NEW TECHNIQUES FOR CALCULATING VOLUMES BY CROSS SECTIONS.

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1 NEW TECHNIQUES FOR CALCULATING VOLUMES BY CROSS SECTIONS. Stuart SPROTT. Numerical methods, Surveying, earthworks, volumes. Why investigate volume calculations? The standard method of calculating earthwork volumes for road works has been the end area method. It s main appeal has been it s simplicity. Volume = ( End area 1 + End area 2 )* Length 2 However in certain circumstances the error can be as high as 50%. This type of error occurs when the shape approximates a pyramid or a cone. In real life this situation can occur when we have stockpiles, or in the case of roadworks, when a cut changes to a fill. A Global view. Perhaps the greatest potential of the net is the possibility of cross fertilisation of ideas between totally different disciplines. I had regularly browsed the net for any ideas on this subject without any success. There were plenty of software firms advertising their software; but very little discussion on techniques used. There was one site that discussed the determination of brain volumes. I had skimmed through this site a couple of times, but could not make much sense of it. The terms and methods seemed completely foreign. On the third reading the fog lifted and I could at last see what he was getting at. The paper is interesting, but it It is obvious that the author has little, if any, contact with Surveyors or Engineers. At first reading it did appear that his methods were completely different. However I can now see that he really uses the same formula as we do; but in a different manner. His approach is a global approach. He uses numerical integration to obtain the total volume. In contrast, we Surveyors have concentrated on individual sections. We find the volume of each section then sum to get the total. A subtle difference, however an important difference. It has meant we have thought in linear terms. Our prismoids have to be bounded by plane surfaces. This is the reason why the rigourous prismoidal formula has been avoided. Our A Spatial Odyssey : 42 nd Australian Surveyors Congress

2 DTMs are formed on the basic assumption that the terrain is made up of countless flat triangles, joined to each other by their sides. I have long thought that the above DTM assumption is not complete. In fact I have been working on a non-linear DTM for quite some time. (See diagram 2) Theory Suppose we have the function y = f(x) Then from calculus the area under the curve from x = a to x = b is as shown below. b Area = (Integral)f(x)dx a Now suppose that the above area is a cross section. Let the area A correspond to the chainage g. Then the volume between chainage s and chainage t will be as shown below. t Volume = (Integral)f(g)dg where A = f(g) s The second equation is really in the same form as the first. Usually the whole operation is written as a double integral. By using the above convention it can easily be seen that Simpsons Rule can be applied to find the final volume. Now Simpsons Rule for three evenly spaced offsets can be written as shown below. Area = h{ first offset + 4( middle offset) + end offset } 3 Now for a volume the above can be applied as shown below. Volume = h{ first area + 4( middle area) + end area } 3 ( See diagram 3) Where h is half the distance between the two end sections. Therefore it can be immediately seen, that the above is exactly the same as the prismoidal formula. This also implies that A Spatial Odyssey : 42 nd Australian Surveyors Congress

3 the prismoidal formula can be applied to curved surfaces. For some reason or other Surveyors are taught that the prismoidal formula should only be used in cases where the outside surfaces are planes. We are usually shown cases where the formula works for simple curved surfaces ( cones, spheres ), but when it comes to earth works it has always been implied that the surfaces must be planes. As Simpsons Rule and by implication the prismoidal formula are examples of numerical integration, then generally speaking they apply when the curve f(g) is continuous. For numerical curves the definition of continuous is not so well defined. ( For well defined functions it is.) However for practical people the intuitive approach is probably all that is needed. The vertical curves in roads are continuous. The outline of a hill against the sky is continuous. Where two straight lines intersect at an angle is an example of a noncontinuous curve. In the case of volumes, if the plot of cross section areas against chainage takes the form of a smooth curve, then it can be assumed that it is continuous. Therefore the numerical integration should cover curved surfaces as well as plane surfaces. The brain volume article suggests fitting a parabola between successive groups of three sections. His reason for this approach is that he is trying to get away from evenly spaced sections. His method of parabola fitting is simple. It seems to work well in the context of the brain volume. However I have reservations about this idea when applied to earthworks. My first approach here is to interpolate a mid cross section between two sections, then apply the prismoidal formula. Interpolation by successive differences. This method was originally developed for interpolation of mathematical tables of non linear functions. It is capable of high accuracy. Because of modern electronic calculators with hard wired functions the method has been consigned to the history dust bin. Essentially it is "a before, during and after" approach. See example below & diagram 4. AREA CHAINAGE Before During After A Spatial Odyssey : 42 nd Australian Surveyors Congress

4 The above ties in with volume work. In the above we are interested in interpolating the area at chainage 30. Area at chainage 30 = 9( ) - ( ) 16 = In general Area interpolated = 9(sum mid terms) - (sum end terms) 16 (See diagram 5) When the spacing is even the above works well and is highly accurate for mathematical functions. By using a different approach it can be made to work when the spacing is not even. For the time being I will keep it simple and stick to even spacing. In the above situation the volume between chainage 20 and 40 would be as shown below. Volume(20-40) = (58 + 4* )*20 6 = There is of course still the problem of determining the volumes of the first and last sections. This can be overcome by treating the first three sections as a parabola and solving for the mid section area for section 1. From the mid section area the volume can be calculated by the prismoidal formula. The same can be applied to the end section. (See diagram 6 for the formula needed & diagram 7 for a worked example.) Uneven Spacing The brain volume article suggested the use of parabolas for the purpose of integration. Take three cross sections; treat the chainage and area as co-ords and solve for a parabola passing through all points. Next by standard integration find the area under the curve; hence the volume. The attraction with this method is that the sections no longer need to be evenly spaced. The negative aspect of this method is that at the junctions of each different parabola the join may not be smooth and continuous. In order to overcome the above I propose to use a third degree curve instead of the parabola and use four points to solve instead of just three. In order to overcome the junction problem I propose to perform integration over the middle section only. Really a follow on from the " before during and after" idea used in the interpolation method A Spatial Odyssey : 42 nd Australian Surveyors Congress

5 This method also misses the first and last section just the same as the interpolation method In order to fix this problem; extend the integration from the nearest middle section to the portion missed. Not a 100% solution; but about the best that can be done. Maths involved. ( See below ) The third degree equation I have used is shown below. ax 3 + bx 2 + cx + d = A where x is the chainage and A the area. We have to solve for a b c & d, four unknowns therefore four points are needed. From the four points we form the following matrix equation. ( See diagram 8 ) Numerical example of the uneven spacing method. AREA CHAINAGE Volume from the above figures computes Using the data on diagram 7 the volume computes 4098, practically the same as the even spacing method. Conclusions. Both methods derived in this paper will deliver volumes estimates that will be more accurate than current methods. It is assumed that cross sections have been taken at intervals that reflect the true nature of the terrain. Cross sections taken too far apart will yield inaccurate volumes. Normal DTM Surveying procedures should be followed. The uneven spacing method gives the Surveyor greater flex ability and should be the first preference. If the volume curve is a straight line then the above methods will yield exactly the same result as the end area method. (This can easily be proved by simple geometry) If the volume curve is concave as in diagram 5, then the end area method over estimates. Again, easily proved. Finally if the volume curve is convex as in diagram 7, then the end area method under estimates. Again, easily proved. This last result is unexpected. Conventional wisdom states that the end area method almost always over estimates. The above suggests that under estimation may be as frequent as the over estimation. Conventional wisdom is also derived from the concept of linear prismoids. I have talked about a curvilinear approach throughout this paper. However the above cannot be considered complete unless the cross sections are considered as a mixture of straight lines and curves. I have looked into this aspect and have developed methods. However the methods are not as simple as the ideas outlined above and are beyond the scope of this paper A Spatial Odyssey : 42 nd Australian Surveyors Congress

6 As mentioned before the inspiration for this paper came from outside the Surveying profession. I firmly believe that we should be looking outside our profession for fresh ideas. The net provides scope for this approach. There is a lot of rubbish on the net; however if you search hard and long the occasional gem will come up. References. Clark. D. C, 1963, Interpolation by Successive Differences, Plane and Geodetic Surveying for Engineers, Volume 2, Pages 30 to 36, London, Constable & Company LTD Foxall H. G. 1970, Estimation of Volumes, Handbook for Practising Land and Engineering Surveyors, Pages 129 to 149, Sydney, The Institution of Surveyors, Australia N.S.W. Division. Rosen G. R., 1999, Brain Volume Estimation from Serial Section Measurements: A comparison of Methodologies, Pages 1 to 13, Originally published by Elsevier Press, Found on the Net under ( ) Diagrams A Spatial Odyssey : 42 nd Australian Surveyors Congress

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