Canopy Tree Height Estimation Using Quantile Regression: Modeling and Evaluating Uncertainty in Remote Sensing

Karsten Schrödter; Jan Pauls; Fabian Gieseke

cs.CVApr 8, 2026

Canopy Tree Height Estimation Using Quantile Regression: Modeling and Evaluating Uncertainty in Remote Sensing

Authors: Karsten Schrödter, Jan Pauls, Fabian Gieseke

Organizations: University of Münster · University of Copenhagen

Abstract

Accurate tree height estimation is vital for ecological monitoring and biomass assessment. We apply quantile regression to existing tree height estimation models based on satellite data to incorporate uncertainty quantification. Most current approaches for tree height estimation rely on point predictions, which limits their applicability in risk-sensitive scenarios. In this work, we show that, with minor modifications of a given prediction head, existing models can be adapted to provide statistically calibrated uncertainty estimates via quantile regression. Furthermore, we demonstrate how our results correlate with known challenges in remote sensing (e.g., terrain complexity, vegetation heterogeneity), indicating that the model is less confident in more challenging conditions.

Figures & tables

Panel of Figure 1 — Figure 1: Tree height estimation: An input image (a) can be used to produce a tree height map (b) using a tree height model (intensity of green increases with higher tree height estimation). Most studies focus on point estimates. In Figure (c), associated uncertainty estimates are provided (intensity of red increases with higher uncertainty). It can be seen that the model is less confident at forest borders and along cutoff stripes.

Model	MSE $\downarrow$	MAE $\downarrow$	$R^{2}$ $\uparrow$	$EC$
Lang et al.	38.32	4.25	0.55	0.47
Pauls et al.	20.64	1.90	0.76	0.58
Ours	20.81	1.90	0.76	0.50

Table 1: Results on point-estimators, measured using Mean Squared Error (MSE,

{\mathrm{m}}^{2}

), Mean Absolute Error (MAE,

\mathrm{m}

R^{2}

and Empirical Coverage (EC).

Metric	MPIW					PICP
$\alpha$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
Lang et. al.	5.54	6.87	8.39	10.22	12.76	0.37	0.45	0.52	0.58	0.64
Ours	2.48	3.07	3.79	4.77	6.47	0.48	0.58	0.67	0.78	0.88

Table 2: Comparison of Mean Prediction Interval Width (MPIW) and Prediction Interval Coverage Probability (PICP) for multiple uncertainty levels

\alpha

for Lang et al. ’s model and our model.

Figure 2: Empirical Coverage vs Nominal Quantile for Lang et al. ’s and our model.

Figure 3: Empirical Coverage vs Nominal Quantile for multiple target bins for our model.

Figure 4: Boxplot of distance of quantile predictions to point-estimate, i.e. median prediction. Per confidence level

\alpha\in[0.5,0.6,0.7,0.8,0.9]

, the prediction interval at level

\alpha

is written as

PIW_{\alpha}=[\hat{y}_{low},\hat{y}_{high}]\subset\mathbb{R}

, while the median prediction is denoted

\hat{y}\in\mathbb{R}

Panel of Figure 5 — Figure 5: Scatter plot of GEDI height against predicted height with color dependent on the prediction interval width at level $\alpha=0.8$ . All heights are clipped into $[$ 0\text{,}\mathrm{m} $,$ 50\text{,}\mathrm{m} $]$ . Left: Lang et al. ’s model. Right: Our model.

Figure 6: Noisy GEDI labels marked in red: $90\%$ -quantile prediction below $10\text{\,}\mathrm{m}$ and GEDI labels above $30\text{\,}\mathrm{m}$ . Left: Satellite image, Middle: $90\%$ -quantile canopy height prediction, Right: GEDI labels

Figure 7: Boxplot of Prediction Interval Width at level

\alpha=0.8

in dependence of being at a forest border. Note that less than

5\%

of the pixels are located at forest borders.

Figure 8: Boxplot of Prediction Interval Width at level

\alpha=0.8

in dependence of average slope within a

140\text{\,}\mathrm{m}

surrounding.

Figure 9: Sample satellite image with point estimates and $80\%$ Prediction Interval Width of Lang et al. ’s model and our model. Left: Satellite image (yearly median composite). Top Right: Point-estimates of canopy height. Bottom Right: $80\%$ Prediction Inverval Width.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 10 — Figure 10: Architecture of the 3D UNet. (a) Each Encoder Layer applies MaxPool3d for $2{\times}$ spatial downsampling followed by a DoubleConv block (two Conv3d + GroupNorm + ReLU); the initial DoubleConv omits the MaxPool. Each Decoder Layer applies ConvTranspose3d for $2{\times}$ spatial upsampling, concatenates skip features (dashed), then applies a DoubleConv block. (b) Overall architecture. Encoder features pass through Temporal Aggregation Conv3d layers with temporal-only kernels $(T{,}1{,}1)$ that collapse the time dimension to 1 before entering the decoder as skip connections. The solid path at the bottom shows the bottleneck flow. Tensor shapes are shown in magenta as $C\times T\times H\times W$ .

Model	MSE $\downarrow$	MAE $\downarrow$	R2 $\uparrow$	$EC_{0.5}$
Lang et. al.	38.32	4.25	0.55	0.47
Pauls et. al.	20.64	1.90	0.76	0.58
Gaussian w/o Shift	22.11	2.17	0.74	0.58
Gaussian w/ Shift	22.54	2.17	0.74	0.58
Log Gaussian w/o Shift	21.86	2.05	0.74	0.54
Log Gaussian w/ Shift	21.86	2.04	0.74	0.54

Appendix

Table 3: Results on point-estimators, measured using Mean Squared Error (MSE,

{\mathrm{m}}^{2}

), Mean Absolute Error (MAE,

\mathrm{m}

R^{2}

and Empirical Coverage (EC).

Metric	MPIW					PICP
$\alpha$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
Lang et. al.	5.54	6.87	8.39	10.22	12.76	0.37	0.45	0.52	0.58	0.64
Gaussian w/o Shift	3.93	4.91	6.04	7.47	9.59	0.70	0.78	0.85	0.90	0.94
Gaussian w/ Shift	3.79	4.72	5.82	7.19	9.23	0.66	0.75	0.82	0.88	0.93
Log Gaussian w/o Shift	3.67	4.62	5.76	7.24	9.61	0.63	0.74	0.82	0.89	0.94
Log Gaussian w/ Shift	3.35	4.22	5.25	6.59	8.73	0.58	0.69	0.79	0.86	0.92

Appendix

Table 4: Comparison of Mean Prediction Interval Width (MPIW) and Prediction Interval Coverage Probability (PICP) for multiple uncertainty levels

\alpha

	VRAM		Time (ms)		FLOPS	Params
Model	Inference	Training	Inference	Training	(G)	(M)
Ours	1668MiB	1672MiB	2.79 $\pm$ 0.72	3.82 $\pm$ 0.18	430.531	85.0659
Ours ${}_{\textit{full}}$	1668MiB	3810MiB	2.79 $\pm$ 0.72	7.54 $\pm$ 0.25	430.531	85.0659
Pauls et al. ${}_{\textit{head\_only}}$	1476MiB	1478MiB	2.55 $\pm$ 0.97	3.00 $\pm$ 0.17	425.623	85.0653
Pauls et al. ${}_{\textit{full}}$	1476MiB	3900MiB	2.55 $\pm$ 0.97	6.77 $\pm$ 0.26	425.623	85.0653

Appendix

Table 5: VRAM usage and compute time during inference and training for a single input, FLOPs and number of parameters from our proposed method and Pauls et al. (2025) , both with either frozen backbone and trainable head (Ours and Pauls et al.

{}_{\textit{head\_only}}

) or fully trainable (Ours

{}_{\textit{full}}

and Pauls et al.

{}_{\textit{full}}

Figure 11: Empirical Coverage vs Nominal Quantile for multiple prediction bins for our model. Across all prediction bins, the conditional quantiles are well-calibrated.

Panel of Figure 12 — Figure 12: Empirical Coverage vs Nominal Quantile for our model trained without shift loss.

Panel of Figure 13 — Figure 13: Boxplot of Prediction Interval Width at level $\alpha=0.8$ for model trained without shift loss in dependence of either Left: being at forest border or Right: average slope within a $140\text{\,}\mathrm{m}$ square. Note that the model without shift loss is more uncertain at forest borders compared to the model with shift loss (compare Figures 7 and 8 ).

Figure 14: Sample satellite image with point estimates and $80\%$ Prediction Interval Width of our model, trained with or without shift loss. Left: Satellite image (yearly median composite). Top Right: Point-estimates of canopy height. Bottom Right: $80\%$ Prediction Inverval Width. It is clearly visible that the shift loss reduces uncertainty at forest borders.

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Remote sensing uses satellite, aerial, and ground-based observations to measure and map Earth and other planetary surfaces.

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Ritu Yadav, Andrea Nascetti and Yifang Ban is with Division of Geoinformatics, KTH Royal Institute of Technology, Sweden.

Uncertainty Quantification Forest Aboveground Biomass Estimation Remote Sensing Regression Prediction Intervals

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Earth Observation regression tasks such as building height, canopy height, and above-ground biomass estimation underpin critical applications in urban planning, forest monitoring, and climate policy, where both accuracy and reliability are critical. Yet most deep learning models yield only deterministic predictions, providing no indication of per-pixel reliability. These regression tasks are inherently challenging due to heterogeneous land surfaces, skewed target distributions, sensor noise, and signal saturation at high target values, making uncertainty (UC) estimation essential for reliable inference. We address this gap by modeling aleatoric uncertainty using year-long Sentinel-1 SAR and Sentinel-2 MSI time series, proposing two complementary approaches: (i) Gaussian UC, which jointly predicts mean and standard deviation under a Gaussian assumption, and (ii) Quantile UC, which estimates the 10th, 50th, and 90th quantiles to capture asymmetric and heteroscedastic error distributions. Both models are evaluated on three representative EO regression tasks at 10 m spatial resolution. Results show that both approaches match or surpass deterministic benchmarks and existing global products, while delivering well-calibrated, interpretable, and operationally useful confidence estimates. Notably, both models outperform the current 10 m state-of-the-art uncertainty-aware model for canopy height estimation. Our implementation will be available at: https://github.com/RituYadav92/EO-Regression-Uncertainty-Estimation

Abstract

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