Juha Lappi

Estimation of standard error of impedance‐estimated frost resistance

The magnitude of measurement errors of the specific impedance difference was estimated and a formula to approximate the variance of the estimated frost resistance was derived. The measurements of specific impedance difference include the measurement errors of impedance before and after frost treatment and cross-sectional area. These errors in connection with the population variation cause variation in the estimated frost resistance. The frost resistance is estimated by first expressing the specific impedance difference values as a logistic sigmoid function of the treatment temperature, and then evaluating the inverse function at a given value of the specific impedance difference. The error variance between the estimated and the measured frost resistance was calculated using the estimated parameters, their standard deviations and correlations. In an example the impedance estimated frost resistance (LT.i0Qm) was - 10.9°C and its standard deviation 0.8°C.

A non-linear hierarchical mixed model to describe tree height growth

A non-linear hierarchical mixed model approach is used to describe height growth of Norway spruce from longitudinal measurements. The parameter variation in the model was divided into unknown random effects, fixed effects and covariate-dependent effects in order to model tree height growth. The values for fixed effect parameters and the variance–covariance matrix of random effects were estimated. Covariates could only explain up to 10% of parameter variability. Height curves were calibrated by means of BLUPs for the unknown random effects using prior height measurements and evaluated using a separate dataset. The resulting curves had a small error variance and plausible shapes.

Estimation of height–diameter curves through multilevel models with special reference to even-aged teak stands

Abstract The use of a multilevel model for estimation and prediction of height–diameter curves in planted teak stands is discussed in the context of analyzing data from a stratified two-stage sample survey. Plantations selected randomly from different age groups within each Territorial Division in the State of Kerala, India, served as first stage units; circular plots along a transect within each selected plantation formed the second stage units. Girth at breast-height was measured on all the trees within the selected plots and height was measured on a sub-sample of trees within each selected plot. Plantations, plots within plantations, and trees within plots formed the three levels in the multilevel model for the height–diameter relation. The overall height–diameter model included fixed effects of age and mean diameter of the stands and also random effects on the deviations of the expected values of the parameters of the height–diameter curve at the plantation and plot levels. Differences in the height–diameter curves among the different Territorial Divisions were also investigated. The age along with the mean diameter of the stands could explain only a small part of the variation in the height–diameter curves. Random plantation and plot effects were prominent, but predictable using height measurements on a few sample trees from plots in individual plantations. The models are useful in generating accurate (localized) predictions of tree height which would eventually lead to better tree volume predictions and evaluation of site quality.

Integration of a nonlinear function in a changing environment: Estimating photosynthesis using mean and variance of radiation

Abstract If the mean (integral) of a convex or concave function is estimated by evaluating the response function at the mean of the environmental variable, the estimate is necessarily biased. This study computes from empirical radiation data the errors obtained when mean photosynthesis is estimated using mean radiation. Furthermore, three different methods of estimating mean photosynthesis are compared in a case where the second power of the irradiance is also integrated and the variance is thus known. The smallest errors are obtained when the irradiance distribution is approximated by a two-point distribution: the bias is reduced to one-tenth and the root mean square error to one-third compared to the situation when only mean radiation is used. The results indicate that, if accurate estimates are needed for integrals of nonlinear responses, the second power of radiation or any other fluctuating environmental variable should also be measured.

Joint effect of angular distribution of radiation and spatial pattern of trees on radiation interception

Abstract A simulation study on the combined effects of sun angle and spatial pattern of trees on the amount of intercepted photosynthetically active radiation (PAR) is presented. Different spatial patterns of trees were generated starting from a square lattice. Tree locations were first rotated by an angle α =0, 22.5 or 45°, and thereafter x -coordinates were multiplied by a constant c x and the y -coordinates were divided by c x . The values of c x were from 0.6 to 1.6 with the increment of 0.2. Simulations were made using constant stand density (400/ha), and crown size and shape. The crown envelope was described as the upper half of an ellipsoid with a height of 6 m and a radius of 1.5 m at the bottom. Only direct radiation was taken into account, and the direct solar irradiance was assumed to decrease exponentially within tree crowns. Simulations were made separately for latitudes 40°N (Thessaloniki) and 60°N (Helsinki). For the whole growing season (April to October at latitude 40°, and from May to September at latitude 60°), the intercepted PAR was largest when α =45° and c x =1.4 (wide east-west spacing) and smallest when α =0° and c x =0.6 (tight east-west spacing). At latitude 40° the optimum was 4.4% larger and the minimum 13.0% smaller than the intercepted radiation for the square pattern ( α =0° and c x =1). At latitude 60° the corresponding figures were 2.0 and 10.0%. The patterns for α =22.5° were nearly optimal for all c x . Reversed relations were found later in fall.

A flexible regression model for diameter prediction

We present a functional regression model for diameter prediction. Usually stem form is estimated from a regression model using dbh and height of the sample tree as predictor. With our model additional diameter observations measured at arbitrary locations within the sample tree can be incorporated in the estimation in order to calibrate a standard prediction based on dbh and height. For this purpose, the stem form of a sample tree is modelled as a smooth random function. The observed diameters are assumed as independent realizations from a sample of possible trajectories of the stem contour. The population average of the stem form within a given dbh and height class is estimated with the taper curves applied in the national forest inventory in Germany. Tree deviation from the population average is modelled with the help of a Karhunen–Loève expansion for the random part of the trajectory. Eigenfunctions and scores of the Karhunen–Loève expansion are estimated through conditional expectations within the methodological framework of functional principal component analysis (FPCA). In addition to a calibrated estimation of the stem form, FPCA provides asymptotic pointwise or simultaneous confidence intervals for the calibrated diameter predictions. For the application of functional principal component analysis modelling the covariance function of the random process is crucial. The main features of the functional regression model are discussed informally and demonstrated by means of practical examples.

A linear programming algorithm and software for forest-level planning problems including factories

Combining stand simulation and forest-level optimization is an efficient way to study harvest scenarios of a forest area. A simulator first generates for each treatment unit a number of treatment schedules. Linear programming (LP) can then be used to study how stand-level schedules can be combined at the forest level with respect to alternative goals and constraints. The special structure of the obtained LP problems can be utilized using the generalized upper-bound technique which takes care of the so-called area constraints. JLP software was based on this technique. Later J software was developed to replace JLP. Now J is developed to deal with factory problems where the transportations costs and capacities of factories are included in the problem definition. The generalized upper-bound technique was modified to handle transportation constraints which tell that each timber unit produced is transported to some of the factories. The number of these constraints is very large. This paper describes the basic features of the algorithm and its implementation in the J software.

Characterizing photosynthetic radiation response or other output function as a mean of element responses

An organism (or any higher level unit) is first assumed to consist of small elements each having a Blackman limiting response curve. Any set of Blackman curves is equivalent to a set of Blackman curves with fixed initial slope and varying saturation points. The initial slope of the response function and the distribution of saturation points uniquely define a possible aggregated response function. Conversely, any smooth (twice differentiable), increasing and concave aggregated response function can be described as a mean of Blackman curves having the same initial slope and varying saturation points. The density function of saturation points is proportional to the second derivative of the aggregated response function. An alternative characterization is obtained by assuming that each element has the same Blackman response but the effective proportion of an environmental variable varies. If the effective proportion has a finite upper bound (e.g. one), the aggregated response function is initially linear. The mean response is then analyzed for the case where the effective proportion varies and each element has a general fixed response function. An unknown element response or an unknown distribution of the effective proportion are solutions of an integral equation. A solution method for the element response is derived. Examples given relate to photosynthetic radiation response.

A comment on «a note on estimating the mean level of photosynthesis from radiation measurements» by Chris Koen

Abstract Koen (1987) derived several approximate formulae for the calculation of mean photosynthesis from radiation measurements. We present a critical review of his equations and suggest that most of his equations provide logically inconsistent estimates. We claim also that his exact algebraic equation is based on a misconception of the real estimation problems.

Estimation of taper curve using stand variables and sample tree measurements

The aim of the study was to develop methods for estimating the taper curves for trees tallied in a forest inventory. The average stem form in a stand was described by the principal components of the stand effects in the stem dimensions measured in the polar coordinate system. Measurements of diameter at breast height, diameter at a height of 6 meters, and height taken from trees on the sample tree plots were used for determining the first four principal components. Regression models were derived to predict the principal components from the site and growing stock variables. These models were used to estimate the taper curves of the tallied trees. Use of the principal components estimated by the regression models gave less reliable results than use of the principal component estimates based on measurement of the height of one randomly chosen tree on the sample plot. The best result was found with combined use of the principal component estimates and one height measurement per sample plot.