# Linking Remote Sensing with APSIM through Emulation and Bayesian Optimization to Improve Yield Prediction

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Overall Workflow

#### 2.2. Study Sites

#### 2.3. Data Collection

#### 2.3.1. Weather and Soil

#### 2.3.2. LAI and NDVI Retrieval and Data Processing

- Randomly sampled 400,000 combinations of PROSAIL input parameters with bounds shown in Table A2 and weights for the soil model.
- Using PROSAIL to compute top of canopy reflectance with the randomly sampled inputs and the simulated soil background.
- The simulated reflectance is convolved with the Sentinel-2 relative spectral response functions to simulate surface reflectance at different bands (addB02, B03, B04, B05, B06, B07, B08, B8A, B11, B12).
- Train NN for LAI by using the simulated reflectance over Sentinel-2 bands as inputs.
- The trained NN is then used to map from satellite surface reflectance to LAI.

#### 2.3.3. Crop Growth Model

#### 2.4. Model and Parameter Selection

#### 2.5. Emulator Development

_{i}represents the ith most sensitive parameter. The emulators were fitted independently at each site and predict an LAI time series at days with available observations as a function of the most sensitive parameters.

#### 2.6. Optimization Schemes

## 3. Results

#### 3.1. Sensitivity Analysis and Emulator Performance

#### 3.2. Within Sample Prediction

#### 3.3. Comparing Posterior Distributions

#### 3.4. Out-of-Sample Prediction

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Library | Type | Reference |
---|---|---|

USGS v7 | Measurements | [59] |

ICRAF-ISRIC | Measurements | [60] |

Price | Soil model | [61] |

Prosail | Soil model | [36] |

Traits Type | Parameter | Symbol | Range | Unit |
---|---|---|---|---|

Canopy structure | leaf area index | $LAI$ | 0–8 | m${}^{2}$/m${}^{2}$ |

leaf angle distribution function | $ALA$ | 0–90 | ${}^{\circ}$ | |

Leaf optical | chlorophyll a and b content | ${C}_{ab}$ | 0–120 | $\mathsf{\mu}$g/cm${}^{2}$ |

Carotenoid content | ${C}_{ar}$ | 0–25 | $\mathsf{\mu}$g/cm${}^{2}$ | |

Anthocyanin content | ${C}_{an}$ | 0 | $\mathsf{\mu}$g/cm${}^{2}$ | |

leaf dry matter per leaf area | ${C}_{m}$ | 0.002–0.01 | $\mathsf{\mu}$g/cm${}^{2}$ | |

leaf water content per leaf area | ${C}_{w}$ | 0–0.04 | mg/cm${}^{2}$ | |

brown pigment content | ${C}_{brown}$ | 0–1 | - | |

mesophyll structure coefficient | N | 1–2.5 | - | |

angles | solar zenith angle | sza | 0–80 | ${}^{\circ}$ |

viewing zenith angle | vza | 0–15 | ${}^{\circ}$ | |

relative azimuth angle | raa | 0–360 | ${}^{\circ}$ |

Parameter | Upper Bound | Lower Bound |
---|---|---|

tt_flower_to_maturity | 1100 | 700 |

tt_emerg_to_endjuv | 500 | 200 |

tt_flower_to_start_grain | 400 | 100 |

leaf_app_rate1 | 65 | 40 |

leaf_app_rate2 | 50 | 20 |

leaf_app_rate3 | 50 | 20 |

largestLeafParams1 | −1 | −2 |

largestLeafParams2 | 0.05 | 0.03 |

y_lai_sla_max1 | 55,000 | 45,000 |

y_lai_sla_max2 | 30,000 | 20,000 |

leaf_init_rate | 30 | 15 |

rue | 3 | 1.6 |

grain_gth_rate | 11 | 8 |

initial_tpla | 450 | 300 |

**Figure A2.**Difference in the posterior distribution of parameters with LAI only, and both LAI and NDVI data constraints in HPDA.

**Figure A3.**95 % confidence interval for NDVI predictions made by the emulators across all the sites using the optimized parameters obtained from different optimization schemes.

**Figure A4.**Density plots comparing observed yield versus APSIM predicted maize yield for two optimization schemes.

**Figure A5.**Difference in the posterior distribution of parameters with LAI only and both data constraint in HPDA.

## References

- Gill, H.S.; Halder, J.; Zhang, J.; Brar, N.K.; Rai, T.S.; Hall, C.; Bernardo, A.; Amand, P.S.; Bai, G.; Olson, E.; et al. Multi-Trait Multi-Environment Genomic Prediction of Agronomic Traits in Advanced Breeding Lines of Winter Wheat. Front. Plant Sci.
**2021**, 12, 1619. [Google Scholar] [CrossRef] - Su, W.; Zhang, M.; Bian, D.; Liu, Z.; Huang, J.; Wang, W.; Wu, J.; Guo, H. Phenotyping of corn plants using unmanned aerial vehicle (UAV) images. Remote Sens.
**2019**, 11, 2021. [Google Scholar] [CrossRef] [Green Version] - Dokoohaki, H.; Gheysari, M.; Mehnatkesh, A.; Ayoubi, S. Applying the CSM-CERES-Wheat model for rainfed wheat with specified soil characteristic in undulating area in Iran. Arch. Agron. Soil Sci.
**2015**, 61, 1231–1245. [Google Scholar] [CrossRef] - Dietze, M. Ecological Forecasting; Princeton University Press: Princeton, NJ, USA, 2017. [Google Scholar]
- Kivi, M.S.; Blakely, B.; Masters, M.; Bernacchi, C.J.; Miguez, F.E.; Dokoohaki, H. Development of a data-assimilation system to forecast agricultural systems: A case study of constraining soil water and soil nitrogen dynamics in the APSIM model. Sci. Total Environ.
**2022**, 820, 153192. [Google Scholar] [CrossRef] [PubMed] - Dokoohaki, H.; Gheisari, M.; Mousavi, S.F.; Mirlatifi, S.M. Estimation soil water content under deficit irrigation by using DSSAT. Water Irrig. Manag.
**2012**, 2, 1–14. [Google Scholar] - Holzworth, D.P.; Huth, N.I.; deVoil, P.G.; Zurcher, E.J.; Herrmann, N.I.; McLean, G.; Chenu, K.; van Oosterom, E.J.; Snow, V.; Murphy, C.; et al. APSIM–evolution towards a new generation of agricultural systems simulation. Environ. Model. Softw.
**2014**, 62, 327–350. [Google Scholar] [CrossRef] - Jones, J.W.; Hoogenboom, G.; Porter, C.H.; Boote, K.J.; Batchelor, W.D.; Hunt, L.; Wilkens, P.W.; Singh, U.; Gijsman, A.J.; Ritchie, J.T. The DSSAT cropping system model. Eur. J. Agron.
**2003**, 18, 235–265. [Google Scholar] [CrossRef] - Archontoulis, S.V.; Castellano, M.J.; Licht, M.A.; Nichols, V.; Baum, M.; Huber, I.; Martinez-Feria, R.; Puntel, L.; Ordóñez, R.A.; Iqbal, J.; et al. Predicting crop yields and soil-plant nitrogen dynamics in the US Corn Belt. Crop Sci.
**2020**, 60, 721–738. [Google Scholar] [CrossRef] [Green Version] - Saddique, Q.; Zou, Y.; Ajaz, A.; Ji, J.; Xu, J.; Azmat, M.; ur Rahman, M.H.; He, J.; Cai, H. Analyzing the Performance and Application of CERES-Wheat and APSIM in the Guanzhong Plain, China. Trans. ASABE
**2020**, 63, 1879–1893. [Google Scholar] [CrossRef] - Rai, T.; Kumar, S.; Nleya, T.; Sexton, P.; Hoogenboom, G. Simulation of maize and soybean yield using DSSAT under long-term conventional and no-till systems. Soil Res.
**2022**, 60, 520–533. [Google Scholar] [CrossRef] - Dokoohaki, H.; Kivi, M.S.; Martinez-Feria, R.A.; Miguez, F.E.; Hoogenboom, G. A comprehensive uncertainty quantification of large-scale process-based crop modeling frameworks. Environ. Res. Lett.
**2021**, 16, 084010. [Google Scholar] [CrossRef] - Sun, R.; Duan, Q.; Huo, X. Multi-Objective Adaptive Surrogate Modeling-Based Optimization for Distributed Environmental Models Based on Grid Sampling. Water Resour. Res.
**2021**, 57, e2020WR028740. [Google Scholar] [CrossRef] - Wallach, D.; Palosuo, T.; Thorburn, P.; Hochman, Z.; Gourdain, E.; Andrianasolo, F.; Asseng, S.; Basso, B.; Buis, S.; Crout, N.; et al. The chaos in calibrating crop models: Lessons learned from a multi-model calibration exercise. Environ. Model. Softw.
**2021**, 145, 105206. [Google Scholar] [CrossRef] - Fer, I.; Gardella, A.K.; Shiklomanov, A.N.; Campbell, E.E.; Cowdery, E.M.; De Kauwe, M.G.; Desai, A.; Duveneck, M.J.; Fisher, J.B.; Haynes, K.D.; et al. Beyond ecosystem modeling: A roadmap to community cyberinfrastructure for ecological data-model integration. Glob. Chang. Biol.
**2021**, 27, 13–26. [Google Scholar] [CrossRef] - Post, H.; Vrugt, J.A.; Fox, A.; Vereecken, H.; Hendricks Franssen, H.J. Estimation of Community Land Model parameters for an improved assessment of net carbon fluxes at European sites. J. Geophys. Res. Biogeosci.
**2017**, 122, 661–689. [Google Scholar] [CrossRef] [Green Version] - Fer, I.; Shiklomanov, A.N.; Novick, K.A.; Gough, C.M.; Arain, M.A.; Chen, J.; Murphy, B.; Desai, A.R.; Dietze, M.C. Capturing site-to-site variability through Hierarchical Bayesian calibration of a process-based dynamic vegetation model. bioRxiv
**2021**. [Google Scholar] [CrossRef] - Dokoohaki, H.; Morrison, B.D.; Raiho, A.; Serbin, S.P.; Zarada, K.; Dramko, L.; Dietze, M. Development of an open-source regional data assimilation system in PEcAn v. 1.7. 2: Application to carbon cycle reanalysis across the contiguous US using SIPNET. Geosci. Model Dev. Discuss.
**2022**, 15, 1–28. [Google Scholar] [CrossRef] - Huang, J.; Gómez-Dans, J.L.; Huang, H.; Ma, H.; Wu, Q.; Lewis, P.E.; Liang, S.; Chen, Z.; Xue, J.H.; Wu, Y.; et al. Assimilation of remote sensing into crop growth models: Current status and perspectives. Agric. For. Meteorol.
**2019**, 276, 107609. [Google Scholar] [CrossRef] - Jacquemoud, S.; Baret, F. PROSPECT: A model of leaf optical properties spectra. Remote Sens. Environ.
**1990**, 34, 75–91. [Google Scholar] [CrossRef] - Shiklomanov, A.N.; Dietze, M.C.; Fer, I.; Viskari, T.; Serbin, S.P. Cutting out the middleman: Calibrating and validating a dynamic vegetation model (ED2-PROSPECT5) using remotely sensed surface reflectance. Geosci. Model Dev.
**2021**, 14, 2603–2633. [Google Scholar] [CrossRef] - Dokoohaki, H.; Miguez, F.E.; Archontoulis, S.; Laird, D. Use of inverse modelling and Bayesian optimization for investigating the effect of biochar on soil hydrological properties. Agric. Water Manag.
**2018**, 208, 268–274. [Google Scholar] [CrossRef] - Akhavizadegan, F.; Ansarifar, J.; Wang, L.; Huber, I.; Archontoulis, S.V. A time-dependent parameter estimation framework for crop modeling. Sci. Rep.
**2021**, 11, 1–15. [Google Scholar] - Dietzel, R.; Liebman, M.; Ewing, R.; Helmers, M.; Horton, R.; Jarchow, M.; Archontoulis, S. How efficiently do corn-and soybean-based cropping systems use water? A systems modeling analysis. Glob. Chang. Biol.
**2016**, 22, 666–681. [Google Scholar] [CrossRef] [PubMed] - Sheng, M.; Liu, J.; Zhu, A.X.; Rossiter, D.G.; Liu, H.; Liu, Z.; Zhu, L. Comparison of GLUE and DREAM for the estimation of cultivar parameters in the APSIM-maize model. Agric. For. Meteorol.
**2019**, 278, 107659. [Google Scholar] [CrossRef] - Malone, R.W.; Huth, N.; Carberry, P.; Ma, L.; Kaspar, T.C.; Karlen, D.L.; Meade, T.; Kanwar, R.S.; Heilman, P. Evaluating and predicting agricultural management effects under tile drainage using modified APSIM. Geoderma
**2007**, 140, 310–322. [Google Scholar] [CrossRef] [Green Version] - Archontoulis, S.V.; Miguez, F.E.; Moore, K.J. Evaluating APSIM maize, soil water, soil nitrogen, manure, and soil temperature modules in the Midwestern United States. Agron. J.
**2014**, 106, 1025–1040. [Google Scholar] [CrossRef] - Puntel, L.A.; Sawyer, J.E.; Barker, D.W.; Dietzel, R.; Poffenbarger, H.; Castellano, M.J.; Moore, K.J.; Thorburn, P.; Archontoulis, S.V. Modeling long-term corn yield response to nitrogen rate and crop rotation. Front. Plant Sci.
**2016**, 7, 1630. [Google Scholar] [CrossRef] - Ojeda, J.J.; Volenec, J.J.; Brouder, S.M.; Caviglia, O.P.; Agnusdei, M.G. Modelling stover and grain yields, and subsurface artificial drainage from long-term corn rotations using APSIM. Agric. Water Manag.
**2018**, 195, 154–171. [Google Scholar] [CrossRef] - Basche, A.D.; Archontoulis, S.V.; Kaspar, T.C.; Jaynes, D.B.; Parkin, T.B.; Miguez, F.E. Simulating long-term impacts of cover crops and climate change on crop production and environmental outcomes in the Midwestern United States. Agric. Ecosyst. Environ.
**2016**, 218, 95–106. [Google Scholar] [CrossRef] [Green Version] - USDA National Agricultural Statistics Service. NASS—Quick Stats. USDA National Agricultural Statistics Service. Available online: https://data.nal.usda.gov/dataset/nass-quick-stats (accessed on 20 November 2021).
- Kang, Y.; Özdoğan, M. Field-level crop yield mapping with Landsat using a hierarchical data assimilation approach. Remote Sens. Environ.
**2019**, 228, 144–163. [Google Scholar] [CrossRef] - Hengl, T.; Mendes de Jesus, J.; Heuvelink, G.B.; Ruiperez Gonzalez, M.; Kilibarda, M.; Blagotić, A.; Shangguan, W.; Wright, M.N.; Geng, X.; Bauer-Marschallinger, B.; et al. SoilGrids250m: Global gridded soil information based on machine learning. PLoS ONE
**2017**, 12, e0169748. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hersbach, H.; Bell, B.; Berrisford, P.; Hirahara, S.; Horányi, A.; Muñoz-Sabater, J.; Nicolas, J.; Peubey, C.; Radu, R.; Schepers, D.; et al. The ERA5 global reanalysis. Q. J. R. Meteorol. Soc.
**2020**, 146, 1999–2049. [Google Scholar] [CrossRef] - Bolton, D.K.; Friedl, M.A. Forecasting crop yield using remotely sensed vegetation indices and crop phenology metrics. Agric. For. Meteorol.
**2013**, 173, 74–84. [Google Scholar] [CrossRef] - Jacquemoud, S.; Verhoef, W.; Baret, F.; Bacour, C.; Zarco-Tejada, P.J.; Asner, G.P.; François, C.; Ustin, S.L. PROSPECT+ SAIL models: A review of use for vegetation characterization. Remote Sens. Environ.
**2009**, 113, S56–S66. [Google Scholar] [CrossRef] - Müller, C.; Elliott, J.; Kelly, D.; Arneth, A.; Balkovic, J.; Ciais, P.; Deryng, D.; Folberth, C.; Hoek, S.; Izaurralde, R.C.; et al. The Global Gridded Crop Model Intercomparison phase 1 simulation dataset. Sci. Data
**2019**, 6, 50. [Google Scholar] [CrossRef] [Green Version] - Martinez-Feria, R.; Nichols, V.; Basso, B.; Archontoulis, S. Can multi-strategy management stabilize nitrate leaching under increasing rainfall? Environ. Res. Lett.
**2019**, 14, 124079. [Google Scholar] [CrossRef] - Keating, B.A.; Carberry, P.S.; Hammer, G.L.; Probert, M.E.; Robertson, M.J.; Holzworth, D.; Huth, N.I.; Hargreaves, J.N.; Meinke, H.; Hochman, Z.; et al. An overview of APSIM, a model designed for farming systems simulation. Eur. J. Agron.
**2003**, 18, 267–288. [Google Scholar] [CrossRef] [Green Version] - Birch, C.; Hammer, G.; Rickert, K. Improved methods for predicting individual leaf area and leaf senescence in maize (Zea mays). Aust. J. Agric. Res.
**1998**, 49, 249–262. [Google Scholar] [CrossRef] - Jones, C.A. CERES-Maize; a Simulation Model of Maize Growth and Development; Number 04; SB91. M2, J6.; Texas A&M University Press: College Station, TX, USA, 1986. [Google Scholar]
- Keating, B.; Wafula, B. Modelling the fully expanded area of maize leaves. Field Crop. Res.
**1992**, 29, 163–176. [Google Scholar] [CrossRef] - Machwitz, M.; Giustarini, L.; Bossung, C.; Frantz, D.; Schlerf, M.; Lilienthal, H.; Wandera, L.; Matgen, P.; Hoffmann, L.; Udelhoven, T. Enhanced biomass prediction by assimilating satellite data into a crop growth model. Environ. Model. Softw.
**2014**, 62, 437–453. [Google Scholar] [CrossRef] - Duan, S.B.; Li, Z.L.; Wu, H.; Tang, B.H.; Ma, L.; Zhao, E.; Li, C. Inversion of the PROSAIL model to estimate leaf area index of maize, potato, and sunflower fields from unmanned aerial vehicle hyperspectral data. Int. J. Appl. Earth Obs. Geoinf.
**2014**, 26, 12–20. [Google Scholar] [CrossRef] - Richter, K.; Atzberger, C.; Vuolo, F.; D’Urso, G. Evaluation of sentinel-2 spectral sampling for radiative transfer model based LAI estimation of wheat, sugar beet, and maize. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens.
**2010**, 4, 458–464. [Google Scholar] [CrossRef] - Fer, I.; Kelly, R.; Moorcroft, P.R.; Richardson, A.D.; Cowdery, E.M.; Dietze, M.C. Linking big models to big data: Efficient ecosystem model calibration through Bayesian model emulation. Biogeosciences
**2018**, 15, 5801–5830. [Google Scholar] [CrossRef] [Green Version] - De Valpine, P.; Turek, D.; Paciorek, C.J.; Anderson-Bergman, C.; Lang, D.T.; Bodik, R. Programming with models: Writing statistical algorithms for general model structures with NIMBLE. J. Comput. Graph. Stat.
**2017**, 26, 403–413. [Google Scholar] [CrossRef] [Green Version] - Dokoohaki, H.; Gheysari, M.; Mousavi, S.F.; Hoogenboom, G. Effects of different irrigation regimes on soil moisture availability evaluated by CSM-CERES-Maize model under semi-arid condition. Ecohydrol. Hydrobiol.
**2017**, 17, 207–216. [Google Scholar] [CrossRef] [Green Version] - Pasley, H.; Nichols, V.; Castellano, M.; Baum, M.; Kladivko, E.; Helmers, M.; Archontoulis, S. Rotating maize reduces the risk and rate of nitrate leaching. Environ. Res. Lett.
**2021**, 16, 064063. [Google Scholar] [CrossRef] - Martinez-Feria, R.A.; Licht, M.A.; Ordóñez, R.A.; Hatfield, J.L.; Coulter, J.A.; Archontoulis, S.V. Evaluating maize and soybean grain dry-down in the field with predictive algorithms and genotype-by-environment analysis. Sci. Rep.
**2019**, 9, 7167. [Google Scholar] [CrossRef] [Green Version] - Dos Santos, C.L.; Abendroth, L.J.; Coulter, J.A.; Nafziger, E.D.; Suyker, A.; Yu, J.; Schnable, P.S.; Archontoulis, S.V. Maize Leaf Appearance Rates: A Synthesis From the United States Corn Belt. Front. Plant Sci.
**2022**, 13, 872738. [Google Scholar] [CrossRef] - Clark, J.S. Why environmental scientists are becoming Bayesians. Ecol. Lett.
**2005**, 8, 2–14. Available online: http://xxx.lanl.gov/abs/https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1461-0248.2004.00702.x (accessed on 20 November 2021). [CrossRef] - Van Oijen, M. Bayesian methods for quantifying and reducing uncertainty and error in forest models. Curr. For. Rep.
**2017**, 3, 269–280. [Google Scholar] [CrossRef] [Green Version] - Oberpriller, J.; Cameron, D.R.; Dietze, M.C.; Hartig, F. Towards robust statistical inference for complex computer models. Ecol. Lett.
**2021**, 24, 1251–1261. [Google Scholar] [CrossRef] [PubMed] - Cressie, N.; Calder, C.A.; Clark, J.S.; Hoef, J.M.V.; Wikle, C.K. Accounting for uncertainty in ecological analysis: The strengths and limitations of hierarchical statistical modeling. Ecol. Appl.
**2009**, 19, 553–570. [Google Scholar] [CrossRef] [PubMed] - Sexton, J.; Everingham, Y.; Inman-Bamber, G. A theoretical and real world evaluation of two Bayesian techniques for the calibration of variety parameters in a sugarcane crop model. Environ. Model. Softw.
**2016**, 83, 126–142. [Google Scholar] [CrossRef] - Makowski, D.; Wallach, D.; Tremblay, M. Using a Bayesian approach to parameter estimation; comparison of the GLUE and MCMC methods. Agronomie
**2002**, 22, 191–203. [Google Scholar] [CrossRef] - Gao, F.; Zhang, X. Mapping crop phenology in near real-time using satellite remote sensing: Challenges and opportunities. J. Remote Sens.
**2021**, 2021, 8379391. [Google Scholar] [CrossRef] - Pearson, N.C.; Livo, K.E.; Driscoll, R.L.; Lowers, H.A.; Hoefen, T.M.; Swayze, G.A.; Klein, A.J.; Kokaly, R.F.; Wise, R.A.; Benzel, W.M.; et al. USGS Spectral Library Version 7 Data. 2017. Available online: https://dx.doi.org/10.5066/F7RR1WDJ (accessed on 3 March 2020).
- Garrity, D.; Bindraban, P. A Globally Distributed Soil Spectral Library Visible Near Infrared Diffuse Reflectance Spectra; ICRAF (World Agroforestry Centre)/ISRIC (World Soil Information) Spectral Library: Nairobi, Kenya, 2004. [Google Scholar]
- Price, J.C. On the information content of soil reflectance spectra. Remote Sens. Environ.
**1990**, 33, 113–121. [Google Scholar] [CrossRef]

**Figure 1.**The overall workflow of linking APSIM-Maize model with radiative transfer model (PROSAIL) for developing the emulators, optimization procedure and model validation.

**Figure 2.**(

**a**) Geographical location and (

**b**) distribution of basic soil properties of calibration and validation sites. SOC: Soil Organic Carbon.

**Figure 3.**Graphical representation of different optimization schemes explored in this study to constrain model parameters, as adopted from [17].

**Figure 4.**Average sensitivity index for maize leaf area index (LAI) corresponding to 14 model parameters in APSIM across all sites for years 2018 and 2019.

**Figure 5.**Density plots comparing emulator and APSIM estimates of LAI and NDVI across calibration sites. The lighter colors represent higher density of points and darker color represent lower density of points.

**Figure 6.**95% confidence interval for emulator predicted LAI using optimized parameters corresponding to each optimization scheme across all calibration sites.Title for each subplot corresponds to the site ID in Beck’s dataset.

**Figure 7.**95% Confidence interval for within sample yield comparison between different optimization schemes.

**Figure 8.**95% Confidence interval for various crop model parameters across different optimization schemes and sites.

**Figure 9.**Density plots comparing yield estimates for out of sample sites (

**top**), and yield variability (

**bottom**) for global and HPDA optimization schemes.

**Table 1.**Model definition for different optimization schemes explored in this study. Across all schemes, $\mu $ represents the vector of model parameters, k represents observations or parameters for site k, Y represents LAI and NDVI observations, and f represents the emulator.

Optimization Scheme | Model Definition | Hyperprior |
---|---|---|

Site-level |
$${\mu}_{k}\sim N({\mu}_{0},\tau )$$
$${Y}_{k}^{LAI}\sim N({f}_{LAI}\left({\mu}_{k}\right),{\sigma}_{LAI})$$
$${Y}_{k}^{NDVI}\sim N({f}_{NDVI}\left({\mu}_{k}\right),{\sigma}_{NDVI})$$
| ${\mu}_{0}$ |

Hierarchical/HPDA |
$$\mu \sim N({\mu}_{0},\tau )$$
$${\alpha}_{k}\sim N(0,{\tau}_{\alpha})$$
$${\mu}_{k}=\mu +{\alpha}_{k}$$
$${Y}_{k}^{LAI}\sim N({f}_{LAI}\left({\mu}_{k}\right),{\sigma}_{LAI})$$
$${Y}_{k}^{NDVI}\sim N({f}_{NDVI}\left({\mu}_{k}\right),{\sigma}_{NDVI})$$
| ${\mu}_{0}$, ${\tau}_{\alpha}$ |

Global/Joint |
$$\mu \sim N({\mu}_{0},\tau )$$
$${Y}_{k}^{LAI}\sim N({f}_{LAI}\left(\mu \right),{\sigma}_{LAI})$$
$${Y}_{k}^{NDVI}\sim N({f}_{NDVI}\left(\mu \right),{\sigma}_{NDVI})$$
| ${\mu}_{0}$ |

Month | Site-Level | Global | HPDA | ||||||
---|---|---|---|---|---|---|---|---|---|

d-Index | RMSE ($\frac{\mathit{Kg}}{\mathit{ha}}$) | ME ($\frac{\mathit{Kg}}{\mathit{ha}}$) | d-Index | RMSE ($\frac{\mathit{Kg}}{\mathit{ha}}$) | ME ($\frac{\mathit{Kg}}{\mathit{ha}}$) | d-Index | RMSE ($\frac{\mathit{Kg}}{\mathit{ha}}$) | ME ($\frac{\mathit{Kg}}{\mathit{ha}}$) | |

May | 0.48 | 0.52 | −0.43 | 0.48 | 0.51 | −0.32 | 0.49 | 0.51 | −0.42 |

June | 0.96 | 0.51 | −0.24 | 0.95 | 0.58 | −0.17 | 0.97 | 0.46 | −0.1 |

July | 0.91 | 0.52 | 0.07 | 0.84 | 0.75 | −0.02 | 0.91 | 0.55 | 0.19 |

Aug | 0.94 | 0.53 | −0.03 | 0.92 | 0.65 | 0.02 | 0.93 | 0.58 | 0.04 |

Sep | 0.96 | 0.44 | −0.18 | 0.92 | 0.69 | 0.07 | 0.95 | 0.53 | −0.1 |

Average | 0.85 | 0.5 | −0.16 | 0.82 | 0.63 | −0.08 | 0.85 | 0.52 | −0.07 |

**Table 3.**Comparing model performance in simulating crop yield on out of sample sites for outside (2014–2017) and within the (2018–2019) the spatiotemporal extent of training dataset.

Year | HPDA | Global | ||||
---|---|---|---|---|---|---|

d-Index | RMSE ($\frac{\mathit{Kg}}{\mathit{ha}}$) | ME ($\frac{\mathit{Kg}}{\mathit{ha}}$) | d-Index | RMSE ($\frac{\mathit{Kg}}{\mathit{ha}}$) | ME ($\frac{\mathit{Kg}}{\mathit{ha}}$) | |

2014 (n = 61) | 0.52 | 1731 | −982 | 0.62 | 1260 | −154 |

2015 (n = 67) | 0.47 | 2017 | 465 | 0.54 | 1878 | 994 |

2016 (n = 50) | 0.27 | 4061 | −2670 | 0.35 | 2359 | −861 |

2017 (n = 64) | 0.49 | 2322 | −1428 | 0.49 | 1855 | −715 |

Average | 0.44 | 2532 | −1153 | 0.50 | 1554 | −184 |

2018 (n = 51) | 0.7 | 1822 | −571 | 0.59 | 1625 | −72 |

2019 (n = 39) | 0.66 | 1823 | −448 | 0.71 | 1629 | 413 |

Average | 0.68 | 1822 | −509 | 0.65 | 1627 | 170 |

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**MDPI and ACS Style**

Dokoohaki, H.; Rai, T.; Kivi, M.; Lewis, P.; Gómez-Dans, J.L.; Yin, F.
Linking Remote Sensing with APSIM through Emulation and Bayesian Optimization to Improve Yield Prediction. *Remote Sens.* **2022**, *14*, 5389.
https://doi.org/10.3390/rs14215389

**AMA Style**

Dokoohaki H, Rai T, Kivi M, Lewis P, Gómez-Dans JL, Yin F.
Linking Remote Sensing with APSIM through Emulation and Bayesian Optimization to Improve Yield Prediction. *Remote Sensing*. 2022; 14(21):5389.
https://doi.org/10.3390/rs14215389

**Chicago/Turabian Style**

Dokoohaki, Hamze, Teerath Rai, Marissa Kivi, Philip Lewis, Jose L. Gómez-Dans, and Feng Yin.
2022. "Linking Remote Sensing with APSIM through Emulation and Bayesian Optimization to Improve Yield Prediction" *Remote Sensing* 14, no. 21: 5389.
https://doi.org/10.3390/rs14215389