Precision Agriculture

, Volume 8, Issue 1–2, pp 75–93 | Cite as

Using a robust variogram to find an adequate butterfly neighborhood size for one-step yield mapping using robust fitting paraboloid cones

  • Martin Bachmaier


The yield map is generated by fitting the yield surface shape of yield monitor data mainly using paraboloid cones on floating neighborhoods. Each yield map value is determined by the fit of such a cone on a neighborhood that looks like a huge butterfly flying along the harvest track. Wide wings of the butterfly guarantee that the map is sufficiently smoothed out across the tracks. The coefficients of regression for modeling the paraboloid cones and the scale parameter are estimated using robust weighted M-estimators where the weights decrease with the distance from one to zero; the latter is at the border of the selected neighborhood. The robust way of estimating the model parameters supersedes a procedure for detecting outliers. For a given neighborhood size, this yield mapping method is implemented by the Fortran program butterflymap.exe , which can be downloaded from the web. To obtain the appropriate size of the selected neighborhood, the variance of the yield map values should equal the variance of the true yields, which is the difference between the variance of the raw yield data and the error variance of the yield monitor. It is estimated using a robust variogram on data that have not had the trend removed. Based on investigating butterfly neighborhoods the yield map was optimized if the search radius across the harvest tracks was eight times the swath width. One reason for this wide neighborhood is that the regression used for modeling the paraboloid cones is based on weights that decrease linearly from 1 in the middle to zero at the border of the neighborhood, so only data points close to the middle have a large weight.


Precision agriculture Raw data correction Yield map Paraboloid cone Weighted M-estimate Butterfly neighborhood Variance comparison 



I thank Margaret A. Oliver, University of Reading, for her helpful and detailed remarks on this paper and Hermann Auernhammer, Technische Universität München-Weihenstephan, for supporting this work.


  1. Arslan S, Colvin TS (2002) Grain yield mapping: yield sensing, yield reconstruction, and errors. Precision Agricult 3:135–154CrossRefGoogle Scholar
  2. Bachmaier M (2000) Efficiency comparison of M-estimates for scale at t-distributions. Stat Papers 41:53–64CrossRefGoogle Scholar
  3. Bachmaier M (2006) FORTRAN programs for download. Raw yield data correction and yield mapping: butterflymap.
  4. Bachmaier M (2007) Finding adequate neighborhoods for robust yield mapping.
  5. Bachmaier M, Auernhammer H (2004) A method for correcting raw yield data by fitting paraboloid cones. In: Proceedings of the AgEng 2004 conference – engineering the future. Session 10: Precision agriculture, 8 p on CD-ROMGoogle Scholar
  6. Bachmaier M, Auernhammer H (2005) Yield mapping based on robust fitting paraboloid cones in butterfly and elliptic neighborhoods. In: Stafford JV, Werner A (eds) Precision agriculture ’05: proceedings of the 5th European conference on precision agriculture. Wageningen Academic Publishers, Wageningen, the Netherlands, pp 741–750Google Scholar
  7. Blackmore BS, Moore M (1999) Remedial correction of yield map data. Precision Agricult 1:53–66CrossRefGoogle Scholar
  8. Cressie NAC (1991) Statistics for spatial data. In: Wiley series in probability and mathematical statistics. Applied probability and statistics section. John Wiley & Sons, Inc, New York, USAGoogle Scholar
  9. Cressie NAC, Hawkins D (1980) Robust estimation of the variogram. Math Geol 12:115–125CrossRefGoogle Scholar
  10. Dowd PA (1984) The variogram and kriging: robust and resistant estimators. In: Verly G et al (eds) Geostatistics for natural resources characterization, part 1. D. Reidel, Dordrecht, pp 91–106Google Scholar
  11. Genton MG (1998) Highly robust variogram estimation. Math Geol 30:213–221CrossRefGoogle Scholar
  12. Hampel FR, Rousseeuw PJ, Ronchetti EM, Stahel WA (1986) Robust statistics. John Wiley & Sons Inc, New York USA, section 2.6Google Scholar
  13. Huber PJ (1981) Robust statistics. John Wiley & Sons Inc, New York, USA, section 78CrossRefGoogle Scholar
  14. Kleinjan J, Chang J, Wilson J, Humburg D, Carlson G, Clay D, Long D (2002) Cleaning yield data. At: Publications.htm, South Dakota State University, BrookingsGoogle Scholar
  15. Lark RM (2000) A comparison of some robust estimators of the variogram for use in soil survey. Eur J Soil Sci 51:137–157CrossRefGoogle Scholar
  16. Noack PH, Muhr T, Demmel M (2003) An algorithm for automatic detection and elimination of defective yield data. In: Stafford JV, Werner A (eds) Precision agriculture. Wageninen Academic Publishers, Wageningen, pp 445–450Google Scholar
  17. Shearer SA, Higgins SG, McNeil SG, Watkins GA, Barnhisel RI, Doyle JC, Leach JH, Fulton JP (1997) Data filtering and correction techniques for generating yield maps from multiple combine harvesting systems. ASAE paper no. 97–1034, St. Joseph, MIGoogle Scholar
  18. Simbahan GC, Dobermann A, Ping JL (2004) Site-specific management. Screening yield monitor data improves grain yield maps. Agron J 96:1091–1102CrossRefGoogle Scholar
  19. Steinmayr T (2003) Fehleranalyse und Fehlerkorrektur bei der lokalen Ertragsermittlung im Mähdrescher zur Ableitung eines standardisierten Algorithmus für die Ertragskartierung. Dissertation at the Technische Universität München-Weihenstephan, pp 24–27Google Scholar
  20. Thylén L, Algerbo PA, Giebel A (2001) An expert filter removing erroneous yield data. In: Robert PC et al (eds) Precision agriculture 2000: proceedings of the 5th international conference on precision agriculture. ASA, CSSA and SSSA, Madison, WI, CD-ROMGoogle Scholar
  21. Thylén L, Jürschik P, Murphy DPL (1997) Improving the quality of yield data. In: Stafford JV (ed) Precision agriculture ’97: proceedings of the 1st European conference on precision agriculture. BIOS Scientific Publishers Ltd, Oxford UK, pp 743–750Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Life Science Engineering, Crop Production EngineeringTechnische Universität MünchenFreising-WeihenstephanGermany

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