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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

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

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Scales of Variability

Particles or Pore

Aggregate

Molecules

Column or Horizon

Field or Watershed

Regional

Pedosphere

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Variability

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

Temporal: variability with increasing duration/time

We will limit our discussion to field scale

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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

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Sources

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

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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

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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

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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

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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

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Normal (Gaussian) Distribution

Mean

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

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Histogram for Sand Content

Sigma Plot 8.0

Normal distribution

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Histogram for Saturated Hydraulic Conductivity

Skewed distribution- Positive

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Skewed distribution- Positive

Skewed distribution- negative

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

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Different Data Structures

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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

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Case Study

- Formulate objectives
- Formulate hypotheses
- Design a sampling scheme
- Collect data
- Data Interpretation

Objective:

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

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Sampling Design?

- Simple random
- Stratified
- Two-stage
- Cluster
- Systematic

4

1

2 -3

5

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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 ????

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Students t-table

df = degree of freedom; p is probability level

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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

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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

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E(Yi)s = Ym

Var(Yi) =0

Deterministic parameters

Variation in properties

Stochastic parameters

Mean value and an uncertainty statistics

Variance

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

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

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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

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Statistical variability of soil properties at local scale

Water Transmission

Textural

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)

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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

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Range (a)

Partial Sill (C1)

ArC View

Variowin

Nugget (C0)

Lag (h; m)

Pannatier, 1996

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- 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

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Variogram

- 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

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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

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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

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Isotropic Models

Linear Model

Spherical Model

Exponential Model

Gaussian Model

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Sill

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

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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

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- 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 ……

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- 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 …………………..

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- 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

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Before you start spatial analysis: Check for normal distribution

WSA- water stability of aggregates (%)

sand- sand content (%)

Ic- saturated hydraulic conductivity (cm/h)

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Use of descriptive statistics

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

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Plot the data to see the structure

Y

Saturated Hydraulic Conductivity

X

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Estimator variance

Example

Variance = 13.7

Variance = 15.5

Variance = 16.1

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Sand Content

Saturated Hydraulic Conductivity

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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

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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

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Parameters for spherical variogram model for soil properties

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Spatial variability: nugget – total sill ratio (NSR)

Lower NSR – higher spatial dependence

Water Transmission

Textural

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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