.INCLINED POINT QUADRATS

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1 .INCLINED POINT QUADRATS / BY J. WARREN WILSON Botany Department, University of Reading (Received io May 1958) (With 5 figures in the text) SUMMARY 'Relative frequency' recorded by point quadrats measures not the actual area of foliage but the area projected in the direction in which the quadrat lies. Accordingly the relative frequency varies both with the slope of the foliage and also when inclined quadrats are used with the inclination of the quadrat. A theoretical study reveals that variation in relative frequency resulting from difi^erences in foliage angle is greatest for vertical quadrats, is considerably reduced when (as suggested by Tinney, Aamodt and Ahlgren) quadrats are inclined at 45, and is least when quadrat inclination is Accordingly the usual, vertical position for point quadrats is the worst possible one, since it results in the most erroneous estimates of percentage contribution (area basis); while with quadrats inclined at 32.5 errors are greatly reduced and are of an order acceptable in general survey work. INTRODUCTION In the point quadrat method of vegetational analysis introduced by Levy and Madden (1933) t^i^ needles are passed vertically through grassland or other low-growing vegetation and the number of contacts between needles and foliage is recorded. The number of contacts per hundred quadrats (i.e. 'relative frequency') is a measure of the area of foliage, expressed as a percentage of the area of ground within which quadrats are being placed. Unfortunately, relative frequency measures not the actual area of foliage, but the area in vertical projection; consequently, its value varies with the slope of the foliage. Estimates by this method of 'percentage contribution' underestimate erect-leaved species and exaggerate the contribution of species having more nearly horizontal leaves, as compared with their true area contributions. These errors are often large, and attempts to apply correction factors are not satisfactory since growth habit varies within species according to age and environment. In 1937 Tinney, Aamodt and Ahlgren proposed the use of quadrats inclined at 45 to the ground. They considered that a sloping needle was more clearly visible in vegetation, and that the longer path of each quadrat resulted in increased accuracy. In fact, the improved visibility is denied by Winkworth (1955), and the increased accuracy is obtained through increased recording labour. Thus neither of these advantages is significant. However, inclined quadrats offer a real advantage which does not seem to have been appreciated. Whereas vertical quadrats can in theory record any proportion of actual foliage area between 100% (for horizontal foliage) and o % (for vertical foliage), inclined

2 2 - J. WARREN WILSON quadrats never record such extreme values assuming, of course, that foliage slopes towards all points of the compass. Consequently it is to be expected that percentage contribution (area basis) will be estimated more accurately by inclined quadrats than by vertical ones. Some support for this expectation is given by the observations of VanKeuren and Ahlgren (1957), who found that percentage contribution (dry matter basis) was in general more highly correlated with inclined than with vertical point quadrat estimates. Although it is clear that estimates of relative frequency will vary with the slope of the foliage and with the angle of inclination of the quadrat, the quantitative relationships are not immediately apparent, Winkworth (1955) has attempted a discussion of this subject, but with little success because he chose to restrict his attention to either (i) only those quadrats which contact foliage, or (ii) only the first contact made by each quadrat with each species. In the present work these arbitrary restrictions are not made. If all contacts with all quadrats are considered, it is possible to develop a theoretical treatment. THEORETICAL STUDY Terminology The symbols and terms used are as follows: a ('foliage angle') is the inclination of the foliage to the horizontal. Curved fohage is regarded as being composed of many plane elements, P ('quadrat inclination') is the inclination of the point quadrats to the horizontal. ('foliage denseness') is the total area of foliage per unit volume of space, measured in sq, cm/cu, cm. Since both foliage angle and foliage denseness vary with height above ground, each is normally expressed as a mean value for a horizontal layer of limited depth usually, i cm deep, <^ ('apparent foliage denseness') is the total area of the projections of all the foliage in a unit volume of space on to a plane perpendicular to a direction making an angle p with the horizontal. It is assumed* that the foliage slopes equally to all points of the compass; consequently the horizontal alignment of the direction of projection is immaterial. The term 'apparent foliage denseness' is used since ^ is in effect the apparent area of foliage per unit volume of space when viewed in a direction making an angle p with the horizontal, p is measured in sq, cm/cu. cm. 'Contact frequency' is the number of contacts with foliage per centimetre of point quadrat. It is assumedf that when quadrats are inclined at an angle (3 the contact frequency is equal to the apparent foliage denseness p, I OOP, The relative frequency is given by the sum of values of for all horizontal centism (3 metre layers through the entire depth of the vegetation; the term sin p is introduced here to allow for the increased length of quadrat resulting from inclination, * The foliage in natural \'cf,'etation may not face all points of the conipass in equal amounts. It is therefore desirable, as Winkwnrth (1955) has pointed out, to align inclined quadrats to different points of the compass (at intervals of, say, 30 ) in equal numbers, so as to compensate for any directional bias that may be present in the foliage, f or this assliniption to be \"ahd it is necessary: (i) That quadrat thickness be negligible. This requirement can be met by recording contaets with the piiint and not the sides of an advancinfi needle (Warren Wilson, 1059), (ii) 'l'hat, for statistical reasons, a sufficient number of contacts be recorded.

3 Inclined point quadrats -j Theory In the Appendix there are derived formulae whieh give the apparent foliage denseness corresponding to any value of 3 in the case of foliage which is inclined at a fixed angle a to the horizontal and which slopes non-preferentially to all points of the compass. In addition to this purely mathematical treatment there is given here a less abstract discussion that is in some ways less satisfactory but has the advantage of enabling the situation to be visualized. Consider first the case of a single piece of 'foliage' that is flat, has no thickness, and is of areayi sq, cm. or simplicity we may think of it as being square in shape (ig. i). a=0 0 = 30,=45 = 60 > = 90 ig. I, (Explanation in text.) Areas projected vertically and at 45' by a Hat piece of 'foliage' i cm square rotating about its horizontal midline, at five foliage angles (a). When P is 45' there are two possible projections at each foliage angle (other than o" and 90) depending on whether the foliage lies more nearly perpendicular or parallel to the direction of projection. When it lies horizontally (a = 0 ) the area projected vertically is A sq. cm; but as the 'foliage' rotates about its horizontal midline the vertically projected area decreases being equal to J cos a and it vanishes when the foliage is vertical (a = 90 ). The area projected at 45 to the horizontal (and in a direction perpendicular to the line about which the 'foliage' rotates) varies similarly, but in this case the maximum and minimum values occur when a = 45" and the 'foliage' is perpendicular or parallel respectively to the direction of projection. Consider now a centimetre cube of space inside which the foliage all faces the same direction and is of denseness. The foliage within this cube may be regarded effectively as a single flat piece of foliage of area ; consequently, all that has been said in the previous paragraph applies here equally. The denseness represents in this case the actual area, and the vertical apparent denseness /'',, represents the actual projected area on to a horizontal plane. urthermore, in the case when the foliage in the centimetre cube under consideration faces not in a single direction but equally to all points of the compass whilst being inclined at an angle a to the horizontal, the apparent foliage denseness ^ will be the actual area projected on to a plane at right angles to a direction making an angle p with the horizontal. In the same way that foliage facing in a single direction was thought of effectively as a fiat square surface, so can foliage facing all points of the compass be thought of effectively as the curved surface of a frustrum of a right circular cone. The slope of the cone is a, and in the extreme cases of a = 90 and a = o" the surface becomes cylindrical and planar respectively. ig, 2 shows the projected areas of frusta foi various values of a, and with p = 90 and 45. Such areas can be measured by construction and planimetry, and can be used to find the ratio of apparent foliage denseness, ^,

4 4 J. WARREN WILSON to actual foliage denseness,. Values of ^ can, however, be determined more accurately by calculation from the formulae given in the Appendix. P = 90" a = 0 0 = 30 a = 45 0 = " = 90- ig. 2. Areas projected vertically and at 45' by a piece of 'foliage' that has an area of i sq. cm and the form of the curved surface of a frustum of a cone; five foliage angles are represented, with the height of the frustum kept constant. COMPARISON O VERTICAL AND INCLINED (45") QUADRATS P, apparent foliage denseness^ 1 j 1 In ig. ^ calculated values of -^ (i.e. -^-^-p j ^ ) are plotted against the ^ -^ foliage denseness foliage angle a for several values of p. Since contact frequency is equal to apparent fohage denseness the graph can alternatively be read as giving the contact frequency (for various values of a and P) at unit foliage denseness. It is seen that for each value of P examined, the contact frequency varies with foliage angle; however, the character of the variation is different for the several values of (3. Since at any particular quadrat inclination the contact frequency is proportional to the relative frequency, the total variation of (expressed as a percentage of its mean value) is equal to the total variation of relative frequency (also expressed as a percentage of its mean value). If there were a quadrat inclination at which relative frequency was independent of foliage angle, it " would be represented on the graph by a horizontal line. In ig. 3 the curves for 3 = 45 and 3 = 90 are distinguished by heavy lines. They show that variation in -^ (and hence in relative frequency) associated with foliage angle r is considerable when inchned (45 ) point quadrats are used, but is greater still with vertical quadrats. or medium values of a, such as generally occur, variation at 3 = 45 is about one-third of that at 3 = 90. Similarly, figures for percentage contribution (area basis) will be less inconsistent when inclined (45 ) quadrats are used in preference to vertical ones. In order to compare the magnitudes of relative frequencies yielded by vertical and inclined quadrats, consider the simple case of foliage confined to a single, horizontal, 100 -f B centimetre layer. The relative frequency is ^ z. When a < p, it is shown in the Appen- ^ sin p dix that p = cos a sin P; thus so long as the foliage angle is not greater than the quadrat inclination, relative frequency is equal to 100 cos a, and is independent of quadrat inclination. When a > p, however, this simple relationship does not apply.

5 Inclined point quadrats 5 ig. 4 compares relative frequencies yielded by vertical and inclined (45") point quadrats: so long as a < p, identical values are obtained; but as the foliage angle increases above 45 the relative frequencies yielded by inclined quadrats do not fall off so sharply as do those from vertical quadrats. INCLINATION (P) 0' 15' 30' " 0 I. 90" OLIAGE ANGLE (-) ig, 3. Apparent foliage denseness (= contact frequency) expressed as a fraction of foliage denseness (i.e. _P) and plotted, for a number of incli- nations (p) against foliage angle (a). QUADRATS INCLINED AT ANGLES OTHER THAN 45 The choice by Tinney, Aamodt and Ahlgren of 45 as the angle for inclined quadrats appears to have been quite arbitrary. It is possible, therefore, that at some other inclination the apparent foliage denseness will be affected even less by foliage angle. rom

6 6 J. WARREN WILSON ig. 3 it is clear that variation in ^^ with foliage angle is less when P = 30" than when P ^ 45", and is minimal at some intermediate inclination: in fact, at approximately 3Z.5. At th.s mchnafon, variat.on m apparen^liag^denseness ^ -^ foliage denseness frequency (ig. 4) is less than half of that at p = 45. P = 32 5' u z LU o 60 p = BO" 60 OLIAGE ANGLE (-I 90 P = 90'' ig. 4. Relative frequencies recorded by vertical and inclined (32.5 ' and 45 ) point quadrats passing through vegetation of unit leaf area index (i sq. cm foliage per sq. cm ground) and various foliage angles. While 32.5' is thus the most satisfactory quadrat inclination having regard to all values of a, other inclinations may be preferable where values of a fall within certain limited ranges. or example, when foliage angle is between 50"' and 90", a quadrat 0.6 INCLINATION (P) 0, ,4 -L. 20" 30" 40 SO" OLIAGE ANGLE (a) B ig. 5. Variation in ^ with foliage angle, at P = 32.5, 42.5 and 45'.

7 Inclined point quadrats 1- ^- r o 1... apparent foliage denseness lnchnation of 42.5 results in variation in -^^, and hence in foliage denseness relative frequency ~- of less than ±0.5 % (ig. 5). Unfortunately, accuracy of this order is not given for all values of a by any single quadrat inclination (though combinations of two or three inclinations can do so, as will be shown in a later paper). In practice, however, estimates obtained by quadrats inclined at 32.5 do not take up the maximum error, since foliage angle is never uniformly o", 37, or 90. Errors in percentage contribution (area basis) will in fact seldom exceed ±5 %. This may be sufficiently accurate for general survey work, and is a marked improvement on the accuracy obtained with quadrats inclined at 45 ; while the often-used vertical position is the worst possible quadrat inclination in that it is associated with maximum effect of foliage angle on estimates of relative frequency and percentage contribution. y ACKNOWLEDGMENT I am grateful to Dr. J. E. Reeve for helpful criticism and advice. REERENCES LEVY, E. B. & MADDEN, E. A. (1933). The point method of pasture analysis. N.Z. J. Agric, 46, TINNEY,. W., AAMODT, O. S. & AHLGREN, H. L. (1937). Preliminary report of a study on methods used in botanical analyses of pasture swards. J. Amer. Soc. Agron., 29, 83S-40. VANKEUREN, R. W. & AHLGREN, H. L. (1957). A statistical study of several methods used in determining the hotanical composition of a sward. I. A study of established pastures. Agron. J., 49, WARREN WILSON, J. (1959). Analysis of the spatial distribution of foliage by two-dimensional point quadrats. Nezv Phytol., 58, WINKWORTH, R. E. (1955). The use of point quadrats for the analysis of heathland. Aust.J. Bot., 3, APPENDIX ON DERIVATION O ORMULAE BY J. E. REEVE University of Reading Suppose, in order to fix our ideas, that point quadrats are being placed in a direction from north to south and inclined at an acute angle p to the horizontal. Referred to a system of rectangular cartesian co-ordinates OXYZ in which the axes OX, OY and OZ point respectively to the east, to the north, and vertically upwards the direction cosines of the direction in which the quadrats are being placed are proportional to (o, cos p, sin p). Let an element of foliage of area A be inclined at an acute angle a to the horizontal in a direction 6 degrees west of north, where we think of 6 as being able to have any value between o and 360. The direction cosines of the normal to this element of foliage are proportional to (sin a sin 0, sin a cos 9, cos a) and thus its apparent area when viewed along the direction in -which quadrats are being placed is the (positive) numerical value of A(cos a sin p sin a cos p cos 6), which we denote by j A(cos asinp sin a cos p cos 6). It thus follows that for foliage which slopes non-preferentially towards all points of the compass the apparent foliage denseness p is related to the actual foliage denseness by the equation

8 8 J. WARREN WILSON J-. j^ [The average value of I cos a sin p sin a cos p cos 9 ] / > [ as 9 vanes from o to 360 J In what follows we write, for the sake of brevity, So that (i) can now be written in the form o(9) = cos a sin p sin a cos p cos 9. P,, = Tbe average value of ^(9) as 9 varies from o to 360. (2) In order to calculate the value of -^ explicitly it is convenient to consider two cases. Case (i) a :^ p. In this case it is not diflicult to show that ^(9) is never negative and hence, using (2), = cos a sin p, (3) r where, again for the sake of brevity, we have written 360 k{q) = 6.cos a sin p sin a cos p sin 9. 2Tr Case {ii) a > p. In this case, if 9o is tbe value of 0 between o and 90 which satisflesthe equation cos 9 = cot a tan p (4) then ^(9) is never negative when 9 lies in the range 9o ^ 9 < 360-9,;, and is always negative when 9 lies in either of the ranges o^9<9o or 360 9o<9^36o. Thus, using (2), we obtain in this case ^ 360 Performing the integrations we obtain where h{q) has the same meaning as before. On simpliflcation this gives ^ = j sin a cos p sin 9,, + ( i -" J cos oc sin p >, (5) \jt \ 90/ J &, In conclusion, what we have shown is that the value of - is given by (3) when a ^ P and by (5) when a > p where, in the latter case, 9,, is the angle between o and 90 (measured in degrees) which satisfles equation (4).

UNIT NUMBER 8.2. VECTORS 2 (Vectors in component form) A.J.Hobson

UNIT NUMBER 8.2. VECTORS 2 (Vectors in component form) A.J.Hobson JUST THE MATHS UNIT NUMBER 8.2 VECTORS 2 (Vectors in component form) by A.J.Hobson 8.2.1 The components of a vector 8.2.2 The magnitude of a vector in component form 8.2.3 The sum and difference of vectors