A Growth and Yield Projection System (GYPSY) for Natural and Post-harvest Stands in Alberta

Transcription

1 A Growth and Yield Projection System (GYPSY) for Natural and Post-harvest Stands in Alerta Prepared y: Shongming Huang Shn X. Meng Yuqing Yang May, 9

2 Technical Report Pu. No.: T/6 ISBN No.: (Printed Edition) ISBN No.: (On-line Edition) For additional copies of this pulication, ease contact: Forest Management Branch Alerta Sustainale Resource Development 8 th Floor, Great West Life Building 99 8 Street Edmonton, Alerta, Canada T5K M4 Tel: (78) Fax: (78) This pulication will e availale on Alerta Government wesite in the coming weeks. Please check the latest information at: Copyright 9 y Forestry Division, Alerta Sustainale Resource Development. All rights reserved. Edmonton, Alerta, Canada May 9 ii

3 Tale of Contents Page. Introduction.... Spatial Versus Non-Spatial GYPSY.... Linking GYPSY to ARS Variales Top Height Models Density Models (Non-Spatial) Basal Area Increment Models (Non-Spatial) Percent Stocking Models Density Models (Spatial) Basal Area Increment Models (Spatial).... Gross Total Volume Models...5. Merchantale Volume Models...6. Height-Diameter Models...7 Acknowledgements... Appendix. A generalized program for predicting site index from top height and total age... iii

4 . Introduction GYPSY (short for Growth and Yield Projection System) is a forest growth model composed of a numer of su-models or functions. Since the inception of the FRIAA-GYPSY project, several rounds of model development and testing have een cometed. Fairly extensive validations were conducted y the model developers within Forest Management Branch as well as external consultants during the last six months. As a result of these validations and susequent discussion, some of the GYPSY su-models were re-formulated and re-fitted. This document lists the final su-models effective May, 9. Unless a sustantial deficiency is detected with a particular su-model during the apication and use of GYPSY, this version of GYPSY will not e modified prior to March. Since most readers are already familiar with the su-models, model structure and model usage, only the final su-models and a rief description of each are provided here. A detailed technical document descriing all su-models is anned for release in March. An Excel version of GYPSY along with a user s manual is anned for release y January. It is expected that many of the su-models and/or the methodologies used to develop the su-models will e pulished in scientific journals through anonymous peer-reviews.

5 . Spatial Versus Non-Spatial GYPSY The non-spatial version of GYPSY utilizes the following models: ). Top height models (Section 4) ). Non-spatial density models (Section 5) ). Non-spatial asal area increment models (Section 6) 4). Gross total and merchantale volume models (Sections and ) The spatial version of GYPSY utilizes the following models: ). Top height models (Section 4) ). Percent stocking models (Section 7) ). Spatial density models (Section 8) 4). Spatial asal area increment models (Section 9) 5). Gross total and merchantale volume models (Sections and ) In oth cases, volume compilations use the nonlinear mixed-effects height-diameter models (Section ) and the standard Alerta taper equations (994). Only the non-spatial version of GYPSY should e used for fire-origin stands. Both the spatial and the non-spatial versions of GYPSY can e used for post-harvest stands. However, it is recommended that the spatial version of GYPSY e used whenever possile. GYPSY was developed for four main Alerta tree species grown in pure and mixed-species stands: aspen (AW), lodgepole pine (PL), white spruce (SW), and lack spruce (SB). All GYPSY su-models are species-specific. In the cases where other species are present, they can in most cases e grouped with the four leading species as follows: AW AW + PB SB SB SW SE + Fir PL PJ + other pine species + LT Based on the availale data from Alerta government PSPs, grouping white irch (BW) with AW distorted the AW models sustantially in most cases (due to the highly variale measurements for BW). Therefore, until more BW data are collected and more tests are done, it is recommended that BW should not e included in GYPSY runs. In the rare case where BW is managed as the leading species, BW could e considered as AW and its yields generated using the AW models, ut the results otained in this manner should e interpreted with caution.

6 . Linking GYPSY to ARS Variales One of the predominant reasons for developing GYPSY is to link the growth and yield of postharvest stands to Alerta s Regeneration Performance Survey Standards and Alternative Regeneration Standards (ARS). In order to run the spatial version of GYPSY for post-harvest stands, the following variales must at a minimum e otained for each cutlock or opening: ). Top height ). Total age ). Density 4). Percent stocking If alternative or enhanced forest management practices such as thinning, hericiding, fertilization, and/or genetic improvement are apied to an opening, measurement of an additional variale asal area is recommended to enale the impacts of such practices to e assessed, and to achieve a etter prediction accuracy: 5). Basal area A rief description of each variale follows. Readers should consult Alerta s Regeneration Survey manual or relevant ARS manual for detailed field measurement procedures. ). Top height Top height is defined as the average height of the largest DBH (diameter at. m aove ground) trees per ha. For the regeneration survey grid used in Alerta, top height refers to the height of the largest (live) DBH tree in a m ot. If the largest DBH tree in the m ot has a lost or roken top that has not yet een reaced with a new leader, the next largest diameter tree of that species is used as the top height tree. There are to e no sustitution of lower diameter size trees due to health or other vigour conditions of the top height tree. Veterans, advanced growth, or ovious wolf trees must e excluded from top height trees. If necessary, relevant tree condition codes can e assigned to the top height trees (and other trees measured in the surveys), following the specifications given in the Regeneration Survey manual or ARS manual. In the cases where all trees in a ot are shorter than. m (i.e., no DBH measurement), the top height tree is the largest diameter tree at stump height (. m aove ground). ). Total age Total age refers to the years since the point of germination. It is taken on the top height tree, regardless of whether the tree is shorter or taller than. m. Total age is used in conjunction with top height for determining site index. Total age must e determined carefully, following the descriptions given in the Regeneration Survey manual or ARS manual.

7 ). Density GYPSY uses different ways to define stand density for different stand types and species: Post-harvest stands: for deciduous species, density refers to stems/ha of the suject species >. m tall; for coniferous species, density refers to stems/ha of the suject species >. m tall Fire-origin stands: density refers to stems/ha of the suject species >. m tall, regardless of the coniferous or deciduous species. 4). Percent stocking Percent stocking refers to the percentage (%) of the m ots of the regeneration survey grid with at least tree of the suject species that meets the density count requirement. It is inferred from density measurements: no actual variale of percent stocking needs to e measured in the field. 5). Basal area In stands where thinning, hericiding, fertilization, genetic improvement, and/or other alternative or enhanced forest management practices are carried out, the diameters, along with the associated density, of some representative or all trees taller than. m need to e measured in some m ots on the regeneration survey grid, to otain the asal area at. m aove ground a GYPSY input variale recommended when running projections for stands that have gone through these treatments: a). All diameter measurements are to e taken at reast height (. m aove ground). In the cases where the trees have not reached reast height, no DBH measurement is taken; ). The. density (stems/ha for trees >. m tall) associated with the DBH measurements must e clearly recorded and separated from other regular variales. Note that the DBH measurement and the. density are additions to the regular variales for alternative or enhanced forest management stands. The. density must e clearly separated from the regular density, which accounts for trees >. m tall for coniferous species. The recording of the data must allow for a clear separation of the. and. densities. 4

8 4. Top height models Due to the variation in height growth patterns, different forms are used for the top height models for the four GYPSY species: [] [] [] + exp( ln( + 5) [ln( t )] 4 5) H = top t (AW) + exp( ln( + totage) [ln( t )] 4 5) + exp( ln( + 5) ln( t ) 4 5) H = top t (SB and PL) + exp( ln( + totage) ln( t ) 4 5) + exp( ln( + 5 ) [ln( t )] 4 5) H = top t (SW) + exp( ln( + totage ) [ln( t )] 4 5) where: H top = top height (m), i.e., average height of the largest DBH trees per ha t = totage-ased site index, i.e., top height at 5 years total age totage = total age from the point of germination Estimated coefficients for the top height models are listed in Tale. Tale. Estimated coefficients for the top height models. Species Aspen Black spruce Lodgepole pine White spruce Model [] [] [] [] An exame program is provided in Appendix to demonstrate how the top height models can e used to predict site index and to make top height projections. Exact mathematical conversions etween total age and reast height age (hage), t and h ( h is the site index at 5 years hage, which is currently used in Alerta), are emedded in the models (see Appendix ). In the cases where stump age or reast height age needs to e converted to total age, the following average conversion factors can e used: AW: total age = stump age +.5; total age = hage + 4 PL: total age = stump age + ; total age = hage + 8 SW: total age = stump age + 4; total age = hage + SB: total age = stump age + 5; total age = hage + 5 5

9 5. Density models (non-spatial) The density models descrie stand density changes over time. They are functions of age, current or initial density, site quality and species mixtures. All models are species-specific. In GYPSY, stand density is defined differently for different stand types and species (see Section ). Aspen [4] N = + exp[ h ln( + 5)] + exp[ h ln( hage)] 5 = + + ln( + 5) / ln( + 5) = c 4 / ( ) = c + ln( ) [ ] ( + c ) c =.77966, and c = Black spruce [5] N = s + exp[ ln( h ) ln( + 5)] + exp[ ln( h ) ln( + totage)] = c = c ( / + ln( + 5) ) s c = c s / s c = -6.86, c =.6648, and c =

10 White spruce [6] + exp[ ln( h ) ln( + 5)] N = + exp[ ln( h ) ln( + totage)] c = + z + / c ln( ) + ln( + 5) ( ) = c = c / c = -.67, c =.76995, and c =.76. if = then z = ; if > then z =. Lodgepole pine [7] N = + exp[ ln( h ) ln( + 5)] + exp[ ln( h ) ln( + totage)] z ( /) z ( /) s = c = c 4 c ( ) 5 = k c, + c6 k = 4 7 z ( /) 4 c c ( ) c = , c = , c =.867, c 4 =.7479, c 5 = -.547, and c 6 = 4.5. if = then z = ; if = then z = ; if s = then z = ; if > then z = ; if > then z = ; if s > then z =. Where in [4]-[7]: N = density (stems/ha) of the suject species = stand density factor for AW, which is N at 5 years hage s = stand density factor for SB, which is N at 5 years totage = stand density factor for SW, which is N at 5 years totage = stand density factor for PL, which is N at 5 years totage h = site index (m) of the suject species at 5 years hage

11 6. Basal area increment models (non-spatial) The asal area increment models provide annual asal area increment predictions. They are functions of reast height age, site index, density, asal area, and species composition (SC), where SC refers to the ratio of stand densities etween the suject species and all species comined (SC = N species /N all ). All models are species-specific. They can e used to make asal area predictions whether the current stand asal area is availale or not. Aspen [8] (/ + a ) ( a Bhage ) 4 abhage exp [ln( + N + Bhage )] SC BAINC = a ln( + SC ) ( + BA ) + exp k = a 4 ln(.+ Bhage /) a =.75, a =.8847, a =.476, a 4 = -.475, and a 5 = a5 + k Black spruce ( a + a ) 4 a [9] BAINC = a exp( a Bhage ) Bhage SC k ( + k = N + Bhage )ln( + ( + BA ) a s N / 4 )exp White spruce a =.96685, a =.565, and a =.79. [] BAINC = 4 ( a + a a ) a6 a( a + Bhage ) ( + Bhage ) ln + exp + k + ( + N ) exp( abhage ) SC + a ln( + ) BA m = + + N / [ln( N Bhage )] exp m = s k a4z ln a5z ln z ln a =.895, a =.77, a = -.48, a 4 = , a 5 =.75, a 6 =.95. if = then z = ; if = then z = ; if s = then z = ; if > then z = ; if > then z = ; if s > then z =. 8

12 Lodgepole pine [] 4 ln( + Bhage ) abhage exp( abhage ) + m BAINC = + m k N + exp + ln+ a SC + a4 ln( + BA ) a m = ( + a + ) N exp 6 N m = a5 ln(.+ Bhage /) = + z + + z + ln ln + s k z ln a =.9984, a =.575, a =.564, a 4 =.6756, a 5 = -.58, and a 6 = if = then z = ; if = then z = ; if s = then z = ; if > then z = ; if > then z = ; if s > then z =. Where in [8]-[]: BAINC = annual asal area increment for the suject species (m /ha/year) Bhage = current reast height age (years) N = initial density at hage zero (for AW) N = initial density at totage zero (for SB, SW and PL) species = site index ( h, m) at 5 years hage SC = current species composition (N species /N all ) BA = current asal area (m /ha) for the suject species In practice, GYPSY users can assume that for aspen, the initial density at hage zero is equivalent to the initial density at totage zero. 9

13 7. Percent stocking models The percent stocking models descrie the level of stocking and the clumpiness of trees in a stand. They provide a direct linkage etween regeneration survey standards and yield forecasts. The models take the following forms: [] [] [4] [5] PS PS PS PS s = P = P = P = P s + exp( [ln( + 5)] ln( )) + exp( [ln( + totage )] ln( + exp( [ln( + 5)] ln( s)) + exp( [ln( + totages )] ln( + exp( [ln( + 5)] + exp( [ln( + totage + exp( [ln( + 5)] + exp( [ln( + totage s )) )) ln( ) + x( P / ) / 5) )] ln( ) + ( / ) / 5) x P ln( ) + Μ), )] ln( ) + Μ) Μ = 4 xp 5 x P s 6 x P where: PS = percent stocking (%), i.e., % of m ots with at least tree of the suject species P = percent stocking index, which is PS at 5 years total age totage = total age (years) from the point of germination = site index (m) at 5 years hage x, x and x are indicator variales defined as follows: x = if P = ; x = if P s = ; x = if P = ; x = if P > ; x = if P s > ; x = if P >. Estimated coefficients for the percent stocking models are listed in Tale. Tale. Estimated coefficients for the percent stocking models. Species Aspen Black spruce White spruce Lodgepole pine Model [] [] [4] [5]

14 8. Density models (spatial) The spatial density models descrie species-specific stand density changes over time as functions of age, current or initial density, site quality, species mixtures and percent stocking. Aspen [6] N = + exp[ h ln( + 5)] + exp[ h ln( hage)] 5 c ln P + + = + + ln( + 5) / ln( 5) + c c = 4 / ( ) = c ) ( + k k c = ln( ) ln( P ) / Black spruce c =.7, and c = [7] N = s + exp[ ln( h ) ln( + 5)] + exp[ ln( h ) ln( + totage)] = c ln( + P c ( / + ln( + 5) ) + c s s ) = c = c s / s White spruce c = , c =.95, and c = [8] N = + exp[ ln( h ) ln( + 5)] + exp[ ln( h ) ln( + totage)]

15 = + ln( + / + c ( ln( ) + ln( + 5) ) c + z ) = c = c / / ( + ) P c = -54.9, c =.4679, and c =.49. if = then z = ; if > then z =. Lodgepole pine [9] + exp[ ln( h ) ln( + 5)] N = + exp[ ln( h ) ln( + totage)] z( /) z ( /) z( s /) = c m 4 c ln( + P ) m = c + c = ( ) 4 c 4 c ( ) 5 = k c, + c6 k = 4 c = -.97, c = -5.94, c =.498, c 4 =.78, c 5 = , and c 6 = 4.9. if = then z = ; if = then z = ; if s = then z = ; if > then z = ; if > then z = ; if s > then z =. Where in [6]-[9]: N = density (stems/ha) of the suject species = stand density factor for AW, which is N at 5 years hage s = stand density factor for SB, which is N at 5 years totage = stand density factor for SW, which is N at 5 years totage = stand density factor for PL, which is N at 5 years totage h = site index (m) of the suject species at 5 years hage P = percent stocking index of the suject species (i.e., PS at 5 years totage)

16 9. Basal area increment models (spatial) The spatial asal area increment models provide species-specific annual asal area increment predictions as functions of reast height age, site index, density, asal area, species composition (SC = N species /N all ), and percent stocking. They can e used whether or not the current stand asal area is availale. Aspen [] BAINC = 4 a Bhage exp k = a 4 ln(.+ Bhage /) ( a Bhage ) ( + BA ) a [ln( + N + Bhage )] ln( + SC ) + exp a =.5596, a =.546, a =.45, a 4 = -.555, a 5 =.86, and a 6 = SC a 5 P a6 + k Black spruce ( a + a ) 4 a [] BAINC = a exp( a Bhage ) Bhage SC k ( + k = N + Bhage )ln( + ( + BA ) s a N / 4 ) exp P a =.889, a =.47669, a =.65, and a 4 = White spruce a4 s [] 4 ( a + a a ) a7 a( a + Bhage ) ( + Bhage ) BAINC = ln + exp + k + = + + N / m [ln( N Bhage )] exp ( + N ) a6 exp( abhage ) SC m P + a ln( + ) BA = s k a4z ln a5z ln z ln a =.47, a =.67, a = -.6, a 4 = , a 5 =.45, a 6 =.6, a 7 = if = then z = ; if = then z = ; if s = then z = ; if > then z = ; if > then z = ; if s > then z =.

17 Lodgepole pine [] 4 ln( + Bhage ) abhage exp( abhage ) + m P BAINC = k N + exp + ln + a SC + a4 ln( + BA ) a m = ( + a + ) N exp 6 N m = a5 ln(.+ Bhage /) = + z + + z + ln ln + s k z ln + m a =.679, a =.54, a =.767, a 4 = , a 5 = -.45, and a 6 =.84. if = then z = ; if = then z = ; if s = then z = ; if > then z = ; if > then z = ; if s > then z =. Where in []-[]: BAINC = annual asal area increment for the suject species (m /ha/year) Bhage = current reast height age (years) N = initial density at hage zero (for AW) N = initial density at totage zero (for SB, SW and PL) species = site index ( h, m) at 5 years hage SC = current species composition (N species /N all ) BA = current asal area (m /ha) for the suject species P = species percent stocking index (i.e., percent stocking at 5 years totage) In practice, GYPSY users can assume that for aspen, the initial density at hage zero is equivalent to the initial density at totage zero.. 4

18 . Gross total volume models The gross total volume models predict the species-specific gross total volume of a stand at a / utilization standard ased on the top height and asal area of the suject species. White spruce and lack spruce: β β [4] Tvol = β BA H top Aspen and lodgepole pine: [5] Tvol = β BA β H β top β4 exp + Htop + Where in [4]-[5]: Tvol = gross total volume (m /ha) of the suject species at the / utilization standard BA = total asal area (m /ha) of the suject species H top = top height (m) of the suject species Estimated coefficients for the gross total volume models are listed in Tale. Tale. Estimated coefficients for the gross total volume models. Model [4] Model [5] Parameter Black spruce White spruce Aspen Lodgepole pine β β β β Given the oserved asal area and top height of a stand from any inventory, the gross total volume models can e used independent of the other models to otain accurate volume predictions. 5

19 . Merchantale volume models The merchantale volume models predict the species-specific merchantale volume of a stand at any user-defined utilization standard. They take the following general form: [6] Mvol = H top Tvol k StumpDOB StumpHT 4 TopDi 5 k 6, + k k = BA N / where: Mvol = merchantale volume (m /ha) of the species at a specified utilization standard Tvol = gross total volume (m /ha) of the species at / utilization standard BA = total asal area (m /ha) of the suject species H top = species top height (m), i.e., average height of the largest DBH trees per ha StumpDOB = stump diameter outside ark (cm) StumpHT = stump height (m) TopDi = top diameter inside ark (cm) N = density (stems/ha) of the species Variales StumpDOB, StumpHT and TopDi define a utilization standard. For exame, StumpDOB =, StumpHT =. and TopDi = 7 represent a /7 utilization standard with a stump height of. m. The minimum merchantale log length (e.g.,.44,.66, or 4.88 m) variale had an insignificant impact on the merchantale volume models. Estimated coefficients for the merchantale volume model [6] are listed in Tale 4. Tale 4. Estimated coefficients for the merchantale volume model [6]. Species Parameter Aspen Black spruce White spruce Lodgepole pine

20 . Height-diameter models Height-diameter models are used to predict missing tree heights. They were developed using the first-order (FO) method of the nonlinear mixed-effects modeling technique. In addition to the four GYPSY species, height-diameter models were also developed for other tree species. For each species, separate models were fitted: a mixed model, a grand mean model, and a maximum model. The mixed model can e used to make accurate local predictions. The grand mean model is used to predict the population average. The maximum model can e used to constraint unrealistic local predictions (if they occur at all) from the mixed model approach, y increasing it y or % and using the increased heights as the upper limits. For the four main species, suregion-ased grand mean models were also fitted. Mixed models (estimated parameters for the mixed models are listed in Tales 5a and 5) H =. + (β i )[ exp( (β i )DBH )] (β i ) + ε (AW, PJ, PB, BW) β H. + (β )[ exp( (β )DBH )] + i i = ε (SB, PL) where: H H H =. + (β (β i )[ exp( β DBH )] ) + + i (β ε (FD) i =. + + ε (FB) + exp[β + (β )ln(dbh + )] i (β ) i =. + + ε (SW, LT) + exp[(β ) + (β )ln(dbh + )] i i H, DBH = tree height (m) and DBH (cm) for the jth tree in the ith ot β, β and β = fixed parameters common to all ots in the population i, i, and i = random parameters specific to ot i ε = a normally distriuted within-ot error term ) Tale 5a. Estimated parameters for the mixed height-diameter models. Parameter Balsam fir Douglas fir Black spruce Lodgepole pine β β β σ σ σ σ

21 Tale 5. Estimated parameters for the mixed height-diameter models. Parameter White irch Tamarack Jack pine Balsam poar Aspen Note: White spruce β β β σ σ σ σ σ σ σ σ, σ and σ are variances for random parameters, and, respectively, and σ are covariances etween random parameters, and σ is the error variance. Maximum models (used to predict the maximum HT at a given DBH) σ, σ AW: SB: SW: PL: FB: BW: FD: LT: PJ: PB: H = [ exp(.6984dbh)] H = [ exp(.489dbh)] 44.5 H =. + + exp[ ln(DBH + )].58 H = [ exp(.99dbh)] 4.67 H =. + + exp[ ln(D BH + )].65 H = [ exp(.75dbh)] 9.9 H =. + + exp[ ln(D BH + )] H =. + + exp[ ln( DBH + )] H = [ exp(.9577dbh)] H = [ exp(.48dbh)].7.998

22 Grand mean models (used to predict the population average HT at a given DBH) AW: AW: AW: SB: SB: SB: SW: SW: SW: SW: SW: PL: PL: PL: PL: FB: BW: FD: LT: PJ: PB: H H = [ exp(.667dbh)] (suregions 4-).985 H = [ exp(.74dbh)] (suregions -, -).9 = [ exp(.699dbh)] (provincial).9 H H = [ exp(.5dbh)] (suregions -, -, ).8 H =. +.47[ exp(.69dbh)] (suregions 4-).46 = [ exp(.549dbh)] (provincial) H =. + (provincial) + exp[ ln(DBH + )] H =. + (suregion -) + exp[ ln(DBH + )].6474 H =. + (suregion ) + exp[ ln(DBH + )].55 H =. + (suregions 4-9) + exp[ ln(DBH + )] 5.67 H =. + (suregions -) + exp[ ln(DBH + )] H H = [ exp(.68dbh)] (suregion ).4598 H = [ exp(.748dbh)] (suregions -,, -).7 H = [ exp(.869dbh)] (suregions 4-9).89 = [ exp(.54dbh)] (provincial) H =. + + exp[ ln(DBH + )] H = [ exp(.97dbh)]. 9.9 H =. + + exp[ ln(DBH + )] H = exp[ ln(DBH H =. +.54[ exp(.696dbh)] H = [ exp(.466dbh)] + )]

23 Acknowledgements This 5-year project was jointly supported y Alerta Sustainale Resource Development and Forest Resource Improvement Association of Alerta (the FRIAA-GYPSY project). Major FRIAA funding and in-kind support for this project came from the companies identified elow. Contriutions y the project team memers R. Briand, D. Dempster, D. Morgan, and G. Behuniak warrant a special mention. Analytic work and input y many peoe, particularly G. Gulyas, E. McWilliams, W. Fast, Y. Qin, D. Aitkin, and T. Nunifu, are appreciated. Company Financial support In-kind support Alerta Newsprint Company Alerta Pacific Forest Industries Alerta Plywood Blue Ridge Lumer Canadian Forest Products Foothills Forest Products Hinton Wood Products Millar Western Forest Products Sundance Forest Industries Sundre Forest Products Weyerhaeuser Company

24 Appendix. A generalized program for predicting site index from top height and total age. *This program predicts the following variales, from given top height and total age: - t (top height at 5 years totage) - h (top height at 5 years hage) -YBH (years needed to reach the reast height of. m from germination) OPTIONS LS=8 PS=6; data examp; input species $ topht totage; cards; AW. 5 AW. 6 SB. 5 SB. 7 PL. 5 PL. 8 SW. 5 SW. 9 ; run; data examp; set examp; if species='pl' then do; B =.8457; B = ; B = -.9; B4 =.5668; si=; do until(as(si-si)<.); k = exp( *sqrt(log(5+.))+*(log(si))**+4*sqrt(5)); k = si**( ); k = ((si)*(+k )/.-)/(exp()*exp(4*sqrt(5))*k ); YBH_=exp((log(k)/)**) - ; x = (+exp( +*sqrt(log( x = (+exp( +*sqrt(log(5 si=topht*x/x; si=(si+si)/; end; +totage +.)) +*log(si)**+4*sqrt(5))); +.)) +*log(si)**+4*sqrt(5))); _t_ = si; _h_= si_t_*(+exp( *sqrt(log(5 +.))+*(log(si_t_))**+4*sqrt(5) ))/ (+exp( *sqrt(log(5 +YBH_ +.))+*(log(si_t_))**+4*sqrt(5) )); ht_ = si_t_*(+exp( *sqrt(log(5 +.))+*(log(si_t_))**+4*sqrt(5) ))/ (+exp( *sqrt(log(totage +.))+*(log(si_t_))**+4*sqrt(5) )); end; if species='sb' then do; B= 4.566; B= ; B= -.575; B4=.474; si=; do until(as(si-si)<.); k = exp( *sqrt(log(5+.))+*(log(si))**+4*sqrt(5)); k = si**( ); k = ((si)*(+k )/.-)/(exp()*exp(4*sqrt(5))*k ); YBH_s=exp((log(k)/)**) - ; x = (+exp( +*sqrt(log( x = (+exp( +*sqrt(log(5 si=topht*x/x; si=(si+si)/; end; +totage +.)) +*log(si)**+4*sqrt(5))); +.)) +*log(si)**+4*sqrt(5))); _t_s = si; _h_s= si_t_s*(+exp( *sqrt(log(5 +.))+*(log(si_t_s))**+4*sqrt(5) ))/ (+exp( *sqrt(log(5 +YBH_s +.))+*(log(si_t_s))**+4*sqrt(5) )); ht_s = si_t_s*(+exp( *sqrt(log(5 +.))+*(log(si_t_s))**+4*sqrt(5) ))/ (+exp( *sqrt(log(totage +.))+*(log(si_t_s))**+4*sqrt(5) )); end;

25 if species='sw' then do; B =.494; B = -.775; B = -.854; B4 =.6548; si=; do until(as(si-si)<.); k = exp( *sqrt(log(5**+.))+*(log(si))**+4*sqrt(5)); k = si**(*log(si)); k = ((si)*(+k )/.-)/(exp()*exp(4*sqrt(5))*k ); YBH_=sqrt(exp((log(k)/)**) - ); x = (+exp( +*sqrt(log( +totage** +.)) +*log(si)**+4*sqrt(5))); x = (+exp( +*sqrt(log(5** +.)) +*log(si)**+4*sqrt(5))); si=topht*x/x; si=(si+si)/; end; _t_ = si; _h_= si_t_*(+exp( *sqrt(log(5** +.))+*(log(si_t_))**+4*sqrt(5) ))/ (+exp( *sqrt(log((5 +YBH_)** +.))+*(log(si_t_))**+4*sqrt(5) )); ht_ = si_t_*(+exp( *sqrt(log(5** +.))+*(log(si_t_))**+4*sqrt(5) ))/ (+exp( *sqrt(log((totage )** +.))+*(log(si_t_))**+4*sqrt(5) )); end; if species='aw' then do; B = ; B = -.945; B = -.778; B4 =.476; si=; do until(as(si-si)<.); k = exp( *sqrt(log(5+.))+*(log(si))**+4*sqrt(5)); k = si**(*log(si)); k = ((si)*(+k )/.-)/(exp()*exp(4*sqrt(5))*k ); YBH_=exp((log(k)/)**) - ; x = (+exp( +*sqrt(log( x = (+exp( +*sqrt(log(5 si=topht*x/x; si=(si+si)/; end; +totage +.)) +*log(si)**+4*sqrt(5))); +.)) +*log(si)**+4*sqrt(5))); _t_ = si; _h_= si_t_*(+exp( *sqrt(log(5 +.))+*(log(si_t_))**+4*sqrt(5) ))/ (+exp( *sqrt(log(5 +YBH_ +.))+*(log(si_t_))**+4*sqrt(5) )); ht_ = si_t_*(+exp( *sqrt(log(5 +.))+*(log(si_t_))**+4*sqrt(5) ))/ (+exp( *sqrt(log(totage +.))+*(log(si_t_))**+4*sqrt(5) )); end; drop B B B B4 si si k k k x x; proc print data=examp(os=); run; *Output; Os species topht totage YBH t h_ ht_ YBH_s _t_s _h_s AW 5 AW 6 SB SB PL PL SW 5 8 SW 9 Os ht_s YBH t h_ ht_ YBH t h_ ht_