# Stand-level

#### Anika Seppelt

Once all trees were detected from the normalized point cloud and the tree list was generated, metrics and variables at stand level can be estimated. These estimations can be based on different plot designs, that are circular fixed area, k-tree and angle-count plots. The sizes of the plots can be regulated by defining the radius, k and the basal area factor (BAF) respectively.

## Computing stand-level metrics and variables

In order to calculate metrics and variables at stand level, the function `metrics.variables` is applied as follows:

    metrics <- metrics.variables(tree.tls = tree.list,
                      tree.ds = tree.ds, tree.field = tree.field,
                      plot.design = c("fixed.area", "k.tree", "angle.count"),
                      plot.parameters = data.frame(radius = 10, k = 10, BAF = 1),
                      scan.approach = "single", var.metr = NULL,
                      dbh.min = 4, h.min = 1.3, max.dist = Inf,
                      dir.data = dir.data, save.result = FALSE, dir.result = NULL)

The input data frame is introduced in `tree.tls` and should contain information about the trees detected from TLS based data e.g., the output data frame of the functions `tree.detection.single.scan` or `tree.detection.multi.scan`. The path of the directory of the .txt files containing the reduced point cloud data generated by the function `normalize` can be specified in `dir.data` otherwise the current working directory will be assigned to this argument. If `save.result` is set to `save.result = TRUE` (default setting), the output files will be saved to the directory indicated in `dir.result`.

The scan approach, either TLS single-scan (`"single"`) or TLS multi-scan and SLAM point clouds approaches (`"multi"`) has to be specified in the argument `scan.approach`. By default, the argument is set to `"multi"`.

If only a selection of metrics and variables should be calculated, the metrics and variables of interest can be specified as a vector in `var.metr`. If nothing is specified, the argument is set to `NULL` and all possible metrics and variables will be calculated.

#### Distance sampling (`distance.sampling` function)

The distance sampling method is used for the correction of occlusion effects. When the `distance.sampling` function was applied, the list with tree detection probabilities (i.e., the output file of the mentioned function) can be introduced in `tree.ds`. By default, this argument is set to `NULL` and metrics using distance sampling corrections will not be calculated. The `distance.sampling` is applied as follows:

    tree.ds <- distance.sampling(tree.tls = tree.list,
                                 id.plots = NULL,
                                 strata.attributes = NULL)

The `distance.sampling` function computes the probability of detection of trees depending on their distance to the TLS device. Detection functions are fitted to the histogram representing the distribution of the trees relative to their horizontal distances. The computation of the detection functions is based on the data frame of the detected trees, i.e. the output file of the `tree.detection.single.scan` or `tree.detection.multi.scan` (here `tree.list`), which is introduced in the argument `tree.tls`. The plots to be analysed can be specified in `id.plots` by inserting a vector containing the plot identification numbers. If not specified, this argument is set to `id.plots = NULL` and all plots will be considered.

Two detection functions are fitted, that are the half normal and hazard rate functions, with and without \(dbh\) as covariate. These functions describe the decreasing detection probability with increasing distance. The probabilities are used for correcting estimation bias of the stand-level variables caused by the lack of detection of trees due to occlusion.

A list with three elements is generated by the `distance.sampling` function. The data frame `tree` represents the detection probabilities for each tree of all plots according to the four different detection functions. The columns `P.hn` and `P.hn.cov` show the detection probabilities calculated by the half normal function without and with \(dbh\) as covariate, while `P.hr` and `P.hr.cov` contain the probabilities according to the hazard rate function without and with covariate respectively.

    head(tree.ds$tree)
    #>   stratum id tree     P.hn  P.hn.cov      P.hr  P.hr.cov
    #> 1       1  1    1 0.922808 0.9293580 0.9388107 0.9519734
    #> 2       1  1    2 0.922808 0.8922843 0.9388107 0.9199997
    #> 3       1  1    3 0.922808 0.9146826 0.9388107 0.9389643
    #> 4       1  1    4 0.922808 0.9017652 0.9388107 0.9278849
    #> 5       1  1    5 0.922808 0.9277925 0.9388107 0.9505678
    #> 6       1  1    6 0.922808 0.9424095 0.9388107 0.9637861

The data frame `par` shows the parameters of the detection functions and `AIC` the Akaike information criterion (AIC) for every detection function fit.

    head(tree.ds$par)
    #>              P.hn.scale P.hn.cov.scale.intercept P.hn.cov.dbh P.hr.scale
    #> X.Intercept.   35.84341                 64.17593    -2.231185   21.85271
    #>              P.hr.shape P.hr.cov.scale.intercept P.hr.cov.dbh P.hr.cov.shape
    #> X.Intercept.   3.544516                 26.38362   -0.7110327       3.832857

    head(tree.ds$AIC)
    #>       P.hn P.hn.cov     P.hr P.hr.cov
    #> 1 3375.162 3376.521 3374.114 3375.123

#### Additional information about trees through field data

If supplementary field data is available for the sample plots, a data frame containing this information can be included in `tree.field`. Each row of the data frame must represent a tree and the data frame must contain at least the following columns:

- `id`: plot identification number (character string or numeric), which must coincide with those in the `id` column of the `tree.tls` argument i.e., the list of trees yielded by the tree detection functions
- `tree`: number of the tree
- `h.dist`: horizontal distance (in m) of the tree’s center to the plot’s center, which must coincide with the centers of their corresponding TLS plots
- `dbh`: tree diameter (in cm) at breast height (1.3 m)
- `h`: total tree height (in m)
- `dead`: integer value indicating whether the tree is dead (1) or not (NA)

Further optional columns are `h.blc` (height based live crown, in m), `h.bdc` (height based dead crown, in m), `v.user` (tree volume, in m\(^3\)) and `w.user` (tree biomass, in Mg).

#### Selecting trees to be included in the calculations

The arguments `dbh.min`, `h.min` and `max.dist` are used to determine the trees that are included in the calculations. The minimum diameter at breast height (\(dbh\)) and the minimum tree height can be specified in `dbh.min` (in cm) and `h.min` (in m) and are set to 4 cm and 1.3 m respectively by default. In `max.dist`, the maximal horizontal distance (in m) of a tree from the plot’s center to be included in the calculations can be defined. If not specified, no trees are discarded because of their distance.

#### Plot design and parameters

The metrics and variables can be computed for different plot designs which are specified by `plot.design` and `plot.parameters`. There are three different plot designs, which are similar to the procedure used in conventional forest inventories. The plot design, which is to be used can be specified in `plot.design`. By default, all plot designs will be considered.

As the name implies, circular fixed area plots (`"fixed.area"`) are plots with a circular area defined by the plot radius (in m). The size of k-tree plots (`"k.tree"`) is determined by the number of trees (k) that enter the plot. The basal area factor (BAF in m\(^2\)/ha) defines the size of angle-count plots (`"angle.count"`). The parameters `radius`, `k` and `BAF` have to be specified as data frame in `plot.parameters`. If one of these parameters is not specified, the corresponding plot design will not be considered in the calculations. To calculate the dominant height and diameter, the number of dominant trees per ha (trees/ha) can be indicated by the argument `num.tree`. By default, the number of dominant trees is set to 100 trees/ha.

#### Output files

The function `metrics.variables` generates a list in which each element represents one of the considered plot designs (fixed area, k-tree and angle-count plots). These elements are data frames containing the estimated metrics and variables at stand-level as columns (described below). Further columns include information about the plot, such as the identification number (`id`) and the stratum identification (`stratum`) both coinciding with the respective columns in the output file of the `tree.detection` functions. Each row represents a simulated plot i.e., a plot with a certain size (defined by the values for `radius`, `k` and `BAF` shown in the respective columns).

If `save.result = TRUE`, the data frames will be saved as seperate .csv files (one for each plot design) to the directory indicated in `dir.result` (or if not specified to the working directory). The .csv files (without row names) will be created using the `write.csv` function from the utils package.

### Stand-level metrics

    metrics[1:6, -c(3:36)]

id

radius

n.pts

n.pts.est

n.pts.red

n.pts.red.est

P01

P05

P10

P20

P25

P30

P40

P50

P60

P70

P75

P80

P90

P95

P99

mean.z

mean.q.z

mean.g.z

mean.h.z

median.z

mode.z

max.z

min.z

var.z

sd.z

CV.z

D.z

ID.z

kurtosis.z

skewness.z

p.a.mean.z

p.a.mode.z

p.a.2m.z

p.b.mean.z

p.b.mode.z

p.b.2m.z

CRR.z

L2.z

L3.z

L4.z

L3.mu.z

L4.mu.z

L.CV.z

median.a.d.z

mode.a.d.z

weibull_c.z

weibull_b.z

mean.rho

mean.q.rho

mean.g.rho

mean.h.rho

median.rho

mode.rho

max.rho

min.rho

var.rho

sd.rho

CV.rho

D.rho

ID.rho

kurtosis.rho

skewness.rho

p.a.mean.rho

p.a.mode.rho

p.b.mean.rho

p.b.mode.rho

CRR.rho

L2.rho

L3.rho

L4.rho

L3.mu.rho

L4.mu.rho

L.CV.rho

median.a.d.rho

mode.a.d.rho

weibull_c.rho

weibull_b.rho

mean.r

mean.q.r

mean.g.r

mean.h.r

median.r

mode.r

max.r

min.r

var.r

sd.r

CV.r

D.r

ID.r

kurtosis.r

skewness.r

p.a.mean.r

p.a.mode.r

p.b.mean.r

p.b.mode.r

CRR.r

L2.r

L3.r

L4.r

L3.mu.r

L4.mu.r

L.CV.r

median.a.d.r

mode.a.d.r

weibull_c.r

weibull_b.r

1

10

9788.667

1796.620

444.6667

322.3520

1.1763271

4.149327

5.984327

7.555327

7.824327

7.932327

8.176327

8.646327

9.357327

10.00233

10.32033

10.68533

11.39733

12.03533

13.43333

8.686622

8.991724

8.185912

6.949452

8.601327

7.935327

15.11633

0.2503271

5.393706

2.322435

0.2673577

14.866

2.423

4.535413

-0.8176501

48.55133

69.29800

98.07203

51.44867

30.64412

1.927974

0.5746516

112023087

1088745309

10962245432

-1830560027

23849979309

1e-07

1.223295

0.7512949

4.218465

9.554482

4.455202

5.325242

3.203021

1.745464

3.977145

0.1000033

9.999991

0.1000033

8.509383

2.917085

0.6547593

9.899988

5.068693

1.782223

0.2812223

46.60337

99.99993

53.39663

0

0.4455206

39291647

289779246

2301164191

-235377229

1816420654

1e-07

2.562719

4.355199

1.560419

4.957027

10.17625

10.45033

9.878583

9.541126

10.03613

7.939957

18.04629

1.9854031

5.653233

2.377653

0.2336472

16.06089

3.486677

2.864220

0.0953509

48.15005

85.54384

51.84995

14.45609

0.5638970

151314734

1701009907

19924353576

-2918437931

44702069599

1e-07

1.765318

2.2362936

4.892407

11.09717

2

10

11222.000

2815.061

611.1667

506.5021

1.0425898

4.121590

5.898590

7.335590

7.912590

8.448590

9.110590

9.663590

10.223590

10.64959

10.79059

10.87959

11.37959

11.90259

13.43459

9.052322

9.383953

8.475137

7.020125

9.605590

10.788590

16.82859

0.2505898

6.114052

2.472661

0.2731521

16.578

2.933

4.501117

-1.0871433

60.10169

23.93897

97.82422

39.89831

75.97166

2.175783

0.5379133

101884570

1031348422

10753389699

-1735525894

23502278857

1e-07

1.586268

1.7362678

4.119804

9.970272

5.443356

6.210808

4.027388

1.859293

5.913727

0.1000035

9.999995

0.1000035

8.944021

2.990656

0.5494140

9.899991

5.164815

1.754274

-0.2467470

52.36865

99.99991

47.63135

0

0.5443358

44630617

347962930

2851615225

-380857693

3209731874

1e-07

2.607957

5.343352

1.892479

6.133511

11.06990

11.25312

10.857815

10.592588

10.95586

10.778054

18.13256

1.1552180

4.090119

2.022404

0.1826941

16.97734

2.092208

4.720474

-0.3191920

47.73584

57.02047

52.26416

42.97944

0.6104983

146515187

1723625155

20809992902

-3142095776

52214652344

1e-07

1.040121

0.2918422

6.395220

11.89079

3

10

7931.000

3522.606

644.6667

676.3987

0.7535283

3.119528

5.446528

7.362528

7.852528

8.331528

8.914528

9.612528

10.219528

10.81653

11.00853

11.29753

12.37553

13.04753

14.07753

9.177116

9.598251

8.419140

6.533724

9.612528

10.845528

15.92853

0.2505283

7.906972

2.811934

0.3064071

15.678

3.156

3.930470

-0.9284260

56.40392

29.20901

96.89650

43.59608

70.75913

3.103504

0.5761434

92236146

971093853

10589014798

-1568290062

21550099741

1e-07

1.685412

1.6684122

3.627730

10.180048

6.672340

6.885693

6.452606

6.234672

6.568161

1.0608273

9.999985

1.0608273

2.892645

1.700778

0.2548997

8.939158

3.089093

1.845899

0.1775212

48.04518

99.99990

51.95482

0

0.6672350

47469238

356252676

2796808210

-593939497

5968664998

1e-07

1.542566

5.611513

4.446252

7.316667

11.64407

11.81267

11.457447

11.243853

11.75383

1.180774

18.26309

1.1807736

3.954716

1.988647

0.1707862

17.08232

2.547107

3.769785

-0.2945041

52.92946

99.99990

47.07054

0.00000

0.6375741

139705385

1716628700

21577086613

-3163588038

55274110885

1e-07

1.274854

10.4633013

6.878014

12.45916

4

10

14815.833

2391.316

253.5000

228.6277

0.8293213

3.240321

5.588321

7.137321

7.584321

7.896321

8.796321

9.439321

9.999321

10.67532

11.14432

11.66732

13.05332

13.78632

15.08332

9.208836

9.676083

8.445007

6.697945

9.439321

9.490321

35.28932

0.2503213

8.823948

2.970513

0.3225720

35.039

3.560

3.513803

-0.5324590

54.14860

48.89439

97.14021

45.85140

51.07578

2.859793

0.2609525

28567998

308407516

3490877952

-480824942

6666412747

3e-07

1.767485

0.2814852

3.426340

10.246363

4.801007

5.401068

4.067086

3.284440

4.668175

0.6118197

9.999958

0.6118197

6.121887

2.474245

0.5153596

9.388139

3.940113

2.071634

0.2721649

47.95151

99.99967

52.04849

0

0.4801027

8901023

61927806

468289564

-66273596

510018428

5e-07

2.021859

4.189187

2.031767

5.418747

10.78393

11.08143

10.423270

9.950309

10.50525

9.212659

35.48690

2.2008590

6.504908

2.550472

0.2365066

33.28604

3.332113

3.510630

-0.3286589

44.53457

77.13968

55.46543

22.85999

0.3038850

37469021

445207961

5485056120

-766979859

12425003393

3e-07

1.618794

1.5712744

4.827566

11.76894

5

10

13582.333

4682.324

574.6667

445.8058

0.8972930

4.469293

7.796293

8.695293

9.044293

9.561293

10.514293

10.960293

11.478293

12.07729

12.46029

12.85829

13.79829

14.36229

15.39429

10.584187

10.963716

9.863618

7.785157

10.964293

10.945293

16.88629

0.2502930

8.178051

2.859729

0.2701888

16.636

3.426

5.038409

-1.1877711

58.91270

50.64919

97.66575

41.08730

49.31287

2.334252

0.6267916

151746390

1789588757

21722917667

-3028745442

47953862766

1e-07

1.718894

0.3611058

4.169715

11.649419

5.359885

6.080607

4.037565

2.053025

5.693991

0.1000073

9.999996

0.1000073

8.245425

2.871485

0.5357363

9.899989

4.844945

1.909141

-0.2794109

54.17449

99.99992

45.82551

0

0.5359887

46676336

353412044

2820930172

-397127026

3289545553

1e-07

2.451160

5.259878

1.946119

6.044389

12.22845

12.53702

11.850902

11.333547

12.47152

11.346734

18.43135

0.8785378

7.641773

2.764376

0.2260609

17.55282

4.173565

2.992009

-0.5124616

53.34442

63.06284

46.65558

36.93708

0.6634594

198422726

2648674843

36436151565

-4630530899

84906244333

1e-07

2.160321

0.8817201

5.072474

13.30724

6

10

6254.500

1674.809

448.0000

364.1714

0.6060312

3.035031

6.009031

8.205031

8.697031

9.233031

10.052031

10.662031

11.199031

11.60703

11.80803

12.04303

12.86603

13.40603

14.68603

9.948021

10.370678

9.078045

6.626844

10.662031

11.459031

16.45303

0.2500312

8.587836

2.930501

0.2945813

16.203

3.111

4.773276

-1.3195711

61.54501

33.64629

96.60452

38.45499

66.31770

3.395478

0.6046315

85730909

962580012

11098719111

-1595976775

23700818001

1e-07

1.688990

1.5110099

3.789572

11.008726

6.594068

6.949814

6.180656

5.719248

6.507049

0.9677901

9.999993

0.9677901

4.818198

2.195039

0.3328809

9.032203

3.993594

1.867432

-0.0993663

48.84616

99.99987

51.15384

0

0.6594072

38500784

303690374

2521537499

-457939362

4555795906

2e-07

1.978972

5.626277

3.308526

7.350166

12.25927

12.48402

11.988194

11.639040

12.45260

1.071134

18.74465

1.0711337

5.561109

2.358200

0.1923605

17.67352

2.645716

4.167097

-0.6340967

54.22377

99.99987

45.77623

0.00000

0.6540144

124231693

1625048934

21779500092

-2943918051

54116275529

1e-07

1.326611

11.1881392

6.047834

13.20862

The stand-level metrics are statistical descriptive values, such as percentiles, standard deviation or means, as well as the number of points belonging to the normal section of the point cloud.

The values `n.pts`, `n.pts.est`, `n.pts.red` and `n.pts.red.est` indicate the number of points and the estimated number of points respectively that belong to the tree’s normal section (1.3 m +/- 0.05 m). This is calculated for the original point cloud (`n.pts`, `n.pts.est`) and the reduced point cloud (`n.pts.red`, `n.pts.red.est`).

Furthermore, the height percentiles (`P01`, `P05`, `P10`, `P20`, `P25`, `P30`, `P40`, `P50`, `P60`, `P70`, `P75`, `P80`, `P90`, `P95`, `P99`, numbers indicating the k-th percentile) are calculated for the \(z\) coordinate of the TLS point cloud. Therefore they denominate the height (in m) above ground level.

#### Statistics of the z, rho and r

To describe the tendencies and distribution of the spherical coordinates `z`, \(\rho\) (horizontal distance, `rho`) and `r` (euclidean distance), the following statistics are calculated:

- arithmetic (`mean.arit.z/rho/r`), quadratic (`mean.qua.z/rho/r`), geometric (`mean.geom.z/rho/r`) and harmonic means (`mean.harm.z/rho/r`)
- median (`median.z/rho/r`)
- mode (`mode.z/rho/r`)
- variance (`var.z/rho/r`)
- standard deviation (`sd.z/rho/r`)
- coefficient of variation (`CV.z/rho/r`)
- range (`D.z/rho/r`)
- interquartile range (`ID.z/rho/r`)
- maximum (`max.z/rho/r`)
- minimum (`min.z/rho/r`)
- kurtosis (`kurtosis.z/rho/r`)
- skewness (`skewness.z/rho/r`)
- percentage of points above the arithmetic mean (`p.a.mean.z/rho/r`) and the mode (`p.a.mode.z/rho/r`)
- percentage of points below the arithmetic mean (`p.b.mean.z/rho/r`) and the mode (`p.b.mode.z/rho/r`)
- percentage of points above a height of 2 m (`p.a.2m.z`, only for the \(z\) coordinate)
- percentage of points below a height of 2 m (`p.b.2m.z`, only for the \(z\) coordinate)
- L-moments of order 2, 3 and 4 (`L2.z/rho/r`, `L3.z/rho/r` and `L4.z/rho/r`)
- third and forth central L-moments (`L3.mu.z/rho/r` and `L4.mu.z/rho/r`)
- ratio of L1 and L2 moments (`L.CV.z/rho/r`)
- median of the absolute deviation from the overall median (`median.a.d.z/rho/r`)
- mode of the absolute deviation from the overall mode (`mode.a.d.z/rho/r`)
- Canopy relief ratio i.e., the ration of `mean.z/rho/r` to `max.z/rho/r` (`CRR.z/rho/r`)
- scale and shape parameters of the Weibull distribution fitted for the spherical coordinates (`weibull_c.z/rho/r`, `weibull_b.z/rho/r`)

### Stand-level variables

    metrics[1:6, 1:36]

id | radius | N.tls | N.hn | N.hr | N.hn.cov | N.hr.cov | N.sh | G.tls | G.hn | G.hr | G.hn.cov | G.hr.cov | G.sh | V.tls | V.hn | V.hr | V.hn.cov | V.hr.cov | V.sh | d.tls | dg.tls | dgeom.tls | dharm.tls | h.tls | hg.tls | hgeom.tls | hharm.tls | d.0.tls | dg.0.tls | dgeom.0.tls | dharm.0.tls | h.0.tls | hg.0.tls | hgeom.0.tls | hharm.0.tls
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
1 | 10 | 222.8169 | 241.4554 | 237.3396 | 239.9341 | 234.2128 | 230.5601 | 12.83139 | 13.90472 | 13.66770 | 13.81712 | 13.48764 | 13.27729 | 72.26462 | 78.30949 | 76.97464 | 77.81611 | 75.96057 | 74.77589 | 26.63218 | 27.07807 | 26.17896 | 25.72557 | 11.98963 | 12.02086 | 11.95768 | 11.92508 | 31.50910 | 31.60757 | 31.40772 | 31.30362 | 12.77989 | 12.78591 | 12.77382 | 12.76769
2 | 10 | 350.1409 | 379.4298 | 372.9622 | 374.9814 | 366.2799 | 364.2843 | 18.91537 | 20.49762 | 20.14823 | 20.25731 | 19.78724 | 19.67943 | 101.80664 | 110.32268 | 108.44214 | 109.02925 | 106.49921 | 105.91896 | 25.75150 | 26.22655 | 25.29375 | 24.86175 | 11.59380 | 11.66199 | 11.51966 | 11.43902 | 32.68378 | 32.75790 | 32.60870 | 32.53280 | 12.41950 | 12.44332 | 12.39576 | 12.37212
3 | 10 | 381.9719 | 413.9235 | 406.8678 | 414.2445 | 403.9974 | 392.6248 | 23.55858 | 25.52924 | 25.09407 | 25.54904 | 24.91703 | 24.21562 | 115.70695 | 125.38573 | 123.24843 | 125.48298 | 122.37892 | 118.93393 | 27.86976 | 28.02296 | 27.71227 | 27.55093 | 11.38201 | 11.44246 | 11.32332 | 11.26633 | 31.17303 | 31.20949 | 31.13698 | 31.10141 | 11.78541 | 11.93916 | 11.62884 | 11.47255
4 | 10 | 254.6479 | 275.9490 | 271.2452 | 274.7699 | 268.1497 | 265.5833 | 22.15673 | 24.01012 | 23.60084 | 23.90752 | 23.33151 | 23.10820 | 124.40696 | 134.81348 | 132.51548 | 134.23744 | 131.00320 | 129.74938 | 32.71822 | 33.28417 | 32.14873 | 31.58609 | 12.69636 | 12.81602 | 12.56975 | 12.43597 | 38.69937 | 38.87412 | 38.53131 | 38.37101 | 12.55974 | 12.57650 | 12.54186 | 12.52257
5 | 10 | 318.3099 | 344.9362 | 339.0565 | 360.7488 | 349.0381 | 334.1288 | 21.34794 | 23.13368 | 22.73934 | 24.19417 | 23.40878 | 22.40886 | 117.11494 | 126.91149 | 124.74819 | 132.72937 | 128.42070 | 122.93514 | 28.64160 | 29.22187 | 27.98239 | 27.23715 | 12.77420 | 12.99329 | 12.47878 | 12.08340 | 34.98638 | 35.05872 | 34.91178 | 34.83518 | 13.39910 | 13.40467 | 13.39359 | 13.38812
6 | 10 | 254.6479 | 275.9490 | 271.2452 | 276.1346 | 269.3076 | 260.6298 | 15.96468 | 17.30011 | 17.00522 | 17.31175 | 16.88375 | 16.33971 | 89.59139 | 97.08563 | 95.43073 | 97.15093 | 94.74903 | 91.69596 | 28.06921 | 28.25304 | 27.88269 | 27.69453 | 12.82381 | 12.84689 | 12.80020 | 12.77611 | 31.41186 | 31.44841 | 31.37513 | 31.33830 | 12.79379 | 12.81666 | 12.77130 | 12.74923

The variables are estimated based on the tree attributes of the detected trees from the point cloud data (output of the functions `tree.deteection.single.scan` and `tree.deteection.multi.scan`) similar to the procedure of conventional forest inventories. The values are computed at stand-level and are therefore extended to an area of one ha.

#### Stand density (N), volume (V) and basal area (G)

The stand density (`N.tls`, trees/ha) is calculated by the following equations. For the circular fixed area and k-tree plots, `N.tls` is calculated by

\[ N.tls=\frac{10000}{\pi R^2}\cdot n \]

with \(R\) being the radius of the plot (in m) and \(n\) the number of detected trees. The density of angle-count plots is calculated by

\[ N.tls=\sum_{i=1}^{n} \frac{BAF}{g_i} \]

with \(BAF\) being the basal area factor (in m\(^2\)/ha) and \(g_i\) the basal area of the tree \(i\) (in m\(^2\)).

The basal area (`G.tls`, in m\(^2\)/ha) is estimated for circular fixed area and k-tree plots by the following equation:

\[ G.tls=\frac{10000}{\pi R^2}\sum_{i=1}^{n} g_i \]

and for angle-count plots, the following equation is applied:

\[ G.tls=BAF \cdot n \]

The stem volume (`V.tls`, in m\(^3\)/ha) is estimated by modelling the stem profile as a paraboloid. The volume is calculated as the volume of the revolution of the paraboloid function. The total heights of the detected trees are estimated as the 99th percentile of the points of the \(z\) coordinate delimited by Voronoi polygons (\(h_{P_{99}i}\), in m). The equation used for the calculation for circular fixed area and k-tree plots is

\[ V.tls=\frac{10000}{\pi R^2}\sum_{i=1}^{n} \pi \cdot \frac{h_{P_{99}i}^2}{2} \cdot \frac{(\frac{1}{2} \cdot dbh_i)^2}{(h_{P_{99}i} - 1.3)^2} \]

and for angle-count plots is

\[ V.tls=\sum_{i=1}^{n} \frac{BAF}{g_i} \cdot \pi \cdot \frac{h_{P_{99}i}^2}{2} \cdot \frac{(\frac{1}{2} \cdot dbh_i)^2}{(h_{P_{99}i} - 1.3)^2} \]

#### Occlusion correction

The above mentioned calculations of the stand-level variables \(N\), \(V\) and \(G\) do not consider possible occlusions of trees. Therefore, different occlusion correction approaches are included in the function.

For angle-count plots, the occlusion correction is based on the Poisson attenuation model. This approach calculates geometric gap probabilities which decreases with increasing distance from the TLS device and follow a Poisson distribution. The calculations of stand density, volume and basal area are corrected with a factor accounting for the gap probabilities and the corrected variables are called `N.pam`, `V.pam` and `G.pam` respectively. For further details see Strahler et al. (20081) and Lovell et al. (20112).

For the other two plot designs (circular fixed area and k-tree plots) different occlusion correction approaches are applied. One approach is based on distance sampling data and can therefore only be used when the function `distance.sampling` was used (i.e., when the `tree.ds` argument is specified other than `NULL`). Functions based on point transect sampling that describe how the probability of tree detection decreases with increasing distance from the TLS device are used for the calculation. These functions are applied for the calculation of the variables `N.hn`/`V.hn`/`G.hn` and `N.hr`/`V.hr`/`G.hr` and are Half-Normal and Hazard-Rate respectively. Additionally, `N.hn.cov`/`V.hn.cov`/`G.hn.cov` and `N.hr.cov`/`V.hr.cov`/`G.hr.cov` are calculated with expanding the scale component of the function with \(dbh\) as covariate.

The other approach corrects shadowing effects. Tthis method calculates an expansion factor to correct the variables. The expansion factor is based on the percentage of the shaded area i.e., unsample area which is not seen from the TLS device due to masking by trees. The corrected calculated variables are called `N.sh`/`V.sh`/`G.sh`. For more details see Seidel and Ammer (20143).

#### Mean and dominant heights (h) and diameters (d)

The mean heights (in m) and diameters (in cm) are calculated by the arithmetic (`h.tls`, `d.tls`), quadratic (`h.g`, `d.g`), geometric (`h.geom`, `d.geom`) and harmonic means (`h.harm`, `d.harm`). To calculate the dominant heights and diameters, only the \(n\) largest trees per hectar are considered. If not otherwise specified in the argument `num.tree` (see above), the number of dominant trees per hectare is set to 100 trees/ha. Dominant heights and diameters are also calculated as the arithmetic (`h.0.tls`, `d.0.tls`), quadratic (`h.0.g`, `d.0.g`), geometric (`h.0.geom`, `d.0.geom`) and harmonic means (`h.0.harm`, `d.0.harm`).

1. Strahler, A.H., Jupp, D.L.B., Woodcock, C.E., Schaaf, C.B., Yao, T., Zhao, F., Yang, X., Lovell, J., Culvenor, D., Newnham, G., Ni-Miester, W., Boykin-Morris, W., 2008. Retrieval of forest structural parameters using a ground-based lidar instrument (Echidna®). Can. J. Rem. Sens. 34 (Suppl. 2), S426–S440. https://doi.org/10.5589/m08-046.↩︎

2. Lovell, J.L., Jupp, D.L.B., Newnham, G.J., Culvenor, D.S., 2011. Measuring tree stem diameters using intensity profiles from ground-based scanning lidar from a fixed viewpoint. ISPRS J. Photogrammetry Remote Sens. 66 (1), 46–55. https://doi.org/10.1016/j.isprsjprs.2010.08.006.↩︎

3. Seidel, D., Ammer, C., 2014. Efficient measurements of basal area in short rotation forests based on terrestrial laser scanning under special consideration of shadowing. iFor. Biogeosci. For. 7 (4), 227–232. https://doi.org/10.3832/ifor1084-007.↩︎