The assessment of tree row attributes by stratified two-stage sampling

Abstract

Tree row inventories are of increasing interest because tree rows mitigate wind erosion and desertification, protect agricultural crops, enhance rural landscape quality, act as bio-corridors, carbon sinks, and a source for bio-energy. The main objective of tree row inventories is to estimate population parameters such as total tree numbers, total tree numbers by species, the mean stem diameter at breast height, the mean tree height and total wood volume. The estimation of these quantities may be straightforwardly carried out whenever aerial images are available in such a way that tree rows can be counted: in these cases, a two-stage cluster sampling may be performed in which the primary units sampled in the first stage are the tree rows in the study area while the secondary units sampled in the second stage are the trees within the selected rows. This paper proposes two sets of two-stage estimators for the interest parameters, based on the Horvitz–Thompson and ratio criteria, together with the corresponding estimators for their sampling variances. The use of stratification is also considered. The proposed procedure was applied to perform a tree row inventory in the Pontina plain (Central Italy): in this case, the tree rows were enumerated by means of ortho-corrected airborne images and stratification was carried out on the basis of the prevailing species and age classes. The inventory results are interesting from a forestry perspective as well as for checking the effectiveness of the procedure.

Appendices

Appendix 1

The estimator (Eq. 7) may be rewritten as

$${\hat{Y}}_{\rm RAP2} = \frac{\frac{N}{n}{\sum_{i \in S} {M_{i} {\bar{y}}_{i}}}}{\frac{N}{n}{\sum_{i \in S} {x_{i}}}} X = \frac{{\hat{Y}}_{\rm HT2}}{{\hat{X}}_{\rm HT}} X$$

(13)

i.e, as a ratio of the Horvitz–Thompson estimators of Y and X, where the numerator is a two-stage estimator while the denominator needs only of the first stage of sampling.

The expansion of Eq. 13 by means of a Taylor series up to the first order in a neighbourhood of Y and X gives rise to

$${\hat{Y}}_{\rm RAP2} \cong {\hat{Y}}_{\rm HT2} - R ({\hat{X}}_{\rm HT} - X), $$

where R=Y/X. Since ${\hat{X}}_{\rm HT}$ is a constant with respect to second-stage sampling, it follows that

$$\begin{aligned} E_{2} ({\hat{Y}}_{\rm RAP2}) \cong & E_{2} ({\hat{Y}}_{\rm HT2}) - R ({\hat{X}}_{\rm HT} - X) \\ =& \frac{N}{n}{\sum\limits_{i \in S} {M_{i} E_{2} ({\bar{y}}_{i})}} - R({\hat{X}}_{\rm HT} - X) \\ =& \frac{N}{n}{\sum\limits_{i \in S} {Y_{i}}} - R({\hat{X}}_{\rm HT} - X)\\ =& {\hat{Y}}_{\rm HT} - R({\hat{X}}_{\rm HT} - X) \\ \end{aligned}$$

(14)

$$V_{2} ({\hat{Y}}_{\rm RAP2}) \cong V_{2} ({\hat{Y}}_{\rm HT2}) = \frac{N^{2}}{n^{2}}{\sum\limits_{i \in S} {M^{2}_{i} V_{2} ({\bar{y}}_{i})}} = \frac{N^{2}}{n^{2}}{\sum\limits_{i \in S} {M_{i} (M_{i} - m_{i})}}\frac{{\sigma ^{2}_{i}}}{{m_{i}}}, $$

where E ₂ and V ₂ denote expectation and variance with respect to the second stage, while Y _i and σ² _i represent the total and the variance of the interest variable in row i.

From Eq. 14 it is at once apparent that the expectation of ${\hat{Y}}_{\rm RAP2}$ approximately equals the Taylor series approximation of the familiar one-stage ratio estimator

$${\hat{Y}}_{\rm RAP} = \frac{{\hat{Y}}_{\rm HT}}{{\hat{X}}_{\rm HT}} X. $$

(15)

From the well-known properties of Eq. 15 (see e.g., Cochran 1977, chap. 6), it follows that

$$ E ({\hat{Y}}_{\rm RAP2}) = E_{1} \left\{ E_{2} ({\hat{Y}}_{\rm RAP2}) \right\} \cong E_{1} \left\{{\hat{Y}}_{\rm HT} - R({\hat{X}}_{\rm HT} - X)\right\} = Y $$

$$\begin{aligned} V ({\hat{Y}}_{\rm RAP2}) =& V_{1} \left\{ E_{2} ({\hat{Y}}_{\rm RAP2})\right\} + E_{1}\left\{V_{2} ({\hat{Y}}_{\rm RAP2})\right\}\\ \cong & V_{1} \left\{ {\hat{Y}}_{\rm HT} - R({\hat{X}}_{\rm HT} - X) \right\} + E_{1}\left\{ \frac{N^{2}}{n^{2}} \sum\limits_{i \in S} {M_{i} (M_{i} - m_{i})\frac{\sigma^{2}_{i}}{m_{i}}}\right\} \\ = & N(N - n)\frac{\sigma^{2}_{e}}{n} + \frac{N}{n}\sum\limits_{i = 1}^{N} {M_{i} (M_{i} - m_{i})\frac{{\sigma^{2}_{i}}}{m_{i}}} \\ \end{aligned},$$

(16)

where $\sigma^{2}_{e} = \frac{1}{N - 1}{\sum_{i = 1}^{N} e^{2}_{i}}, e_{i} = Y_{i} - RX_{i}$ while E ₁ and V ₁ denote expectation and variance with respect to the first stage.

Now, it is worth noting that an unbiased estimator of $V({\hat{Y}}_{\rm RAP2})$ is given by

$${\tilde{V}}({\hat{Y}}_{\rm RAP2}) = N(N - n) \frac{s^{2}_{{\tilde{e}}}}{n} + \frac{N}{n}{\sum\limits_{i \in S} {M_{i} (M_{i} - m_{i})\frac{{s^{2}_{i}}}{m_{i}}}}, $$

(17)

where $s^{2}_{{\tilde{e}}} = \frac{1}{n - 1} \sum\nolimits_{i \in S} ({\tilde{e}}_{i} - {\bar{e}})^{2} = \frac{1}{n} \sum_{i \in S} {\tilde{e}}^{2}_{i} - \frac{1}{n(n - 1)} \sum\nolimits_{h \neq i \in S} {\tilde{e}}_{i} {\tilde{e}}_{h}, {\tilde{e}}_{i} = {\hat{Y}}_{i} - RX_{i}$ and ${\bar{e}}$ denotes the sample mean of the ${\tilde{e}}_{i}{\text{s}}.$ Indeed, since

$$E_{2} ({\tilde{e}}_{i}) = E_{2} ({\hat{Y}}_{i}) - RX_{i} = Y_{i} - RX_{i} = e_{i} $$

and

$$V_{2} ({\tilde{e}}_{i}) = V_{2} ({\hat{Y}}_{i}) = M_{i} (M_{i} - m_{i}) \frac{{\sigma^{2}_{i}}}{{m_{i}}},$$

then

$$E_{2} ({\tilde{e}}^{2}_{i}) = V_{2} ({\hat{Y}}_{i}) + e^{2}_{i}. $$

Hence

$$\begin{aligned} E_{2} (s^{2}_{{\tilde{e}}}) =& \frac{1}{n}\sum\limits_{i \in S} E_{2} ({\tilde{e}}^{2}_{i}) - \frac{1}{{n(n - 1)}} \sum\limits_{h \neq i \in S} E_{2} ({\tilde{e}}_{i} {\tilde{e}}_{h}) \\ =& \frac{1}{n} \sum\limits_{i \in S} M_{i} (M_{i} - m_{i}) \frac{{\sigma ^{2}_{i}}}{{m_{i}}} + \frac{1}{n} \sum\limits_{i \in S} {e^{2}_{i}} - \frac{1}{{n(n - 1)}} \sum\limits_{h \neq i \in S} e_{i} e_{h} \\ \end{aligned}$$

$$\begin{aligned} E_{2} \left\{ {\tilde{V}} ({\hat{Y}}_{\rm RAP2}) \right\} = & N(N - n)\frac{{E_{2} (s^{2}_{e})}}{n} + \frac{N}{n} \sum\limits_{i \in S} M_{i} (M_{i} - m_{i}) \frac{{E_{2} (s^{2}_{i})}}{{m_{i}}} \\ =& \frac{{N(N - n)}}{n^{2}} \sum\limits_{i \in S} M_{i} (M_{i} - m_{i}) \frac{{\sigma ^{2}_{i}}}{m_{i}} + \frac{{N(N - n)}}{n^{2}} \sum\limits_{i \in S} e^{2}_{i} \\ & - \frac{{N(N - n)}}{n^{2} (n - 1)} \sum\limits_{h \neq i \in S} e_{i} e_{h} + \frac{N}{n} \sum\limits_{i \in S} M_{i} (M_{i} - m_{i}) \frac{{\sigma^{2}_{i}}}{m_{i}} \\ =& \frac{N(N - n)}{n^{2}} \sum\limits_{i \in S} e^{2}_{i} - \frac{{N(N - n)}}{n^{2} (n - 1)} \sum\limits_{h \neq i \in S} e_{i} e_{h} + \frac{{N^{2}}}{n^{2}} \sum\limits_{i \in S} M_{i} (M_{i} - m_{i}) \frac{{\sigma^{2}_{i}}}{m_{i}} \\ \end{aligned}$$

from which

$$\begin{aligned} E \left\{ {\tilde{V}}_{2} ({\hat{Y}}_{\rm RAP2})\right\} =& E_{1} \left[E_{2} \left\{{\tilde{V}}_{2} ({\hat{Y}}_{\rm RAP2}) \right\} \right]\\ =& E_{1} \left\{ \frac{{N(N - n)}}{{n^{2}}} \sum\limits_{i \in S} e^{2}_{i} - \frac{{N(N - n)}}{{n^{2} (n - 1)}} \sum\limits_{h \neq i \in S} e_{i} e_{h} + \frac{{N^{2}}}{{n^{2}}} \sum\limits_{i \in S} M_{i} (M_{i} - m_{i}) \frac{{\sigma^{2}_{i}}}{{m_{i}}} \right\} \\ =& \frac{{N - n}}{n} \sum\limits_{i = 1}^{N} e^{2}_{i} - \frac{{N - n}}{{n(N - 1)}} \sum\limits_{h \neq i = 1}^{N} e_{i} e_{h} + \frac{N}{n} \sum\limits_{i = 1}^{N} M_{i} (M_{i} - m_{i}) \frac{{\sigma^{2}_{i}}}{{m_{i}}} \\ \end{aligned}.$$

Since the sum of e _i extended to all the population units equals 0, it follows that

$$ \sum\limits_{h \neq i = 1}^{N} e_{i} e_{h} = - \sum\limits_{i = 1}^{N} e^{2}_{i} $$

in such a way that

$$\begin{aligned} E \left\{{\tilde{V}}_{2} ({\hat{Y}}_{\rm RAP2})\right\} =& \frac{{N - n}}{n} \sum\limits_{i = 1}^{N} e^{2}_{i} + \frac{{N - n}}{{n(N - 1)}} \sum\limits_{i = 1}^{N} e^{2}_{i} + \frac{N}{n} \sum\limits_{i = 1}^{N} M_{i} (M_{i} - m_{i}) \frac{{\sigma^{2}_{i}}}{{m_{i}}} \\ =& \frac{{N - n}}{n}\frac{N}{{N - 1}} \sum\limits_{i = 1}^{N} e^{2}_{i} + \frac{N}{n} \sum\limits_{i = 1}^{N} M_{i} (M_{i} - m_{i})\frac{{\sigma^{2}_{i}}}{{m_{i}}} \\ =& N(N - n)\frac{{\sigma^{2}_{e}}}{n} + \frac{N}{n} \sum\limits_{i = 1}^{N} M_{i} (M_{i} - m_{i})\frac{{\sigma^{2}_{i}}}{{m_{i}}}\\ =& V({\hat{Y}}_{\rm RAP2}).\\ \end{aligned}$$

However, the quantity (Eq. 17) cannot be computed from the sample information, since R and hence all the residuals ${\tilde{e}}_{i} = {\hat{Y}}_{i} - RX_{i}$ are unknown. Thus, as is customary in ratio estimation, the quantities ${\hat{e}}_{i} = {\hat{Y}}_{i} - {\hat{r}} x_{i}$ may be used as residuals. Since their sample mean equals 0, when used in Eq. 17 they give rise to Eq. 8.

Appendix 2

Eq. 11 constitutes a function of ${\hat{M}}_{l}$ and ${\hat{\mu}}_{l}$ for any l ∈G. Thus, since

$$\frac{{\partial {\hat{\mu}}_{G}}}{{\partial {\hat{M}}_{l}}} = \frac{1}{{{\hat{M}}_{G}}}({\hat{\mu}}_{l} - {\hat{\mu}}_{G})$$

and

$$\frac{{\partial {\hat{\mu}}_{G}}}{{\partial {\hat{\mu}}_{l}}} = {\hat{w}}_{l} $$

then, the Taylor series expansion of ${\hat{\mu}}_{G}$ up to the first order gives rise to

$$\begin{aligned} {\hat{\mu}}_{G} \cong & \mu_{G} + \frac{1}{{M_{G}}} \sum\limits_{l \in G} (\mu_{l} - \mu_{G})({\hat{M}}_{l} - M_{l}) + \sum\limits_{l \in G} w_{l} ({\hat{\mu}}_{l} - \mu_{l}) \\ =& \frac{1}{{M_{G}}} \sum\limits_{l \in G} (\mu_{l} - \mu_{G}) {\hat{M}}_{l} + \sum\limits_{l \in G} w_{l} {\hat{\mu}}_{l} \\ \end{aligned}.$$

(18)

Thus, from Eq. 18 it is straightforward to prove that ${\hat{\mu}}_{G}$ turns out to be approximately unbiased with variance

$$V({\hat{\mu}}_{G}) \cong \frac{1}{{M^{2}_{G}}} \sum\limits_{l \in G} (\mu_{l} - \mu_{G})^{2} V({\hat{M}}_{l}) + \sum\limits_{l \in G} w^{2}_{l} V({\hat{\mu}}_{l}) + 2 \sum\limits_{l \in G} (\mu_{l} - \mu_{G}) w_{l} {\rm COV}({\hat{M}}_{l}, {\hat{\mu}}_{l}). $$

(19)

But, since

$$\begin{aligned} {\rm COV}({\hat{M}}_{l}, {\hat{\mu}}_{l}) =& E({\hat{M}}_{l} {\hat{\mu}}_{l}) - E({\hat{M}}_{l}) E({\hat{\mu}}_{l})\\ \cong & E \left\{ \frac{N}{n} \sum\limits_{i \in S} x_{j} \frac{{\frac{N}{n} \sum_{i \in S} M_{i} {\bar{y}}_{i}}}{{\frac{N}{n} \sum_{j \in S} x_{i}}} \right\} - M_{l} \mu_{l} \\ =& E \left\{ \frac{N}{n} \sum\limits_{i \in S} M_{i} {\bar{y}}_{i} \right\} - Y_{l} \\ =& E({\hat{Y}}_{l}) - Y_{l} = 0 , \\ \end{aligned}$$

quantity (Eq. 19) reduces to

$$V({\hat{\mu}}_{G}) \cong \frac{1}{{M^{2}_{G}}} \sum\limits_{l \in G} (\mu_{l} - \mu_{G})^{2} V({\hat{M}}_{l}) + \sum\limits_{l \in G} w^{2}_{l} V({\hat{\mu}}_{l}) $$

which may be trivially estimated by formula 12.