataset.

Model performance was assessed using the Root Mean Square Error (RMSE), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), and R², defined as follows:

$$RMSE=\sqrt{{\displaystyle \frac{\sum {\left(p_i-o_i\right)}^2}{n}}},$$

(4)

$$MAE={\displaystyle \frac{\sum \left|o_i-p_i\right|}{n}},$$

(5)

$$MAPE={\displaystyle \frac{\sum \frac{\left|o_i-p_i\right|}{o_i} \ast 100}{n}},$$

(6)

$$R^2=1-{\displaystyle \frac{\sum {\left(o_i-p_i\right)}^2}{\sum {\left(o_i-\overline{o}\right)}^2}},$$

(7)

where p_i and o_i represent the predicted and observed Ψ_stem values (MPa) for observation I, respectively; $\overline{o}$ is the mean observed Ψ_stem; and n is the total number of observations used in the evaluation.

Model parsimony was evaluated using the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC), computed as follows:

$$AIC=n\ast \ln \left({\displaystyle \frac{RSS}{n}}\right)+2\ast k,$$

(8)

$$BIC=n\ast \ln \left({\displaystyle \frac{RSS}{n}}\right)+k\ast \ln \left(n\right),$$

(9)

where RSS is the residual sum of squares and k is the number of estimated parameters.

3. Results

3.1. Ψ_stem During 2023 and 2024 Growing Seasons

Figure 6 presents boxplots and histograms of Ψ_stem for 2023 and 2024. In 2023, Ψ_stem ranged from −0.40 to −1.23 MPa, with a seasonal mean of −0.89. In 2024, values ranged from −0.41 to −1.25 MPa, with a mean of −0.82 MPa. Across both growing seasons, Ψ_stem followed a consistent seasonal pattern characterized by less negative values early in the season and progressively more negative values as the canopy developed. More frequent measurements in 2024 confirmed the pattern observed in 2023, including a noticeable shift occurring near DOY 175. Harvest took place on DOY 206–208 in 2023 and DOY 199–201 in 2024. In 2024, post-harvest Ψ_stem measurements exhibited more short-term fluctuations compared to 2023.

Figure 7 compares Ψ_stem values for near-matched DOY pairs between 2023 and 2024. Ψ_stem was less negative early in the season (DOY 169–170) and became more negative during mid-season, corresponding to the onset and progression of pit hardening. Late-season measurements did not show a clear monotonic decline; instead, Ψ_stem values remained within a similar range or were slightly less negative than mid-season measurements. Although the general seasonal pattern was comparable across years, some DOY pairs showed greater variability among sampled trees and slightly lower medians in 2024.

3.2. Soil EC_a and Soil Water Content Relationship with Ψ_stem

The soil apparent electrical conductivity (EC_a) map generated using the data from the EM38 survey is shown in Figure 8. Spatial variation was limited across the orchard, with EC_a values ranging from 14 to 28 mS/m. Within the subset of monitored trees, variability was even smaller (20–25 mS/m), indicating a relatively uniform soil profile in the area.

Figure 9 presents the daily SWC for all sensor depths during the measurement period. In 2023, SWC remained fairly uniform throughout the season. At 15 cm depth, values were stable near 0.20 m³/m³ before harvest, increasing to 0.25 m³ m⁻³ after harvest. At 45 cm, SWC fluctuated minimally between 0.21 and 0.22 m³ m⁻³ pre-harvest and increased to 0.25 m³ m⁻³ post-harvest. Deeper sensors showed similar patterns: SWC at 60 cm decreased slightly from 0.23 to 0.21 m³ m⁻³ before harvest and then increased to 0.23 m³ m⁻³, while at 75 cm, values declined from 0.25 to 0.22 m³ m⁻³ and later increased to 0.23 m³ m⁻³.

In contrast, 2024 exhibited greater SWC variability. At 15 cm, SWC declined from 0.23 to 0.19 m³ m⁻³ before stabilizing at 0.20 m³ m⁻³ post-harvest. At 45 cm, values fluctuated more widely (0.27 to 0.19 m³ m⁻³) during the pre-harvest period before stabilizing near 0.20 m³ m⁻³ post-harvest. SWC at 60 and 75 cm showed a progressive seasonal decline, from 0.26 to 0.19 m³ m⁻³, and from 0.27 to 0.20 m³ m⁻³, respectively.

The relationship between SWC and Ψ_stem is shown in Figure 10 using all individual sensor measurements. Correlations were higher during the pre-harvest period, with R² values of 0.30, 0.31, 0.14, and 0.03 at 15, 45, 60, and 75 cm depths, respectively. Averaging across depths yielded an R² of 0.31.

3.3. Relationship Between Ψ_stem and Meteorological Variables

In 2023, cumulative precipitation totaled 107 mm, with August contributing the highest monthly amount (43 mm), while cumulative ET₀ reached 777 mm, peaking in July at 196 mm. In 2024, cumulative precipitation was 177 mm, with the highest monthly value in May (57 mm). Total seasonal ET₀ was 811 mm, with July again showing the maximum monthly ET₀ (193 mm).

During 2023, T_a ranged from 4 °C (DOY 125) to 39 °C (DOY 203), and VPD ranged from 0.03 to 6.04 kPa, with the maximum also occurring in DOY 203. Solar radiation varied from 3.8 to 31 MJ m⁻², and daily ET₀ ranged from 0.8 to 7.9 mm. In 2024, T_a varied from −0.5 °C (DOY 123) to 39 °C (DOY 192), while VPD ranged from 0.02 to 6.0 kPa. Solar radiation ranged from 4.8 to 31.7 MJ m⁻², and ET₀ from 1.3 to 8.1 mm. Meteorological conditions during flight campaigns reflected a warm, arid environment, with an average T_a of 29 °C and an average VPD of 3.0 kPa. The highest values occurred during flight #18 (DOY 219, 2024) with T_a = 36 °C and VPD = 4.7 kPa, while the lowest T_a (22 °C) and VPD (1.7 kPa) were recorded during flight #24. Relative humidity averaged 27% and generally remained below 32%. Average meteorological conditions during each flight are shown in Appendix D.

The relationship between Ψ_stem and meteorological variables was examined using linear and quadratic regression models. Both instantaneous (flight-time) and daily meteorological values were evaluated for the pre-harvest period and for the entire growing season. Quadratic models were included because scatterplots for solar radiation and ET₀ indicated non-linear patterns (Appendix E).

A regression analysis was conducted to evaluate the linear and quadratic relationships between the four meteorological parameters and Ψ_stem. Statistical significance was assessed, and the data were stratified by year, pre-harvest and post-harvest periods, as well as across the entire season. The full results are presented in Appendix F, while

Table 1. Best performing regression models describing the relationship between Ψ_stem and four meteorological variables: air temperature (T_a), vapor pressure deficit (VPD), solar radiation, and reference evapotranspiration (ET₀). Values represent the highest coefficient of determination (R²) obtained across linear and quadratic models and across instantaneous and daily time scales for each driver. Results are shown for the year and period in which the strongest relationship was observed. Significance levels: T p < 0.001 (***). Full comparison is provided in Table A2.

Year	Period	Driver	Model	Time Scale	R2
2024	Pre-h	Ta	Linear	Daily	0.30 ***
2024	Pre-h	VPD	Linear	Daily	0.30 ***
Both	Pre-h	Solar radiation	Quadratic	Daily	0.30 ***
Both	Pre-h	ET0	Quadratic	Daily	0.33 ***

summarizes the strongest relationships.

The strongest relationships between Ψ_stem and meteorological variables were observed during the pre-harvest period. Across both years, quadratic models using daily averages of solar radiation and ET₀ provided the strongest relationships, which explained up to 33% of the variation in Ψ_stem. T_a and VPD showed comparable R² values regardless of whether instantaneous or daily measurements were used, and differences between linear and quadratic models were minimal.

3.4. Relationship Between Ψ_stem, Multispectral Bands, and VIs

Figure 11 shows the temporal variability of reflectance in the blue, green, red, red-edge, NIR, and LWIR bands across both growing seasons for all 14 sampled trees. Reflectance values on the first flight date were unusually high in the optical bands and low in the LWIR band, likely due to overcast conditions, which were only observed on that particular day. Although onboard light sensors can partially compensate for diffuse illumination, reflectance values collected under fully overcast skies are unreliable for quantitative analysis [42]. Therefore, data from the first flight was excluded. Two trees, located at the edges of the orchard, consistently exhibited higher reflectances in the optical bands across both years. These trees were located 5 m from a gravel road, and their elevated reflectance was likely influenced by dust deposition. To avoid introducing bias into the analysis, data from those trees were removed from subsequent regression modeling.

In 2023, spectral variability between flight dates was generally low, with the exception of the red and NIR bands, which showed higher temporal fluctuation. In 2024, variability between flight dates was more evident in the optical bands. Across both years, the red and blue bands showed a decreasing trend over the season, while the green and red-edge bands exhibited a decline followed by an increase after DOY 232. A clear reduction in reflectance around DOY 230 was followed by a delayed recovery in several bands. Red-band reflectance increased after DOY 170 and peaked on DOY 198, immediately before harvest (DOY 199–201).

Canopy temperatures (T_c) derived from the LWIR band were generally consistent across trees, differing by less than 2 °C on most dates. Larger variability occurred on DOY 228 of 2023, when irrigation had been withheld for more than three weeks, and on DOY 170 of 2024, when irrigation was applied on the same day as the flight.

Across the complete dataset (n = 330), correlations between Ψ_stem and individual spectral bands were weak (Figure 12). The highest R² values were obtained for the green band (0.20), T_c (0.20), NIR (0.19), and red band (0.14). When relationships were evaluated by flight date, the highest R² values in the red and green bands occurred when flights were conducted approximately six days after irrigation.

Pre-harvest data showed substantially stronger relationships between Ψ_stem and reflectance. The red, green, and NIR bands yielded R² values of 0.51, 0.33, and 0.29, respectively. T_c also improved (R² = 0.23). Post-harvest Ψ_stem exhibited no meaningful relationship in any spectral band except LWIR (R² = 0.12). Given the weak post-harvest relationships, regression modeling was conducted using only pre-harvest data (n = 141).

To assess whether early-season measurements disproportionately influenced the pre-harvest relationships due to the presence of higher (less negative) Ψ_stem values, R² values were recomputed after restricting the pre-harvest Ψ_stem range to match the post-harvest period Ψ_stem range (−0.7 to −1.2 MPa). Results are presented in

Table 2. Coefficient of determination (R2) between multispectral bands and Ψ_stem during the pre-harvest and post-harvest periods when restricting Ψ_stem values to their overlapping range (−0.7 to −1.2 MPa).

Period	Blue	Green	Red	Red-Edge	NIR	LWIR
Pre	0.01	0.25	0.19	0.10	0.11	0.08
Post	0.00	0.00	0.00	0.00	0.00	0.12

.

As expected, restricting the Ψ_stem range reduced R² values for the pre-harvest period across all spectral bands. However, pre-harvest relationships remained consistently higher than post-harvest relationships for most bands, with the exception of the LWIR. These results demonstrate that although part of the stronger pre-harvest performance is attributable to the broader early-season Ψ_stem range, seasonal differences persist even within comparable Ψ_stem values.

Among the vegetation indices evaluated, the Red Chromatic Coordinate (RCC) showed the strongest relationship with Ψ_stem (R² = 0.68. Three additional indices (GARI, MSR, and SR) performed similarly, each achieving R² values around 0.66. The thermal metric T_c − T_a also improved model performance compared to T_c alone, reaching an R² of 0.35. The selected VIs and thermal metrics considered the best candidates for estimating Ψ_stem are presented in

Table 3. Vegetation indices and thermal metrics showing the strongest relationships with Ψ_stem. The table includes index names, formulas, R² values, and primary literature sources.

Vegetation Index	Full Name	Formula	R2	Reference
RCC	Red Chromatic Coordinate	R/(R + G + B)	0.68	[43]
GARI	Green Atmospherically Resistant Vegetation Index	(N − (G − (B − R)))/(N − (G + (B − R)))	0.67	[44]
MSR	Modified Simple Ratio	(N/R − 1)/((N/R + 1)0.5)	0.66	[45]
SR	Simple Ratio	N/R	0.66	[46]
Tc − Ta	Canopy Temperature and Air Temperature Difference	LWIR − Ta	0.35	[47]

.

Seasonal dynamics of commonly used vegetation indices and indices that emerged as potential estimators for Ψ_stem differed between years, as shown in Figure 13. In 2023, NDVI, GARI, and MSR showed a gradual increase towards late season, while RCC declined over time. NDRE remained relatively stable, with only minor fluctuations. In contrast, 2024 exhibited stronger intra-seasonal variability. NDVI and GARI declined sharply during mid-season (DOY 180–200) before partially recovering, while NDRE and RCC showed pronounced mid-season peaks.

The canopy and air temperature (T_c − T_a) difference also differed between years. In 2023, T_c − T_a displayed a clear mid-season minimum, while in 2023, values were more variable and characterized by short-term spikes. Overall, 2024 showed greater temporal variability across indices compared to 2023.

3.5. Performance of SR Models for Estimating Ψ_stem

The six equations for estimating Ψ_stem found using the SR approach are presented below:

$${\mathrm{\Psi }}_{stem}=0.2364-3.4979\ast RCC,$$

(10)

$${\mathrm{\Psi }}_{stem}=0.7-3.2069\ast RCC-0.8164\ast {ET}_0,$$

(11)

$${\mathrm{\Psi }}_{stem}=0.0297\ast \left(T_c-T_a\right)-2.96\ast RCC,$$

(12)

$${\mathrm{\Psi }}_{stem}=0.0045\ast {\left(T_c-T_a\right)}^2-2.8789\ast RCC,$$

(13)

$${\mathrm{\Psi }}_{stem}={SWC}_{15\mathrm{c}\mathrm{m}}+0.228\ast MSR+0.0047\ast T_c-1.87,$$

(14)

$${\mathrm{\Psi }}_{stem}={\displaystyle \frac{11.9}{T_a}}+0.0272\ast LWIR\ast MSR\ast {SWC}_{15\mathrm{c}\mathrm{m}}-1.9$$

(15)

where RCC is the Red Chromatic Coordinate index; ET₀ is the reference evapotranspiration; T_c is the canopy temperature; T_a is the air temperature; SWC₁₅ is the soil volumetric water content at 15 cm; MSR is the modified simple ratio Modified Simple Ratio index; and LWIR is the long-wave infrared band.

To confirm that each predictor retained by symbolic regression contributed significantly to model performance, OLS regression was performed using the fixed functional forms of Equations (10)–(15). All retained coefficients were highly significant (p < 0.001), confirming that each variable provides statistically meaningful explanatory power for Ψ_stem prediction (Appendix G).

The evaluation metrics are presented in

Table 4. Complexity and performance metrics for the six Ψ_stem estimation equations developed using symbolic regression.

Equation	Complexity	R2	RMSE (MPa)	MAE (MPa)	MAPE (%)	AIC	BIC
(10)	4	0.67	0.11	0.08	11.4	−615.19	−606.34
(11)	6	0.73	0.10	0.08	11.7	−620.36	−608.56
(12)	10	0.75	0.09	0.07	9.3	−654.27	−645.42
(13)	12	0.77	0.09	0.07	8.9	−661.27	−652.42
(14)	16	0.79	0.08	0.06	8.3	−519.94	−505.20
(15)	13	0.80	0.08	0.06	7.6	−672.81	−661.02

. Models were evaluated under different data availability scenarios, ranging from models using only RGB-derived indices to those incorporating multispectral, thermal, meteorological, and soil water content variables. Across all six models, R² values ranged from 0.67 to 0.80, while RMSE varied only slightly from 0.11 to 0.08 MPa. MAE remained below 0.09 MPa, and MAPE values were all below 12%.

Model suitability varied by data availability. Equation (10), based solely on RCC, provided strong predictive performance when only RGB imagery was available. Equation (9) improved accuracy when meteorological data were included. Equations (12) and (13) further increased accuracy by incorporating thermal information, while Equations (14) and (15), which additionally used soil moisture, achieved the highest performance metrics. These models cover a range of data availability scenarios, from RGB-only inputs to combined spectral, thermal, meteorological, and soil moisture measurements. Equation (15) significantly reduced RMSE compared to Equation (10) (∆RMSE = 0.03 MPa; paired t-test p < 0.001). However, differences between Equation (15) and less complex models such as Equation (10) were not statistically significant (p > 0.05), indicating diminishing returns with increasing model complexity.

Based on information criteria, Equation (15) provided the best fit to the data, exhibiting the lowest AIC (−672.81) and BIC (−661.02) values among all candidate equations. However, Equation (13) also demonstrated good predictive performance (R² = 0.77; RMSE = 0.09 MPa) and low AIC and BIC values, offering a favorable trade-off between model performance and complexity, achieving comparable predictive accuracy with fewer input variables.

Figure 14 shows the agreement between observed and estimated Ψ_stem for the six proposed equations. All models produced a strong linear relationship between measured and estimated Ψ_stem, with residual scattering consistently distributed around the regression line. This pattern was confirmed by Q-Q plots of residuals (Appendix H).

Predictive ability was further confirmed using a leave-one-tree-out cross-validation procedure (

Table 5. Leave-one-tree-out cross-validation performance of the six estimation models, compared with a generalized additive model (GAM) and a null model. Metrics include RMSE, MAE, and MAPE.

Equation	RMSE (MPa)	MAE (MPa)	MAPE (%)
(10)	0.11	0.08	11.5
(11)	0.10	0.08	10.3
(12)	0.09	0.07	9.5
(13)	0.09	0.07	9.0
(14)	0.09	0.07	8.7
(15)	0.09	0.06	8.2
GAM	0.09	0.06	8.5
Null	0.19	0.17	22.7

). All equations performed well, with RMSE values between 0.11 and 0.09 MPa, MAE values between 0.08 and 0.06 MPa, and MAPE values below 12%, very similar to what was found in Table 3. The null model, which used the overall mean Ψ_stem as a prediction, performed substantially worse (RMSE = 0.19 MPa), demonstrating the value of incorporating physiological and environmental predictors. Among the candidate models, Equation (13) resulted in the lowest overall errors during cross-validation (RMSE = 0.09 MPa, MAPE = 8.2%), with performance similar to the generalized additive model (GAM).

When coefficients were recalibrated using the 2024 dataset and applied to 2023 observations, predictive performance remained within the same error range observed during cross-validation, demonstrating temporal stability of the SR-derived structures. Importantly, coefficient estimates retained both magnitude and sign consistency relative to the original formulations (Equations (10)–(15)), further supporting the temporal robustness of the identified mathematical structures. The recalibrated coefficients and validation performance are summarized in

Table 6. Coefficients re-calibrated using 2024 data and validation performance on 2023 observations for SR-derived equations.

Equation	Coef_1	Coef_2	Intercept	RMSE	MAE
(10)	3.2326	--	0.1730	0.11	0.08
(11)	2.9069	−0.9606	0.7718	0.11	0.09
(12)	0.0287	−2.9099	--	0.10	0.08
(13)	0.0042	−2.8160	--	0.10	0.07
(14)	0.2530	−0.0159	−1.4334	0.11	0.08
(15)	1.1201	0.02714	−1.9337	0.09	0.07

.

To assess model generalizability, the six SR-derived equations were externally validated using an independent dataset collected from an orchard in Santaquin, UT (

Table 7. Performance metrics of the six Ψ_stem equations evaluated using an external validation dataset collected from an orchard in Santaquin, UT, USA.

Equation	RMSE (MPa)	MAE (MPa)	MAPE (%)
(10)	0.12	0.10	12.5
(11)	0.13	0.11	13.4
(12)	0.06	0.05	6.09
(13)	0.07	0.05	6.61
(14)	0.09	0.07	8.38
(15)	0.09	0.06	6.85

). During Ψ_stem measurements, the hourly mean T_a, VPD, and ET₀ were 27.2 °C, 2.74 kPa, and 0.65 mm h⁻¹, respectively.

Despite differences in site conditions and the smaller sample size, most models maintained good predictive performance. Equation (12) exhibited the best overall performance, followed closely by Equation (13). In contrast, Equations (10) and (11) showed higher error levels (RMSE > 0.12 MPa). Overall, models incorporating RCC and T_c − T_a exhibited superior performance in the external orchard dataset.

The agreement between predicted and observed Ψ_stem values over the external validation dataset is illustrated in Figure 15. Equations (12) and (13) show a strong linear correspondence with limited dispersion around the regression line, with consistent predictive behavior across the evaluated range. In contrast, Equations (14) and (15) exhibit greater scatter and deviation from the 1:1 relationship. No systematic over- or under-prediction was observed for the models, supporting their robustness under independent site conditions.

3.6. Spatial Variability of Generated Ψ_stem Maps

Figure 16 presents the spatial distribution Ψ_stem across the Kaysville orchard 2 July 2023 (DOY 183), as estimated by the six SR-derived equations. Equations (10) and (11) produced more heterogeneous spatial patterns, with pronounced areas of more negative Ψ_stem values, particularly in the central and western portions of the orchard. In contrast, Equations (12) and (13) generated smoother spatial gradients, with fewer extreme negative and less negative values, exhibiting a moderate overall distribution and similar spatial distribution. Equation (14) tended to generate higher Ψ_stem values, resulting in extensive areas with higher Ψ_stem.

Figure 17 illustrates the Ψ_stem maps generated for 16 July 2024 (DOY 198). Equations (10)–(13) produced similar spatial patterns, consistently identifying zones of more negative Ψ_stem values in the western portion of the orchard and comparatively higher Ψ_stem values toward the eastern rows. In contrast, Equations (14) and (15) generated overall less negative Ψ_stem estimates, resulting in more uniform spatial distributions across the orchard. Across all models, the eastern section of the orchard exhibited higher Ψ_stem estimates than the western portion. Despite differences in magnitude among equations, the general spatial patterns were largely consistent.

4. Discussion

4.1. Environmental and Soil Factors Influencing Ψ_stem

The experimental site exhibited minimal spatial heterogeneity, which helped isolate physiological and spectral drivers of Ψ_stem. A linear relationship between clay content and soil EC_a has been documented [42,43]. The range of soil EC_a values was narrow across the study area, indicating uniform soil texture, consistent with the USDA soil survey classification of a single fine sandy loam soil type. Because EC_a is strongly associated with clay content [48] and, consequently, water-holding capacity, its spatial uniformity supports the assumption that variation in Ψ_stem was not driven by soil-type differences.

Soil water content patterns are also aligned with expected root-zone dynamics. Tart cherry roots are concentrated between 10 and 50 cm [49], and the strongest relationships between SWC and Ψ_stem occurred at 15 and 45 cm, particularly during the pre-harvest period. Soil moisture affects root signals that regulate stomatal conductance (g_s) [50], and g_s is highly related to Ψ_stem [51]. In our study, SWC had a positive linear relationship with Ψ_stem. A similar relationship was observed in grapevines [13]. The relationship was stronger during the pre-harvest period.

Meteorological variables showed moderate relationships with Ψ_stem, with the strongest association observed between Ψ_stem and daily ET₀ values during pre-harvest (R² = 0.36). This suggests that Ψ_stem becomes more negative as ET₀ increases until reaching a threshold, beyond which changes in Ψ_stem occur more gradually. Ref. [52] argues that in prune trees, under fully irrigated conditions, a maximum Ψ_stem can be expected under any given environmental condition. Negative linear relationships between Ψ_stem and instantaneous T_a have also been reported in young olive trees [53]. Together, these findings suggest that increasing atmospheric demand triggers stomatal regulation in tart cherry, consistent with responses observed in other woody perennial crops [54].

4.2. Seasonal Physiological Drivers Affecting Spectral and Environmental Responses

Seasonal patterns in Ψ_stem and temporal shifts in predictor relationships likely reflect underlying physiological changes. Harvest represents a major physiological event, substantially altering source-sink relationships. The removal of fruits, a dominant sink, typically decreases transpiration [55], as sink strength influences stomatal conductance [56]. In tart cherries, mechanical harvesting may further affect post-harvest water relations. Bark damage, which is common during mechanical harvest [57], can temporarily reduce the flow of water and nutrients before callus formation restores vascular flow [58]. These combined effects likely contributed to the weakening of relationships between Ψ_stem and spectral and environmental variables observed after harvest. In contrast, ref. [59] reported increased correlations between Ψ_stem and vegetation indices during post-harvest in sweet cherries. Sweet cherries are hand-harvested. Seasonal shifts in the relationship between Ψ_stem and reflectance have also been reported in other crops. In citrus, ref. [26] found no significant correlation during the fruit cellular division stage, while stronger relationships appeared during the rapid fruit growth stage. Additionally, tart cherry trees produce two types of leaves: early-season spur leaves and later-emerging shoot leaves. Leaf age strongly impacts spectral reflectance [60] and, if unaccounted for, can lead to misinterpretations in remote sensing analyses [61].

4.3. Spectral and Thermal Indicators of Ψ_stem

Among individual spectral bands, the red band exhibited the strongest relationship with Ψ_stem (R² = 0.51). Reflectance in this region is closely associated with chlorophyll concentration [62]. Although plant stress has a tendency to reduce leaf chlorophyll, identifying the specific cause using remote sensing might be difficult due to the generality of the leaf optical response to stress [63]. Similar relationships between Ψ_stem and optical bands reflectance collected with UAVs have been reported in previous studies. In grapevines under deficit irrigation, ref. [23] found that the red and green bands showed stronger relationships with Ψ_stem than red-edge and NIR bands, with R² values of 0.64 and 0.62, respectively, during the pre-harvest period. Likewise, ref. [64] reported that the red bands showed the strongest relationship with Ψ_stem in citrus, particularly during the fruit maturation stage. At a coarser spatial resolution, ref. [23] demonstrated that Sentinel 2 optical bands were more significant to explain the variability of Ψ_stem in grapevines, especially the red, red-edge, and NIR bands.

Among the evaluated vegetation indices, the RCC outperformed NIR-based indices in predicting Ψ_stem (R² = 0.68) during the pre-harvest period. RCC has been previously associated with canopy photosynthesis [65] and phenological dynamics [66]. Because RCC is derived exclusively from visible bands and represents relative dominance of red reflectance within the RGB spectrum, it is particularly sensitive to shifts in chlorophyll concentration and canopy color dynamics that may precede structural degradation detectable in the NIR region. Additionally, during the pre-harvest period, the presence of red fruit likely contributes to increased red reflectance within the canopy. Because fruit development and pigmentation are linked to plant-water status [67,68], RCC may indirectly capture stress-related changes associated with fruit maturation. Together, these factors may explain the stronger relationship observed between RCC and Ψ_stem under the orchard conditions of this study.

Thermal imagery alone showed a weak relationship with Ψ_stem, consistent with findings in sweet cherries [35]. Although the T_c − T_a improved performance relative to T_c, the relationship remained modest. This limited sensitivity may reflect physiological behavior similar to that observed in mature sweet cherry trees, where water stress is regulated primarily through stomatal closure, reducing the sensitivity of T_c − T_a [69]. While the Crop Water Stress Index (CWSI) remains the standard thermal-based indicator [47], its implementation requires defining upper and lower canopy temperature baselines, and the methods to do so vary widely among studies [70,71,72,73], remaining operationally challenging.

4.4. Performance, Complexity, and Interpretability of Symbolic Regression Models

Symbolic regression produced accurate and interpretable equations for estimating Ψ_stem. Model performance (R² = 0.67–0.80 and RMSE = 0.11–0.09 MPa) is comparable to previously published machine learning models for almonds [30], pistachios [29], vineyards [32], and sweet cherries [35]. However, as noted by [29], many ML models remain difficult to deploy due to limited interpretability and cannot be utilized by end-users. A key advantage of SR over conventional ML approaches is model transparency. The resulting equations explicitly describe underlying functional relationships among variables, such as the added explanatory power of T_c − T_a, when combined with spectral and environmental predictors. In this study, model complexity increased as additional inputs (thermal, meteorological, and SWC) were incorporated, but gains in accuracy beyond the multispectral + thermal combinations were marginal. These limited improvements in performance do not justify the added cost and operational burden of additional measurements. The interpretability and reproducibility of SR equations enhance their potential for validation, adoption, and trust among growers and irrigation managers.

4.5. Practical Implications for Irrigation Management

A major challenge in implementing regulated deficit irrigation in fruit trees is effectively monitoring plant water stress. Spatially explicit Ψ_stem maps offer valuable insights into within-orchard variability and can support site-specific irrigation decisions. UAV-based imagery can be acquired at flexible temporal resolutions, particularly in arid regions such as Utah, where cloud cover is minimal during summer months. Ψ_stem maps generated using the equations developed in this study could assist growers in optimizing irrigation by assessing water stress levels and making informed irrigation decisions.

Equation (10) is particularly attractive for operational applications where meteorological or soil moisture data are unavailable. Although Equations (12) and (13) achieved lower MAPE values on the external validation orchard (~6%) compared to Equation (10) (12.5%), the improvement in RMSE was relatively modest (0.06 MPa). This suggests that acceptable predictive performance can still be achieved using a substantially simpler model structure. Importantly, Equation (10) relies exclusively on RGB imagery, eliminating the need for thermal data acquisition. Thermal sensors can significantly increase system costs and require additional calibration procedures, introducing greater operational complexity. In contrast, radiometrically calibrated RGB imagery is more accessible, lower in cost, and easier to collect via standard aerial surveying protocols. For example, a combined RGB and radiometric thermal sensor camera can cost up to ~$20,000 USD, whereas individual bands of RGB sensors typically range from $2000 to $5000 USD.

4.6. Limitations

Several factors likely contributed to unexplained variability in the models. A limitation of the present study is that meaningful relationships between Ψ_stem and weather, SWC, thermal, and multispectral imagery were only detectable during the pre-harvest. Physiological decoupling appears to occur after harvest, limiting model applicability in the post-harvest phase. Future research should focus on characterizing post-harvest physiological processes and their influence on plant water relations. Nutritional variability within orchards can also alter leaf chlorophyll concentrations, affecting reflectance in ways unrelated to water status. High nitrogen levels typically result in greater chlorophyll content [74], leading to more absorption in visible reflectance [75].

The operational use of UAV-derived Ψ_stem maps for irrigation scheduling requires the definition of crop-specific water-stress thresholds for tart cherry. While thresholds have not yet been established for tart cherries, reference values have been proposed for sweet cherry cultivars, such as −1.3 MPa for ‘Prime Giant’ [18] and −1.5 MPa for ‘Summit’ [76]. Moderate water stress has been shown to have no significant impact on tart cherry [77] and sweet cherry [35] yields.

The predictive uncertainty of the proposed models should be considered when interpreting Ψ_stem values near water stress thresholds. For example, if an operational irrigation trigger were set at −1.3 MPa, the model error that corresponds to approximately 0.08 MPa represents roughly 6% of the threshold magnitude. In practice, this level of uncertainty is relatively small compared to the typical seasonal range of Ψ_stem (approximately −0.4 to −1.25 MPa in this study) and is unlikely to alter irrigation decisions except when predicted values fall very close to the selected threshold. In such cases, growers may incorporate safety margins or repeated measurements to reduce decision uncertainty.

5. Conclusions

This study demonstrates that Ψ_stem can be reliably estimated using high-resolution UAV multispectral imagery integrated with environmental data through symbolic regression. Strong relationships between Ψ_stem and optical reflectance were observed during the pre-harvest period. Among the evaluated spectral predictors, RCC was the most informative vegetation index, outperforming individual spectral bands and thermal metrics. Symbolic regression produced six equation-based models covering a range of data-availability scenarios, from RGB imagery alone to combinations including thermal, meteorological (T_a and ET₀), and soil volumetric water content at 15 cm. Across all models, predictive performance was high (R² = 0.67 to 0.80; RMSE 0.11 to 0.08 MPa), where small performance gains were achieved by increasing model complexity. External validation using an independent orchard dataset confirmed model robustness and transferability, particularly for equations combining thermal, multispectral, and weather information. A key contribution of this work is the demonstration that symbolic regression can achieve performance comparable to ‘black box’ machine learning models while retaining full mathematical interpretability. The derived equations explicitly reveal the functional relationships between spectral, thermal, and environmental variables with Ψ_stem, enabling transparence, reproducibility, and practical deployment. This interpretability is critical for the application of models to map Ψ_stem for irrigation decision support. The ability to generate orchard-scale Ψ_stem maps offers a promising alternative to labor-intensive measurements with the pressure chamber. Future studies should therefore investigate the combined effects of plant nutrition and Ψ_stem on canopy reflectance to better separate nutritional and water-stress signals. Finally, crop-specific water-stress threshold Ψ_stem values for tart cherries are needed to enable practical irrigation scheduling. Overall, this study establishes symbolic regression as a powerful and interpretable framework for plant-based water status estimation and supports its integration into precision irrigation management for specialty crops.