Boris Zeide wrote:
> DIAMETER JUMP: A NEGLECTED COMPONENT OF STAND DYNAMICS
> We all know that average diameter of forest stands changes as a result
> of two processes: tree growth and thinning, natural or silviculatural.
> Unlike growth, thinning changes diameter abruptly, in a jump. I did not
> find much more information in the literature. First of all, it would be
> interesting to know how much diameter jump contributes to the total
> diameter change.Please share with me what do you think about this matter.
Interesting topic, although I am not sure that it has been "neglected".
I suppose that almost everybody building a growth model for managed
stands has modelled this in one form or another. I'll try to explain how
we did handle it New Zealand.
I am a little hesitant pretending to lecture to all of you on this here.
But I have seen people (including us?) rediscovering the same things
again and again, or worse, using models that are logically inconsistent.
If not interested, or if you find the tone too patronizing, just hit the
DELETE button. Sorry for the waste of bandwidth!
Let D be the (quadratic) mean dbh, B the basal area, and N the number of
trees per unit area. When necessary, attach a subindex b for before
thinning, a for after thinning, and r for removed in thinning. Obvious
relationships:
Nb - Nr = Na, Bb - Br = Ba, B = c D^2 N,
where c = pi/40000 if B is m2/ha, D in cm and N in trees per hectare, or
some other constant for those still using units from the Dark Ages :-)
Juggling these, one can disguise equivalent models writing them in
messier (or more "interesting") ways. Without needing anything else, we
can produce things like this "explanation" of the diameter jump:
(Da/Db)^2 = 1 + (Nr/Na)[1 - (Dr/Db)^2]
I won't go into all the combinations and permutations, rather try to use
the simplest expressions.
J.Beekhuis ("Prediction of yield and increment in Pinus radiata stands
in New Zealand", N.Z.Forest Service Technical Paper No.49, 1966) shows,
in his Table 4, percentages of basal area removed in thinning for
various percentages of number of trees removed. Slightly different
values are given for different stand dominant height classes, and for
first, second, or later thinnings, but a reasonable fit overall is achieved
with
Br/Bb = 1 - (1 - Nr/Nb)^k ,
or
Ba/Bb = (Na/Nb)^k , (1)
with k around 0.75 (k is dimensionless). Many seem to have used models
equivalent to this one. The constant k represents the thinning
selectivity, with k=1 if the thinning is systematic (e.g. row thinning),
k<1 for thinning from below, and k>1 for thinning from above. Ratios such
as Da/Db or Dr/Db have often been proposed as indices of thinning
selectivity or "thinning type". These are functionally related to k for a
given thinning intensity Nr/Nb, but can otherwise be expected to vary with
intensity (if we remove all the trees, necessarily Dr/Db = 1).
Suppose we thin N1 trees and calculate the remaining basal area with (1).
Then we decide to thin another N2 trees, and apply (1) again. As should be
expected, this sequential calculation gives exactly the same result as
using (1) once for a removal of N1+N2 trees. Other thinning models
proposed in the literature, however, do NOT have this property, and are
therefore unsatisfactory on consistency and logical grounds.
As usual, the simplest way of ensuring this kind of consistency is to start
with a differential equation. Equation (1) results from assuming that each
small reduction in the relative number of trees always causes a
proportional reduction in relative basal area:
dB/B = k dN/N . (2)
Or, remembering that d ln x = dx/x,
d ln B / d ln N = k .
(3)
Integration gives (1). This seems to be a good enough approximation for
many purposes. More generally, one could take k as some function of top
height, as Beekhuis did, and the form of (1) does not change. More
generally still, k in (3) could depend also of the current B and/or N. For
example, I have started from the following general form:
d ln B / d ln N = k H^p B^q N^r (4)
(N.Z.Jour.For.Sci. 14:65-88, 1984, equation (10)). This has been used in
about a dozen regional stand growth models, in each instance testing the
parameters p, q and r for significance. Equation (4) is separable, always
easily solved by integrating both sides in
B^(-q-1) dB = k H^p N^(r-1) dN
Two examples: In the paper just cited (for radiata pine in Golden Downs
forest), q=0, and the fitted integrated model was
Ba = Bb exp[-31.9 H^-0.283 (Na^-0.0946 - Nb^-0.0946)]
Just to change species, for Douglas-fir plantations we found
Ba = [Bb^-0.270 + 2.03 H^-0.157 (Ba^-0.151 - Bb^-0.151)]^-3.70
(K.R.N.Law, "A growth model for Douglas fir grown in the South Island",
Forest Research Institute, FRI Project Record 2488, 1990 (unpublished)).
It is also popular to model all this through the manipulation of diameter
distributions, bringing about another full bunch of potential pitfalls.
Won't get into that.
As Boris says, a similar "diameter jump" effect (above what would be
expected from tree growth only) arises from natural mortality occurring
among the smaller trees. Of course, it gets a bit more complicated, being
entangled with the growth happening simultaneously. And there are
interesting connections with "self-thinning laws", but that's another
story...
Sufficient to point out an interesting coincidence in Beekhuis' paper.
On page 15 he gives an approximate relationship between mortality in
basal area and in number of trees:
Delta B = 0.004158 D^2 Delta N
Delta B and Delta N are annual mortalities computed over one- or two-year
intervals, with everything in British units. Substituting D in terms of
B and N, after some simplification we get
Delta B = 0.7624 B/N Delta N ,
or
Delta B / B = 0.7624 Delta N / N ,
which is pretty close to (2).
--
Oscar Garcia - ogarcia@inia.es
Xunta de Galicia, Centro de Investigacions Forestais de Lourizan
Apartado 127, 36080 Pontevedra (Spain)
Fax (+34) 986 856420. Tel. 986 856400, home: 986 840866
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