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##### Law and Order

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**Law and Order**2003 Western Mensurationists July 3, 2003 Greg JohnsonWeyerhaeuser Company**Law and Order**• Assmann’s 1970 edition of “The Principles of Forest Yield Study” cites a number of “Laws” and “Rules” of interest to biometricians and mensurationists. • How do some of these Laws and Rules stand up with our species and modern measurements?**Some Definitions**• Law: • A statement describing a relationship observed to be invariable between or among phenomena for all cases in which the specified conditions are met: e.g., the law of gravity. • Rule: • A generalized statement that describes what is true in most or all cases: e.g., In this office, hard work is the rule, not the exception.**Under Scrutiny**• Backman’s Growth Law (p. 42)* • Pressler’s Law (p. 58) • Eichhorn’s Rule (p. 161) • Mitscherlich’s Law (p. 20) *page reference from Assmann (1970)**The Backman Growth Law(1943)**• Developed by Gaston Backman (1883-1964) a Swedish physiologist and student of eugenics. • First published in “Growth and Organic Time” (1943). • The growth process is the basis of life, expresses its inner nature and is closely related to other main biological processes and phenomena of the organism. • The conceptual basis of Backman’s function of growth is the postulate that the logarithm of growth rate is negatively proportional to the square of time’s logarithm. • “I consider that living organisms develop in a logarithmical world, where the spatial and temporal values have a logarithmical scale”.**Where K is the constant of growth and T is a measure of**time. • “normal time” (the moment of time, when the growth rate is the greatest). • “normal rate” (the growth rate at the normal time) The Backman Growth Law • Weck (1953) found good correspondence with Backman’s Law in his forest - biological investigations. H. Thomasius (1965) had success for biomass, but not with height or diameter. • What is the “Law”?**Yield can be estimated through integration:**• Organic time is expressed as a function of physical time: The Backman Growth Law • where: x is “organic” time and t is “physical” time.**Testing Backman’s Law**• Bruce’s (1981) site 120 Douglas-fir site curve:**Maximum growth: 2.96 ft/year**Testing Backman’s Law • Differentiated, the height-age curve generates Bruce’s growth curve:**Testing Backman’s Law**• When organic time is 1.0, growth is at its maximum and Backman’s Law yields a growth value of 1.0, thus Backman’s Law apparently describes relative growth rate. • If we use 2.96 as a growth scaling factor, we are left with estimating K, C1 and C2 -- note that Backman’s formulation indicates that physical time is logarithmically related to organic time. • Estimates of K, C1 and C2 using Backman’s formulation applied to Bruce’s curve were unsuccessful, however, if the relationship between physical time and organic time is linear (C1t + C2) the growth law works well:**Defined to yield 2.96 at**normal time Testing Backman’s Law**Testing Backman’s Law**• Holding K, C1 and C2 constant, and changing the growth scaling factor for each site index, Bruce’s site curves are amazingly consistent with Backman’s Law across the range of site indices 80 - 140:**Pressler’s Law**• Red shift of emission line spectra from extragalactic cluster nebula is directly proportional to the radial distance to that emitting object; the red shifting is caused by the phenomenon of a gravitational field, the magnitude of which value is equal to the force per unit mass contained in the total volume of a sphere with that radius, and is directed toward said emission source. • In short, a radiating body twice as far away will demonstrate twice the red shift.**Pressler’s Law(1865)**• Max Robert Pressler (1815-1886) • “Law of stem formation”: The area increment on any part of the stem is proportional to the foliage capacity in the upper part of the tree and therefore is nearly equal in all parts of the stem which are free from branches. • Called the “Pipe theory” by tree physiologists.**Live crown**Pressler’s Law**Exaggerated basal area**growth Basal area growth below the live crown is constant Pressler’s Law**Testing Pressler’s Law**• The Stand Management Cooperative stem dissected 15 trees from two plots, collecting radial increment measurements at various points along the bole. • DBH ranged from 12.2 - 40.3 cm. • Total height ranged from 14.4 - 25.4 m. • Crown ratios ranged from 0.32 to 0.56. • Section heights are expressed as a percentage of height to crown base with height at crown base = 1.0. • Two-year basal area growth (BAG) is expressed as a percentage of BAG at crown base.**Testing Pressler’s Law**Crown base**Testing Pressler’s Law**Crown base**Testing Pressler’s Law**• The Law does not account for 9 of 15 trees that had significant butt swell (in growth). • Nearly half the trees showed a decline in BAG as distance from crown base increased. • Most trees experienced their greatest BAG at or near crown base. • All trees showed a strong decline in BAG as distance above crown base increased.**Testing Pressler’s Law**• We can test the proportionality of stem growth to foliage capacity above crown base: • Use Hann’s (1997, 1999) equations for crown width above crown base for Douglas-fir to compute crown diameter at any point in the crown. • If Pressler’s Law holds, andcrown surface area is a good proxy for foliage capacity, there should be a linear relationship between crown surface area above a given point and stem basal area growth at that point.**Testing Pressler’s Law**BAG does not appear to be proportional to crown surface area.**Testing Pressler’s Law**BAG appears to be closer to proportional to crown width for some trees.**Eichhorn’s Rule(1904)**• Eichhorn, F., 1904: Beziehungen zwischen Bestandeshöhe und Bestandesmasse. Allg. Forst- und Jagdztg. 80: 45-49. • “A given mean height of stand is matched by the same volume in all site classes.” • The rule has been called the “basic law of forestry.” • “This relationship was observed first for fir and then also for beech; it applies only as long as stand treatments are extremely weak and the resources correspond approximately to the total volume achievement.”**Testing Eichhorn’s Rule**• Stand Management Cooperative data from 1519 plots were used. • To minimize treatment artifacts, data from thinned and fertilized plots were not used. • Focused on two site classes “100” (100-124, n=979) and “125” (125-149, n=540). Other site classes were available, but the sample size was variable and small.**Testing Eichhorn’s Rule**• A Chapman-Richards function was fit to both site classes separately and combined. • Eichhorn’s Rule suggests that there should be no benefit to separate regressions for each site class. An F test (F=0.5159) indicated no benefit to fitting site classes separately, confirming Eichhorn’s Rule.**Testing Eichhorn’s Rule**• The Growth Model User’s Group ran an experiment (the Growth Model Run-Off) where members were asked to project 3 stands with their favorite growth model. • 7 models were used. 11 version-model combinations were tested. • Do they conform to Eichhorn’s Rule?**Growth Model Run-Off Results at Age 60**Stand 1 = Site 115 Stand 2 = Site 99 Stand 3 = Site 124 stand 1 stand 3 stand 2**Mitscherlich’s Physiological Dependence (Efficacy)**Law(1909) • Mitscherlich, E.A. 1909. Des Gesetz des Minimums und das Gesetz des abnehmended Bodenertrages. Landwirsch. Jahrb. 3: 537-552 • “Yield increase in response to fertilizer is proportional to the difference between the present and maximum yields.” • Where: y=yield at a given nutrient amount x, A=maximum obtainable yield under perfect non-limiting conditions, c=“response coefficient**Mitscherlich’s Law**• Also known as the law of diminishing returns. • Probably first stated by Johann Heinrich von Thünen (1783-1850) • Yield is estimated by integration: • Mitscherlich developed his law using potted plant studies under highly controlled environments and carefully metered fertilization regimes.**Testing Mitscherlich**• Used two-year basal area growth data from RFNRP installations. • Only 200 and 400 lbs N /acre applications considered. • Response is defined as percent treated basal area growth over control growth. • Did not adjust for differences in initial condition (and they do exist). • Average King 50-year site index: 112.4 feet (range: 55 - 154 feet) • Initial basal area: 183.0 feet2/acre (range: 7.2 - 362.2)**Median =14.5%**26.7% Testing Mitscherlich**Testing Mitscherlich**• Assuming site index is a reasonable proxy for indigenous nitrogen availability, then Mitscherlich’s (A - y) can be approximated by an installation’s site index. • Adding fertilizer at two rates (200 and 400 lbs N) to the same site index should, under the law, result in: • A declining response with increasing site index [(A - y) gets smaller] • A smaller increment of response to the addition of the second 200 lbs N (the 400 lbs N application)**Smaller response to incremental increases in N**Response declines as site index increases Testing Mitscherlich**Testing Mitscherlich**• The Efficacy Law predicts that there is a site index where no response to N would be observed. • Linear regressions through the data suggest that the site index where this occurs lies between 158 and 169 feet (albeit with a wide confidence interval).**Testing Mitscherlich**• Using the Law’s concept of diminishing returns, we can compute the maximum N application rate that will demonstrate an incremental increase in response: