PRINCIPLES AND PRACTICAL APPLICATIONS OF

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PRINCIPLES AND PRACTICAL APPLICATIONS OF

• POINT SAMPLINGinFORESTRY
• Presented by
• SONNEY GEORGE
• College of Forestry

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Definition

• Sampling that uses a critical angle as an
  expression of the ratio between the dimensions of
  the objects sampled and their distance from the
  sampling point.
• When the critical angle is exceeded, the object
  is included in the sample.
• Types of point sampling (Angle Count Sampling )
• Horizontal point sampling
• Vertical point sampling

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• Genesis
• Bitterlich,1931

Diameter of a tree and its distance to a
neighboring tree defined a measurable angle
?
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Comparison of point sampling with Fixed Plot
Method

• Fixed plot method
• Select sample plots. No and size of the plots
  determined according to Statistical principles
  based on the variability of the stand
• Mark boundaries
• Measure dimensions of the plot
• Calculate the plot area
• Measure the dbh or gbh of each tree .

• Horizontal Point Sampling
• Select sample points. No of points decided as per
  the variability of the stand. No fixed size
• No fixed boundaries
• No dimensions to be measured
• No calculation of area
• Count the no. Of trees whose BH cross section
  exceeds the critical angle.

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Comparison of point sampling with Fixed Plot
Method - Continued

• Calculate basal area of each tree using BA ?
  d2/4 or g2/4?
• Find the total for all the trees. If a 0.1 ha
  plot is used, there may be 250 trees at a spacing
  of 2m X 2m.
• Divide this total basal area with the total stand
  area and then multiply with 10,000 to get the
  basal area density in m2/ha.

• Multiply this number with the Basal Area Factor
  associated with the fixed angle device. This
  gives the BA per ha.

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Procedure Tally Half Tally
Non Tally
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Principle The historical development of the
method and a geometrical explanation is available
in Bitterlich (1984). Here the method will be
explained in terms of the probability concept of
Grosenbaugh (1952b, 1955).

• Plan of a forest area with the horizontal
  projection of basal areas

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Land area with tree stems cut at BH
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Explanation

• Select a number of points N in the plan.
• Let n points falls on tree cross sections at BH.
• Then n/N gives an estimate of Basal area per ha.
• Analogous with selecting balls from a bag.
• Let the bag contains n black balls and m white
  balls.
• where total of mn N balls.
• The proportion of black balls expected in a good
  sample will be n/N.

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Selecting points Vs balls

• Select 1000 points
• Let 5 points fell on areas occupied by trees
• Then basal area per ha would be 5/1000 ha
• (5 /1000) ha per ha (5/1000)x 10000 m2 per
  ha
• 50 m2
  / ha .5
• Ratio of basal area to stand area actually
  observed in the field is also about .2 to .7
• The variability is too much
• Example of Saw logs
• Hence the above scheme is impractical because
• We cant find a practical way of selecting points
  like that
• The variability is too high

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The plan of a forest area with a horizontal
projection of the basal areas and imaginary
circular zones around them.

• Let di be the diameter of a tree.
• Then diameter of the circular Zone around it will
  be Di.
• Where Di f X di and f a constant factor.

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Circular zones - relation to basal areas

• Let the diameter of the individual tree be
  denoted by di for i 1,2,...
• Their respective basal areas will be equal to
  (?di2)/4
• Total basal area ? gi ??(?/4) di2 for i 1 to
  n
• (?/4) ? di2
  for i 1 to n
• Diameter of the corresponding circular zones are
  fdi for i 1,2,3...
• Total area of the circular zones is
• ? Ai ? (?/4)(f di)2 for i 1 to n

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Circular zones - relation to basal areas -
contd.

• ? (?/4)f2( di)2 for i 1 to
  n
• f2(?/4) ? (
  di)2 for i 1 to n
• f2 X Total basal
  area
• OR Total area of zones/ f2 Total Basal area
• Or ratio of the total area of zone
  to the total basal area will
  be f2 1
• Inherent variation is very very less.
• The efficiency of sampling is greatly increased.
• A few sample points are sufficient.

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Estimation of zonal area

• Count no. Of Zones.
• Let the average number of zones counted is n.
• Total area of zones will be n m2/m2 of the
  stand.
• or n ha/ha of the stand.
• Hence Zonal area corresponding to a count of n is
  given by.
• Zonal area n m2 per m2 of stand area.
• n X 10,000 m2 per 10,000 m2 of stand area.

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Estimation of basal area from zonal area

• We know Zonal area n X 10,000 m2 per ha of
  stand area.
• ? Basal area n X 10,000 X (1/ f2 ) m2 per
  ha of stand area.
• Since Total area of zones/ f2 Total Basal
  area
• ?Basal area per ha average number of zones
  n counted X 10,000 X (1/ f2 ) m2.
• Where 10,000 X (1/ f2 ) is the Basal Area
  Factor referred earlier

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Angle subtended by a tree at the boundary of its
circular zone.

• Then in ? OPQ
• sin ?/2 d/2 1/f
  fd/2
• or ?/2 sin-1(1/f) a constant
• or ? a constant
• i.e.. Angle subtended by the BH cross section of
  a tree at the periphery of its circular zone is a
  constant

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Angle subtended by the tree inside, outside and
at the periphery of its circular zone.

• ? is the angle subtended by the tree at any
  point
• ?? is the angle subtended by the tree at the
  periphery of its circular zone

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Sides of the fixed angle ???when the vertex is
inside, outside and on the periphery of the
circular zone

• For a point inside the periphery the two arms of
  the angle passes through the tree stem
• For a point on the periphery they just touches
  the stem
• For a point outside the periphery they are wide
  apart from the stem

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Wedge prism - principle.
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Procedure Tally Half Tally
Non Tally
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Field problems equipments

• Half tally trees
• Relascopes- a general term for Instruments used
  for point sampling, incorporating a fixed angle
  ?.
• The simplest Relascope consists of a distance
  piece d and a cross piece C as shown below

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Calibrations

• Let Basal area (in m2) per ha G kz
• where z the no of tallied trees.
• Then k 10,000 X (1/f2)
• Relationship between BAF and angle ?
• Earlier we have proved that
• Sin ?/2 1/f
• Also BAF 10,000 X (1/f2)
• ?BAF, k 10,000 X (sin ?/2 )2
• 10,000 X sin 2(?/2 )

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Practical Applications of ACS Basic
Considerations - Inventory layout

• Usually systematic sampling on a rectangular
  grid.
• Any sample gives information about the area
  immediately surrounding the sample itself.
• Results from a circular plot, for example, will
  be a good predictor of values in the concentric
  circle immediately outside the first, and so on.
• In ACS the important results, i.e.. those
  relating to the large trees, are gathered over
  wide areas.
• In a close grid these large tree circular areas
  overlaps.
• High accuracy by extrapolation and interpolation
  from neighboring sample plots.

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Considerations of Bias

• Choose points on the map and transfer to the
  field.
• This kind of a restricted selection is not
  biased.
• Total sampling area is subdivided into fairly
  equal parts.
• (Only difference from Stratified Random
  Sampling.)
• In this respect systematic sampling is a
  special kind of stratification.

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Density of sampling points.

• Dealt with in a voluminous literature (e.g..
  Loestsch, Zohrer and Haller, 1973. )
• For a Homogenous stand.
• (1) if A lt 5 ha, n 2(?k) A
• (2) If A ? 5 ha, n 2(?k) (5 log A)/ log A
• Where n is the total number of sampling
  points.
• a, the corresponding distance(in M) on a square
  grid.
• A, the total area (in ha).
• So that a 100 ?(A/n)

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Density of sampling points - Table(K4).
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Basal area factor number of trees per sample

• There is a large literature on this important
  item, as for example Arvanitis and ORegan
  (1969) Beers and Miller (1964) Bitterlich
  (1973) Fountain, Hunt and Hassler (1983)
  Grosenbaugh (1955) Nyssonen and Vuokila (1963)
  Pflugbeil (1964) Seiber(1973) Spurr (1962)
  Stohr(1959) wensel,Leviatan and Barber (1980)
  Zohrer (1973).

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Basal area factor number of trees per sample -
continued

• When the basal area factor is too small.
• Number of sample trees selected will be very
  high.
• Hidden trees may affect the estimates.
• When the basal area factor is too large.
• Number of trees selected will be too small.
• Variability in the number of circular zones
  becomes high.
• Because there wont be any proper overlap of
  these zones.
• Thumb rule
• A basal area factor giving 10 - 15 sample trees
  is considered ideal by many authors.

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Percentage sampled.

• We know ? Ai/As ???gi/(GAs) where
  Ai Circular Zone area of tree i in ha ,
  gi basal area of tree i in m2 , G k X 1 is
  the basal area per ha for a single tree sampled
  and As is the area of the stand in hectares.
• We know basal area in ha (gi) Zonal area in
  ha (Ai) x (1/f2) for each tree tallied per ha of
  land area
• or giAi (1/f2) 10,000 m2 per ha Ai X k
• or Ai gi/k in ha per ha of land area for each
  tallied tree or Ai gi/G
• ?The estimate of percentage sampled is then given
  by A 100 ? gi/(GAs)

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PRECISION

• The following table lists some of the related
  techniques and compares their accuracy derived
  from a simulation study. The actual volume
  measured using fixed plot technique is 619 M3 .

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Applications

• Determination of Basal area
• Determination of Volume
• Estimates derived mainly from basal area density
  
• Using Volume/basal-area tables
• Occularly classifying trees into basal diameter
  groups
• ACS with diameter obviation

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Applications - contd.

• Stem number density from diameter measurements
• Other features to be measured on selected trees
• General classifications
• Dominant ,suppressed etc.
• Elements for volume
• DBH, ht, dia at other heights
• Elements for quality

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Other applications

• Determination of thinning needs
• Measurement of stand homogeneity
• Determination of yield optimum
• Crown cover density
• Direct angle count
• Marshalls method
• Determination of dominant height
• Determination of Stacked wood density

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Determination of Basal area

• The most important and primary application of
  ACS is in the estimation of Basal area/ha or the
  basal area density. Multiplication of the total
  number of tallied trees with the BAF gives an
  unbiased estimation of the Basal area/ha.

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Determination of VolumeEstimates derived mainly
from basal area density

• An ingenious and practical proposal was made by
  Ivanyuta (1962) and Strand (1964) making use of
  suggestions by Nyssonen (1956) uses stand Form
  Height along with Basal Area. We know V GFH
  or V/G FH
• Btterlich (1973a) - average tree height and
  form factor for each tree species.
• Shanmugasundaram (1989) - felling and actual
  measurement of the mean basal area tree. Mean
  basal area estimated using ACS, by dividing the
  total basal area with the number of trees (both
  estimated by ACS).
• Using Volume / basal area tables

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Determination of VolumeOccularly Classifying
Trees Into Broad Diameter Groups

• According to Dilworth and Bell (1973), ACS
  requires only approximate diameters.
• Ns12732 k (1/d1- 1/d1s)/s (Bitterlich,
  1960a)
• (provided that an even distribution within the
  classes can be assumed)
• Ns average number of trees per hectare
  represented by one in-tree falling in a diameter
  class between d1 and d1s
• k basal-area factor in m2/ha

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ACS WITH DIAMETER OBVIATIONBeers (1964),
following Grosenbaugh (1955).

• We know N k (ft2/acre)/g(ft2)
• where N is the No. Of trees represented by one
  in-tree and g its basal area
• If d is in inches N k /(?d2)/( 4 12 12
  k 4 144/(?d2)
• If the single-tree volume is expressed in cubic
  feet by the formula v b d2 h, where b is the
  regression coefficient, then
• V/acre Nv k 4 144 /(?d2) b d2 h
  bhk 4 144/? Ebh Where E k 4 144/?
  183.35 k
• i.e... we have to measure only the heights of in
  trees

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ADDITIONAL MEASUREMENTS ON ALL SELECTED TREES -
STEM NUMBER DENSITY FROM DIAMETER MEASUREMENTS

• Grosenbaugh (1952b) constructed stand table using
  ACS.
• We know as per ACS Basal Area per ha. G kz
  where z no. Of trees tallied.
• Each in-tree contributes k basal area/ha
• ? k Ni gi or Ni k/gi
• z and N
  ? (k/gi ) I 1

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STEM NUMBER DENSITY FROM DIAMETER MEASUREMENTS
-contd.

• Extending this to diameter classes.
• Each tree counted in class s represents Ns
  (k/gs) where gs basal area of mid diameter of
  class and k basal area factor
• If zs trees are counted in class s
• Then Ns (k/gs)zs and???N ? Ns where N
  stem number per hectare and Ns stem number per
  hectare for diameter class s

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OTHER FEATURES TO BE MEASURED ON SELECTED TREES

• ACS gives unbiased estimate of stem no. Per ha Ni
• Hence any quantity/quality measured on tallied
  trees can be expanded to per ha estimates.
• This is done by multiplying the parameter
  measured, Yi, by Ni.

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DETERMINATION OF STACKED-WOOD DENSITY

• Estimated by measurement of the conversion
  factor.
• In the diagram below the angle gauge has a ratio
  of sample area to cross section of 100 1.

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Principle

• Volume of circular zone area ? AB2
• ? (5d)2 25 ? (d)2 25 4 ?
  (d)2/4
• 100 area of circle A i.e... for a count
  of n n 1/100 gives the average tree cross
  sectional area per unit stacked area which is the
  conversion factor.

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Procedure

• Count cross sections exceeding the arms of the
  gauge n
• which gives unit zonal area in unit stacked area
• ??c s area in unit stacked area n (1/100)
  Conversion factor

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HIRATAS VERTICAL ANGLE COUNT - Principle

• Total circular zone area per ha z X 10,000 m2
  where z is the count of tally trees.
• Total ht circular area/ha Gh z X 10,000 x
  (ht circular area/circular zone area)

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Derivation

• z X 10,000 X (1/f2)
• ( since ratio of ht circular area to circular
  zone area is constant f2)
• z X 10,000 (?h2)/(?R2)
• z X 10,000 (h2)/(R2)
• z X 10,000 tan2? -- 2
• Hence mean ht circular area gh Gh/N
  --- 3 Where N ( no. of trees) is
  determined by horizontal point sampling.

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Derivation - contd.

• Now mean height circular area can be expressed in
  terms of the corresponding ht. As gh ? HH2
  - 4
• Where HH is Hiratas height or the height
  corresponding to gh
• From 3, 4 and 2 ? HH2 Gh/N z X 10,000
  tan2?/N
• or HH2 z X 10,000 tan2?/(N?)
• or HH z X 10,000tan2?/(N?)1/2
• 100 tan ? X? (z/N?)
• 100 ? (z/N) because
• when ? 60 34 tan (60 34) ? ?

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Vertical ACS compared with Horizontal ACS
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Instruments Used in Point sampling

• Cross staff
• Blade instruments
• Thumb as Relascope
• Rectangular openings round a disc.
• Wedge prism
• Spiegel Relascope
• Vertical slit angle gauge
• Automatic slope adjusting wedge prism
• Telerelascope

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Wedge prism - principle.
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