# Prediction of Site Index and Age Using Time Series of TanDEM-X Phase Heights

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{SI}). The top height is defined as the mean height of the 100 trees with the largest diameter at breast height per hectare. This definition of top height is sometimes called H100, and is meant to represent the upper height of tree crowns in the forest. Top height has successfully been estimated using different remote sensors. Examples include estimating top height with the maximum airborne laser scanning (ALS) canopy model height in a 10 m × 10 m window, or the maximum height in 500 m

^{2}plots of an aerial stereo-image-based canopy height model (CHM) [3,4].

^{2}plots. Kandare et al. [6] used an individual tree crown (ITC) approach for predicting SI in boreal forests using airborne laser scanning (ALS) and hyperspectral data. They estimated the age, height, and diameter at breast height of the dominant trees from ALS and hyperspectral metrics. These were then used in age-height curves to predict SI. When predicting both SI and age, the method by Kandare et al. achieved RMSEs of 4.3 m and 34 years, respectively. When the age from field data was used in the prediction, the RMSE of SI predictions dropped to 1.18 m [6]. Solberg et al. used age-independent equations of top height growth and single tree ALS data to predict SI by matching single dominant trees in repeated ALS measurements six years apart [7]. They estimated SI values very close to field-based values for individual sample trees (bias 0.27 m, RMSE about 2.8 m, as interpreted from a figure). Penner et al. [8] used two successive ALS collections, acquired 13 years apart, to estimate SI with an RMSE of 2.5 m and a bias of 0.3 m on 400 m

^{2}field plots.

^{2}plots. They used TanDEM-X image pairs from three growth seasons calibrated using ALS data. Persson and Fransson [11] used four TanDEM-X acquisitions covering three growth seasons, calibrated using ALS data or Lorey’s height from field data. They predicted SI with 4.4 m RMSE and age with 17.8 years RMSE on 0.5 ha plots. The need for calibration, however, hampers the scalability of the methods, as it relies on local high-resolution ALS data or field data. Furthermore, the usefulness of calibration data decreases with the time between data collection and the TanDEM-X acquisition date due to forest growth and other changes. Because of this, longer time series may often need calibration data from multiple time points.

## 2. Materials and Methods

#### 2.1. Test Site and Field Data

#### 2.2. SAR Data

#### 2.3. Top Height Estimation

#### 2.4. Site Index

_{2}at stand age A

_{2}, given a measured top height H

_{1}at stand age A

_{1}:

_{2}are previously determined tree species-specific fixed parameters, and A

_{SI}is the reference age. Equation (4) is commonly used to calculate SI given field measurements of top height and age, as was done with the field data in this study. By setting A

_{2}to the preferred SI reference age and H

_{1}and A

_{1}to the measured height and age, H

_{2}equals the SI. The HDCs were developed from multiple measurements on sets of field plots in even-aged forests, the predominant forest type in Sweden, and are therefore valid in such forests.

_{1}is set to a specific SI value instead of a measured height and A

_{1}is set to the corresponding reference age, H

_{2}gives the expected top height at any age A

_{2}. For illustration, the resulting HDCs of Norway spruce for a few values of SI are shown in Figure 1.

#### 2.5. Site Index Estimation

_{1}and A

_{1}in Equation (4) to SI and the corresponding reference age, respectively, and substituting A

_{0}+ GP (growth period) for A

_{1}, allows us to express TanDEM-X top height H as a function of SI and GP, explicitly

_{0}were determined by applying a weighted non-linear least squares regression of Equation (7) to the time series of TanDEM-X top heights, leaving initial age (age at the time of the first TanDEM-X measurement, A

_{0}) and/or SI as parameters. The function was fit to each field plot using dominant species information from the field data to select the correct fixed parameters, and two different prediction cases were applied; (a) estimates of both SI and A

_{0}for each plot, and (b) estimates of only SI, assuming that the initial age is known. In case (b), A

_{0}in the fitting was supplied from the field data. Figure 2 illustrates prediction case (a). In this figure, the process of fitting A

_{0,}can be considered as a translation in time of the time series of TanDEM-X top heights to find the optimal fit, while the fitting of SI corresponds to the choice of optimal curve out of the family defined by Equation (7).

_{0}(in prediction case (a)) were initialized to 25 and 75 and constrained to the intervals [4, 60] and [4, 200], respectively. This algorithm was chosen because of the possibility of setting bounds for the parameters. Otherwise, the algorithm tended to diverge or produce implausible parameter values for plots where TanDEM-X phase heights decreased over time. In case the fitting did not converge, it was restarted and initialized using the parameter values obtained in the non-converging fit.

_{0}predicted by parameter estimation were visually inspected via plots of the fitted HDC alongside the TanDEM-X top heights and the HDC expected from the field-data-based SI and age. The quality of predictions of A

_{0}and SI were evaluated by comparisons with the corresponding field-data-based values, and prediction results were further visually evaluated through plots to investigate possible correlations between prediction errors and SI, stand age, species, or treatment groups. The Root Mean Square Error (RMSE) and bias were calculated for each treatment group k as

_{k}is the number of field plots in group k. Additionally, the coefficient of determination between predictions and reference values was calculated.

## 3. Results

#### 3.1. Predicting Both SI and Stand Age

_{0}, the RMSEs and biases (as defined in Equations (8) and (9)) tended to increase with the intensity (in terms of expected relative biomass reduction) of the treatment. Table 2 shows the evaluation results for this case. For the untreated plots, the predicted SI had an RMSE of 6.9 m. The RMSE of pre-commercially thinned plots was 9.5 m, increasing to 16.1 m for the thinned group. The clear-cut group, however, had a slightly lower RMSE of 13.2 m.

^{2}of the clear-cut plots is not reported, as it yields no information with only two observations. A plot of the field measured vs. predicted SI is shown in Figure 3.

_{0}had an RMSE of 38 years for the untreated group, 22 years for the pre-commercially thinned group, and then increased with the intensity of treatment to 82 for the thinned group and 137 years for the clear-cut plots.

^{2}between predicted and reference age are similar to those for SI, 0.4 for untreated plots, and decreasing with treatment intensity. The R

^{2}of the clear-cut plots is not reported, as it yields no information for only two observations.

#### 3.2. Predicting SI Assuming Known Age

^{2}, between true and predicted SI were relatively high, with R

^{2}values between 0.6 and 0.8. A plot of the field measured vs. predicted SI is shown in Figure 7.

#### 3.3. Error Characteristics

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Roach, W.J.; Simard, S.W.; Defrenne, C.E.; Pickles, B.J.; Lavkulich, L.M.; Ryan, T.L. Tree Diversity, Site Index, and Carbon Storage Decrease With Aridity in Douglas-Fir Forests in Western Canada. Front. For. Glob. Change
**2021**, 4, 682076. [Google Scholar] [CrossRef] - Skovsgaard, J.P.; Vanclay, J.K. Forest Site Productivity: A Review of the Evolution of Dendrometric Concepts for Even-Aged Stands. For. Int. J. For. Res.
**2008**, 81, 13–31. [Google Scholar] [CrossRef] - Kugler, F.; Schulze, D.; Hajnsek, I.; Pretzsch, H.; Papathanassiou, K.P. TanDEM-X Pol-InSAR Performance for Forest Height Estimation. IEEE Trans. Geosci. Remote Sens.
**2014**, 52, 6404–6422. [Google Scholar] [CrossRef] - Stepper, C.; Straub, C.; Pretzsch, H. Assessing Height Changes in a Highly Structured Forest Using Regularly Acquired Aerial Image Data. For. Int. J. For. Res.
**2015**, 88, 304–316. [Google Scholar] [CrossRef] - Véga, C.; St-Onge, B. Mapping Site Index and Age by Linking a Time Series of Canopy Height Models with Growth Curves. For. Ecol. Manag.
**2009**, 257, 951–959. [Google Scholar] [CrossRef] - Kandare, K.; Ørka, H.O.; Dalponte, M.; Næsset, E.; Gobakken, T. Individual Tree Crown Approach for Predicting Site Index in Boreal Forests Using Airborne Laser Scanning and Hyperspectral Data. Int. J. Appl. Earth Obs. Geoinf.
**2017**, 60, 72–82. [Google Scholar] [CrossRef] - Solberg, S.; Kvaalen, H.; Puliti, S. Age-Independent Site Index Mapping with Repeated Single-Tree Airborne Laser Scanning. Scand. J. For. Res.
**2019**, 34, 763–770. [Google Scholar] [CrossRef] - Penner, M.; Woods, M.; Bilyk, A. Assessing Site Productivity via Remote Sensing—Age-Independent Site Index Estimation in Even-Aged Forests. Forests
**2023**, 14, 1541. [Google Scholar] [CrossRef] - Persson, H.J.; Fransson, J.E.S. Analysis of Tree Height Growth with TanDEM-X Data. In Proceedings of the 35th EARSeL Symposium, Stockholm, Sweden, 15–19 June 2015; Volume 1, pp. 1–6. [Google Scholar]
- Wallerman, J.; Nyström, K.; Bohlin, J.; Persson, H.J.; Soja, M.J.; Fransson, J.E.S. Estimating Forest Age and Site Productivity Using Time Series of 3D Remote Sensing Data. In Proceedings of the 2015 IEEE International Geoscience and Remote Sensing Symposium (IGARSS), Milan, Italy, 26–31 July 2015; pp. 3321–3324. [Google Scholar]
- Persson, H.J.; Fransson, J.E.S. Estimating Site Index From Short-Term TanDEM-X Canopy Height Models. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens.
**2016**, 9, 3598–3606. [Google Scholar] [CrossRef] - Fiedler, H.; Krieger, G.; Zink, M.; Younis, M.; Bachmann, M.; Huber, S.; Hajnsek, I.; Moreira, A. The TanDEM-X Mission: An Overview. In Proceedings of the 2008 International Conference on Radar, Adelaide, Australia, 2–5 September 2008; pp. 60–64. [Google Scholar]
- Krieger, G.; Zink, M.; Bachmann, M.; Bräutigam, B.; Schulze, D.; Martone, M.; Rizzoli, P.; Steinbrecher, U.; Walter Antony, J.; De Zan, F.; et al. TanDEM-X: A Radar Interferometer with Two Formation-Flying Satellites. Acta Astronaut.
**2013**, 89, 83–98. [Google Scholar] [CrossRef] - Karila, K.; Vastaranta, M.; Karjalainen, M.; Kaasalainen, S. Tandem-X Interferometry in the Prediction of Forest Inventory Attributes in Managed Boreal Forests. Remote Sens. Environ.
**2015**, 159, 259–268. [Google Scholar] [CrossRef] - Persson, H.J.; Olsson, H.; Soja, M.J.; Ulander, L.M.H.; Fransson, J.E.S. Experiences from Large-Scale Forest Mapping of Sweden Using TanDEM-X Data. Remote Sens.
**2017**, 9, 1253. [Google Scholar] [CrossRef] - Persson, H.J.; Fransson, J.E.S. Comparison between TanDEM-X- and ALS-Based Estimation of Aboveground Biomass and Tree Height in Boreal Forests. Scand. J. For. Res.
**2017**, 32, 306–319. [Google Scholar] [CrossRef] - Chen, H.; Cloude, S.R.; Goodenough, D.G. Forest Canopy Height Estimation Using Tandem-X Coherence Data. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens.
**2016**, 9, 3177–3188. [Google Scholar] [CrossRef] - Kugler, F.; Hajnsek, I. Forest Characterisation by Means of TerraSAR-X and TanDEM-X (Polarimetric and) Interferometric Data. In Proceedings of the 2011 IEEE International Geoscience and Remote Sensing Symposium, Vancouver, BC, Canada, 24–29 July 2011; pp. 2578–2581. [Google Scholar]
- Olesk, A.; Praks, J.; Antropov, O.; Zalite, K.; Arumäe, T.; Voormansik, K. Interferometric SAR Coherence Models for Characterization of Hemiboreal Forests Using TanDEM-X Data. Remote Sens.
**2016**, 8, 700. [Google Scholar] [CrossRef] - Praks, J.; Antropov, O.; Hallikainen, M.T. LIDAR-Aided SAR Interferometry Studies in Boreal Forest: Scattering Phase Center and Extinction Coefficient at X- and L-Band. IEEE Trans. Geosci. Remote Sens.
**2012**, 50, 3831–3843. [Google Scholar] [CrossRef] - Schlund, M.; Magdon, P.; Eaton, B.; Aumann, C.; Erasmi, S. Canopy Height Estimation with TanDEM-X in Temperate and Boreal Forests. Int. J. Appl. Earth Obs. Geoinf.
**2019**, 82, 101904. [Google Scholar] [CrossRef] - Soja, M.J.; Persson, H.J.; Ulander, L.M.H. Modeling and Detection of Deforestation and Forest Growth in Multitemporal TanDEM-X Data. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens.
**2018**, 11, 3548–3563. [Google Scholar] [CrossRef] - Schlund, M.; Kukunda, C.B.; Baumann, S.; Wessel, B.; Kiefl, N.; von Poncet, F. Potential of Forest Monitoring with Multi-Temporal TANDEM-X Height Models. In Proceedings of the IGARSS 2020–2020 IEEE International Geoscience and Remote Sensing Symposium, Waikoloa, HI, USA, 26 September–2 October 2020; pp. 308–311. [Google Scholar]
- Huuva, I.; Persson, H.J.; Wallerman, J.; Fransson, J.E.S. Detectability of Silvicultural Treatments in Time Series of Penetration Depth Corrected Tandem-X Phase Heights. In Proceedings of the IGARSS 2022–2022 IEEE International Geoscience and Remote Sensing Symposium, Kuala Lumpur, Malaysia, 17–22 July 2022; pp. 5909–5912. [Google Scholar]
- Solberg, S.; May, J.; Bogren, W.; Breidenbach, J.; Torp, T.; Gizachew, B. Interferometric SAR DEMs for Forest Change in Uganda 2000–2012. Remote Sens.
**2018**, 10, 228. [Google Scholar] [CrossRef] - Solberg, S.; Næsset, E.; Gobakken, T.; Bollandsås, O.-M. Forest Biomass Change Estimated from Height Change in Interferometric SAR Height Models. Carbon Balance Manag.
**2014**, 9, 5. [Google Scholar] [CrossRef] - Askne, J.I.H.; Persson, H.J.; Ulander, L.M.H. Biomass Growth from Multi-Temporal TanDEM-X Interferometric Synthetic Aperture Radar Observations of a Boreal Forest Site. Remote Sens.
**2018**, 10, 603. [Google Scholar] [CrossRef] - Goldstein, R.M.; Werner, C.L. Radar Interferogram Filtering for Geophysical Applications. Geophys. Res. Lett.
**1998**, 25, 4035–4038. [Google Scholar] [CrossRef] - Dall, J. InSAR Elevation Bias Caused by Penetration Into Uniform Volumes. IEEE Trans. Geosci. Remote Sens.
**2007**, 45, 2319–2324. [Google Scholar] [CrossRef] - Schlund, M.; Baron, D.; Magdon, P.; Erasmi, S. Canopy Penetration Depth Estimation with TanDEM-X and Its Compensation in Temperate Forests. ISPRS J. Photogramm. Remote Sens.
**2019**, 147, 232–241. [Google Scholar] [CrossRef] - Elfving, B.; Kiviste, A. Construction of Site Index Equations for Pinus Sylvestris L. Using Permanent Plot Data in Sweden. For. Ecol. Manag.
**1997**, 98, 125–134. [Google Scholar] [CrossRef] - Eriksson, H.; Johansson, U.; Kiviste, A. A Site-index Model for Pure and Mixed Stands of Betula Pendula and Betula Pubescens in Sweden. Scand. J. For. Res.
**1997**, 12, 149–156. [Google Scholar] [CrossRef] - Johansson, U.; Ekö, P.M.; Elfving, B.; Johansson, T.; Nilsson, U. Rön Från Sveriges Lantbruksuniversitet; Swedish University of Agricultural Sciences: Uppsala, Sweden, 2014. [Google Scholar]
- R Core Team. R: A Language and Environment for Statistical Computing; R Core Team: Vienna, Austria, 2013. [Google Scholar]
- Dennis, J.E.; Gay, D.M.; Welsch, R.E. Algorithm 573: NL2SOL—An Adaptive Nonlinear Least-Squares Algorithm [E4]. ACM Trans. Math. Softw.
**1981**, 7, 369–383. [Google Scholar] [CrossRef] - Li, F.K.; Goldstein, R.M. Studies of Multibaseline Spaceborne Interferometric Synthetic Aperture Radars. IEEE Trans. Geosci. Remote Sens.
**1990**, 28, 88–97. [Google Scholar] [CrossRef] - Vastaranta, M.; Niemi, M.; Wulder, M.A.; White, J.C.; Nurminen, K.; Litkey, P.; Honkavaara, E.; Holopainen, M.; Hyyppä, J. Forest Stand Age Classification Using Time Series of Photogrammetrically Derived Digital Surface Models. Scand. J. For. Res.
**2016**, 31, 194–205. [Google Scholar] [CrossRef]

**Figure 2.**An illustration of the prediction of initial age A

_{0}and SI by fitting a HDC to a time series of TanDEM-X top heights. The fitting of A

_{0,}indicated by a dashed line in the figure, can be thought of as horizontal translation of the time series of data points, while the fitting of SI corresponds to the choice of curve.

**Figure 3.**Plot of Predicted vs. field measured SI, obtained via simultaneous prediction of age. Colored by treatment. In the bottom left panel, thinned plots are shown in blue, while clear-cut plots are shown in red.

**Figure 7.**Predicted vs. field measured SI, using field measured ages in the fitting. Colored by treatment. In the bottom left panel, thinned plots are shown in blue, while clear-cut plots are shown in red.

**Figure 8.**Time series untreated pine dominated plot 169. TanDEM-X top heights superimposed with predicted HDC and reference HDC. Predicted SI: 39.2, field SI: 36.1 predicted age: 32.2 field age: 43.

Treatment | Top Height [m] Min/Mean/Max | Age [Years] Min/Mean/Max | SI Min/Mean/Max | n |
---|---|---|---|---|

Untreated | 14/25/32 | 25/52/140 | 13/34/45 | 26 |

Pre-commercially thinned | 14/20/28 | 15/25/50 | 27/35/44 | 7 |

Thinned | 12/21/32 | 20/35/105 | 16/38/50 | 45 |

Clear-cut | 25/26/28 | 60/70/80 | 30/30/30 | 2 |

Undocumented | 15/24/29 | 25/49/96 | 16/34/40 | 11 |

**Table 2.**Treatment-wise summary statistics of SI and age predictions from HDC fitting to time series of TanDEM-X phase heights. P-c thinned = Pre-commercially thinned.

Treatment | SI RMSE [m] | SI R^{2} | SI Bias [m] | Age RMSE [Years] | Age R^{2} | Age Bias [Years] | n |
---|---|---|---|---|---|---|---|

Untreated | 6.9 | 0.46 | −1.6 | 38 | 0.406 | 9.6 | 26 |

P-c thinned | 9.5 | 0.24 | −7.8 | 22 | 0.321 | 11.9 | 7 |

Thinned | 16.1 | 0.06 | −12.6 | 82 | 0.015 | 52.5 | 45 |

Clear-cut | 13.2 | - | −13.2 | 137 | - | 136.8 | 2 |

Treatment | SI RMSE [m] | SI R^{2} | SI Bias [m] | n |
---|---|---|---|---|

Untreated | 4.0 | 0.80 | −0.8 | 26 |

P-c thinned | 5.3 | 0.63 | −1.21 | 7 |

Thinned | 3.3 | 0.73 | 0.47 | 45 |

Clear-cut | 2.2 | — | −1.99 | 2 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Huuva, I.; Wallerman, J.; Fransson, J.E.S.; Persson, H.J.
Prediction of Site Index and Age Using Time Series of TanDEM-X Phase Heights. *Remote Sens.* **2023**, *15*, 4195.
https://doi.org/10.3390/rs15174195

**AMA Style**

Huuva I, Wallerman J, Fransson JES, Persson HJ.
Prediction of Site Index and Age Using Time Series of TanDEM-X Phase Heights. *Remote Sensing*. 2023; 15(17):4195.
https://doi.org/10.3390/rs15174195

**Chicago/Turabian Style**

Huuva, Ivan, Jörgen Wallerman, Johan E. S. Fransson, and Henrik J. Persson.
2023. "Prediction of Site Index and Age Using Time Series of TanDEM-X Phase Heights" *Remote Sensing* 15, no. 17: 4195.
https://doi.org/10.3390/rs15174195