Forest Mensuration (Wollega University, Gimbi Campus) - PDF

Wollega University Gimbi Campus College of Natural Resource and Environmental Science Forestry Department Program: BSc. Degree in Forestry Academic Year III, Semester I Course title /code: For...

Wollega University Gimbi Campus College of Natural Resource and Environmental Science Forestry Department Program: BSc. Degree in Forestry Academic Year III, Semester I Course title /code: Forestry Inventory and Mensuration (Frst 3052) Course credit: 3 Chr, 5 ETC Gimbi, Ethiopia. Year, 2021/22/23 1  1. Introduction 1.1. Definition of terms commonly applied in forest mensuration and inventory. 2. Techniques of different tree height measuring 3. Volume estimation of a standing tree, 4. Tree biomass estimation theories and techniques, 5. Methods and techniques for estimating number of stems per hectare (stocking density) 6. Average stand diameter estimation techniques and diameter distribution 7. Estimating volume of wood per hectare 8. factors considered during site quality assessment and Methods of site quality assessment. 9. Application of growth and yield models in forest management 10. Forest inventory planning principles and procedures. 11. Sampling designs in forest inventory 2 1. Introduction 1.1. Forest Mensuration 1.1.1. What is Forest Mensuration?  Mensura means “measure” in Latin word.  Forest Mensuration is that branch of forestry which deals with the determination of dimensions (i.e. diameter, height, volume), form, age and increment of single trees, stands or whole woods, either standing or after felling.  It includes measurements of felled and standing trees, sawn wood and round logs and various other products i.e. bamboos, charcoal, bark, fruits.  It concerns with linear, area, volume and weight (biomass) measurements. 3  It is “the art and science of providing the quantitative information about trees and forest stands necessary for forest management, planning and research”.  It is the application of measurement principles to obtain quantifiable information for forest management decision- making.  It is concerned with obtaining information about forest resources and conditions.  It deals with the measurement of trees and stands and the analysis of the resultant information. 4  applied to forest vegetation and forest products holds value for basic ecology as well as sustainable forest management.  As demands on the world’s forests have grown, scientists and professionals are increasingly called on to quantify forest composition, structure, and the goods and services forests provide.  Grounded in geometry, sampling theory, and ecology as well as practical field experience, forest mensuration offers opportunities for creative problem solving and critical thinking. 5  It includes coverage of traditional and emerging topics, with attention to SI and Imperial units throughout.  specifically address the integration of remotely sensed data in the forest inventory process, and inventory methods for dead and downed wood.  Not only for traditional forestry but for the non-timber inventory and for remote sensing, is the use of covariates to make sampling more efficient and spatially explicit.  This is introduced in the introductory of statistics and the sampling designs that has been restructured to highlight approach and lay the foundation for further learning. 6  The Role of Forest Mensuration in Forest Management  The ultimate objective of forest mensuration is to provide:  quantitative information about the forest and its resources that will allow making reasonable decisions on its destiny, use, and management.  It is one of the cornerstones in the foundation of forestry.  in the broadest sense it is a management activity involving:  forest land,  the plants and animals on the land, and  humans as they use the land. 7 These plans require foresters to make detailed predictions about:  the growth and yield of forest resources, and how harvesting and  other forest management activities influence the flow of timber and other resources.  Based on the outputs from these models, forest managers make decisions about: ? where, ? when, ? how, and ? how much forest land should be treated. 8  Elsewhere, management planning may reflect shorter time horizons, but the decisions are no less critical.  Good forest management decisions require good tools to analyze the impacts of management activities on the quantities and  flows of the various forest resources and on the state of the forest itself.  These tools require good models and, ultimately, these models require good data. 9  The following questions are examples of the problems that must be solved for a particular forest: 1. What silvicultural treatment will result in best regeneration and growth? 2. What species is most suitable for reforestation? 3. Is there sufficient timber to supply a forest industry and for an economical harvesting operation? 4. What is the value of the timber and land? 5. What is the recreational potential? 6. What is the wildlife potential? 7. What is the status of biodiversity on the area? 8. What is the status of the forest as a carbon sink? 10 1.2. Forest Inventory 1.2.1. What is Forest inventory?  Forest inventory is the systematic collection of data and forest information for assessment or analysis.  An estimate of the value and possible uses of timber that was an important part of the broader information required to sustain ecosystems.  In inventory the followings are important things to measure:  species,  diameter at breast height (DBH),  height,  site quality,  age, and defects. 11  From the data collected one can calculate:  the number of trees per acre (density),  the basal area,  the volume of trees in an area, and  the value of the timber.  Inventories can be done for other reasons than just calculating the value.  A forest can be cruised to visually assess timber and determine potential fire hazards and the risk of fire.  The results of this type of inventory can be used in preventive actions and also awareness. 12  Wildlife surveys can be undertaken in conjunction with:  timber inventory to determine the number and  type of wildlife within a forest.  The aim of the statistical forest inventory is to provide comprehensive information about the state and dynamics of forests for:  strategic and  management planning.  ‘Inventory’ is originally, a commercial term meaning the record showing the quantity and value of articles in a store. 13  Forest inventory therefore has to necessarily provide information about:  the size and shape of the area and  the qualitative and/or quantitative information of the growing stock.  “Forest Inventory is the tabulated, reliable and satisfactory tree information, related to the required unit, respectively units, of assessment in hierarchic order”.  In a sense, forest inventory is an attempt to describe quantity, quality, diameter distribution of forest trees and many of the characteristics of land upon which the trees are growing. 14  Objectives of Forest Inventory  to determine volume of timber growing in the forest with a view to determine the yield.  required for assessing the value for purposes of the sale of exchange or for estimating the return to be expected from clear-felled coupes.  for determining the current periodic annual increment and  to prepare map of the area showing regions of high or low volume production per unit of area to help decision making in setting up of industries. 15  to supply information for forest management planning and for pre-investment decision on forest industry establishment or expansion.  to assess the feasibility of a project from economic, social or any other considerations.  Basis for sale  Basis of management  Measurement for research  Measurement for planning 16  Scope:  Provides foundations of measurement principles applicable to any forest measurement problems.  Application of statistical theory and use of computer for data processing.  Forecast of future yields.  Measurement not only standing trees but also felled timber and their conversion. 17 Methods of tree stand measurements depends on:  Purpose of measurement;  Form of the product i.e., log, chips, etc.  Quality and total value of product  Relative precision and cost of measurement for different methods. 18 1.2.2. Bias, Accuracy and Precision  Bias (systematic error) is the lack of accuracy and can be reduced by identifying the causes of this systematic error.  Bias implies systematic distortion arising from such sources as a flaw in measurement or an incorrect method of sampling.  The difference between true value and the measured value.  Inaccurate  systematic error affecting all the measurements. 19 Causes of Bias Eg.1. Instrument not properly adjusted 2. Measure of 100- m units with a tape only of 99-m long 3. Under estimation of tree height by timber cruisers consistently 4. Arbitrary shift of field plot locations to capture typical samples. 20  Accuracy is a relative measure of the exactness of the value of an inferred variable for a population.  It means how close the estimates are to the actual true value.  True value can be determined only by very careful measurements with accurate instruments and formula used.  As to the Forest Mensuration, although mensuration is a branch of mathematics, It does not attempt to secure absolute mathematical accuracy. 21  F. Mensuration aims at reasonable and relative accuracy, i.e. maximum accuracy which is profitable and possible to obtain in practice.  The reasons for inaccurate measurements are: a) Characteristics of trees b) Varying methods and conditions of felling and conversion c) Personal bias of the estimator d) Biological character of the forest e) The use to which the measurements are to be put 22  Precision is the closeness of agreement between independent results of measurements obtained under stipulated conditions.  It is the inverse of uncertainty in the sense that the more precise something is, the less uncertain.  It gives consistent results from repeated measurements,  There are three types of precision: A. Theoretical precision:  precision which is justified on theoretical grounds; eg the internal angles in a triangle theoretically add up to exactly 180°. 23 B. Required precision:  precision of the measurement for a particular purpose; e.g., the precision required when machining two metal parts so they can fit together. C. Achievable precision:  precision of the measurement which can actually be achieved from the measurement equipment.  This achievable precision is shown in the number of decimal places reported in the measurement.  The precision of the measurement is the number of units that can be reasonably reported from the measurement. 24  If target is directly hit the center, that shows the absolute accurate.  If it hits to the middle circle (>50% to 99.99%), it is precise.  If target hits in outer circle then it is bias (< 50%) 25   Uncertainty  Uncertainty is the lack of knowledge of the true value of a variable that can be described as a probability density function (PDF) characterizing the range and likelihood of possible values.  Where: μ (“Mu”) is the mean of the distribution  An uncertainty analysis aims to provide quantitative measures of the uncertainty and includes random errors and bias.  Random error is inversely proportional to precision and describes a random variation below or above a mean value. 26  Unit of measurement in forest mensuration 12 inch = 1 foot 1 cft = 0.0283 cu m 3 feet = 1 yard 66 feet or 22 yards 1 cft/acre = 0.070 cu m/ha = 100 links (1chain ) 1 ha = 2.47105 acres 8 furlongs or 1760 yard = 1 mile 1 cu m/ha = 14.291 cft/acre 10 chains or 220 yards = 1 furlong 1 cubic meter = 35.3147 cubic ft I inch = 25.4 mm 1 foot = 30.48 cm 1 kilogram = 2.20462 pounds 1 yard = 0.9144m 1 metric = 0.98420 ton 1 acre = 0.40468 ha 1 mile = 1.609km 27 Quiz (10%) 1. What is Forest inventory? (2 pts) 2. What are objectives of forest inventory? (2 pts) 3. Define forest Mensuration (2 pts) 4. Why we measure forests ? (2 pts) 5. Write three types of precision. (2 pts) 28 2. Measurement of Trees  The main objective of measurement of individual trees is to estimate the quantity of timber, firewood or any other forest product which can be obtained from them.  It covers: 2.1 Diameter and Girth Measurement 2.2 Height Measurement 2.3 Measurement of Logs and Fuel wood 29 2.1 Diameter Measurement of Trees  The linear measurement, the main object of which is to estimate the volume of the trees.  The volume of a tree is dependent on diameter or girth at breast- height, total height and form factor.  It is not only necessary for calculation of volume of logs, but also necessary for making as well as to correlate height, volume, age, increment of trees. 30 2.1.1 DBH Measurement & its significance  DBH is simply the average stem diameter outside bark at point, 1.37m above ground.  Universally adopted standard height for measuring girth, diameters and basal area of standing trees recommended by FAO is 1.37 m above ground. 31  Why tree diameter is taken at breast height?  Convenient height for taking measurement  Avoids the fatigue caused unnecessarily  Saves extra expenditure from clearing the base  Abnormalities , eg. Usually root swell appear below breast- height  Standardizes diameter measurement giving a uniform point of measurement.  Diameter measurement at stump height is preferred, but standardization is lost because height of stump depends upon skill of the labor and the commercial value of the tree. 32  DBH = Diameter at Breast Height = 2 r Where, r is radius of stem  Girth = Perimeter of Stem at Breast Height = 2πr Thus, Gbh = Dbhπ  DBH DUB = DOB –2 t Girth g = g’ –2πt Where, g = girth under bark. g’ = girth over bark. t = bark thickness. 33 Ratio of UB and OB 34 Basic Level  The basic depth of the bark can be estimated by cutting through the bark to the wood or observing the bark of at least five recently felled trees. Advanced Level  Bark thickness can be measured with a specifically designed bark gauge to an accuracy of one millimeter.  It relies heavily on feel; so to minimize error the following guidelines are suggested. Place the chisel end of the gauge against the bark and apply pressure to the handle until the chisel penetrates to the wood interface. 35  The operator musty "feel" the change in pressure as the he / she attempts to push the chisel through the wood. avoid twisting the gauge when it is embedded in the tree - the chisel tip is easily fractured; for trees with rough and fissured barks, measure bark thickness on the ridges; ensure the flange of the gauge is flush with the bark before reading; if calipers are used for measuring diameter, measure bark thickness at the points of calipering; because bark thickness tends to vary from one side of a tree to another, take 3-4 readings around the stem and average. 36 2.1.2 Rules of DBH Measurement and instruments used Tree On Level Ground Tree On Slope 37 TREE FORKING AT OR TREE FORKING ABOVE 1.3 m BELOW 1.3 M 38 BUTTRESSED TREE 39 TREE DEFORMED TREE LEANING TREE AT 1.3 m 40 Points to be considered at the time of measuring diameter o Breast height point should be marked by intersecting vertical and horizontal lines 12 cm long , painted with white color o Breast height should be marked by means of a measuring stick on standing trees at 1.37m or 4 ft. o On the sloppy ground, the diameter at breast height should be measured on the uphill side, o In the case of leaning tree, dbh is measured along the tree stem and not vertically. 41  Breast height mark should be shifted up or down as little as possible to a more normal position of the stem and then diameter measured if stem is abnormal.  Buttress is formed due to edaphic factor so if buttress is seen, the dbh should be taken a little above the buttress formed 42 Instruments for DBH measurement Wedge prism 43 Instruments for DBH measurement… 44 Instruments for DBH measurement… 45  Mostly used to measure dbh/girth in developing countries are wooden scale, caliper and tape.  Biltmore stick, Penta-prism, Bar and Dendrometer are used to measure the diameter & girth of trees in developed countries  Their use depends upon the condition of trees(Felled or standing) & the degree of required (research, business, etc.) 46 Caliper  It is generally made of wood, consists of a graduated ruler and two arms. One is fixed at right angles to one end of the ruler and the other movable parallel to the fixed ruler.  Size more than120cm in length is rarely used. The use depends on the desired accuracy.  Calipers marked are painted to differentiate the reading.  In some calipers, the arms are made parallel by screw adjustment. Metal calipers made of aluminum alloy are in use.  They are not heavier than wooden calipers and ease to keep clean & adjustment 47 Advantages  Diameters can be read directly in cm and mm thus making the instrument applicable for precise scientific work.  The point arms touching the tree are always insight and irregularities if any can be avoided.  By firmly pressing the arms against the tree bole, the loose swollen bark is crushed out &irregularity from this source is avoided.  It is adaptable for use by unskilled labor.  The errors are both +ve & -ve and therefore the chances are that they may neutralize to give more accurate results than the tape which gives only +ve errors. 48 Disadvantages  They are not accurate when not in adjustment  Calipers sufficient in size to measure large trees are very awkward to carry & handle.  Two measurements have to be taken on every tree to get the correct diameter.  In steep hilly terrain, measurement of second diameter in correct orientation is often very difficult  Movable arm often sticks when the scale is wet or dirty, thus wasting a lot of time. 49 Precautions:  The calipers must be placed on the tree with movable arms well-opened and must not be forced on the tree, thereby causing stress or damage to the arms.  The reading must be taken before the caliper is removed from the tree.  If the cross-section of stem is more or less elliptical, it is necessary to measure two diameters which corresponds to the major and minor axes of ellipse.  Since axes of an ellipse are at right angles to each other, diameter is then understood to the average of two measurements 50  Suppose, d1 & d2 are two dia. of elliptical section of tree, Basal area=(π/4)d1x d2  Calipers must be placed at right angles to the axis of the tree. 51 2.2 Height Measurement Definition  Total height of the tree is the straight line distance from the tip of the leading shoot /or from the highest point of the crown where there is no leader to the ground level, usually measured on slopes from the uphill side of the tree.  Objective  To find out tree volume  To read volume tables, form factor table, yield tables, etc.  To find out productive capacity of the site  To find out the site quality of a locality. 52 Different Heights of Tree Measurement 1. Bole Height  The distance between ground level and Crown Point is called bole height 2. Crown Point  The position of the first crown forming living or dead branch. 53 3. Commercial Bole Height If the height of bole that is usually fit for utilization as timber called commercial bole height 4. Height of Standard Timber Bole Height of the standard timber bole is from the ground level to diameter over bark 20 cm or 10 cm 5. Stump height  The height of the top of the stump above ground which gives the tree stem is lift attached to the ground after felling ( 20 to 30 cm ), called Stump height.  Value of standing tree before felling is called stumpage value. 54 Crown Length/crown depth.  The vertical measurement of the crown of the tree from the tip to the point half way between the lower green branches, forming green crown all round and the lowest green branch on the bole, called crown length. Crown height  The height of the crown as a measured vertically from the ground level to the point half way between the lowest green branches forming green crown all round, called crown height. 55 2.2.1 Principles of Height Measurement 2.2.1.1 Trigonometric Principles a. Tangent Law b. Sine Law 2.2.1.2 Principles of similar triangle 56 Basics of Tree Height Measurement 57 2.2.1.2 Trigonometric Principles  The height of the tree is calculated with the help of the tangents of the angles to the top and base of the tree and the distance of the observer from the tree.  Methods of measurement employ simple principles of Geometry or Trigonometry and assume that trees are perpendicular to the ground.  Instruments used for measuring tree heights are collectively known ashypsometers.  The basic trigonometric principle used in hypsometers is illustrated in the accompanying diagram. 58 2.2.1.2 Trigonometric Principles….. height measurement on a level ground  The observer is stationed at a fixed horizontal distance D from the base of the tree. 59  On a level ground or on gentle terrain, his horizontal (level) line of sight will usually divide the tree height into two segments A and B.  When the Observer views the base and the top, the line of sight makes angles a (angle of depression) and b (angle of elevation) respectively with the horizontal line.  Then Tree Height = A + B = D x (tan a + tan b)  Abney level and several altimeters operate on this principle. Using the above relation these instruments give height readings directly at fixed horizontal distances (D) from the tree. 60 Accuracy of Measurement: Accuracy of height measurement based on formula (A) above depends on the following factors. a) Accuracy with which it is possible to determine the horizontal distanceD b) The Observer finds a position from which tree top and foot of the tree are clearly visible.  In trigonometry, in any triangle, sines of angles are proportional to the opposite sides. Thus Sin ∠AOB = Sin∠ABO AB AO AB = AO * ∠Sin AOB Sin ∠ABO 61  Principles of Similar Triangle  Similar Triangle B B A E D C Where, DE and AE are known and AC can be measured in ground 62  2.2.2 Height of Vertical trees On slope: Case I: Observer at the middle of the tree, AB = DO (Tan u + Tan d) Case II: Observer below bottom of the tree, BD = AB-AD = AO (Tan u –Tan d) 63 2.2.2 Height of Vertical trees… Case III:  In mountainous terrain, the observer’s position may be either above the top of the tree  Observer is above the base and top of the tree.  Height of the tree, AB= ED. (Tanβ –Tanα) = EB. Cosβ.(Tanβ -Tanα) 64  What about the Principles of Measuring Tree Height of Leaning trees? 65 Methods of Tree Height Measurement… Ocular estimate  In estimating height of trees by eye, a height scale has to be fixed in mind.  This is easily done by measuring the heights of a few trees with some instruments before the start of the work and that of a few trees again in the middle of the work.  With this standard in mind, the estimator judges the heights of trees to be measured and records them. 66  To make it more reliable, a pole of 3 m length may be placed against the tree and then the tree is imagined to be divided in 3 m sections and the height is calculated.  If estimator has no practice in dividing the tree in imagination, he can make use of a pencil and divided in to sections.  Non-Instrumental Methods  All non-instrumental and instrumental methods are based on the assumption that the tree is vertical.  So, the calculated height does not give the actual height of height because all trees lean on one side or the other some of non-instrumental methods are given below: 67 a) Shadow Method  In this method, a pole of convenient length is fixed upright in the ground and its height above the ground is measured Let, A AB is the tree, ab is the pole, BD is length of shadow of tree bd is the shadow of ab  The height of the tree can then be calculated by simple proportion as follows: 68  It can be applied on clear sunny days.  It gives accurate results in the early mornings or late evening , when tree cast long shadow.  It is difficult to apply in well-stocked forests. AB/ab = BD/bd 69 b) Single Pole Method  Let AB be the tree and  ab be a pole about 1.5m long, held at b vertically so that the distance from observer’s eye E to b is equal to ab.  Eb = ab, then a  AB/ab = EB eb  AB= EB*ab eb (Since, ab = Eb, AB = EB) 70  Instrumental Contemporary Instruments Hypsometers, altimeters and clinometers are used to measure height. Hypsometers:  Used for determining the height of standing tree from observations taken at some distance from the tree. Altimeters:  Generally, altitude measuring instruments, which can be devised to determine heights of tree 71 Contemporary Instruments… Clinometers:  It measures angle of slope.  Any instrument which measures angles of slope can be used for height of tree by trigonometrical methods.  Some clinometers designed for this purpose called hypsometers are based on geometrical principles of similar triangles or based on relations between the sides of right angled triangles. 72 A. Trigonometrical Principles 1. Abney’s Level 2. Relaskop 3.Topographical Abney’s Level 4. Haga Altimeter 5. Blume-leiss Hypsometer 73 Abney’s Level 74 3b. Diameter and Height Measuring Instrument 1. SPIEGEL -RELASKOP  It is a specialized instrument for measuring: Directly 1. Distances (range-finding) 2. Heights 3. Slopes 4. Diameters 5. Basal area/ha Indirectly 1. Number of stems/ha 2. Volumes of individual trees 3.Form factors and form heights 4. Volumes of stands 75  SPIEGEL –RELASKOP…..  How to use it, calibration and maintenance The instrument consists of: 1. An eyepiece on the top and back of the instrument 2. A clear sighting window at the front 3. A visor which can be lifted to shade the clear window and the scales in the instrument 4. Three ground glass windows which let light into the instrument 5. A brake knob which, when pushed in, frees the scale wheel: the wheel is locked when the button is released 6. A tripod socket for mounting the instrument 76 SPIEGEL –RELASKOP …..  Angle-count sampling (ACS) provides a direct measurement of the forest basal-area density at a particular sample point (in square meters per hectare or other appropriate units) by simply counting all the trees which have a diameter exceeding a given “critical angle”. 77 SPIEGEL –RELASKOP ….. Basal-area Factor (k), RU-Width, Relaskop - Unit (RU), Distance Factor (Df)  In a full 360 degree sweep, the basal-area factor k is the ratio of the basal area of each tree to its marginal circle in hectares:  k (in m²/ha) = (π d²/4) / (π R²/104) = (50 d/R)² where d and R are measured in the same units, e.g. meters. 78 The width of band 1 is called a Relaskop-Unit (RU), because for any number n of such RU-widths we have the simple formula: k (in m²/ha) = n² where n can also be any fraction you decide to use. The square of the number of Relaskop-Units n used for angle count sampling is the basal area factor k in m²/ha.  The critical distance R, given a particular tree diameter d, and basal-area factor k is: R = d x 50/√k = d x Df where Df = 50/√k is called the distance factor. 79  In regard to k=1 (band 1 on the scale), we have the special case already mentioned, where Df = 50. In this case 1 RU corresponds to a critical angle with a width of 2% of the horizontal distance, that is: A width of 1 RU corresponds to 2% of the horizontal distance to the measured object. 80 2.3. Measuring Tree Volume 2.3.1.Tree shape: For the purpose of measurement of wood, the portion of tree which is above ground, that is, the stem and crown, is of consideration.  The size and height of the crown and its distinction from the stem depend on many factors like species, age, stem density etc.  Trees take different forms and shapes depending on the conditions under which they are grown.  Again trees of the same species of different ages will have different shapes. 81 2.3.1.1 Taper:  Trees are found to taper.  In other words, the diameter and girth of a tree gradually reduce from base to top.  However, as one goes along the height of a tree, the taper is not uniform.  The degree of taper varies from part to part and thus leads to different shapes of the various parts of the stem. 82 2.3.1.2. Complex Form: Since the tree form or shape is complex, measuring tree volume with accuracy becomes a difficult task. The only accurate means of measuring tree volume is to fell a tree, submerge the various parts in water by turn, and calculate the total volume of displaced water. It is, however, obvious that for the purpose of management this method of submerging is not feasible and not advisable either. 83 2.3.2.Tree parts resemble some geometrical solids:  Forest managers try to estimate the tree volume by taking linear measurements and applying the properties of geometrical solid bodies to which the tree parts tend to approximate.  It may be borne in mind that calculation of tree volume based on such linear measurements is only an estimation of tree volume. 84 2.3.2.1.Parts of a tree stem tend to approximate truncated parts of some known geometrical shapes.  The base of the tree tends to be neiloid while the tip tends to be conoid.  The main part of the bole tends to be paraboloid. 85 2.4. Volume of Stem wood:  In order to find the volume of stem wood, volume of individualsegments may be calculated and the sum of such volumes will give the total volume.  Thus from Fig., the total stem volume or trunk volume of the tree may be computed from the following relation.  Trunk volume = Volume of segment AB (Neiloid frustum) + Volume of segment BC (Paraboloid frustum) + Volume of segment CD (cone). 86  Formula of geometrical solids:  While calculating the volume of individual sections, formula for volume of the corresponding geometrical solids may be employed.  Simplified Volume computation :  In practice, for the purpose of measurement, a stem or log is presumed to have the form of a truncated paraboloid , that is, frustum of a paraboloid.  Thus formula of paraboloid frustum can be employed to find the stem volume, taking the stem as one section in the shape of a paraboloid frustum. 87 88 Measurement of Stem Volume by Form Factors  Form factors may be determined, with sufficient accuracy.  If form factors are evaluated for sufficient number of felled trees of different dimensions, then average form factors may be determined for various girth and height classes.  Generally it will be seen that for a species in a given location, form factors of trees of different dimensions vary within narrow limits.  In other words same form factor will apply to considerable ranges of height and diameter. 89  A table may thus be prepared for average form factors for trees of different girth and height classes.  Such table may be utilized for estimation of volume of standing trees from measurements of diameter or girth at breast height and height and by applying the formula.  V= F x A x H 90