Growth intercept models for black spruce, jack pine and balsam fir in Quebec by Daniel Mailly1,2 and Mélanie Gaudreault1 ABSTRACT

The objective of this study was to develop variable growth intercept models for coniferous species of major importance in Quebec using Nigh’s (1997a) modelling technique. Eighty-three, 68, and 70 stem analysis plots of black spruce (Picea mariana [Mill.] BSP), jack pine (Pinus banksiana Lamb.) and balsam fir (Abies balsamea (L.) Mill) were used, respectively. The growth intercept models for black spruce were the most precise, followed by those for jack pine and finally by those for balsam fir, based on the root mean square errors. Results indicated that the accuracy of the models was good, relative to those previously published for other species in Canada. Interim testing of the models revealed a low mean error for all three species that may not be of practical significance for site index determination, although more data should be obtained to further test the models. Key words: balsam fir, black spruce, growth intercept, jack pine, model, nonlinear regression, site index RÉSUMÉ

L’objet de cette étude était de mettre au point des modèles de croissance internodale variable pour les principales essences résineuses du Québec, selon l’approche méthodologique de Nigh (1997a). Au total, 83 placettes d’analyse de tiges d’épinette noire (Picea mariana [Mill.] BSP), 68 de pin gris (Pinus banksiana Lamb.) et 70 de sapin baumier (Abies balsamea (L.) Mill) ont été utilisées. En se basant sur la racine carrée des erreurs quadratiques moyennes, les modèles de croissance internodale pour l’épinette noire se sont avérés les plus précis; venaient ensuite ceux du pin gris et finalement ceux du sapin baumier. La précision obtenue se compare très bien à celle des modèles déjà publiés pour plusieurs autres essences que l’on retrouve au Canada. Une validation provisoire a également démontré que l’erreur d’estimation moyenne de l’indice de qualité de station était relativement faible pour les trois essences et qu’elle n’entraînait pas de répercussions pratiques sur le terrain. Une collecte de données additionnelle devrait toutefois être réalisée afin de poursuivre les tests de validation. Mots clés : croissance internodale, épinette noire, indice de qualité de station, modèle, pin gris, régression non linéaire, sapin baumier

Introduction

de la recherche forestière, Ministère des Ressources naturelles, de la Faune et des Parcs, 2700, rue Einstein, Sainte-Foy, Québec G1P 3W8. 2Author to whom all correspondence should be addressed. E-mail: [email protected]

Site index has traditionally been estimated using height over age curves in the province of Quebec (e.g., Vézina and Linteau 1968, Boudoux 1978, Pothier and Savard 1998). While such curves tend to be effective tools for estimating site index in mature stands, they are unreliable for estimating site index in juvenile stands, i.e., stands that are younger than 20 years old at breast height (Nigh 1995). Conversely, variable growth intercept models are being increasingly used outside Quebec to give reliable site index estimates for young stands by relating the average annual height growth of trees to site index. These models have the advantages of being: 1) developed specifically for estimating site index, not height; 2) intended for young stands; 3) not constrained to pass through the site index at index age; and 4) less sensitive to small deviations from the mean height when compared to height-age models (Nigh 1996). Because of the commercial importance of black spruce, jack pine and balsam fir as major wood supply species in Quebec, a need existed for growth intercept models for these species. The only available growth intercept models have been developed in British Columbia for coastal Douglas-fir (Pseudotsuga menziesii [Mirb.] Franco var. menziesii; Nigh

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Daniel Mailly

1Direction

Mélanie Gaudreault

Fig. 1. Location of the sample plots for: a) black spruce (n = 83), b) jack pine (n = 68) and c) balsam fir (n = 70) in Quebec.

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Table 1. Summary information about the study plots Black spruce (n = 83)

Jack pine (n = 68)

Balsam fir (n = 70)

Age (yrs)

Height (m)

Site index (m)

Age (yrs)

Height (m)

Site index (m)

Age (yrs)

Height (m)

Site index (m)

Mean Standard deviation Minimum Maximum

75 25 46 154

16.5 2.5 10.5 22.3

13.1 2.8 5.7 17.1

67 16 46 115

Age at 1.0 m 18.2 2.7 12.2 22.5

16.5 2.5 11.6 21.8

65 13 46 103

16.2 2.4 10.3 21.5

13.5 3.1 5.7 19.6

Mean Standard deviation Minimum Maximum

74 24 45 151

16.5 2.5 10.5 22.3

13.5 2.7 6.4 17.2

66 16 45 114

Age at 1.3 m 18.2 2.7 12.2 22.5

16.7 2.5 11.7 21.9

63 12 45 98

16.2 2.4 10.3 21.5

13.8 3.0 6.3 20.2

1997a), lodgepole pine (Pinus contorta Dougl. ex Loud. var. latifolia Engelm; Nigh 1997b), interior Douglas-fir (Pseudotsuga menziesii var. glauca [Beissn.] Franco; Nigh 1997c), interior western hemlock (Tsuga heterophylla [Raf.] Sarg.; Nigh 1998), interior spruce (Picea glauca [Moench] Voss, Picea engelmannii Parry ex Engelm. and Picea glauca x engelmannii; Nigh 1999), coastal western hemlock (Tsuga heterophylla [Raf.] Sarg.; Nigh 1999), Sitka spruce (Picea sitchensis [Bong.] Carrière; Nigh 1999), western larch (Larix occidentalis Nutt.; Nigh et al. 1999), western redcedar (Thuja plicata Donn ex D. Don; Nigh 2000), black spruce (Picea mariana [Mill.] BSP; Nigh and Klinka 2001), Ponderosa pine (Pinus ponderosa P. Laws. ex C. Laws.; Nigh 2002) and white spruce (Picea glauca [Moench] Voss; Nigh 2004). The purpose of this study was to develop growth intercept models for black spruce, jack pine and balsam fir in Quebec. Another objective was to provide an interim validation of the models based on an independent data set. In general, the modelling technique closely follows that of Nigh (1997a), which should be consulted for additional details.

height to verify if the tree had experienced a suppression period during its early life. If so, another dominant tree having a similar D130 but with no suppression was selected as a replacement. The selected sample trees were cut down and sections from the stem were taken at stump height (0.15 m), 0.60 m, 1.00 m, 1.30 m, 2.00 m and at each subsequent 1.0 m or 2.0 m length. The WINDENDRO™ system (Guay et al. 1992) was used to measure annual increment rings and COFECHA (Holmes 1983) was used for crossdating the tree sections. Section data were converted into heightage data by linear interpolation using ANATI (Tardif 2001) and were analyzed using SAS (SAS Institute Inc. 1999). Characteristics of the stands sampled are presented in Table 1. Growth intercept modelling

The data for this study consisted of 83, 68 and 70 sample plots of black spruce, jack pine and balsam fir, respectively. The plots were located throughout the distribution range of each species in the province (Fig. 1). The plots were remeasured or established in accordance with the recommended procedures for permanent sample plots in Québec (Ministère des Ressources naturelles du Québec 2001). The plots were circular, 0.04 ha in size, and were located in areas that were ecologically homogeneous and where the target species was predominant. Within each plot, the three largest-diameter trees (diameter measured at breast height, D130, height = 130 cm, Brokaw and Thompson 2000) of the target species were chosen as sample trees. In cases where the permanent plot was remeasured from the network of plots of the Inventory Branch (Ministère des Ressources naturelles, de la Faune et des Parcs du Québec), the trees were chosen outside the plot at a minimum distance equivalent to the dominant tree height and had diameters (D130) similar to the three largest D130 trees found inside the plot. Prior to felling a tree, an increment core was taken at 0.75 m

The method used for growth intercept data analysis was similar to the one presented in Nigh (1997a). The main difference in the application of the method lies in the reference height for age which is set at 1.0 m in Quebec instead of 1.3 m (breast height) generally used in other provinces. In this study, both reference heights were used and two sets of equations were provided. For clarity in the text, age measured at 1.0 m (100 cm) is hereafter referred to as “age100” and age measured at 1.3 m (130 cm) as “age130.” The height–age data were plotted for each tree, by plot. Data for ages above 50 years were deleted, as they were not used in the data analysis. Trees displaying anomalous growth were deleted from the analysis, as were plots with fewer than two suitable trees. In general, this included trees that have not been able to maintain a consistent height growth (typical of dominant trees) or trees that appeared to have suffered from the last spruce budworm epidemics (balsam fir and black spruce only). In addition, the heights of some trees that were between 45 and 50 years old were extrapolated using a Chapman-Richards function (Fekedulegn et al. 1999) based on the last ten years of growth. For calculations made for ages100, this procedure allowed us to salvage 6, 12 and 14 black spruce, jack pine and balsam fir plots, respectively. For calculations made for ages130, 9, 13 and 15 black spruce, jack pine and balsam fir plots were salvaged, respectively, using this procedure. Next, the mean height growth was calculated by plot. The site index was obtained by calculating the top height

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Material and Methods Data collection

reached at 50 years of age taken at 1.0 m and 1.3 m. Growth intercepts were calculated according to equation 1 for each age100 and age130 from one to 50: GIA =

HA – RH

. 100

[1]

A –Ao

where: GIA is the growth intercept (cm/yr) for age (A) = 1, 2, . . ., 50 (yrs), HA is the average height (m) at age A, RH is the reference height for age (1.0 or 1.3 m) and A0 is an adjustment factor for age to account for the first year of height growth above the reference height, which is a partial year’s growth, H0 and H1 are the heights of the tree at the nodes immediately below and above reference height: Ao =

RH – Ho

[2]

H1 –Ho

The growth intercepts of all plots were related to the site index for each age100 or age130 using Nigh’s (1997a) variable growth intercept model: SI = RH + e b1 . (GI A)b2

[3]

where SI is the estimated site index, GIA is the growth intercept (cm/yr) for age A at reference height (HR), and b1 and b2 are model parameters. One set of parameters was estimated for each age100 or age130 between 1 and 50 using SAS NLIN procedure. An analysis of the residuals was done to confirm the regression assumptions of unbiasedness, normality, and homoscedasticity (Ratkowski 1983, Sen and Srivastava 1990). Each model was analyzed for bias (a t-test to verify whether the mean of the residuals was significantly different from zero), normality with the W statistic (Shapiro and Wilk 1965), homoscedasticity using the Fk statistic (Endrenyi and Kwong 1981), intrinsic and parameter effects nonlinearity (Bates and Watts 1980) and parameter bias (Box 1971). Interim validation of the models

Many plots inventoried in this study had to be rejected because they did not meet all the selection criteria (e.g., some plots had trees with anomalous height growth). As a consequence, all inventory plots were used in the calibration phase leaving no plots for the validation. However, an independent data set for each species was assembled to test the growth intercept models. These data were collected in the same manner as the model development data. Although originally collected to develop new height-age curves for these species as a function of ecosystem site type (J.P. Saucier, personal communication), these data were adequate for interim model testing. The independent data set consisted of 30 black spruce, 5 jack pine and 40 balsam fir stem analysis plots, established in the Balsam Fir – Yellow Birch and the Balsam Fir – White Birch bioclimatic subdomains in 2002 (Saucier et al. 1998). To test the model, the following calculations were made for each species studied at ages100 or ages130. Firstly, the actual site index of each plot from the independent data set was

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computed, i.e., the height reached at 50 years. Secondly, an estimated site index was calculated using the fitted growth intercept models for each age from one to 50. Thirdly, the errors in the estimates (actual site index — estimated site index) were calculated and averaged for each age to observe the variation in estimated site index when applying the models (Nigh and Martin 2001). For a specific age, an error greater than zero indicates that the model is underestimating site index.

Results Tables 2–3 present the analysis results of model [3] for ages 1 to 50 for black spruce, jack pine and balsam fir, respectively. For each species, the table includes the parameter estimates and the root mean square error (RMSE), which is a measure of model accuracy. The results of the tests for bias, normality, homoscedasticity, intrinsic and parameter-effects nonlinearity, and parameter bias are summarized below: • Bias: none of the models showed any evidence of bias. • Normality: the models for ages100 1, 10, 11 (black spruce), 50 (jack pine), and for ages130 2 to 7 (black spruce), 50 (jack pine) and 44 to 47 (balsam fir) showed slight (0.004 < p < 0.05) evidence of non-normality. • Homoscedasticity: the models for ages100 2, 31 to 35 (black spruce), 5 (jack pine) and 50 (balsam fir) and for ages130 28 to 31 (black spruce) and 50 (balsam fir) showed some evidence of heteroscedasticity based on the Fk test, but the residual plots indicated that it was not serious. • Intrinsic and parameter-effects nonlinearity: the measures of nonlinearity and parameter bias were small, which indicated that the parameter estimates were virtually linear (Ratkowsky 1983). • Parameter bias: parameter bias was less than 1% except for ages100 28 to 30 (black spruce), 23 (jack pine) and 31, 32 (balsam fir) and for ages130 26 to 29 (black spruce), 21 (jack pine) and 29, 30 (balsam fir). The tests for bias, normality, and homoscedasticity showed that the usual least-squares regression assumptions (Sen and Srivastava 1990) were in general, satisfactorily met. Fig. 2 shows graphically the fitted growth intercept models at age 25 overlaid on the data points for black spruce, jack pine and balsam fir, respectively. Fig. 3 shows the mean error (dashed line) of the variable growth intercept models for black spruce, jack pine and balsam fir for the interim test data plotted against age from 1 to 50. Confidence intervals at 95% level (solid lines) for the mean error are also shown and indicate when potential bias may occur, i.e., where a confidence line crosses the mean error value of zero. Testing of the black spruce models revealed that for ages100 under 12 (Fig. 3a), the mean error was significantly different from zero and that the models may be biased. However, the estimated bias was greater than 1 m only for ages100 of less than 5. The mean error for ages130 under 8 (Fig. 3d) was also significantly different from zero; in these cases, the models overestimated the true site index by approximately 50 cm. On average, the magnitude of the error was between 15 and 30 cm for both reference heights. The model testing for jack pine showed that the mean error for ages100 (Fig. 3b) between 17 and 30 and for age100 50 was significantly different from zero. Similarly,

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Table 2. Growth intercept models and their root mean squared error (RMSE) for ages 1–50 at 1.0 m height b1

Black spruce b2 RMSE

b1

Jack pine b2

RMSE

b1

0.3003 0.3928 0.3929 0.4151 0.4439 0.4657 0.4857 0.5029 0.5145 0.5254 0.5386 0.5527 0.5659 0.5808 0.5963 0.6117 0.6301 0.6490 0.6699 0.6934 0.7166 0.7395 0.7613 0.7823 0.7984 0.8093 0.8191 0.8301 0.8424 0.8551 0.8653 0.8730 0.8817 0.8898 0.8974 0.9051 0.9112 0.9170 0.9236 0.9300 0.9381 0.9481 0.9580 0.9676 0.9764 0.9833 0.9893 0.9937 0.9979 1.0028

2.2085 1.9462 1.8052 1.7209 1.6436 1.5543 1.4530 1.3676 1.3194 1.2827 1.2468 1.2201 1.1998 1.1900 1.1801 1.1612 1.1342 1.1025 1.0773 1.0532 1.0284 0.9992 0.9677 0.9351 0.8993 0.8699 0.8460 0.8239 0.7959 0.7611 0.7252 0.6891 0.6530 0.6169 0.5800 0.5447 0.5124 0.4778 0.4412 0.4066 0.3723 0.3368 0.3014 0.2624 0.2201 0.1787 0.1390 0.1000 0.0635 0.0454

1.5451 1.4724 1.4941 1.5017 1.4965 1.4582 1.3943 1.3354 1.2869 1.2352 1.1743 1.1146 1.0543 0.9946 0.9319 0.8678 0.8111 0.7609 0.7153 0.6624 0.6012 0.5370 0.4700 0.3964 0.3308 0.2729 0.2199 0.1699 0.1190 0.0700 0.0217 -0.0338 -0.0939 -0.1551 -0.2131 -0.2662 -0.3172 -0.3630 -0.4017 -0.4329 -0.4600 -0.4839 -0.5083 -0.5341 -0.5618 -0.5934 -0.6208 -0.6441 -0.6681 -0.6939

Balsam fir b2 RMSE

Age at 1.0 m (years) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

108

1.4487 1.2960 1.1842 1.0451 0.9463 0.8754 0.8134 0.7673 0.7357 0.7165 0.6884 0.6467 0.6036 0.5610 0.5203 0.4789 0.4386 0.4015 0.3647 0.3250 0.2853 0.2490 0.2133 0.1772 0.1467 0.1155 0.0800 0.0444 0.0094 -0.0254 -0.0610 -0.0968 -0.1354 -0.1777 -0.2181 -0.2565 -0.2952 -0.3321 -0.3674 -0.4025 -0.4357 -0.4686 -0.5030 -0.5362 -0.5671 -0.5950 -0.6218 -0.6466 -0.6713 -0.6953

0.3676 0.4092 0.4383 0.4760 0.5026 0.5213 0.5374 0.5484 0.5547 0.5573 0.5633 0.5738 0.5851 0.5967 0.6078 0.6194 0.6307 0.6412 0.6518 0.6633 0.6749 0.6856 0.6963 0.7073 0.7166 0.7262 0.7373 0.7484 0.7595 0.7707 0.7823 0.7940 0.8067 0.8205 0.8338 0.8465 0.8594 0.8716 0.8835 0.8952 0.9065 0.9177 0.9294 0.9408 0.9516 0.9615 0.9712 0.9802 0.9892 0.9980

1.9063 1.7134 1.5287 1.4053 1.3605 1.3290 1.2958 1.2709 1.2508 1.2362 1.2116 1.1791 1.1468 1.1172 1.0941 1.0701 1.0471 1.0309 1.0184 1.0010 0.9763 0.9520 0.9297 0.9037 0.8764 0.8466 0.8137 0.7789 0.7438 0.7119 0.6826 0.6552 0.6268 0.5965 0.5655 0.5322 0.4935 0.4511 0.4100 0.3710 0.3342 0.3001 0.2666 0.2309 0.1932 0.1561 0.1196 0.0857 0.0535 0.0321

1.6884 1.3427 1.3201 1.2334 1.1269 1.0483 0.9747 0.9100 0.8661 0.8251 0.7757 0.7242 0.6757 0.6208 0.5644 0.5076 0.4406 0.3718 0.2958 0.2113 0.1281 0.0462 -0.0309 -0.1046 -0.1604 -0.1965 -0.2281 -0.2638 -0.3043 -0.3461 -0.3788 -0.4021 -0.4285 -0.4525 -0.4738 -0.4951 -0.5098 -0.5231 -0.5388 -0.5531 -0.5731 -0.5997 -0.6252 -0.6493 -0.6701 -0.6843 -0.6951 -0.7007 -0.7053 -0.7121

0.3393 0.3568 0.3448 0.3379 0.3368 0.3470 0.3651 0.3813 0.3943 0.4080 0.4243 0.4404 0.4567 0.4732 0.4906 0.5084 0.5241 0.5376 0.5498 0.5643 0.5817 0.6004 0.6203 0.6422 0.6618 0.6795 0.6959 0.7114 0.7272 0.7423 0.7574 0.7747 0.7935 0.8127 0.8309 0.8478 0.8641 0.8789 0.8917 0.9025 0.9122 0.9210 0.9299 0.9393 0.9493 0.9604 0.9703 0.9790 0.9880 0.9976

2.3553 2.3920 2.3624 2.3486 2.3616 2.3500 2.2992 2.2464 2.2082 2.1606 2.0968 2.0286 1.9520 1.8781 1.8034 1.7235 1.6539 1.5876 1.5237 1.4475 1.3661 1.2959 1.2263 1.1446 1.0674 1.0044 0.9415 0.8794 0.8288 0.7902 0.7545 0.7100 0.6656 0.6270 0.5904 0.5524 0.5119 0.4759 0.4412 0.4053 0.3687 0.3344 0.3022 0.2675 0.2297 0.1893 0.1499 0.1097 0.0689 0.0356

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Table 3. Growth intercept models and their root mean squared error (RMSE) for ages 1–50 at 1.3 m height b1

Black spruce b2 RMSE

b1

Jack pine b2

RMSE

b1

0.3915 0.3901 0.4086 0.4383 0.4658 0.4848 0.5014 0.5135 0.5257 0.5405 0.5531 0.5660 0.5802 0.5974 0.6124 0.6288 0.6474 0.6680 0.6930 0.7167 0.7385 0.7611 0.7837 0.8005 0.8117 0.8210 0.8308 0.8425 0.8551 0.8648 0.8721 0.8788 0.8854 0.8925 0.8997 0.9055 0.9112 0.9179 0.9243 0.9330 0.9436 0.9534 0.9626 0.9713 0.9779 0.9830 0.9874 0.9919 0.9972 1.0033

1.8974 1.7552 1.6841 1.6134 1.5198 1.4200 1.3506 1.3046 1.2677 1.2349 1.2152 1.2017 1.1956 1.1871 1.1674 1.1434 1.1173 1.0947 1.0688 1.0390 1.0081 0.9738 0.9408 0.9077 0.8788 0.8586 0.8390 0.8130 0.7797 0.7458 0.7123 0.6760 0.6400 0.6028 0.5672 0.5355 0.5017 0.4658 0.4326 0.3996 0.3649 0.3312 0.2947 0.2546 0.2158 0.1783 0.1400 0.0985 0.0588 0.0361

1.3848 1.4829 1.4947 1.4991 1.4395 1.3484 1.2686 1.2159 1.1656 1.1132 1.0578 0.9963 0.9346 0.8768 0.8256 0.7761 0.7266 0.6727 0.6162 0.5526 0.4833 0.4118 0.3420 0.2816 0.2244 0.1693 0.1134 0.0583 0.0129 -0.0323 -0.0877 -0.1464 -0.2044 -0.2559 -0.3023 -0.3462 -0.3805 -0.4091 -0.4338 -0.4562 -0.4810 -0.5092 -0.5380 -0.5637 -0.5857 -0.6070 -0.6290 -0.6538 -0.6804 -0.7047

Balsam fir b2 RMSE

Age at 1.3 m (years) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

0.8756 0.7715 0.7269 0.7066 0.6742 0.6449 0.6340 0.6322 0.6340 0.6115 0.5717 0.5309 0.4945 0.4612 0.4246 0.3889 0.3555 0.3226 0.2871 0.2516 0.2215 0.1926 0.1612 0.1284 0.0927 0.0570 0.0206 -0.0165 -0.0565 -0.0998 -0.1435 -0.1866 -0.2252 -0.2595 -0.2934 -0.3279 -0.3609 -0.3943 -0.4269 -0.4582 -0.4898 -0.5207 -0.5495 -0.5761 -0.6002 -0.6229 -0.6440 -0.6656 -0.6870 -0.7063

0.5171 0.5414 0.5543 0.5598 0.5688 0.5763 0.5774 0.5756 0.5727 0.5775 0.5879 0.5991 0.6092 0.6184 0.6288 0.6391 0.6487 0.6583 0.6690 0.6796 0.6887 0.6977 0.7075 0.7177 0.7289 0.7402 0.7517 0.7636 0.7765 0.7904 0.8046 0.8187 0.8315 0.8431 0.8546 0.8662 0.8774 0.8887 0.8998 0.9106 0.9215 0.9323 0.9425 0.9521 0.9610 0.9694 0.9774 0.9856 0.9937 1.0013

1.5386 1.4193 1.4046 1.3913 1.3621 1.3302 1.3051 1.2822 1.2661 1.2371 1.2000 1.1672 1.1414 1.1230 1.0994 1.0738 1.0575 1.0479 1.0302 1.0039 0.9819 0.9611 0.9356 0.9059 0.8708 0.8343 0.8010 0.7705 0.7386 0.7063 0.6745 0.6426 0.6115 0.5805 0.5473 0.5086 0.4675 0.4282 0.3907 0.3556 0.3225 0.2875 0.2525 0.2190 0.1855 0.1523 0.1188 0.0842 0.0492 0.0272

1.3075 1.2953 1.2308 1.1231 1.0250 0.9561 0.8949 0.8499 0.8045 0.7502 0.7042 0.6573 0.6054 0.5431 0.4884 0.4286 0.3609 0.2861 0.1963 0.1110 0.0333 -0.0463 -0.1253 -0.1832 -0.2201 -0.2499 -0.2812 -0.3193 -0.3602 -0.3910 -0.4124 -0.4313 -0.4497 -0.4695 -0.4887 -0.5021 -0.5151 -0.5307 -0.5453 -0.5674 -0.5956 -0.6204 -0.6431 -0.6636 -0.6764 -0.6840 -0.6891 -0.6948 -0.7030 -0.7132

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0.3660 0.3311 0.3267 0.3248 0.3429 0.3698 0.3928 0.4074 0.4208 0.4348 0.4499 0.4669 0.4839 0.4999 0.5142 0.5279 0.5415 0.5565 0.5723 0.5906 0.6110 0.6322 0.6530 0.6715 0.6891 0.7062 0.7236 0.7407 0.7549 0.7692 0.7867 0.8051 0.8236 0.8401 0.8552 0.8696 0.8812 0.8913 0.9002 0.9085 0.9176 0.9278 0.9381 0.9475 0.9559 0.9641 0.9724 0.9817 0.9916 1.0008

2.3793 2.4508 2.4687 2.4715 2.4256 2.3528 2.2814 2.2256 2.1633 2.0937 2.0194 1.9351 1.8502 1.7802 1.7233 1.6613 1.5978 1.5242 1.4448 1.3672 1.2919 1.2079 1.1175 1.0431 0.9716 0.9024 0.8397 0.7860 0.7503 0.7209 0.6809 0.6419 0.6096 0.5796 0.5483 0.5138 0.4827 0.4512 0.4197 0.3868 0.3564 0.3271 0.2955 0.2648 0.2325 0.1977 0.1578 0.1125 0.0637 0.0297

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Fig. 2. Site index – growth intercept relationship for age 25 for black spruce (a, d), jack pine (b, e) and balsam fir (c, f). Parts a, b, and c are based on a reference height for age of 1.0 m, parts d, e, and f are based on a reference height for age of 1.3 m.

the mean error for ages130 between 15 and 30 (Fig. 3e) was significantly different from zero. On average, the magnitude of the error was between 20 and 30 cm for both reference heights. Finally, the model testing for balsam fir showed that the mean error for ages100 (Fig. 3c) between 37 and 50, and for ages130 between 37 and 45 (Fig. 3f) were significantly different from zero. On average, the magnitude of the error was between 15 and 20 cm for both reference heights.

Discussion

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The deliverables of this study were growth intercept models for black spruce, jack pine and balsam fir with ages measured at two reference heights (1.0 m and 1.3 m). The models for black spruce were the most precise, followed by those for jack pine and finally by those for balsam fir (see Fig. 3 and the root mean square errors reported in Tables 2 and 3). The accuracy of the models was good, relative to those for other species. Overall, the growth intercept models had the

Fig. 3. Mean error (dashed line) and 95% confidence interval for the mean error (solid lines) for black spruce (a, d), jack pine (b, e) and balsam fir (c, f). Parts a, b, and c are based on a reference height for age of 1.0 m, parts d, e, and f are based on a reference height for age of 1.3 m.

same accuracy as those for black spruce in British Columbia (Nigh and Klinka 2001), lodgepole pine (Nigh 1997b) and western redcedar (Nigh 2000), but were slightly more accurate than those for interior spruce (Nigh 1999), interior western hemlock (Nigh 1998), coastal Douglas-fir (Nigh 1997a), coastal western hemlock (Nigh 1999), interior Douglas-fir (Nigh 1997c), Sitka spruce (Nigh 1999), western larch (Nigh et al. 1999) and Ponderosa pine (Nigh 2002). As

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is typical for growth intercept models, the accuracy of the model increased as tree age increased (Nigh et al. 1999). This study has shown that the variable growth intercept model proposed by Nigh (1997a) was applicable to black spruce, jack pine and balsam fir in Quebec. The model produced well-behaved equations (Fig. 2 and Nigh 1999), characteristic of variable growth intercept models. The relationship between site index and the growth intercept lengths

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Fig. 4. Inverted growth intercept models for black spruce in Quebec (solid line) and British Columbia (dashed line) for site indices 8, 12 and 16.

as shown in Fig. 2 is slightly curvilinear, which indicated that a unit increase in growth intercept length was met with a progressively smaller increase in site index. This also suggested that trees growing on good sites reached a maximum rate of height growth earlier than trees growing on poor sites (Thrower 1987). It should be noted that foresters will want to know the age at which the error in estimated site index will be under a certain threshold. An idea of this precision can be obtained by looking at the root mean squared error from Tables 2–3, which gives an approximate range of the magnitude of the error in the estimated site index that can be expected 95% of the time (Nigh 1998). To obtain a site index estimate with a precision of 2 m, stands should not be surveyed until they are at least 20, 22 and 26 years old at 1.0 m height for black spruce, jack pine and balsam fir, respectively. For measures taken at 1.3 m height, they should not be surveyed until they are at least 20, 21 and 25 years-old for black spruce, jack pine and balsam fir, respectively. The model testing for black spruce showed that the mean error decreased with age at both reference heights, which indicated that the models can be used in young stands but that the estimates of site index for those ages may be less reliable. The interim testing revealed some evidence of bias in the growth intercept models for all species. The mean error, however, was in general in the order of ± 30 cm for all three species and as such may not be of practical significance (Ministère des Ressources naturelles du Québec 2000). The model testing is considered interim because the test data used came from only two bioclimatic subdomains (Balsam Fir — Yellow Birch and the Balsam Fir — White Birch), even though a graphical analysis showed that the model development and test data were somewhat similar across all ages. Also, the sample size for jack pine was extremely small and hence the growth intercept models for this species requires further validation with a larger independent data set. Since growth intercept models for black spruce are also available in British Columbia (Nigh and Klinka 2001), the

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models developed in Quebec and in British Columbia were inverted so that height could be plotted by site index and age for comparison purposes (Fig. 4). The inverted models were graphed up to 50 years and compared for site indices 8, 12 and 16. The comparison revealed that there was little difference in the shape of the height over age curves, especially between 35 and 50 years and confirmed that the height growth pattern for black spruce is relatively similar across different regions of Canada (Nigh et al. 2002). The juvenile height growth rate was slightly higher in British Columbia than in Quebec, however, and this may be due to climatic dissimilarities between the two regions or to some level of growth plasticity of black spruce in relation to climate. The application of growth intercept models requires trees with a height growth that reflects site productivity (B.C. Ministry of Forests 1995). The fact that balsam fir and black spruce are reputed to be able to withstand long periods of suppression (Crossley 1976) and to be vulnerable to spruce budworm epidemics (Sanders et al. 1985) may complicate the process of tree selection. As Monserud (1985) pointed out, trees that are believed to have suffered episodes of suppression or defoliation, or have had top damage, should not be used as site trees. Using trees that do not reflect site productivity may underestimate site index. Since suppression may be an important problem, there is a risk that several growth intercept sample plots may have to be rejected in some areas. An alternative to the rejection of such trees is to substitute a free-growth equivalent period to the suppression years (Seymour and Fajvan 2001). With this method, an “adjusted age” not including the suppression years can be determined and used for site index determination instead of the true age. Although useful for estimating the site index of juvenile stands, the growth intercept models developed in this study are not readily compatible with Quebec’s current site index equations for growth and yield predictions (Pothier and Savard 1998). The latter are based on temporary sample plot data and are more asymptotic than those based on individual dominant trees (Raulier et al. 2003). Making the transition from the growth intercept models to the height-age models causes the estimated site index to jump. A conversion equation3 is thus necessary to make the transition between the two systems of equations.

Conclusion Growth intercept models for black spruce, jack pine and balsam fir that are applicable across the province of Québec are now available. Sampling for these models covered the major ecosystems that support these three species in the province. Furthermore, the models are applicable at two reference heights for age determination (1.0 and 1.3 m), so that they can be tested and applied not only in Quebec but also in other areas (e.g., eastern Canada). An interim testing of the models revealed a low mean error for all three species that may not be of practical significance, although more data should be obtained to further test the models. 3

Mailly, D. and M. Gaudreault. Application des modèles de croissance internodale pour les principales essences résineuses du Québec. Note de recherche forestière, Direction de la recherche forestière, Ministère des ressources naturelles, de la Faune et des Parcs, Québec. In preparation.

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Acknowledgements Special thanks to Gordon Nigh from the B.C. Ministry of Forests, Research Branch, for his statistical guidance and for his many helpful comments throughout this study. We thank the staff of the Ministère des Ressources naturelles, de la Faune et des Parcs (MRNFP) du Québec in many regions of the province for assistance in locating stands suitable for this study. We also thank Hervé Lortie, Johanne Claveau, Guy Brousseau, Carl Lemieux, Pierre Grondin, Stéphane Tremblay, Jean-Pierre Saucier, Philippe Racine and Claude Gagné for their help in collecting the data; Jolène Lemieux, Sylvain Turbis, Jean Noël and Denis Hotte for their help in database analysis; and Patrice Tardif, Louis Blais and Isabelle Auger for their help with the statistical analyses. Comments by Peter Marshall of the Faculty of Forestry (University of British Columbia), by Stéphane Tremblay of the Direction de la recherche forestière (Forêt Québec) and those of an anonymous reviewer improved the manuscript. This study was financed by the Direction de la recherche forestière (Forêt Québec) through projects 311-1620 and 311-3065.

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