Geostatictics for soil nutrient mapping

Doctoral Seminar
Spatial Interpolation Methods for Soil Nutrient
Mapping

Nirmal Kumar , Ph. D Scholar, Department of Soil Science and Agricultural Chemistry
College of Agriculture, Indira Gandhi Krishi Vishwavidyalaya, Raipur
A key feature of soil is the variation with depth in soil properties
Soil properties varies across the landscape
also

We need SPATIAL DATA
About:
•Soil fertility like – NPK, Ca, Mg, etc…..
•Water Table Depths
•Height
•
•
•
•
For:
• Planning, risk assessment, and
decision making in environmental
management
• Scientists need accurate spatial
continuous data across a region to
make justified interpretations
• Precision Agriculture

Sources for SPATIAL DATA
Soil Survey Maps??
• Field surveys are typically from
point sources
• Soil variability at the field scale
cannot be completely described
by soil mapping units
• Not on soil fertility status
• Small scale maps are made of
polygons with many components

Sources for SPATIAL DATA
Remote Sensing Images??
• Gives Spatial Data in different
Scales
• But, only surface features
• Limited number of bands in RS
images
• Hyperspectral imageries show
promises

Blanket on spikes
Other options??

Estimating a point
here: interpolation
Sample
data
Interpolation
 Estimating the attribute values of locations that
are within the range of available data using known
data values
Estimating a point
here: extrapolation
Sample
data
Extrapolation
 Estimating the attribute values of locations
outside the range of available data using
known data values
Interpolation VS Extrapolation

How to??
Spatial Interpolation
Non-geostatistical methods
Geostatistical methods
Combined
Triangular Irregular Network
Natural Neighbours
Inverse Distance Weighting
Trend Surface Analysis
Thin Plate Splines
.
.
.
Simple Kriging
Ordinary Krigging
Block Krigging
Indicator Krigging
Cokrigging
.
.
.
Trend Surface Analysis
Combined with Krigging
Regression krigging
Lapse Rate Combined with
Krigging

• Unknown values are predicted with a mathematical
formula that uses the values of nearby known points.
• Things closer together have similar properties.

Tobler’s first law of geography
• “All things are similar, but nearby things are more
similar than distant things.”
• Or, more poetically
"All things by a mortal power,
Near or far
Hiddenly
To each other linked are,
That thou canst not stir a flower
Without the troubling of a star"
Francis Thompson, "The Mistress of Vision" 1897

The Concept
Nearest Neighbour
Average
Average Depth
5 feet
Inverse Distance Weight
Not only the distance, but direction is also important
We need Geostatistics

Nearly all the methods of prediction, including the simpler
forms of kriging, can be seen as weighted averages of data.
Where,
x0 = target point for which we want a value;
z*(xi), i = 1; 2; . . . ;N, at places xi are the measured data;
and i are the weights assigned to them

Nearest Neighbours
• Predicts the value of an attribute at an unsampled point based on the value of the
nearest sample by drawing perpendicular bisectors between sampled points (n),
forming such as Thiessen (or Dirichlet/Voronoi) polygons (Vi, i=1,2,…, n).
• produces one polygon per sample and the sample is located in the centre of the
polygon.
• The estimations of the attribute at unsampled points within polygon Vi are the
measured value at the nearest single sampled data point xi that is zˆ (x0) = z(xi).
• Weights are given by:
All points (or locations) within each polygon are assigned the same value

Thiessen polygons have the unique property that each polygon contains only one input point,
and any location within a polygon is closer to its associated point than to the point of any other
polygon.
The Thiessen polygons are constructed as follows:
• All points are triangulated into a triangulated irregular network (TIN) that meets the
Delaunay criterion.
• The perpendicular bisectors for each triangle edge are generated, forming the edges of the
Thiessen polygons. The location at which the bisectors intersect determine the locations of
the Thiessen polygon vertices.
Thiessen Polygon

IDW
assumes that the rate of correlations and similarities between neighbours are proportional to
the distance among them. So, that can be defined as a distance reverse function of every point
from neighbouring points
unknown value of a point is influenced by a closer points than away points

Inverse Distance Weighted
Local method
Exact
Can be linear or non-linear
The weight (influence) of a sampled data
value is inversely proportional to its
distance from the estimated value

(Example)
4
3
2
100
160
IDW:
Closest 3
neighbors,
p = 2
200
  
 





 








1),(
1
),(
1
1
i
with
ii
n
i
p
i
n
i
p
i
i
zyxzor
d
d
z
yxz

(Example)
4
3
2
A = 100
B = 160
C = 200
1 / (42) = .0625
1 / (32) = .1111
1 / (22) = .2500
Weights
A
B
C

(Example)
1 / (42) = .0625
1 / (32) = .1111
1 / (22) = .2500
.0625 * 100 = 6.25
.1111 * 160 = 17.76
.2500 * 200 = 50.00
Weights Weights * Value
A
B
C
74.01 / .4236 = 175
Total = .4236
6.25 +17.76 + 50.00 = 74.01
4
3
2
A = 100
B = 160
C = 200

Why Geo-statistics??
Estimation of the unknown permeability Z0 based
on a set values of permeability at n locations
Estimation of the unknown permeability Z0
given 7 known values. Numbers in
parenthesis are weights assigned to the
known values based on inverse distance

• Ordinary,
• Simple,
• Universal,
• Indicator,
• Probability,
• And Disjunctive Kriging.
assume things that are close are more alike.
The semivariogram allows you to explore this assumption.
The process of fitting a semivariogram model while capturing the spatial relationships is
known as variography
A 2 step procedure
Geostatistical analysis of data occurs in two phases:
• modeling the semivariogram or covariance to
analyze surface properties, and
• kriging. A number of kriging methods are available
for surface creation in Geostatistical Analyst,

To create an empirical semivariogram,
• determine the squared difference between the values for all pairs of locations.
• Plot half the squared difference on the y-axis and the distance that separates the
locations on the x-axis,
• it is called the semivariogram cloud.
Variography
More similar
More dissimilar
1. Create empirical semivariogram
2. Fit a model

Binning
To reduce the number of points in the empirical
semivariogram, the pairs of locations will be
grouped based on their distance from one another.
This grouping process is known as binning.
• For omnidirectional semivariograms, use bins that include the closest separation distances in
the sample design so that the maximum amount of information is obtained about the
semivariogram at small separation distances.
• Avoid making bins so narrow that the number of pairs in any bin is less than 30.
• For directional semivariograms, it is necessary to have bins for both separation distance and
direction.
• Narrower directional bins could also be selected, but wider ones should probably be
avoided, since they would lead to a loss of information about directional effects.

Creating Variography

A linear model
A spherical model
An exponential model

nugget effect is simply the sum of measurement error and microscale variation and, since
either component can be zero, the nugget effect can be comprised wholly of one or the other.
The height that the semivariogram
reaches when it levels off
discontinuity at the
origin
microscale variation
measurement error
distance at which the semivariogram
levels off to the sill

Kriging
Like IDW interpolation, kriging forms weights from surrounding measured values to predict
values at unmeasured locations.
As with IDW interpolation, the closest measured values usually have the most influence.
However, IDW uses a simple algorithm based on distance, but kriging weights come from a
semivariogram that was developed by viewing the spatial structure of the data.
1. Create the Г – matrix and g-vector,
2. Calculate weights ʎ, and
3. Make a prediction
The key step in kriging is to estimate the n weighting factors for locations that neighbor the
unsampled location where interpolation is to occur.
a set of N+1 simultaneous equations with N+1 unknowns (the N weighting factors and the
undetermined Lagrangian multiplier).
Where,
[Г] is an N+1 by N+1 matrix of semivariogram values between measured locations,
[ʎ] is an N+1 by 1 matrix of weighting factors and the Lagrangian multiplier, and
[g] is an N+1 by 1 matrix of semivariogram values between the interpolated location and its neighboring
locations.

Kriging
ordinary kriging
Simple kriging
Universal kriging
Punctual Kriging
Block kriging
COKRIGING
Cokriging is an interpolation technique that uses information about the spatial patterns of two
different, but spatially correlated properties to interpolate only one of the properties.

Concerns when comparing methods and models
1. optimality
2. validity. Two issues
For example, the root-mean-squared prediction error may be smaller for a particular model. Therefore, you
might conclude that it is the "optimal" model. However, when comparing to another model, the root-mean-
squared prediction error may be closer to the average estimated prediction standard error. This is a more
valid model because when you predict at a point without data, you have only the estimated standard errors
to assess your uncertainty of that prediction. When the average estimated prediction standard errors are
close to the root-mean-squared prediction errors from cross-validation, you can be confident that the
prediction standard errors are appropriate.

GEOSTATISTICAL SAMPLING
simple random sampling stratified random sampling
systematic sampling design random sampling within cells
geostatistical sampling design
systematic sampling for the purposes of
accurate interpolation by kriging to
produce spatial pattern maps
sampling for accurate estimation
of the semivariogram

Before Starting
Check for
• Normality, Outliers
• Global trends
• Local Trends (Anisotropy)
Softwares

Geostatictics for soil nutrient mapping

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