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ES 470 SAMPLING AND ANALYSIS OF HYDROLOGICAL DATA Manoj K. Shukla, Ph.D. Assistant Professor PowerPoint PPT Presentation

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ES 470 SAMPLING AND ANALYSIS OF HYDROLOGICAL DATA Manoj K. Shukla, Ph.D. Assistant Professor Environmental Soil Physics. FEBRUARY 09, 2006, (W147, 3 - 5 PM). J. H. Dane G.C. Topp (Editors) Methods of Soil Analysis- Part 4, Physical Methods. ES-470. Scales of Variability.

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ES 470 SAMPLING AND ANALYSIS OF HYDROLOGICAL DATA Manoj K. Shukla, Ph.D. Assistant Professor

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ES 470


Manoj K. Shukla, Ph.D.

Assistant Professor

Environmental Soil Physics

FEBRUARY 09, 2006, (W147, 3 - 5 PM)

J. H. Dane G.C. Topp (Editors)

Methods of Soil Analysis- Part 4, Physical Methods


Scales of Variability

Particles or Pore



Column or Horizon

Field or Watershed





Spatial: variability with increasing distance (space) from a location

Temporal: variability with increasing duration/time

We will limit our discussion to field scale


Agriculture Field ???

  • In situ soil exhibits large degree of variability or heterogeneity

  • Changes in soil types need to be accounted for in the composite sampling

  • The composite sample must maintain the heterogeneity of the insitu soil



Intrinsic Factors:

Soil forming factors, time, soil texture, mineralogy, pedogenesis (geological, hydrological, biological factors)

The intrinsic variables have a distinct component that can be called regionalized, i.e., it varies in space, with nearby areas tending to be alike

Extrinsic Factors:

Land use and management, fertilizer application, other amendments, drainage, tillage


Structure of Variability

Random sampling is done to ensure that estimates are unbiased

Meet the criterion of independent sampling under identical conditions

Yi = m + ei

where Yi is the realization of a soil attribute at location i, m is the mean value for the spatial domain, and ei is a random error term


An attribute (i.e., bulk density, nitrate concentration, etc.) is described through two statistical parameters

E [Yi] = m

First moment or Mean

E [(Yi - m)2] = s2

Second moment or Variance


E [Yi] = m

E [(Yi - m)2] = s2

Mean and variance or first and second moment are often assumed to be the parameters of a normal (Gaussian) probability distribution function; and

Allow for a series of sophisticated statistical analysis

Arithmetic mean = m = (x1 + x2 + x3) / 3

Geometric mean = m = (x1* x2* x3)1/3

Harmonic mean = m = (1/x1 + 1/x2 + 1/x3)* (1/n)

Variance (s2) = (1/n) * ∑(xi – xm)2


Soil N content data

Mean = 1.35 g kg-1

Variance = 0

E [Yi] = 1.35

Mean = 1.339 g kg-1

Variance = 0.0003

E [Yi] = 1.339 ± (0.0003)0.5


Normal (Gaussian) Distribution


The function is symmetric about the mean, it gains its maximum value at the mean, the minimum value is at plus and minus infinity


Histogram for Sand Content

Sigma Plot 8.0

Normal distribution


Histogram for Saturated Hydraulic Conductivity

Skewed distribution- Positive


Skewed distribution- Positive

Skewed distribution- negative

One of the tail is longer than other- Distribution is skewed


Different Data Structures


So in place of

E [Yi] = m

E [Yi] = m + b(xi) + ei

An Appropriate model

Where b(xi) can be a constant or a function, both dependent on a spatial or temporal scale

Therefore, simple randomization may not be sufficient

Stratified sampling will be better

Stratified sampling- the area is divided into sub areas called strata


Case Study

  • Formulate objectives

  • Formulate hypotheses

  • Design a sampling scheme

  • Collect data

  • Data Interpretation


Determine the relative magnitude of statistical and spatial variability at Field scale


Sampling Design?

  • Simple random

  • Stratified

  • Two-stage

  • Cluster

  • Systematic



2 -3



How many samples?

Sample size for simple random sampling

Relative error should be smaller than a chosen limit (r)

Where m1-a/2 = (1-a/2) quartile of the standard normal distribution; S- standard deviation of y in the area; is mean

Standard deviation or coefficient of variation is known

Absolute error to be smaller than a chosen limit d

Time and Resources ????


Students t-table

df = degree of freedom; p is probability level


Example data of N concentration: 1.10, 1.11, 1.12, 1.13, 1.13, 1.14, 1.16, 1.17, 1.19, 1.20, 1.23, 1.24, 1.25

Relative error = 0.01 g kg-1

Mean of Y = 1.17 g kg-1

Standard deviation = 0.05

Alpha = 0.05

Degree of freedom = 13-1 = 12

t Students (table) = 1.782


Relative error (r) = 0.02 g kg-1

Alpha = 0.10

Degree of freedom = 13-1 = 12

T Students (table) = 1.782

r = 0.02

r = 0.01


E(Yi)s = Ym

Var(Yi) =0

Deterministic parameters

Variation in properties

Stochastic parameters

Mean value and an uncertainty statistics


Semi variogram function

Var(Yi)s = s2s

Var[(Yi)s-(Yi+h)s]= 2g(h)

  • It is always implied:

  • Domain is first- or second- order stationary

  • Process is adequately characterized by a mean value and an uncertainty statistics


We will use a data collected on a grid of 20 x 20 cm in a field seeded to grass for last 20 years


Variability can be expressed by coefficient of variation

  • Where:

    • x = an individual value

    • n = the number of test values

    • = the mean of n values

Standard deviation of two independent sets

where: n1 = number of values in the first set; s1 = standard deviation of the first set of values; n2 = number of values in second set; s2 = standard deviation of second set of values


Statistical variability of soil properties at local scale

Water Transmission


Coefficient of variation (CV)

AWC- Available water content (cm)

VTP - Volume of transport pores (qs-q6) (%)

VSP - Volume of storage pores (%)

ic - Steady state infiltration rate (cm/min)

Ks - Sat. hydraulic conductivity (cm/min)

I - Cumulative infiltration (cm)

I5 - Infiltration rate at 5 min (cm/min)


Shukla et al. 2004

Descriptive statistics (or CV) cannot discriminate between intrinsic (natural variations) and extrinsic (imposed) sources of variability

Geostatistical analysis- grid based or spatial sampling

For example-20 m x 20 m


Range (a)

Partial Sill (C1)

ArC View


Nugget (C0)

Lag (h; m)

Pannatier, 1996


  • Note:

  • g increases with increasing lag or separation distance

  • A small non-zero value may exist at g = 0

  • This limiting value is known as nugget variance

  • It results from various sources of unexplained errors, such as measurement error or variability occurring at scales too small to characterize given the available data

  • At large h, many variograms have another limiting value

  • This limiting value is known as sill

  • Theoretically, it is equal to the variance of data

  • The value for h where sill occurs is known as range



  • The most common function used in geostatistical studies to characterize spatial correlation is the variogram

  • The variogram, g(h), is defined as one-half the variance of the difference between the sample values for all points separated by the distance h

where var [ ] indicate variance and E { } expected value


Estimator for the variogram is calculated from data using

where N(h) is total number of pairs of observations separated by a distance h.

Caution- variograms can be strongly affected by outliers in the data


Variogram Model

  • Variogram model is a mathematical description of the relationship between the variance and the separation distance (or lag), h

  • There are four widely used equations


Isotropic Models

Linear Model

Spherical Model

Exponential Model

Gaussian Model



C0, Nugget

a, Range

Linear Model

Spherical Model

Does not have a sill or range and the variance is undefined

Precisely defined sill or range


b ~ a/3

b ~ a/30.5

Exponential Model

Gaussian Model

Range is 1/3 of the range for spherical model

Range is 1/sqrt(3) of the range for spherical model


  • Variogram is constructed by

  • Calculating the squared differences for each pair of observations (xj - xk)

  • Determining the distance between each pair of observation

  • Averaging the squared differences for those pairs of observations with the same separation distance

If observations are evenly spaced on a transect, separation distances are multiple of the smallest distance

h1 = 2 m; h2 = 4m; h3 = 6 m ……


  • When observations are placed on an irregular pattern, variograms are :

  • constructed by assigning appropriate lag interval

  • Binning procedure

  • B ins are created with interval centers at distances

  • h1 = (1-2) m; h2 = (2-4) m; h3 =(4-6) m …………………..


  • Important considerations when calculating a variogram:

  • As separation distance becomes too large, spurious results occur because fewer pairs of observation exist for large separations due to finite boundary

  • Width of lag interval can affect the sample variogram due to number of samples and variation in the separation distances that fall into a particular lag interval

  • Uncorrelated and correlated data show different nugget effects

  • Number of datasets used influence on variogram


Before you start spatial analysis: Check for normal distribution

WSA- water stability of aggregates (%)

sand- sand content (%)

Ic- saturated hydraulic conductivity (cm/h)


Use of descriptive statistics

Mean, median (most middle), skewness, etc.


Plot the data to see the structure


Saturated Hydraulic Conductivity



Estimator variance


Variance = 13.7

Variance = 15.5

Variance = 16.1


Sand Content

Saturated Hydraulic Conductivity


Modeling of Variogram

Sand Content

Spherical Model

SS = 0.04598

Nugget = 0

Range = 37.92 m

Sill = 16.0

Spherical Model

SS = 0.00994

Nugget = 3.04

Range = 49.77 m

Sill = 16.0


Saturated Hydraulic Conductivity

Spherical Model

SS = 0.0494

Nugget = 0

Range = 19.8 m

Sill = 0.0384

Spherical Model

SS = 0.04938

Nugget = 0.004

Range = 19.8 m

Sill = 0.0384


Parameters for spherical variogram model for soil properties


Spatial variability: nugget – total sill ratio (NSR)

Lower NSR – higher spatial dependence

Water Transmission


Nugget to total sill ratio

NSR < 0.25 highly spatial variable

NSR > 0.75 less spatial variable

Cambardella et al., 1994

Shukla et al. 2004


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