La loi tronquée de de Liocourt

Résumé - La loi tronquée est une nouvelle loi de distribution des nombres d'arbres par catégorie de diamètres qui s'exprime par la formule: où : N (*, i) est le nombre d'arbres dans la catégorie de diamètres i (i entier prenant des valeurs de 0 à + ∞); M est un coefficient de proportionnalité; a et β sont les 2 paramètres de la loi tronquée. Quand β tend vers l'infini, (1 + a) est égal au paramètre de la loi de de Liocourt, avec N (*, i) = N (*, i - 1) / (1 + a). Les propriétés de cette loi tronquée semblent devoir expliquer diverses «anomalies» observées par les forestiers: - excès fréquent de bois moyens et déficit fréquent de gros bois; - relation mal vérifiée entre le site et le paramètre de de Liocourt. Cette loi tronquée permet de donner une valeur «raisonnable» au diamètre d'exploitabilité : DM = 5 β.

Abstract - The de Liocourt's truncated law. The truncated law is a new diameter distribution which fits real data in the diameter classes better than de Liocourt's law does, for regular stands as well as irregular ones. The formula of the truncated law is: where: N (*, i) = is the number of trees in the diameter class i (i integer varying from 0 to +∞); M is a proportional coefficient; a and β are the 2 parameters of the truncated law. When β tends to infinity, 1 + a is the usual de Liocourt's q-ratio with N (*, i) =N (*, i-1) / (1 +α). This truncated law may be written in a different way with an accessory variable which gives a meaning to some of its properties: where: a is a continuous real variable in [0, + ∞[; c is a parameter; N (a, i ) is the number of trees for the value a, in the diameter class i; M is the total number of trees when a takes all values in [0, + ∞[, and i all values in [0, + ∞[. When am is the maximum value given to the variable a: The normal form of the truncated law, given above, is obtained with β = am/c. a may be seen as the age; a_m is the maximum value given to the age a, for instance the exploitable age of the stands. The total number N(a,*), of trees of age a decreases when the age a increases, according to a negative exponential function. The number of trees, N(a,i), of age a in the diameter class i follows a Poisson distribution with parameter a/c. The average diameter d(a) of the trees which have an age equal to a is: d(a) = u a/c, where u is the diameter class width. The parameter c is a length of time related to the average diameter growth; it depends on silviculture and environmental conditions. So: DM = d(a_m) = u am / c; and β = a_m/c = DM/u The parameter β can be related to the maximum value am of the age of the stands, or to the average diameter DM of the trees which have the maximum age a _m, known or not. The value DM may be a useful index for irregular stands, instead of the exploitable age am used for regular stands. The average age a(i) of the trees in the diameter class i is: The average diameter growth of the trees in the diameter class i is: Δd(i) : Several examples, with figures, are given to show that this truncated law fits real data well (numbers of trees, volumes, and average ages) in the diameter classes, for balanced regular stands as well as for irregular stands. The value of the parameter a can be related to regeneration. A negative value of a seems to indicate that the regeneration has not been sufficient in the +period preceding the survey. This truncated law seems to explain several drawbacks of the original de Liocourt's law: - An excess number of medium size trees, and a shortage in the number of large trees; - Discrepancy between site and de Liocourt's parameter.