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Improvement in Precision of Crop Acreage Estimation by Remote Sensing Using Frequency Distribution Based Stratification

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Abstract

A study was conducted to improve precision of crop acreage adopting stratified random sampling approach. Remotely sensed data was used to classify mustard crop for the states of Rajasthan, Madhya Pradesh, Uttar Pradesh, Gujarat and Haryana covering 81% of mustard area of India. A grid of size 5 × 5 km was super-imposed on classified image of study area and proportion of mustard crop within the grid was ascertained. Crop proportion was used to determine strata. Stratification was done based on equal interval of proportion, equal sample number and cumulative square root of frequency method. Cumulative square root of frequency method gave highest precision in all the cases.

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Acknowledgement

Authors are thankful to Dr. R.R Navalgund, Director, Space Applications Centre for his support and encouragement. Authors are grateful to Dr. J.S. Parihar, Deputy Director, Earth, Ocean, Atmosphere, Planetary Sciences and Applications Area (EPSA) for his keen interest and encouragement. Authors also thank Dr. Sushma Panigrahy, Group Director, Agriculture, Terrestrial Biosphere and Hydrology Group (ABHG) for her guidance and support in carrying out this work. This work was carried out under FASAL project sponsored by Department of Agriculture and Co-operation.

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Annexure

Annexure

Mathematical basis for construction of strata by frequency method (Cochran 1992b)

The variance of estimate in stratified random sampling is given by

$$ {\text{variance}}\left( {{\text{V}}\left( {{{\overline {\text{y}} }_{\text{st}}}} \right)} \right) = \left( {1/{{\text{N}}^2}} \right)*\sum\limits_{{{\text{h}} = 1}}^{\text{L}} {{{\text{N}}_{\text{h}}}\left( {{{\text{N}}_{\text{h}}} - {{\text{n}}_{\text{h}}}} \right){{\text{S}}_{\text{h}}}^2/{{\text{n}}_{\text{h}}}} $$

where Nh is population size and nh is sample size in stratum h, N is total population value given by \( {\text{N}} = \sum\limits_{{{\text{h}} = 1}}^{\text{L}} {{{\text{N}}_{\text{h}}}} \)

$$ {{\text{S}}_{\text{h}}}^{{2}} = \sum\limits_{{h = 1}}^L {{{{{{\left( {{y_{{hi}}} - {{\overline y }_h}} \right)}^2}}} \left/ {{{n_h} - 1}} \right.}} $$

where yhi is the ith observation in the stratum h, yh is mean of stratum h and Sh 2 is the variance within the stratum h

Under Neyman allocation and ignoring finite population correction \( {\text{V}}\left( {{{\overline {\text{y}} }_{\text{st}}}} \right) \) is given as \( {\text{V}}\left( {{{\overline {\text{y}} }_{\text{st}}}} \right) = \left( {{1}/{\text{n}}} \right)*{\left( {\Sigma {{\text{W}}_{\text{h}}}{{\text{S}}_{\text{h}}}} \right)^{{2}}} \) where \( {{\text{W}}_{{{\text{h}} = }}}{{\text{N}}_{\text{h}}}/{\text{N}} \) is stratum weight, S 2h is stratum variance, Minimizing \( {\text{V}}\left( {{{\overline {\text{y}} }_{\text{st}}}} \right) \) amounts to minimizing WhSh

$$ {{\text{W}}_{\text{h}}} = \int\limits_{{{{\text{y}}_{{{\text{h}} - 1}}}}}^{{{{\text{y}}_{\text{h}}}}} {\left( {{\text{f}}\left( {\text{t}} \right)} \right){\text{ dt}}} $$

Under assumptions of large number of strata and rectangular distribution within stratum \( {{\text{W}}_{\text{h}}}\sim {{\text{f}}_{\text{h}}}\left( {{{\text{y}}_{\text{h}}} - {{\text{y}}_{{{\text{h}} - {1}}}}} \right)\quad {{\text{S}}_{\text{h}}} = \left( {{{\text{y}}_{\text{h}}} - {{\text{y}}_{{{\text{h}} - {1}}}}} \right)/\surd \left( {{12}} \right)\,{{\text{y}}_{\text{h}}} \) and yh−1 are bounds & fh is frequency

$$ \surd \left( {{12}} \right)\;\left( {\sum\limits_{{{\text{h}} = 1}}^{\text{L}} {{{\text{W}}_{\text{h}}}{{\text{S}}_{\text{h}}}} } \right) = \sum\limits_{{{\text{h = }}1}}^{\text{L}} {{{\text{f}}_{\text{h}}} {{{ \left( {{{\text{y}}_{\text{h}}} - {{\text{y}}_{{{\text{h}} - {1}}}}}\right)} }^{{2}}}} = \sum\limits_{{{\text{h = }}1}}^{\text{L}} {\left( {\surd {{\left( {{{\text{f}}_{\text{h}}}\left( {{{\text{y}}_{\text{h}}} - {{\text{y}}_{{{\text{h}} - {1}}}}} \right)} \right)}^{{2}}}} \right.} $$
$$ {\text{G}}\;\left( {\text{h}} \right) = \int\limits_{{{{\text{y}}_{{{{\text{h}}^{ - }}{1}}}}}}^{{{{\text{y}}_{\text{h}}}}} {\surd \;\left( {{\text{f}}\left( {\text{t}} \right)} \right)\;{\text{dt}}} \quad \quad {\text{G}}\left( {\text{h}} \right){\text{ is a function of h}} $$
$$ {\text{G}}\left( {\text{h}} \right) - {\text{G}}\;\left( {{\text{h}} - {1}} \right)\sim \surd \left( {{{\text{f}}_{\text{h}}}} \right)\left( {{{\text{y}}_{\text{h}}} - {{\text{y}}_{{{\text{h}} - {1}}}}} \right) $$

\( \surd \left( {{12}} \right)\;\left( {\sum\limits_{{{\text{h}} = 1}}^{\text{L}} {{{\text{W}}_{\text{h}}}{{\text{S}}_{\text{h}}}} } \right) = \sum\limits_{{{\text{h}} = 1}}^{\text{L}} {{{\left( {{\text{G}}\left( {\text{h}} \right) - {\text{G}}\left( {{\text{h}} - {1}} \right)} \right)}^{{2}}}} \) where \( \sum\limits_{{{\text{h}} = 1}}^{\text{L}} {{{\left( {{\text{G}}\left( {\text{h}} \right) - {\text{G}}\left( {{\text{h}} - {1}} \right)} \right)}^{{2}}}} \) is constant \( \left( {\sum\limits_{{{\text{h}} = 1}}^{\text{L}} {{{\text{W}}_{\text{h}}}{{\text{S}}_{\text{h}}}} } \right) \) is minimized when G (h)-G (h-1) is constant i.e. Variance is minimized when \( {\text{cum}}.\surd \;\left. {\left( {{{\text{f}}_{\text{h}}}} \right)\left( {{{\text{y}}_{\text{h}}} - {{\text{y}}_{{{\text{h}} - {1}}}}} \right)} \right) \) is constant

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Bhagia, N., Rajak, D.R. & Patel, N.K. Improvement in Precision of Crop Acreage Estimation by Remote Sensing Using Frequency Distribution Based Stratification. J Indian Soc Remote Sens 39, 153–160 (2011). https://doi.org/10.1007/s12524-011-0098-y

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