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Metrics for Evaluating Interreplicate Variability of Irrigation Scheduling Sensors

Tsz Him Lo1,*, Jacob P. Rix1, H. C. (Lyle) Pringle III2, Daran R. Rudnick3, Drew M. Gholson1, Hope Njuki Nakabuye4, Abia Katimbo4

Published in Journal of the ASABE 67(1): 115-126 (doi: 10.13031/ja.15513). Copyright 2024 American Society of Agricultural and Biological Engineers.

1National Center for Alluvial Aquifer Research, Mississippi State University, Stoneville, Mississippi, USA.

2Delta Research and Extension Center, Mississippi State University, Stoneville, Mississippi, USA.

3Biological Systems Engineering, University of Nebraska-Lincoln, Nebraska, USA.

4West Central Research, Extension and Education Center, University of Nebraska-Lincoln, North Platte, Nebraska, USA.

*Correspondence: himmy.lo@msstate.edu

Submitted for review on 29 December 2022 as manuscript number NRES 15513; approved for publication as a Research Article and as part of the “Digital Water: Computing Tools, Technologies, and Trends” Collection by Associate Editor Dr. Brian Leib and Community Editor Dr. Kati Migliaccio of the Natural Resources & Environmental Systems Community of ASABE on 23 October 2023.

Highlights

Abstract. Much of the research on irrigation scheduling sensors, especially soil water sensors, assesses and refines the accuracy of sensor calibrations. However, a sensor with an accurate calibration but high variability among replicates may require a larger-than-acceptable number of replicates for informing recommendations of optimal irrigation timing. To compare the interreplicate variability of sensors across types and calibration accuracy levels, this study presented eight metrics: (1) absolute spread-to-change ratio, (2) shifted spread-to-change ratio, (3) coefficient of change variation, (4) standard deviation of relative value, (5) standard deviation of relative change, (6) standard deviation of absolute triggering date, (7) standard deviation of shifted triggering date, and (8) standard deviation of relative triggering date. These metrics enabled comparisons either by nondimensionalizing sensor measurements or by expressing interreplicate variability in terms of time. For demonstrating their usage and their particularities, the metrics were applied to two datasets that included soil water sensor types such as neutron probe (503DR), dielectric sensor (TDR-315, CS616, CS655, HydraProbe II, 5TE, TEROS 12, Drill & Drop), and granular matrix sensor (Watermark 200SS). The neutron probe in the single-depth dataset and the granular matrix sensor in the multi-depth dataset generally displayed less interreplicate variability than other evaluated sensor types over multiple drying cycles. Future research is suggested to calculate and improve the eight metrics for identifying combinations of sensor types, deployment methods, and data interpretation techniques that minimize interreplicate variability and maximize irrigation scheduling precision.

Keywords. Assessment, Comparison, Index, Nondimensionalization, Precision, Reliability, Soil water, Standardization, Uncertainty, Variation.

For irrigation scheduling, sensors can provide science-based information to support decisions about irrigation amount and timing (Gu et al., 2020; Andrade et al., 2020; Taghvaeian et al., 2020; Bwambale et al., 2022). One category of irrigation scheduling sensors is soil water sensors (Yoder et al., 1998; Topp and Ferré, 2002; Evett and Parkin, 2005; Evett, 2007). When researchers evaluate the performance of soil water sensors, the primary criterion tends to be absolute accuracy (Fares et al., 2011; Sui et al., 2019; Kukal et al., 2020). According to the conventional view, the best sensor is the one making soil water measurements that are numerically closest to those made by a reference method (e.g., volumetric soil sampling; Chow et al., 2009; Vaz et al., 2013; Ojo et al., 2015).

The importance of absolute accuracy is paramount if sensor measurements are to be compared against independently determined soil water limits (Leib et al., 2003; Liang et al., 2016; Datta et al., 2018). However, in situ factors can hinder such comparisons even for highly accurate sensors. For instance, soil water limits are sometimes established while assuming a measurement volume that differs in size and position from the actual measurement volume of the sensor. Comparisons between these two measurement volumes could be distorted whenever volumetric water content (?) varies significantly with distance from the plant and from the placement of irrigation water (Stieber and Shock, 1995; Coelho and Or, 1996; Soulis et al., 2015). Additionally, vertical and horizontal nonuniformity in soil properties can cause incompatibility between the ? values reported by sensors and the soil water limit values obtained by laboratory techniques or pedotransfer functions (Zettl et al., 2011; Evett et al., 2019; Vories and Sudduth, 2021).

Given the challenges of comparing against independently determined soil water limits, comparing against irrigation triggers (i.e., a preestablished value at which irrigation is recommended) that were determined by the same irrigation scheduling sensor may be more practical. These triggers can be set by relating sensor measurements to local experiences of crop yield response (Bryant et al., 2023), sensor data patterns (Starr and Paltineanu, 1998), visual stress detection (Kacira et al., 2002), and existing scheduling methods such as those that account for weather and canopy/phenology (Jensen, 1969) or use other sensors (Thompson et al., 2007). When irrigation is scheduled based on comparing against such sensor-specific triggers, absolute sensor accuracy ceases to be the chief concern. Instead, sensor measurements can be viewed as a relative indicator of crop water status.

Consequently, the reliability of this relative indicator in signaling irrigation timing can become the principal criterion by which the performance of an irrigation scheduling sensor is judged. In contrast, the quality of sensor information on irrigation amount can be assigned a lower priority because sensors influence only the application timing but not the application amount in many irrigation systems. One aspect of reliability is repeatability over time—whether a particular sensor-reported value can be interpreted identically across all drying cycles (Zhu et al., 2019). Another aspect of reliability is repeatability over replicates—whether a particular sensor-reported value can be interpreted identically across all comparable units of the same sensor at all comparable locations (Schmitz and Sourell, 2000).

The latter aspect of reliability, hereafter referred to as interreplicate variability, is the focus of this article. Owing to phenomena such as hardware inconsistencies (Kelleners et al., 2005), environmental differences (Wilson et al., 2004), installation disturbances (Rothe et al., 1997), and microscale heterogeneities (Evett et al., 2012), replicates are not guaranteed to report similar values despite being co-located in a seemingly homogeneous area. What one replicate reports can be described as a random observation of a probability distribution whose spread increases with the magnitude of interreplicate variability (Lo et al., 2020b). In other words, the replicates of a sensor with high interreplicate variability are likely to report values deviating substantially from the true mean and to reach the irrigation trigger at vastly different times. Deploying more replicates improves the confidence with which the true mean and its triggering dates are estimated (Tollner et al., 1991), yet many users deploy just one or two replicates because of cost, convenience, and/or unawareness. If these users chose an irrigation scheduling sensor with high interreplicate variability, they would be prone to being misled into suboptimal irrigation timing.

Past studies tend to evaluate the interreplicate variability of irrigation scheduling sensors in terms of standard deviation (Evett et al., 2006; Evett et al., 2009; Rosenbaum et al., 2010). Yet, this metric suffers from multiple limitations. Retaining the dimensions of the sensor values, the standard deviation does not permit comparisons among sensors across categories (e.g., soil water sensors vs. dendrometers, porometers, leaf pressure chambers, infrared thermometers, etc.). Even within a category, standard deviation values for one soil water sensor cannot be compared quantitatively with those for another soil water sensor in the absence of highly accurate calibrations (Lo et al., 2020a). Some studies introduced a locally calibrated sensor (e.g., neutron probe) as the reference for cross-calibrations, but measurement volume differences among sensors can be a concern (Rudnick et al., 2018; Chen et al., 2020; Sharma et al., 2021). Eliminating the need for volumetric soil sampling to standardize sensor measurements prior to comparisons would be desirable.

An ideal approach for evaluating interreplicate variability of irrigation scheduling sensors would allow quantitative comparisons among sensors, irrespective of category and calibration accuracy. This study takes a step toward the ideal by presenting eight interreplicate variability metrics. These metrics are related to reliability in signaling irrigation timing and are applicable to diverse combinations of sensors and conditions. Two datasets were employed as case studies that illustrate the computation of and the inferences from these metrics and that clarify the strengths and weaknesses of these metrics.

Materials and Methods

Data Sources

Dataset A was collected in 2018 at University of Nebraska–Lincoln West Central Research, Extension and Education Center near North Platte, Nebraska, USA. Eight units each of TDR-315 (hereafter TDR315; Acclima, Meridian, Idaho, USA), CS616 (Campbell Scientific, Logan, Utah, USA), CS655 (Campbell Scientific, Logan, Utah, USA), HydraProbe II (hereafter HP2; Stevens Water, Portland, Oregon, USA), 5TE (METER Group, Pullman, Washington, USA), and TEROS 12 (hereafter TEROS12; METER Group, Pullman, Washington, USA) dielectric soil water sensors—as well as eight aluminum access tubes for one unit of the CPN 503DR (hereafter 503DR; Instrotek, Concord, California, USA) neutron thermalization soil water sensor—were all installed (fig. 1a) in an area entirely classified as Cozad silt loam (Typic Haplustolls; NRCS, 2022). Subsequently, Hoegemeyer 2511NRR (Corteva Agriscience, Indianapolis, Indiana, USA) soybean (Glycine max L.) was hand-planted at 40 seeds m-1 with 0.19-m row spacing on 10 May. All sensor measurements were centered at a depth of 0.30 m. All dielectric sensors were oriented horizontally, whereas the same 503DR unit was used throughout the dataset. Defining each unit as a replicate for the dielectric sensors while defining each access tube as a replicate for 503DR, there were eight replicates of seven soil water sensors (table 1). No irrigation was applied.

Dataset B was collected in 2022 at the Mississippi State University Delta Research and Extension Center near Stoneville, Mississippi, USA. In an area entirely classified as Sharkey clay (Chromic Epiaquerts; NRCS, 2022), Pioneer P47A64X (Corteva Agriscience, Indianapolis, Indiana, USA) soybean was mechanically planted at 18 seeds m-1 in twin rows on 30 April. The spacing within a pair of twins was 0.20 m, whereas the spacing between the closest rows of two adjacent pairs was 0.81 m. Subsequently, six replicates each of Drill & Drop (hereafter Drill&Drop; Sentek, Stepney, South Australia, Australia) and Watermark 200SS (hereafter WM200SS; Irrometer, Riverside, California, USA) soil water sensors were installed (fig. 1b; table 1). Each replicate of Drill&Drop was one multisensor capacitance probe unit whose measurements were centered at the depths of 0.05, 0.15, 0.25, 0.35, 0.45, 0.55, 0.65, 0.75, 0.85, 0.95, 1.05, and 1.15 m. A hole matching the tapered shape of the probe was prepared using a product-specific installation kit, and then the probe was vertically inserted into the hole after moistening the probe with tap water from a spray bottle. Each replicate of WM200SS included five units whose granular matrices were centered at the depths of 0.10, 0.30, 0.50, 0.70, and 0.90 m, respectively. Before installation, each WM200SS unit was cemented to an end of a polyvinyl chloride (PVC) pipe section and was preconditioned through a wet-dry-wet process (Rix et al., 2021). A hole was excavated by a 22-mm auger powered by a cordless drill, and then the sensor-pipe assembly was vertically inserted into the hole after pouring in a slurry of tap water and local soil. All soil water sensors were positioned halfway between a crop row and its twin. Each of the five furrow irrigation applications was a few days later than the irrigation date that was expected to maximize crop yield. These intentional delays allowed the soil to dry more than is typical in irrigated production.

Figure 1. Layout of sensor replicates in a) dataset A and b) dataset B; please see the sensor-specific definition of replicate in the text.

Interreplicate Variability Metrics

Eight alternate metrics were computed for comparing the interreplicate variability of irrigation scheduling sensors, regardless of category and calibration accuracy. These metrics accounted for three distinct techniques of interpreting sensor data to monitor crop water status and impose irrigation triggers. The first technique interpreted the data in terms of sensor values (e.g., ?, or tension for soil water sensors), which would be the conventional practice. The second technique interpreted the data in terms of shifted values by subtracting from each sensor value the replicate-specific value corresponding to wet conditions. This wet benchmark might be near field capacity for a well-drained soil or near field satiation (i.e., predominant yet incomplete saturation because of inevitable air entrapment under field conditions; ASABE Standards, 2019) for a poorly drained soil. The third technique interpreted the data in terms of relative values by linearly scaling sensor values into a dimensionless index ranging from 0 under wet conditions to 1 under dry conditions, much like the Crop Water Stress Index (Jackson et al., 1988).

Table 1. Soil water sensors in datasets A and B.
DatasetSensorsReplicatesDepths
ATDR315, CS616, CS655, HP2, 5TE, TEROS12, 503DR80.3 m only
BDrill&Drop, WM200SS60.0-1.0 m

Each alternate metric in this study enumerated interreplicate variability in dimensionless value, in dimensionless change, or in triggering date for one of the three interpretation techniques (table 2). Just like traditional metrics such as standard deviation (SD) and coefficient of variation (CV), the score of every alternate metric equaled zero in the absence of interreplicate variability and rose towards positive infinity as interreplicate variability increased. Because the true mean would usually be unknown in practice, it was approximated by the sample mean among the studied replicates in the equations of the metrics. The absolute spread-to-change ratio (ASC; eq. 1) produced a dimensionless SD of sensor value, nondimensionalizing it by the absolute value of mean temporal change in sensor value. The shifted spread-to-change ratio (SSC; eq. 2) followed the form of ASC but first converted sensor values into shifted values. The coefficient of change variation (CCV; eq. 3) was almost exactly the CV (always dimensionless) of the temporal change in sensor value (and in shifted value), except for the insertion of the absolute value function in the denominator. The standard deviation of relative value (SDRV; eq. 4) first converted sensor values into relative values before entering them into the SD function. Employing the same conversion, the standard deviation of relative change (SDRC; eq. 5) represented the SD of the temporal change in relative values. The standard deviation of the absolute triggering date (SDAT; eq. 6) quantified the SD of when the replicates, respectively, reached the mean sensor value at a particular time of interest (simulating a shared irrigation trigger in terms of sensor value). The standard deviation of the shifted triggering date (SDST; eq. 7) expressed the SD of when the replicates, respectively, reached the mean shifted value at a particular time of interest (simulating a shared irrigation trigger in terms of the shifted value). The standard deviation of the relative triggering date (SDRT; eq. 8) was equated to the SD of when the replicates, respectively, reached the mean relative value at a particular time of interest (simulating a shared irrigation trigger in terms of relative value).

Table 2. Classification of the eight interreplicate variability metrics by enumerated quantity and by interpretation technique; sensor data can be interpreted in terms of replicate-reported values (“sensor”), in terms of the difference between replicate-reported values and a replicate-specific wet benchmark (“shifted”), or in terms of a dimensionless index (“relative”) between 0 (at wet conditions) and 1 (at dry conditions).
Enumerated QuantityInterpretation Technique
SensorShiftedRelative
Dimensionless ValueASC
(eq. 1)		SSC
(eq. 2)		SDRV
(eq. 4)
Dimensionless ChangeCCV (eq. 3)SDRC
(eq. 5)
Triggering DateSDAT
(eq. 6)		SDST
(eq. 7)		SDRT
(eq. 8)

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

where

t = index (dimensionless) of time

i = index (dimensionless) of replicate

n = total number (dimensionless) of replicates

vi,t = value (variable dimensions) reported by replicate i at time t

vi,W = value (variable dimensions) reported by replicate i under wet conditions

vi,D = value (variable dimensions) reported by replicate i under dry conditions

xi,t = date (decimal days) on which the sensor value reported by replicate i reached the mean sensor value at time t

yi,t = date (decimal days) on which the shifted value reported by replicate i reached the mean shifted value at time t

zi,t = date (decimal days) on which the relative value reported by replicate i reached the mean relative value at time t

ASCt = absolute spread-to-change ratio (dimensionless) at time t

SSCt = shifted spread-to-change ratio (dimensionless) at time t

CCVt = coefficient (dimensionless) of change variation at time t

SDRVt = standard deviation (dimensionless) of relative value at time t

SDRCt = standard deviation (dimensionless) of relative change at time t

SDATt = standard deviation (decimal days) of absolute triggering date at time t

SDSTt = standard deviation (decimal days) of shifted triggering date at time t

SDRTt = standard deviation (decimal days) of relative triggering date at time t.

Implementation

To compute the eight metrics for dataset A, factory-calibrated single-depth ? was treated as the sensor value for dielectric sensors, whereas locally calibrated single-depth ? was treated as the sensor value for 503DR. Dataset A contained 19 dates when 503DR measurements were taken during the growing season. ASC and CCV were calculated over the 18 intervals between these measurement dates. Replicate-specific values from the commonly wettest date (29 May; corresponding to slightly wetter than field capacity) were selected as the wet benchmark, while replicate-specific values from the commonly driest date (16 July; corresponding to slightly wetter than permanent wilting point) were selected as the dry benchmark. Accordingly, SDRV was calculated on the other 17 dates, and SDRC was calculated over the same 18 intervals as ASC and CCV. SSC was calculated over 17 intervals, skipping the interval ending on the wet date (29 May). SDAT was calculated on 7 July because it was the only date on which xi,t could be identified for all replicates of all sensors. SDST and SDRT were calculated on 15 June and 7 July (corresponding approximately to 18% and 76% depletion of plant-available soil water, respectively) to span a wide range of soil water triggers.

Figure 2. Interreplicate mean of volumetric water content (?) for seven sensors in dataset A, with error bars and shaded areas marking 1 standard deviation above and below each mean; rainfall was measured by the North Platte 3SW automated weather station (Nebraska State Climate Office; https://mesonet.unl.edu) 1 km away.

To compute these eight metrics for dataset B, factory-calibrated values at depths between 0.0 and 1.0 m were averaged as profile ? for Drill&Drop and as profile soil water tension for WM200SS. Profile ? for WM200SS was also calculated by averaging depth-specific ? estimates across the top 1 m. The sand, silt, and clay proportions of the dominant soil component at the experimental site (i.e., component #23276326) were first obtained from NRCS (2022) and were then entered into a web tool (https://dsiweb.cse.msu.edu/rosetta/) implementing the Rosetta3 pedotransfer function (Zhang and Schaap, 2017). Finally, the predicted parameter values (?r = 0.137 m3 m-3; ?s = 0.536 m3 m-3; a = 1.2 m-1; and n = 1.244) for the van Genuchten-Mualem soil water characteristic curve (van Genuchten, 1980) were used to estimate depth-specific ? from depth-specific tension. The analysis of interreplicate variability metrics considered exclusively the midnight (0:00 local daylight time) measurements on the 67 consecutive days from 16 June to 22 August (inclusive). ASC and CCV were calculated over the 44 daily intervals without rain and irrigation. Selecting replicate-specific extrema as the wet and dry benchmarks, SDRV was calculated on all 67 dates, and SSC and SDRC were calculated over the 44 unwetted daily intervals. SDAT, SDST, and SDRT were calculated on every date on which xi,t, yi,t, or zi,t could be identified for all replicates of a sensor. Therefore, these triggering date metrics were calculated on more dates for a sensor with lower interreplicate variability.

Results and Discussion

Dataset A

Dataset A highlighted the problem of SD and CV for comparing the interreplicate variability of different ?-reporting soil water sensors. During the experimental period, 5TE exhibited an interreplicate mean ? range of merely 0.09 m3 m-3, while all other sensors exhibited an interreplicate mean ? range of around 0.16 m3 m-3 (fig. 2). If the sensors were compared in terms of SD and were found to display equal SD, interreplicate variability would actually be highest for 5TE considering its lower sensitivity to ?. Likewise, CS616 consistently reported an interreplicate mean ? at least 0.11 m3 m-3 higher than all other sensors (fig. 2). If the sensors were compared in terms of CV and were found to display equal CV, interreplicate variability would actually be highest for CS616 considering its presumable overestimation of ?. These examples portrayed the need for alternate metrics that support fair comparisons of interreplicate variability among sensors.

The eight alternate metrics (eqs. 1-8) helped characterize the interreplicate variability of various sensors. The ASC (fig. 3a) and SDAT (fig. 3f) scores for TDR315 were among the lowest of its peers, indicating comparatively small interreplicate differences in sensor values. The SSC (fig. 3b), CCV (fig. 3c), SDRC (fig. 3e), and SDST (fig. 3f) scores for CS616 were among the lowest of its peers, indicating comparatively small interreplicate differences in shifted values and in value changes. The SDRV (fig. 3d) and SDRT (fig. 3f) scores for HP2 were among the lowest of its peers, indicating comparatively small interreplicate differences in relative values. Such conclusions were not and could not be drawn definitively by Lo et al. (2020a) because only 503DR was locally calibrated and because these metrics were not used.

The metrics also confirmed major findings by Lo et al. (2020a). The first finding was that infiltrated rainfall increased interreplicate variability in dataset A. Likely because of preferential flow, value changes were especially different among replicates as the wetting front reached the sensors, sparking spikes in CCV and SDRC. Such spikes explained the particularly strong right skew of these two change-focused metrics (figs. 3c and 3e). Consequently, the scores of other metrics also tended to increase with the arrival of wetting fronts. The second finding was that value changes were less variable than original values and shifted values. Scores of CCV (fig. 3c) were generally lower than scores of ASC (fig. 3a) and SSC (fig. 3b) for every sensor, but scores of SSC (fig. 3b) were distinctly lower than scores of ASC (fig. 3a) for CS616 and CS655 only. This observation revealed that, in dataset A (and dataset B), the interreplicate differences in sensor values were not solely composed of replicate-specific constant offsets from the interreplicate mean. On the contrary, the accumulation of small yet systematic interreplicate differences in value changes comprised a significant part of interreplicate variability. Therefore, shifting sensor values based on the replicate-specific wet benchmark did not remove all interreplicate variability because replicates also differed in responses to wetting and drying.

Figure 3. Scores of the (a) ASC; (b) SSC; (c) CCV; (d) SDRV; (e) SDRC; and (f) SDAT, SDST, and SDRT metrics for dataset A; subfigures a-e used boxes to identify 1st quartile, median, and 3rd quartile scores and used whiskers to identify minimum and maximum scores.

Summarizing all metrics, 503DR displayed the lowest interreplicate variability and appeared most reliable for recommending irrigation timing. 5TE displayed the highest interreplicate variability and appeared least reliable for recommending irrigation timing. Among dielectric sensors, both TDR315 and CS616 achieved comparatively low scores of SDAT, SDST, and SDRT (fig. 3f), signifying similar interreplicate differences in triggering dates despite the apparent overestimation of ? by CS616. This result exemplified the possibility of a sensor with low calibration accuracy but high scheduling reliability, as long as suitable sensor-specific irrigation triggers can be determined.

Dataset B

Drill&Drop and WM200SS wetted and dried in synchrony, but the interreplicate mean of the two sensors did not share a one-to-one relationship (fig. 4). According to WM200SS, each drying cycle tended to begin and end at a drier state than the previous cycle. Drill&Drop did not follow such a pattern, so the same Drill&Drop interreplicate mean corresponded to a progressively drier WM200SS interreplicate mean as time proceeded. This phenomenon might be attributed to a preferential response of Drill&Drop to wet zones within the soil (Logsdon, 2009) and to a lag of WM200SS in approaching equilibrium with the surrounding soil (McCann et al., 1992). Both sensors experienced higher SD as the soil dried, but the trend was much steeper for WM200SS tension than for Drill&Drop ? (fig. 4). As compared with SD on the wettest dates, SD on the driest dates tended to be a third higher for Drill&Drop ? but three times higher for WM200SS tension.

Despite intraseasonal changes in the vertical distribution of soil water, WM200SS ? shared a virtually one-to-one relationship with WM200SS tension. However, this profile ? was consistently wetter than the ? equivalent of this profile tension according to the soil water characteristic curve. Owing to the shape of the curve, the same difference in tension corresponded to a progressively smaller difference in ? as the soil dried. Profile ?, as the average of depth-specific ?, would be consequently less sensitive to the driest depths and would convey a wetter soil water status than profile tension, as the average of depth-specific tension. The diminishing slope of the soil water characteristic curve also affected WM200SS ? in two other ways. The SD of WM200SS ? displayed no discernible trend with soil water status, and WM200SS ? dried at a more rapidly decelerating rate than WM200SS tension (fig. 4). Greater curvature in the drying rate of WM200SS than in the drying rate of Drill&Drop ? (fig. 4) may also be contributed by a preferential response of Drill&Drop to wet zones within the soil (Logsdon, 2009) and to a lag of WM200SS in approaching equilibrium with the surrounding soil (McCann et al., 1992).

Figure 4. Interreplicate mean of volumetric water content (?; left and right panels) and of soil water tension (center panel) for two sensors in dataset B, with shaded areas marking 1 standard deviation above and below each mean; rainfall was measured by Stoneville SW automated weather station (Delta Agricultural Weather Center; http://deltaweather.extension.msstate.edu) 1 km away.

Overall, the metrics revealed that, regardless of data interpretation technique, WM200SS exhibited lower interreplicate variability than Drill&Drop (fig. 5). The advantage of the WM200SS over Drill&Drop was greatest when interpreting in terms of sensor values. The median ASC score for Drill&Drop was three times the median ASC score for WM200SS (fig. 5a). Scores of SDAT could be calculated on six times the number of dates for WM200SS as for Drill&Drop (fig. 5f). Given an equal number of replicates, WM200SS would be more reliable for recommending irrigation timing based on sensor values. Yet, the uncertainty of those recommendations tended to increase as the soil dried, with SDAT scores for WM200SS being around 2 d on the dates when irrigation would be typically considered to maximize crop yield (fig. 5f). The median and quartile scores of CCV were comparable between the two sensors (fig. 5c) and were lower than those of ASC for both sensors (fig. 5a). These observations showed that interreplicate differences in sensor value changes were similar in magnitude between the two sensors and were smaller than interreplicate differences in sensor values.

Switching to interpreting in terms of shifted values produced a different impact on the two sensors. Scores of SSC (fig. 5b) and SDST (fig. 5g) tended to be respectively worse than scores of ASC (fig. 5a) and SDAT (fig. 5f) for WM200SS. Additionally, SDST scores (fig. 5g) could be calculated on slightly fewer dates than SDAT scores (fig. 5h) for WM200SS. The bias of the wettest value for each WM200SS replicate was observed to be neither similar to nor associated with the bias of other values for the same replicate. Therefore, the shifted interpretation technique was not beneficial for WM200SS. The opposite was true for Drill&Drop, narrowing its gap with WM200SS. The median SSC score for Drill&Drop (fig. 5b) was 0.6 times the median ASC score for Drill&Drop (fig. 5a) and merely 1.5 times the median SSC score for WM200SS (fig. 5b). Drill&Drop SDST scores (fig. 5g) could be calculated on three times the number of days as Drill&Drop SDAT scores (fig. 5f) and on half the number of days as WM200SS SDST scores (fig. 5g). A core reason for such improvement was that one Drill&Drop replicate consistently reported values wetter than the interreplicate mean. Because shifting compensates for replicate-specific constant offsets, the shifted interpretation technique would enhance the reliability of Drill&Drop for recommending irrigation timing.

Among the three techniques, the relative interpretation technique maximized the reliability of both sensors for recommending irrigation timing. For Drill&Drop, SDRT scores (fig. 5h) could be calculated on five times the number of dates as SDAT scores (fig. 5f). For WM200SS, SDRT scores (fig. 5h) could be calculated on 1.2 times the number of dates as SDAT scores (fig. 5f). Nonetheless, interreplicate variability remained higher for Drill&Drop than for WM200SS when interpreting both in terms of relative values. The median SDRV score for Drill&Drop was two times the median SDRV score for WM200SS (fig. 5d), and SDRT scores were 0.5 d higher on average for Drill&Drop than for WM200SS (fig. 5h). In contrast to the gradual rises in SDAT (fig. 5f) and SDST (fig. 5g) scores over each drying cycle, the gradual decline in SDRT scores for WM200SS over the last two drying cycles (fig. 5h) was an artifact of the analysis. The driest value for every WM200SS replicate occurred at the end of those two cycles. While the relative values for all WM200SS replicates were converging to 1, the SDRT (and SDRV) values for WM200SS were artificially deflated. Such an artifact was absent for Drill&Drop because the driest value for every Drill&Drop replicate occurred at the end of different drying cycles. Across the eight metrics, SDRC was the only one at which Drill&Drop outperformed WM200SS (fig. 5e), indicating that WM200SS exhibited larger interreplicate differences in relative value changes than Drill&Drop.

Figure 5. Scores of the a) ASC; b) SSC; c) CCV; d) SDRV; e) SDRC; and f) SDAT, SDST, and SDRT metrics for dataset B; subfigures a-e used boxes to identify 1st quartile, median, and 3rd quartile scores and used whiskers to identify minimum and maximum scores.

Interreplicate variability metric scores were generally similar between WM200SS ? and WM200SS tension. However, this profile ? tended to be less variable than this profile tension when interpreting in terms of sensor values. The median ASC score for profile ? was 0.7 times the median ASC score for profile tension (fig. 5a), whereas SDAT scores were 0.2 d lower on average for profile ? than for profile tension (fig. 5f). This difference was likely attributed to less skewing of profile ? than profile tension by depths that were dry outliers. On farms where irrigation is scheduled based on profile tension, the above hypothesis suggests that a slight but easy reduction in scheduling uncertainty might be achieved by calculating profile ? using an approximate soil water characteristic curve and then converting back to the equivalent tension using the same curve.

Further Considerations

To clarify, the purpose of the eight alternate metrics in this study is to characterize the interreplicate variability of sensing systems intended for irrigation scheduling. The scores of interreplicate variability metrics do not denote crop water status and thus do not inform users about optimal irrigation timing. On the contrary, sensing systems achieving lower scores of the interreplicate variability metrics are expected to be more reliable in recommending irrigation dates with the same number of replicates.

The key strength of ASC, SSC, and CCV is a low data requirement. ASC and CCV require just two measurement times to be shared by the sensors under comparison. SSC requires an additional measurement time representing wet conditions, which are prevalent early in the season for humid and subhumid climates. The weaknesses of ASC, SSC, and CCV include conceptual abstraction and sensitivity to near-zero changes. When computed over daily intervals, the scores can be highly right-skewed and difficult to understand without much context. To mitigate these weaknesses, one idea would be to focus on intervals far from rain events, with typical evapotranspiration rates, and at intermediate dryness—within the range where irrigation might be considered. Longer intervals are also an option.

Requiring both a measurement time representing wet conditions and a measurement time representing dry conditions, SDRV and SDRC are more demanding. The selection of the dry benchmark is especially important. Because relative values are constrained between 0 and 1, the dry benchmark must be dry enough so that the range where irrigation might be considered would be assigned intermediate relative values. Otherwise, relative values would be clustered around 1 at these times of interest, distorting SDRV and SDRC scores and reducing the utility of these metrics. The datasets in this study would suggest that withholding rain (using rainout shelters or other controlled environments as befitting) and irrigation for at least an extra week beyond these times of interest may be ideal for observing appropriate dry benchmarks. In cases where observing these dry benchmarks in situ may be too challenging, methods to estimate the dry benchmark would be beneficial both for the computation of SDRV and SDRC and for irrigation scheduling based on relative value triggers. Without the knowledge of appropriate dry benchmarks, relative values cannot be computed, which in turn prevents the establishment and adoption of relative value triggers. Strengths of the SDRV and SDRC metrics include the simplicity of the standard deviation and most users’ greater comprehension of relative value and changes than sensor value and changes.

The greatest strength of SDAT, SDST, and SDRT is ease of understanding. The spread in time when the replicates, respectively, reached a shared irrigation trigger is an intuitive way to express interreplicate variability. However, waiting for all replicates to reach a particular trigger inherently requires drying cycles that extend beyond the point when the interreplicate mean reaches this trigger. In other words, computing these three metrics at intended triggers necessitates specially collecting sensor data while implementing drier-than-intended triggers. For assessing SDAT, SDST, and SDRT, datasets implementing intended triggers are unsuitable, whereas datasets with less frequent or no irrigation are preferred, just as described in the earlier discussion on assessing SDRV and SDRC.

In this study, the interreplicate variability of irrigation scheduling sensors was compared at shared measurement times during a shared experiment. However, pooling together separate datasets to conduct a metaanalysis can be of interest. All eight metrics are compatible with metaanalyses, but intercomparisons should be performed exclusively when all datasets are at similar crop water status. Furthermore, intercomparisons of ASC, SSC, and CCV should focus on intervals when all datasets are undergoing similar magnitudes of change in crop water status. Likewise, intercomparisons of SDRV and SDRC should employ a standardized definition of wet and dry conditions, such as field capacity/satiation and permanent wilting point (ASABE Standards, 2019).

Regrettably, the eight metrics are not completely free from dependence on sensor category and calibration accuracy. These metrics assume that every replicate of every sensor reports values that change somewhat linearly with crop water status. Whenever this assumption is substantially invalid, performing nonlinear transformations (e.g., logarithmic transformation) prior to using these metrics may be helpful. In dataset B, the tension-versus-? slope of the assumed soil water characteristic curve was almost an order of magnitude steeper at the wet end of the observed range (~10 kPa) than at the dry end of the observed range (~100 kPa). Nevertheless, the present analyses of this dataset did not notice dramatic differences in interreplicate variability metric scores between WM200SS tension and WM200SS ? (fig. 5). Computing these metrics over daily intervals may have masked some of the nonlinear effects because small daily changes in a nonlinear variable can be closely approximated as changes in a linear variable. Analogously, the magnitude of nonlinear effects may increase with higher interreplicate variability because small interreplicate differences in a nonlinear variable can be closely approximated as interreplicate differences in a linear variable. Such issues/remedies and other limitations of these metrics will be elucidated as these metrics are applied in additional studies and examined by other colleagues.

Conclusions

This study has applied eight interreplicate variability metrics to compare different collections of irrigation scheduling sensors, including: (1) multiple ?-reporting sensors at a single depth; and (2) a ?-reporting sensor and a tension-reporting sensor, each as a profile average of multiple depths. These cases constitute merely a small subset of the vast opportunities offered by the metrics to evaluate the interreplicate variability of irrigation scheduling sensors across categories and calibration accuracy levels, which has been traditionally challenging. The interreplicate variability metrics were found to be useful not just in ranking sensors by reliability for recommending irrigation timing given an equal number of replicates. These metrics also shed light on the underlying nature of the value differences across replicates and time.

Though the main goal of this study was to demonstrate the use of these metrics, general implications for irrigation scheduling can be drawn from some of the results. One optimistic finding is that, across the two studied datasets, interreplicate differences in triggering dates were shown to be reduced by adopting shifted value triggers and especially relative value triggers rather than sensor value triggers. To take advantage of this finding, more work is needed to investigate the estimation of the dry benchmark and the establishment of relative value triggers. One pessimistic finding is that interreplicate differences in triggering dates can persist to be unacceptably large, even with relative value triggers. Because some users schedule irrigation with one replicate of one sensor, scores of triggering date metrics above 1 d hint at substantial risk of imprecise recommendations of irrigation timing. More work is needed to constrain these recommendations by developing sensors and deployment methods with lower interreplicate variability, by interpreting sensor data in ways more resilient to interreplicate variability (than sensor values, shifted values, and relative values are), and by integrating multiple information sources. Continual innovations to decrease the cost and inconvenience of adding more sensor replicates are also needed.

Acknowledgments

This article is a contribution of the National Center for Alluvial Aquifer Research and the Mississippi Agricultural and Forestry Experiment Station. This study was sponsored by the Agricultural Research Service, United States Department of Agriculture, under Cooperative Agreement 58-6066-2-023. The authors thank Turner Dorr, Jacob Nickel, Deepti Upadhyaya, Italo Pinho de Faria, Jim Nichols, and Todd Russell for assisting with the field experiments. The authors are grateful to the Nebraska State Climate Office and the Delta Agricultural Weather Center for collecting and sharing weather data. This publication is dedicated to the memory of Dr. Freddie Lamm (1955-2022), whose kindness and contributions to ASABE and the irrigation community will be remembered.

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