LiDAR-based reference aboveground biomass maps for tropical forests of South Asia and Central Africa.

Accurate mapping and monitoring of tropical forests aboveground biomass (AGB) is crucial to design effective carbon emission reduction strategies and improving our understanding of Earth's carbon cycle. However, existing large-scale maps of tropical forest AGB generated through combinations of Earth Observation (EO) and forest inventory data show markedly divergent estimates, even after accounting for reported uncertainties. To address this, a network of high-quality reference data is needed to calibrate and validate mapping algorithms. This study aims to generate reference AGB datasets using field inventory plots and airborne LiDAR data for eight sites in Central Africa and five sites in South Asia, two regions largely underrepresented in global reference AGB datasets. The study provides access to these reference AGB maps, including uncertainty maps, at 100 m and 40 m spatial resolutions covering a total LiDAR footprint of 1,11,650 ha [ranging from 150 to 40,000 ha at site level]. These maps serve as calibration/validation datasets to improve the accuracy and reliability of AGB mapping for current and upcoming EO missions (viz., GEDI, BIOMASS, and NISAR).

Evaluation of automated pipelines for tree and plot metric estimation from TLS data in tropical forest areas.

BACKGROUND AND AIMS: Terrestrial LiDAR scanning (TLS) data are of great interest in forest ecology and management because they provide detailed 3-D information on tree structure. Automated pipelines are increasingly used to process TLS data and extract various tree- and plot-level metrics. With these developments comes the risk of unknown reliability due to an absence of systematic output control. In the present study, we evaluated the estimation errors of various metrics, such as wood volume, at tree and plot levels for four automated pipelines. METHODS: We used TLS data collected from a 1-ha plot of tropical forest, from which 391 trees >10 cm in diameter were fully processed using human assistance to obtain control data for tree- and plot-level metrics. KEY RESULTS: Our results showed that fully automated pipelines led to median relative errors in the quantitative structural model (QSM) volume ranging from 39 to 115 % at the tree level and 10 to 134 % at the 1-ha plot level. For tree-level metrics, the median error for the crown-projected area ranged from 46 to 59 % and that for the crown-hull volume varied from 72 to 88 %. This result suggests that the tree isolation step is the weak link in automated pipeline methods. We further analysed how human assistance with automated pipelines can help reduce the error in the final QSM volume. At the tree scale, we found that isolating trees using human assistance reduced the error in wood volume by a factor of 10. At the 1-ha plot scale, locating trees with human assistance reduced the error by a factor of 3. CONCLUSIONS: Our results suggest that in complex tropical forests, fully automated pipelines may provide relatively unreliable metrics at the tree and plot levels, but limited human assistance inputs can significantly reduce errors.

Composition and structure of a stretch of tropical forest in the Eastern Amazon / Composição e estrutura de um trecho de floresta tropical na Amazônia Oriental

ABSTRACT: The aim of this study was to determine the diversity. composition. forest structure of a stretch of dense ombrophilous forest in the state of Amapá. The area is located in the east of the state of Amapá. in the Eastern Amazon. at the following coordinates: 2º 0'0.00 '' N. 14º 0'0.00''O. The sampling process used was systematic. where nine plots of 1 ha (100 m x 100 m) were allocated. 0.5 km apart. within a 13 km long and 0.03 km wide transect. The horizontal structure of the forest was evaluated using phytosociological parameters. One thousand seven hundred and ninety trees were sampled in this study. they were distributed in 131 species. 69 genera and 48 families. The species that presented more individuals were Pouteria guianensis Aubl. (136). Eschweilera coriacea (DC.) S.A.Mori (78). Inga paraensis Ducke (45). Licania paraensis Prance (38). The area is highly diverse and dissimilar with great variability and complexity in vegetable formation.

RESUMO: O objetivo deste estudo foi determinar a diversidade, composição e estrutura florestal de um trecho de floresta ombrófila densa do estado do Amapá. A área está localizada a leste do Estado do Amapá, Amazônia Oriental. Nas seguintes coordenadas: 2º 0'0,00'' N. 14º 0'0.00''O. O processo de amostragem utilizado foi o sistemático em que foram alocadas nove parcelas de 1 ha (100 m x 100 m), distanciadas em 0.5 km entre si, dentro de um transecto de 13 km de comprimento por 0.03 km de largura. A estrutura horizontal da floresta foi avaliada por meio de parâmetros fitossociológicos. 1790 árvores foram amostradas neste estudo. Elas estão distribuídas em 131 espécies, 69 gêneros e 48 famílias. As espécies que apresentaram mais indivíduos foram Pouteria guianensis Aubl. (136), Eschweilera coriacea (DC.) S.A.Mori (78), Inga paraensis Ducke (45), Licania paraensis Prance (38). A área é altamente diversa e dissimilar com grande variabilidade e complexidade na formação vegetal.

Leveraging Signatures of Plant Functional Strategies in Wood Density Profiles of African Trees to Correct Mass Estimations From Terrestrial Laser Data.

Wood density (WD) relates to important tree functions such as stem mechanics and resistance against pathogens. This functional trait can exhibit high intraindividual variability both radially and vertically. With the rise of LiDAR-based methodologies allowing nondestructive tree volume estimations, failing to account for WD variations related to tree function and biomass investment strategies may lead to large systematic bias in AGB estimations. Here, we use a unique destructive dataset from 822 trees belonging to 51 phylogenetically dispersed tree species harvested across forest types in Central Africa to determine vertical gradients in WD from the stump to the branch tips, how these gradients relate to regeneration guilds and their implications for AGB estimations. We find that decreasing WD from the tree base to the branch tips is characteristic of shade-tolerant species, while light-demanding and pioneer species exhibit stationary or increasing vertical trends. Across all species, the WD range is narrower in tree crowns than at the tree base, reflecting more similar physiological and mechanical constraints in the canopy. Vertical gradients in WD induce significant bias (10%) in AGB estimates when using database-derived species-average WD data. However, the correlation between the vertical gradients and basal WD allows the derivation of general correction models. With the ongoing development of remote sensing products providing 3D information for entire trees and forest stands, our findings indicate promising ways to improve greenhouse gas accounting in tropical countries and advance our understanding of adaptive strategies allowing trees to grow and survive in dense rainforests.

Author Correction: Disturbance Regimes Drive The Diversity of Regional Floristic Pools Across Guianan Rainforest Landscapes.

A correction to this article has been published and is linked from the HTML and PDF versions of this paper. The error has been fixed in the paper.

Disturbance Regimes Drive The Diversity of Regional Floristic Pools Across Guianan Rainforest Landscapes.

Disturbances control rainforest dynamics, and, according to the intermediate disturbance hypothesis (IDH), disturbance regime is a key driver of local diversity. Variations in disturbance regimes and their consequences on regional diversity at broad spatiotemporal scales are still poorly understood. Using multidisciplinary large-scale inventories and LiDAR acquisitions, we developed a robust indicator of disturbance regimes based on the frequency of a few early successional and widely distributed pioneer species. We demonstrate at the landscape scale that tree-species diversity and disturbance regimes vary with climate and relief. Significant relationships between the disturbance indicator, tree-species diversity and soil phosphorus content agree with the hypothesis that rainforest diversity is controlled both by disturbance regimes and long-term ecosystem stability. These effects explain the broad-scale patterns of floristic diversity observed between landscapes. In fact, species-rich forests in highlands, which have benefited from long-term stability combined with a moderate and regular regime of local disturbances, contrast with less diversified forests on recently shaped lowlands, which have undergone more recent changes and irregular dynamics. These results suggest that taking the current disturbance regime into account and including geomorphological stratifications in climate-vegetation models may be an effective way to improve the prediction of changes in species diversity under climate change.

Spatial Structure of Above-Ground Biomass Limits Accuracy of Carbon Mapping in Rainforest but Large Scale Forest Inventories Can Help to Overcome.

Precise mapping of above-ground biomass (AGB) is a major challenge for the success of REDD+ processes in tropical rainforest. The usual mapping methods are based on two hypotheses: a large and long-ranged spatial autocorrelation and a strong environment influence at the regional scale. However, there are no studies of the spatial structure of AGB at the landscapes scale to support these assumptions. We studied spatial variation in AGB at various scales using two large forest inventories conducted in French Guiana. The dataset comprised 2507 plots (0.4 to 0.5 ha) of undisturbed rainforest distributed over the whole region. After checking the uncertainties of estimates obtained from these data, we used half of the dataset to develop explicit predictive models including spatial and environmental effects and tested the accuracy of the resulting maps according to their resolution using the rest of the data. Forest inventories provided accurate AGB estimates at the plot scale, for a mean of 325 Mg.ha-1. They revealed high local variability combined with a weak autocorrelation up to distances of no more than10 km. Environmental variables accounted for a minor part of spatial variation. Accuracy of the best model including spatial effects was 90 Mg.ha-1 at plot scale but coarse graining up to 2-km resolution allowed mapping AGB with accuracy lower than 50 Mg.ha-1. Whatever the resolution, no agreement was found with available pan-tropical reference maps at all resolutions. We concluded that the combined weak autocorrelation and weak environmental effect limit AGB maps accuracy in rainforest, and that a trade-off has to be found between spatial resolution and effective accuracy until adequate "wall-to-wall" remote sensing signals provide reliable AGB predictions. Waiting for this, using large forest inventories with low sampling rate (<0.5%) may be an efficient way to increase the global coverage of AGB maps with acceptable accuracy at kilometric resolution.

How do habitat filtering and niche conservatism affect community composition at different taxonomic resolutions?

Understanding how local species assembly depends on the regional biogeographic and environmental context is a challenging task in community ecology. In spatially implicit neutral models, a single immigration parameter, I(k), represents the flux of immigrants from a regional pool that compete with local offspring for establishment in communities. This flux counterbalances the effect of local stochastic extinctions to maintain local species diversity. If some species within the regional pool are not adapted to the local environment (habitat filtering), the migrant flux is reduced beyond that of the neutral model, such that habitat filtering influences the value of I(k) in non-neutral situations. Here, we propose a novel model in which immigrants from the regional pool are filtered according to their habitat preferences and the local environment, while taxa potentially retain habitat preferences from their ancestors (niche conservatism). Using both analytical reasoning and simulations, we demonstrate that I(k) is expected to be constant when estimated based on the community composition at several taxonomic levels, not only under neutral assumptions, but also when habitat filtering occurs, unless there is substantial niche conservatism. In the latter case, I(k) is expected to decrease when estimated based on the composition at species to genus and family levels, thus allowing a signature of niche conservatism to be detected by simply comparing I(k) estimates across taxonomic levels. We applied this approach to three rain forest data sets from South India and Central America and found no significant signature of niche conservatism when I(k) was compared across taxonomic levels, except at the family level in South India. We further observed more limited immigration in South Indian forests, supporting the hypothesis of a greater impact of habitat filtering and heterogeneity there than in Central America. Our results highlight the relevance of studying variations of I(k) in space and across taxonomic levels to test hypotheses about the ecological and evolutionary drivers of biodiversity patterns.

Characterizing the phylogenetic tree community structure of a protected tropical rain forest area in Cameroon.

Tropical rain forests, the richest terrestrial ecosystems in biodiversity on Earth are highly threatened by global changes. This paper aims to infer the mechanisms governing species tree assemblages by characterizing the phylogenetic structure of a tropical rain forest in a protected area of the Congo Basin, the Dja Faunal Reserve (Cameroon). We re-analyzed a dataset of 11538 individuals belonging to 372 taxa found along nine transects spanning five habitat types. We generated a dated phylogenetic tree including all sampled taxa to partition the phylogenetic diversity of the nine transects into alpha and beta components at the level of the transects and of the habitat types. The variation in phylogenetic composition among transects did not deviate from a random pattern at the scale of the Dja Faunal Reserve, probably due to a common history and weak environmental variation across the park. This lack of phylogenetic structure combined with an isolation-by-distance pattern of taxonomic diversity suggests that neutral dispersal limitation is a major driver of community assembly in the Dja. To assess any lack of sensitivity to the variation in habitat types, we restricted the analyses of transects to the terra firme primary forest and found results consistent with those of the whole dataset at the level of the transects. Additionally to previous analyses, we detected a weak but significant phylogenetic turnover among habitat types, suggesting that species sort in varying environments, even though it is not predominating on the overall phylogenetic structure. Finer analyses of clades indicated a signal of clustering for species from the Annonaceae family, while species from the Apocynaceae family indicated overdispersion. These results can contribute to the conservation of the park by improving our understanding of the processes dictating community assembly in these hyperdiverse but threatened regions of the world.

Aboveground biomass mapping of African forest mosaics using canopy texture analysis: toward a regional approach.

In the context of the reduction of greenhouse gas emissions caused by deforestation and forest degradation (the REDD+ program), optical very high resolution (VHR) satellite images provide an opportunity to characterize forest canopy structure and to quantify aboveground biomass (AGB) at less expense than methods based on airborne remote sensing data. Among the methods for processing these VHR images, Fourier textural ordination (FOTO) presents a good method to detect forest canopy structural heterogeneity and therefore to predict AGB variations. Notably, the method does not saturate at intermediate AGB values as do pixelwise processing of available space borne optical and radar signals. However, a regional-scale application requires overcoming two difficulties: (1) instrumental effects due to variations in sun­scene­sensor geometry or sensor-specific responses that preclude the use of wide arrays of images acquired under heterogeneous conditions and (2) forest structural diversity including monodominant or open canopy forests, which are of particular importance in Central Africa. In this study, we demonstrate the feasibility of a rigorous regional study of canopy texture by harmonizing FOTO indices of images acquired from two different sensors (Geoeye-1 and QuickBird-2) and different sun­scene­sensor geometries and by calibrating a piecewise biomass inversion model using 26 inventory plots (1 ha) sampled across very heterogeneous forest types. A good agreement was found between observed and predicted AGB (residual standard error [RSE] = 15%; R2 = 0.85; P < 0.001) across a wide range of AGB levels from 26 Mg/ha to 460 Mg/ha, and was confirmed by cross validation. A high-resolution biomass map (100-m pixels) was produced for a 400-km2 area, and predictions obtained from both imagery sources were consistent with each other (r = 0.86; slope = 1.03; intercept = 12.01 Mg/ha). These results highlight the horizontal structure of forest canopy as a powerful descriptor of the entire forest stand structure and heterogeneity. In particular, we show that quantitative metrics resulting from such textural analysis offer new opportunities to characterize the spatial and temporal variation of the structure of dense forests and may complement the toolbox used by tropical forest ecologists, managers or REDD+ national monitoring, reporting and verification bodies.

Do spatially-implicit estimates of neutral migration comply with seed dispersal data in tropical forests?

Neutral community models have shown that limited migration can have a pervasive influence on the taxonomic composition of local communities even when all individuals are assumed of equivalent ecological fitness. Notably, the spatially implicit neutral theory yields a single parameter I for the immigration-drift equilibrium in a local community. In the case of plants, seed dispersal is considered as a defining moment of the immigration process and has attracted empirical and theoretical work. In this paper, we consider a version of the immigration parameter I depending on dispersal limitation from the neighbourhood of a community. Seed dispersal distance is alternatively modelled using a distribution that decreases quickly in the tails (thin-tailed Gaussian kernel) and another that enhances the chance of dispersal events over very long distances (heavily fat-tailed Cauchy kernel). Our analysis highlights two contrasting situations, where I is either mainly sensitive to community size (related to ecological drift) under the heavily fat-tailed kernel or mainly sensitive to dispersal distance under the thin-tailed kernel. We review dispersal distances of rainforest trees from field studies and assess the consistency between published estimates of I based on spatially-implicit models and the predictions of the kernel-based model in tropical forest plots. Most estimates of I were derived from large plots (10-50 ha) and were too large to be accounted for by a Cauchy kernel. Conversely, a fraction of the estimates based on multiple smaller plots (1 ha) appeared too small to be consistent with reported ranges of dispersal distances in tropical forests. Very large estimates may reflect within-plot habitat heterogeneity or estimation problems, while the smallest estimates likely imply other factors inhibiting migration beyond dispersal limitation. Our study underscores the need for interpreting I as an integrative index of migration limitation which, besides the limited seed dispersal, possibly includes habitat filtering or fragmentation.

Testing pairwise association between spatially autocorrelated variables: a new approach using surrogate lattice data.

BACKGROUND: Independence between observations is a standard prerequisite of traditional statistical tests of association. This condition is, however, violated when autocorrelation is present within the data. In the case of variables that are regularly sampled in space (i.e. lattice data or images), such as those provided by remote-sensing or geographical databases, this problem is particularly acute. Because analytic derivation of the null probability distribution of the test statistic (e.g. Pearson's r) is not always possible when autocorrelation is present, we propose instead the use of a Monte Carlo simulation with surrogate data. METHODOLOGY/PRINCIPAL FINDINGS: The null hypothesis that two observed mapped variables are the result of independent pattern generating processes is tested here by generating sets of random image data while preserving the autocorrelation function of the original images. Surrogates are generated by matching the dual-tree complex wavelet spectra (and hence the autocorrelation functions) of white noise images with the spectra of the original images. The generated images can then be used to build the probability distribution function of any statistic of association under the null hypothesis. We demonstrate the validity of a statistical test of association based on these surrogates with both actual and synthetic data and compare it with a corrected parametric test and three existing methods that generate surrogates (randomization, random rotations and shifts, and iterative amplitude adjusted Fourier transform). Type I error control was excellent, even with strong and long-range autocorrelation, which is not the case for alternative methods. CONCLUSIONS/SIGNIFICANCE: The wavelet-based surrogates are particularly appropriate in cases where autocorrelation appears at all scales or is direction-dependent (anisotropy). We explore the potential of the method for association tests involving a lattice of binary data and discuss its potential for validation of species distribution models. An implementation of the method in Java for the generation of wavelet-based surrogates is available online as supporting material.

Comment on "Global correlations in tropical tree species richness and abundance reject neutrality".

Ricklefs and Renner (Reports, 27 January 2012, p. 464) have argued that the neutral theory of biodiversity and biogeography cannot explain the correlations in family abundances and species richness found between tropical forests from distinct continents. However, we show that such patterns can arise from neutral processes of diversification, migration, and drift over large spatial and temporal scales.

Assessing aboveground tropical forest biomass using Google Earth canopy images.

Reducing Emissions from Deforestation and Forest Degradation (REDD) in efforts to combat climate change requires participating countries to periodically assess their forest resources on a national scale. Such a process is particularly challenging in the tropics because of technical difficulties related to large aboveground forest biomass stocks, restricted availability of affordable, appropriate remote-sensing images, and a lack of accurate forest inventory data. In this paper, we apply the Fourier-based FOTO method of canopy texture analysis to Google Earth's very-high-resolution images of the wet evergreen forests in the Western Ghats of India in order to (1) assess the predictive power of the method on aboveground biomass of tropical forests, (2) test the merits of free Google Earth images relative to their native commercial IKONOS counterparts and (3) highlight further research needs for affordable, accurate regional aboveground biomass estimations. We used the FOTO method to ordinate Fourier spectra of 1436 square canopy images (125 x 125 m) with respect to a canopy grain texture gradient (i.e., a combination of size distribution and spatial pattern of tree crowns), benchmarked against virtual canopy scenes simulated from a set of known forest structure parameters and a 3-D light interception model. We then used 15 1-ha ground plots to demonstrate that both texture gradients provided by Google Earth and IKONOS images strongly correlated with field-observed stand structure parameters such as the density of large trees, total basal area, and aboveground biomass estimated from a regional allometric model. Our results highlight the great potential of the FOTO method applied to Google Earth data for biomass retrieval because the texture-biomass relationship is only subject to 15% relative error, on average, and does not show obvious saturation trends at large biomass values. We also provide the first reliable map of tropical forest aboveground biomass predicted from free Google Earth images.

Ecophylogenetics: advances and perspectives.

Ecophylogenetics can be viewed as an emerging fusion of ecology, biogeography and macroevolution. This new and fast-growing field is promoting the incorporation of evolution and historical contingencies into the ecological research agenda through the widespread use of phylogenetic data. Including phylogeny into ecological thinking represents an opportunity for biologists from different fields to collaborate and has provided promising avenues of research in both theoretical and empirical ecology, towards a better understanding of the assembly of communities, the functioning of ecosystems and their responses to environmental changes. The time is ripe to assess critically the extent to which the integration of phylogeny into these different fields of ecology has delivered on its promise. Here we review how phylogenetic information has been used to identify better the key components of species interactions with their biotic and abiotic environments, to determine the relationships between diversity and ecosystem functioning and ultimately to establish good management practices to protect overall biodiversity in the face of global change. We evaluate the relevance of information provided by phylogenies to ecologists, highlighting current potential weaknesses and needs for future developments. We suggest that despite the strong progress that has been made, a consistent unified framework is still missing to link local ecological dynamics to macroevolution. This is a necessary step in order to interpret observed phylogenetic patterns in a wider ecological context. Beyond the fundamental question of how evolutionary history contributes to shape communities, ecophylogenetics will help ecology to become a better integrative and predictive science.

Deeply gapped vegetation patterns: on crown/root allometry, criticality and desertification.

The dynamics of vegetation is formulated in terms of the allometric and structural properties of plants. Within the framework of a general and yet parsimonious approach, we focus on the relationship between the morphology of individual plants and the spatial organization of vegetation populations. So far, in theoretical as well as in field studies, this relationship has received only scant attention. The results reported remedy to this shortcoming. They highlight the importance of the crown/root ratio and demonstrate that the allometric relationship between this ratio and plant development plays an essential part in all matters regarding ecosystems stability under conditions of limited soil (water) resources. This allometry determines the coordinates in parameter space of a critical point that controls the conditions in which the emergence of self-organized biomass distributions is possible. We have quantified this relationship in terms of parameters that are accessible by measurement of individual plant characteristics. It is further demonstrated that, close to criticality, the dynamics of plant populations is given by a variational Swift-Hohenberg equation. The evolution of vegetation in response to increasing aridity, the conditions of gapped pattern formation and the conditions under which desertification takes place are investigated more specifically. It is shown that desertification may occur either as a local desertification process that does not affect pattern morphology in the course of its unfolding or as a gap coarsening process after the emergence of a transitory, deeply gapped pattern regime. Our results amend the commonly held interpretation associating vegetation patterns with a Turing instability. They provide a more unified understanding of vegetation self-organization within the broad context of matter order-disorder transitions.

Spatial decoupling of facilitation and competition at the origin of gapped vegetation patterns.

Spatially periodic vegetation patterns, forming gaps, bands, labyrinths, or spots, are characteristic of arid and semiarid landscapes. Self-organization models can explain this variety of structures within a unified conceptual framework. All these models are based on the interplay of positive and negative effects of plants on soil water, but they can be divided according to whether they assume the interactions to be mediated by water redistribution through runoff/diffusion or by plants' organs. We carried out a multi-proxy approach of the processes operating in a gapped pattern in southwest Niger dominated by a shrub species. Soil moisture within the root layer was monitored in time and space over one month of the rainy season. Soil water recharge displayed no spatial variation with respect to vegetation cover, but the stock half-life under cover was twice that of bare areas. A kernel of facilitation by the aboveground parts of shrubs was parameterized, and soil water half-life was significantly correlated to the cumulated facilitative effects of shrubs. The kernel range was found to be smaller than the canopy radius (81%). This effect of plants on soil water dynamics, probably through a reduction of evaporation by shading, is shown to be a better explanatory variable than potentially relevant soil and topography parameters. The root systems of five individuals of Combretum micranthum G. Don were excavated. Root density data were used as a proxy to parameterize a kernel function of interplant competition. The range of this kernel was larger than the canopy radius (125%). The facilitation-to-competition range ratio, reflecting the above-to-belowground ratio of plant lateral extent, was smaller than 1 (0.64), a result supporting models assuming that patterning may emerge from an adaptation of plant morphology to aridity and shallow soils by means of an extended lateral root system. Moreover, observed soil water gradients had directions opposite to those assumed by alternative mathematical models based on underground water diffusion. This study contributes to the growing awareness that combined facilitative and competitive plant interactions can induce landscape-scale patterns and shape the two-way feedback loops between environment and vegetation.

Beta diversity in spatially implicit neutral models: a new way to assess species migration.

The spatially implicit neutral model (SINM) of S. P. Hubbell predicts species' abundance distributions at two levels: local communities where extinction balances immigration, characterized by the immigration number I, and the metacommunity, a source pool of migrants where speciation balances extinction. Previously, a plot's I was estimated from its species abundance distribution. Here, we relate neutral theory to the additive partitioning of species diversity and calculate the immigration rate into different plots from the variation in species composition among them. We revisit the G(ST) statistic of population genetics to introduce a new version, G(ST)(k), conditional on each community sample k. We derive an analytical expectation of G(ST)(k) as a function of the local immigration number, I(k), under a generalized version of the SINM, which allows the pool of migrants to deviate from the large-scale speciation-extinction balance. Simulations and field data suggest that G(ST)(k) provides reasonable estimates of immigration numbers, which were compared with the results from alternative likelihood-based estimations.