The study was conducted across three study sites in Tasmania, Australia: Cressy (41°41’9”S, 147°4’49”E); Forthside (41°13′32.16″S, 146°16′26.22″E); and Gunns Plains (41° 17’ 0”S, 146° 3’ 0”E). The three sites were selected as they represent Tasmania’s climatically diverse potato growing regions. Cressy, Forthside and Gunns Plains are classified as low- (~610 mm y^-1), moderate- (~980 mm y^-1) and high-rainfall (~1330 mm y^-1) environments, respectively, based on their cumulative long-term annual rainfall. Owing to climatic, landscape and geological variations, a variety of soil types (Isbell, 2016) exist in Tasmania. Kurosol, Dermosol, Chromosol, Ferrosol and Sodosol are the main soil types which are found in 74.3% of the areas suitable for growing potato (Ojeda et al., 2020).

For this study, we created a complete factorial between soil type and site to explore the impact of soil type on model structural uncertainty. By testing all possible combinations, we can assess how each factor and their interactions contribute to the model structural uncertainty.

Different models of crop (PMF and nPMF), soil water (SoilWat and SWIM3) and irrigation (IM1 and IM2) were used to create eight independent model structures ( Table S1 ) within APSIM Classic (v7.10) (Chapagain et al., 2023). Simulation of each model structure were carried out for the three soil types in the selected sites from 1900 to 2020 (i.e. 3 soils × 3 locations × 8 model structures × 121 years = 8,712 simulated growing seasons).

Two potato models in APSIM were used for this study. In the first potato model, Plant Modelling Framework (PMF) was used for model development (Brown et al., 2011), whereas a legume-based modelling framework (Robertson et al., 2002) was used for model development in the second potato model (nPMF) (Ridwan Saleh, 2009). These models not only differ in their programming languages, but also they differ in internal processes such as: evapotranspiration; dry matter production; phenological stage determination; nitrogen uptake and partitioning (Chapagain et al., 2023). More details about the characteristics and structure of these two potato models can be found in Brown, Huth (Brown et al., 2011) and Ridwan Saleh (2009), respectively.

SoilWat (Probert et al., 1996) and Soil Water Infiltration and Movement v3.0 (SWIM3) (Huth et al., 2012) are two soil models available in APSIM that have different levels of complexity and used different modelling approaches (Vogeler et al., 2022; Chapagain et al., 2023). SoilWat calculates soil water dynamics using a cascading water balance model, whereas SWIM3 uses the Richards equation approach. SWIM3 has been developed in such a way that it uses the same inputs as SoilWat for processes such as soil water retention and runoff which allows SWIM3 to use the existing soil databases for SoilWat. Further, it also helps users of SoilWat to use SWIM3 (Huth et al., 2012; Chapagain et al., 2023). More details about these soil water models can be found in Probert, Dimes (Probert et al., 1996) and Huth, Bristow (Huth et al., 2012), respectively.

This study used two irrigation models (IM1 and IM2) previously described by Chapagain, Huth (Chapagain et al., 2023) which employed manager scripts in APSIM. Following farming practices described by Ojeda, Rezaei (Ojeda et al., 2021b), two models were developed to provide 15 mm of irrigated water during each irrigation schedule between 3^rd November and 7^th February. 15 mm is the required deficit in available soil water to apply irrigation and the amount to irrigate (i.e. 15 mm is added if the deficit is above 15 mm when there is an irrigation opportunity). In the case of IM1, a fixed irrigation schedule was established, occurring every 3 days, but irrigation was only implemented when the soil water deficit (SWD) exceeded 15 mm. IM2, on the other hand, computed SWD daily and initiated irrigation when SWD surpassed 15 mm. The management parameters used for SWD computation, the quantity of irrigation water, and irrigation efficiency were the same for both models, with the variations lying in their respective decision-making logic (Chapagain et al., 2023).

The details of the models are presented in Table 1 . Model complexity is assessed based on the comparison of the variance between different model structures. The crop model, soil water balance model and irrigation model used to create different model structures in this study have different complexities. The soil and crop models have many more state variables, processes affecting these state variables, and parameters and lines of code used to describe the processes. Thus, they have greater complexity as described by Snowling and Kramer (2001). The irrigation model is smaller. As a model, it is similar in size/complexity to some components used as building blocks for the larger crop model.

Out of the five main soils found in more than 70% of potato growing areas (Ojeda et al., 2020), we used the three most dominant soils of Tasmania’s potato regions: Red Ferrosol (RFr), Grey Kurosol (GKr) and Red Dermosol (RDm). Approximate FAO equivalent of these soils are Ferralsols, Alisols and Chernozems respectively (Schad, 2016), whereas USDA equivalent are Oxisols, Vertisols and Borolls respectively. Observed data from Hinton, Harrison (Hinton et al., 2018) were used to create soil profiles in APSIM (see Table S2 for more details) which consists of: bulk density; drained upper limit (DUL); drained lower limit (LL15); saturated volumetric water content (SAT); soil pH; organic carbon; and electric conductivity. Maximum plant available water capacity (PAWC) is computed as the difference between DUL and LL15. Among soil types, PAWC ranged from 122 mm in Red Ferrosol (from 0 to 110 cm soil depth) to 201 mm in Red Dermosol (from 0 to 90 cm soil depth).

The daily weather inputs used in APSIM simulations were: precipitation (mm); potential evapotranspiration (mm); maximum and minimum temperature (°C); solar radiation (MJ m^{− 2}); and vapor pressure (hPa). There were separate timeseries for each of the three sites, retrieved for 1900-2020 from the Scientific Information for Land Owners (SILO) database (longpaddock.qld.gov.au/silo/gridded-data) (Jeffrey et al., 2001). In the SILO dataset, the gaps in observational data are filled using interpolation methods, generating continuous data needed for crop modelling.

The crop management inputs used in this study represents the practices of the potato growing areas in Tasmania based on the mean across 112 commercial farm districts during growing seasons between 2003 to 2007 (Ojeda et al., 2021b). Potatoes were harvested upon reaching full senescence. Detailed description of crop management inputs used for the study is given in Table 2 .

The uncertainty analysis was carried out on the following five outputs: (i) dry tuber yield; (ii) irrigation; (iii) water drainage; (iv) nitrogen leaching; and (v) PGM. Except PGM, all other variables are direct outputs in APSIM. Thus, PGM was computed as follows:

where yield is the dry potato tuber yield (t ha^-1); price is the income generated per ton of tubers (342 USD t^-1, a constant in this study); irrigation is the cumulative irrigation demand from planting to full crop senescence and irrigation cost is the cost associated with irrigation water (35 USD (ML)^-1, a constant in this study). Yield and irrigation values were obtained from APSIM output files, whereas price and irrigation cost were retrieved from AgriGrowth Tasmania (2021). The values were originally reported in AUD and a conversion rate of 1 AUD= 0.7 USD (exchange rate of August 2022) was used to convert it to USD.

Analysis of variance (ANOVA) approach (Iversen et al., 1987; Sawyer, 2009) was applied to quantify the sources of structural uncertainty in different soils. A 3-way ANOVA was carried out with three factors: crop model; soil water model; and irrigation model. The analysis also included the interactions between these elements, to partition the total variance in the selected model outputs resulting from various model structures (Equation 2). The actual uncertainty was expressed in terms of standard deviation.

where, σ² _mo = total variance in model output (yield, irrigation, drainage, nitrogen leaching and PGM for this study) due to cm (crop model), swm (soil water model), im (irrigation model) and int (interactions between them);

where: TSS = total sum of squares; and SS= sum of squares.

The absolute difference in simulated output due to the difference in choice of model is calculated using Equation 7.

where: MO_diff = Absolute difference in simulated model output due to the difference in choice of model; O_A = simulated output due to model A; O_B = simulated output due to model B.

Irrespective of soil types, crop model contributed the most to the mean proportion of variance in most model outputs as compared to other models (see Figure 2 and Table 3 ). Structural uncertainty resulting from choice of crop model was above 60% for yield, drainage, nitrogen leaching and PGM in all soil types (from 78.3% to 92.2% for Red Ferrosol, from 60.2% to 92.2% for Grey Kurosol and from 63.3% to 93.9% for Red Dermosol). A notable exception was irrigation, where the choice of irrigation model contributed the largest percentage of the structural uncertainty (65.6% for Red Ferrosol, 64.9% for Grey Kurosol and 47.0% for Red Dermosol). The contributions of soil water model and irrigation model varied depending on the soil type and model output (see Figure 2 and Table 3 ). For example, soil water model contributed 15.4% and 12.7% for drainage and nitrogen leaching in Red Ferrosol, whereas 3.5% and 6.6% in Grey Kurosol and 30.7% and 23.5% in Red Dermosol for the same model outputs.

First order effects (>88%) due to the choice of model components were found to be dominant over second order interactions (<12%) between model components. Further, interaction between them in Grey Kurosol was always greater than Red Ferrosol and Red Dermosol ( Table 3 ). Additionally, model complexity did not affect uncertainty arising from the choice of model. For example, uncertainty resulting from choice of irrigation model (quite simple models) contributed the largest for irrigation (47-65.6%) as compared to crop model (20.5-49.4%) and soil water model (0.1-2.7%) ( Figure 2 ).

The magnitude of model uncertainty, expressed in terms of standard deviation (SD), differed depending on the output assessed ( Table 4 ). For instance, SD for simulated yield was 0.4-1 t ha^-1 for crop model, 0.3-0.9 t ha^-1 for soil water model and 0.2-0.9 t ha^-1 for irrigation model, whereas the contribution of these three contributors for simulated drainage was 9.9-30.6 mm, 7.5-15.5 mm and 7.1-16.6 mm, respectively. In addition, a significant influence of environmental conditions was observed on the SD of simulated outputs ( Table 4 ).

In absolute terms, crop model was the dominant source of variance in most model outputs ( Figure 3 , Figure S1 , Figure S2 , Figure S3 ), except for irrigation demand where the irrigation model was the largest contributor of uncertainty ( Figure 4 ). Overall, the absolute difference in model outputs was markedly influenced by soil type and site ( Figures 3 , 4 ; Figures S1 – S3 ). For example, for soil types, the variation in simulated nitrogen leaching due to difference in crop models ranged between 0 kg ha⁻¹ and 37.1 kg ha⁻¹ for Red Ferrosol, 0 kg ha⁻¹ and 35.9 kg ha⁻¹ for Grey Kurosol 0 kg ha⁻¹ and 51.5 kg ha⁻¹ for Red Dermosol, followed by soil model and irrigation model. Conversely, the range was from 0 kg ha⁻¹ to 18.5 kg ha⁻¹ for Cressy, from 0 kg ha⁻¹ to 33.7 kg ha⁻¹ for Forthside and from 0 kg ha⁻¹ to 51.5 kg ha⁻¹ for Gunns Plains, followed by soil model and irrigation model in case of sites.

The mean and standard deviation (SD) of simulated model outputs varied for each model structure ( Table S3 ). In general, the model developed using combination of the PMF model with SoilWat and IM2 (P2_SW_M2) resulted the highest absolute values for yield (16.7 t ha^-1), irrigation (346.6 mm) and PGM (5586.9 USD ha^-1) across different types of soil. Conversely, the nPMF model combined with SWIM3 and IM1 (P1_SM_M1) produced the lowest absolute values for yield (9.9 t ha^-1), irrigation (128.1 mm), and PGM (3285.9 USD ha^-1). Similarly, the model developed using combination of the nPMF model with SoilWat and IM2 (P1_SW_M2) yielded the highest absolute values for drainage (110.1 mm) and nitrogen leaching (9.9 kg ha^-1), whereas the PMF model combined with SWIM3 and IM1 (P2_SM_M1) produced the lowest absolute values for drainage (2.9 mm) and nitrogen leaching (0.1 kg ha^-1).

Density distributions, presented in Figure 5 (and also in Figures S4–S7 ), consistently showed differences in model outputs between soil types and site conditions. The density distributions were positively skewed for all outputs. For yield and PGM, variability was inversely proportional to amount of rainfall and PAWC ( Figures 5 , S5 ). Compared to other sites and soil types, Cressy and Red Ferrosol showed a greater degree of variation. This might be as Cressy receives low rainfall compared to the other two sites and Red Ferrosol has the lowest PAWC among the soil types. Under wetter conditions (the lower right quadrant of the four panels in Figure 5 ), variability in yield and PGM was dependent mainly on choice of crop model. However, in case of drainage and nitrogen leaching, the distributions concentrated more towards zero, with higher values occurring during wetter seasons ( Figures S6 , S7 ).

Our research highlights that environment (soil type × site) has a significant impact on the magnitude of structural uncertainty for the different model outputs. This is reflected from the analysis of relative and actual variance. Our findings align with previous research that indicate soil and weather conditions have marked influence on uncertainty of model outputs (Waha et al., 2015; Coucheney et al., 2018; Ojeda et al., 2020; Rettie et al., 2022; Chapagain et al., 2023).

Uncertainty due to the choice of crop model was overall the largest source of uncertainty in simulated outputs. This may be because several differences exist between these crop models ( Table 1 ), including dry matter production, phenological stages, calculation of evapotranspiration and nitrogen uptake (Chapagain et al., 2023). When uncertainty due to crop models is the largest source of uncertainty, it is concerning because it implies that the predictions made using these models can differ substantially, which can have significant implications in choosing appropriate models for decision-making in agriculture, such as determining crop management practices or policy decisions. If the models are predicting differently, the decisions based on them may not be optimal, leading to potential negative impacts on food security, economic outcomes and environmental sustainability. Therefore, it is essential to identify and address the sources of uncertainty in cropping systems. Our results are consistent with previous uncertainty assessment studies (Araya et al., 2015; Li et al., 2015; Yin et al., 2015; Wang et al., 2017; Tao et al., 2018; Rettie et al., 2022). Yin, Tang (Yin et al., 2015) used four crop models with five global climate models (GCMs) to quantify how climate change will affect China’s major crops in the future and found that crop models provide a greater degree of uncertainty in yield than differences between GCMs in most parts of China. Similarly, Araya, Hoogenboom (Araya et al., 2015) found most variations in maize yields were attributed to the choice of crop models while investigating the effect of climate change on maize yield using two crop models and 20 GCMs in Ethiopia. However, the results reported in this paper are probably one of the first analyses which looked at uncertainty of model structure using the same modelling platform in a vegetable crop such as potato across different environments.

The uncertainty due to choice of model was not necessarily affected by model complexity, which was assessed by comparing the variances across different model structures. It is notable that uncertainty from changes to selection of the simple model can have similar impacts to choice regarding the larger models. For example, irrigation model was the most significant source of uncertainty for irrigation results, despite these being simple models (see Figures 2 , 4 ) as compared to soil water models and crop models ( Table 1 ). This result aligns with Ramirez-Villegas, Koehler (Ramirez-Villegas et al., 2017) where the authors assessed model complexity and uncertainty using two versions of GLAM model for Indian groundnut and reported that skill improved only marginally and in small areas as a result of added complexity.

We highlighted the fact that higher mean proportion of variance does not necessarily equate with higher magnitude of uncertainty in actual terms ( Figures 2 – 4 , S1 – S3 and Table 3 ) and should not be the only factor considered in uncertainty assessments. Although the choice of crop model significantly affects both yield and PGM variance (accounting for over 90% of the impact), the narrow ranges of simulated yield (0.2 to 1 t ha^-1) and PGM (50.6 to 374.4 USD ha^-1) standard deviations in relation to the mean values (yield: 14.6 t ha^-1, PGM: 4901 USD ha^-1) indicate relatively low uncertainty in these values. Similarly, while the choice of irrigation model contributes more than 45% to the variance, the relatively small standard deviations (ranging from 11 to 33.3 mm) compared to the overall mean irrigation of 500 mm suggest a low level of actual uncertainty in these values. In contrast, when considering environmental variables such as drainage and nitrogen leaching, the crop model choice accounts for over 60% and 70% of the variance, respectively. However, the comparatively large standard deviations (ranging from 7.1 to 30.6 mm for drainage and 0.6 to 7.7 kg ha^-1 for nitrogen leaching) compared to the overall mean values of drainage (44.4 mm) and nitrogen leaching (3.2 kg ha^-1) indicate a significant level of actual uncertainty in these variables. Thus, it is important to consider the proportion of variance as well as the actual variance (i.e., in the unit of the variable) to provide more accurate and reliable estimates of uncertainty surrounding model outputs. Examining the output variance in relation to the different model structures and variance of inputs can contribute to understanding the model’s sensitivity to changes in those different structures and inputs. However, comparing output variance to the variance of specific inputs is not sufficient for making judgments and should be complemented with a comprehensive evaluation that encompasses multiple validation measures to ensure a robust assessment of the accuracy and reliability of crop models.

Different methods and indicators may be appropriate in different contexts depending on the type of data and the specific uncertainties involved. For example, in some cases, sensitivity analysis, scenario analysis or probabilistic modelling may be useful in assessing uncertainty. In other cases, expert judgment, historical data analysis or qualitative assessments may be more appropriate. By considering a range of methods and indicators, decision-makers can develop a more nuanced understanding of the uncertainties they are facing and take appropriate steps to manage or mitigate those uncertainties. This can ultimately lead to better outcomes and more effective decision-making.

Compared to other sites and soils, Cressy, which is a low rainfall site and Red Ferrosol, which has lowest PAWC, exhibited significantly higher variability in our study which emphasizes the inverse relationship of variability in yield and PGM to the amount of rainfall and PAWC ( Figures 5 , S5 ). However, for drainage and nitrogen leaching metrics, wetter conditions provided the larger uncertainty range. Understanding how the range in variability from model outputs is related to rainfall and PAWC for a target site is important for farmers and land managers because it can help inform model selection, which then will inform crop selection and management decisions. Our results highlight how drier (wetter) conditions expose differences within crop models at various stages of modelling whereas, wetter (drier) conditions mask them. If models cannot perform well under drier (wetter) conditions, decision makers may not be able to accurately predict the impacts of dry (wet) conditions on crop yields, food production or environmental impacts. This can lead to suboptimal decision making. For example, if a region has high rainfall and high PAWC, farmers will need different modelling tools to optimize the selection of crop varieties that are more tolerant to wet conditions or implementing drainage systems to mitigate the effects of excess water. Similarly, in regions with low PAWC, farmers need to target modelling tools that optimize handling for practices such as conservation tillage or crop rotations to improve soil water retention and reduce the risk of yield losses during dry periods. Getting these decisions wrong can lead to inefficient use of resources, such as irrigation water or fertilizers, which can further exacerbate the impacts of dry (wet) conditions. Thus, it is necessary to consider the conditions and factors most important to a user, and test and optimize a modelling system for this use case.

PAWC is a constant value in the model, it is calculated as the difference between DUL and LL15 for each soil layer. What is variable is the soil moisture each day depending on the rainfall, evapotranspiration, runoff, drainage, irrigation, etc. Hence, PAWC is static and rainfall and actual soil moisture are dynamic but highly affected by irrigation time and amount. When actual soil moisture is close to DUL, the maximum PAWC is achieved and there is no water stress. Irrigation’s purpose is to limit damaging water stress to a plant. This inherently reduces the differences between soil types, as if soil moisture is constantly kept at an ideal level using optimal irrigation scheduling. Irrigation will reduce the impact of irrigation on yield through variation in soil PAWC. However, varying PAWC will change hydraulic behavior and therefore directly impact leaching losses etc. Varying PAWC will also affect irrigation timing because soils that retain water will require less irrigation (i.e. affect the irrigation trigger, thus affecting the timing of irrigation, affecting the volume of irrigation, thus affecting leaching losses or possible yield impact).

Soil properties affect the movement of water, the availability of water to plant roots and the overall water-holding capacity of the soil. They influence plant growth, nutrient availability and overall performance of crop models and can still impact simulations even if soil moisture is set close to field capacity. There could still be some water stress if the cumulative water demand exceeds the maximum irrigation amount. In dry environments, there could potentially be brief periods of stress. More importantly, the amount of drainage and therefore, nitrogen leaching can be affected. There is potential for nitrogen loss through leaching from the crop. Similarly, denitrification can change in supply, though the amounts could be smaller. Nitrogen losses impact environmental and economic outcomes through loss of fertilizer. Additionally, drainage losses have potential environmental impacts and economic losses as a result of irrigation water losses, which is directly related to increase in water and electricity costs.

This research investigates how models perform in conditions with varying moisture availability, specifically drier (characterized by low rainfall and low PAWC) or wetter (characterized by high rainfall and high PAWC) conditions, through the use of modelling simulations. However, our study does not include a comparison between the simulated results and observations because our primary focus was solely on understanding structural uncertainty and hence, observation was not included to avoid the introduction of observation uncertainty. This absence of a comparison may hinder a comprehensive evaluation of the models’ performance and reliability, and therefore, it should be incorporated into future assessments.

Making agronomic decisions can be challenging when there are differences in outputs arising from differences in model structures. Crop models can help decision-makers to identify the most appropriate agricultural management practices and technologies that balance the trade-offs between different objectives over the short and long term. Ultimately, this can help to ensure sustainable agricultural production that meets the needs of society while minimizing negative impacts on the environment. For example, compared to the nPMF model, the PMF model overpredicted potato tuber yield, irrigation, and PGM, but underpredicted nitrogen leaching and drainage suggesting the two models have different strengths and weaknesses in our study. PMF model may be better at predicting certain aspects of potato production, such as tuber yield, irrigation demand and PGM, but may not perform as well when it comes to predicting environmental impacts such as nitrogen leaching and drainage. On the other hand, nPMF model may be more accurate in predicting environmental outcomes but may not perform as well in predicting yield, irrigation demand and PGM. Thus, our study highlights the need to carefully weigh the pros and cons of using each model and decide which model to use based on the specific needs and goals of the agricultural production system.

The findings of this study contribute to the understanding of prevalent challenge in uncertainty research - the quantification of structural uncertainty in crop model predictions caused by different modelling structures or configurations within the same platform. Although, there are large-scale research efforts such as AgMIP (Rosenzweig et al., 2013), FACCE-JPI (Gøtke et al., 2015) and ISI-MIP (Warszawski et al., 2014) which aim to improve the accuracy and reliability of crop modelling by comparing and evaluating the results from different models applied to the same experimental data or field conditions, our approach of uncertainty quantification is quiet distinct when compared to the approach by these initiatives. Our method involves quantifying uncertainty in both processes and structure using a single modelling platform to account for uncertainty that stems from model algorithms and equations while preventing the inclusion of other uncertainties that result from variations in modelling platforms or the different ways in which different models take input data and provide outputs (Chapagain et al., 2023). Additionally, we have considered multiple agronomic, environmental, and economic outputs to demonstrate how model performance can differ depending on the output analyzed and a careful analysis is required considering the trade-offs between them for informed decision-making. For example, by considering environmental results like nitrogen leaching and drainage, along with yield and partial gross margin, we can perform a balanced analysis and opt for a model that provides the most suitable outcomes for a particular objective. The techniques described in this article have the potential to be used on multiple modelling systems, settings, and crops to anticipate a single or multiple outcomes.

The results of our study should, however, be interpreted considering some limitations. We acknowledge that this study only considered and analyzed the influence of soil type on structural uncertainty of simulated outputs and there may be other factors such as irrigation strategy, sowing date and genotype which may have more significant impact. Secondly, we have developed the different model structures within the same modelling platform (APSIM) and results may differ depending on the crop model used. Additionally, our investigation was restricted to three soil types and three sites, although these three soils are the most prevalent in Tasmania, and we included the three sites to account for a broad range of rainfall conditions. Furthermore, the scope of this study is focused on investigating the structural uncertainty that arises from different model structures within the same modelling platform across different environments. The aim is to analyze how different model structures can lead to variance in simulated model outputs. In this regard, the study does not involve comparing the simulation results with observations. This decision was made because observations themselves inherently possess a certain level of uncertainty, and examining this uncertainty was beyond the intended scope of the study. Instead, the primary objective is to explore the impact of different model structures on the simulation outcomes across different soils and sites, thus providing insights into the structural uncertainties within the same modelling platform. In the future, the analysis framework could be expanded to various spatial scales and environments and upscaled to regional and national levels by employing gridded data.