# HQ-DiT: Efficient Diffusion Transformer with FP4 Hybrid Quantization

Wenxuan Liu
New York University
<EMAIL_ADDRESS>
&Sai Qian Zhang
New York University
<EMAIL_ADDRESS>

###### Abstract

Diffusion Transformers (DiTs) have recently gained substantial attention in
both industrial and academic fields for their superior visual generation
capabilities, outperforming traditional diffusion models that use U-Net.
However, the enhanced performance of DiTs also comes with high parameter
counts and implementation costs, seriously restricting their use on resource-
limited devices such as mobile phones. To address these challenges, we
introduce the Hybrid Floating-point Quantization for DiT (HQ-DiT), an
efficient post-training quantization method that utilizes 4-bit floating-point
(FP) precision on both weights and activations for DiT inference. Compared to
fixed-point quantization (e.g., INT8), FP quantization, complemented by our
proposed clipping range selection mechanism, naturally aligns with the data
distribution within DiT, resulting in a minimal quantization error.
Furthermore, HQ-DiT also implements a universal identity mathematical
transform to mitigate the serious quantization error caused by the outliers.
The experimental results demonstrate that DiT can achieve extremely low-
precision quantization (i.e., 4 bits) with negligible impact on performance.
Our approach marks the first instance where both weights and activations in
DiTs are quantized to just 4 bits, with only a 0.12 increase in sFID on
ImageNet $256\times 256$.

##  1 Introduction

Diffusion Transformers (DiTs) [1] have garnered increasing attention due to
their superior performance over traditional diffusion models (DMs) that use
U-Net [2] as the backbone DNN. Since their introduction, they have been
extensively researched and applied in both academic and industrial fields [1,
3, 4, 5, 6, 7], with the most notable application being OpenAI’s SoRA [8].
Recent research has demonstrated its impressive generative capabilities across
various modalities [9]. However, the iterative denoising steps and massive
computational demands significantly slow down its execution. Although various
methods [10, 11] have been proposed to reduce the thousands of iterative steps
to just a few dozen, the large number of parameters and the complex network
structure of DiT models still impose a significant computational burden at
each denoising timestep. This hinders their applicability in practical
scenarios with limited resource constraints.

Model quantization is widely recognized as an effective approach for reducing
both memory and computational burdens by compressing weights and activations
into lower-bit representations. Among various quantization methods, Post-
training quantization (PTQ) offers a training-free approach (or minimal
training cost for calibration purposes [12, 13, 14, 15, 16]) for fast and
effective quantization. Compared to Quantization-Aware Training (QAT), which
requires multiple rounds of fine-tuning, PTQ incurs significantly lower
computational costs. This makes it an appealing solution for quantizing large
models like DiT. Existing PTQ methods for DMs [17, 18] primarily employs
fixed-point quantization (i.e., INT quantization); however, significant
quantization errors can occur at low precision. To demonstrate this, we
evaluate several recent quantization methods for large models, including
SmoothQuant [19], FPQ [20], and GPTQ [21] on DiT. As shown in Figure 1,
quantizing both weights and activations to 4-bit precision results in serious
performance degradation, leading to an observed increase in the FID up to 100.
Moreover, the quantization process involved in these methods also incurs high
computational costs due to the calibration process required to search for the
optimal quantization scheme.

Figure 1: Performance of different approaches on ImageNet $256\times 256$.
Both weights and activations are quantized with 4 bits. The x-axis denotes the
runtime for each quantization approach. The size of the circle indicates the
standard deviation.

Floating-point (FP) quantization presents a more flexible alternative to INT
quantization. Compared to the integer numeric format, which uses a fixed
scaling factor, FP quantization is adaptive to different scales in the data
due to the inclusion of an exponent. This adaptability allows it to maintain
precision across different magnitudes [22, 23], making FP an ideal data format
choice for various commercial hardware platforms, such as Nvidia’s Blackwell
(H200) [24]. Unlike INT quantization, which obtains quantized values through
truncation and rounding, the challenge in FP quantization lies in selecting
the appropriate composition, specifically how many bits are assigned to the
exponent and mantissa. Inappropriate selections can result in suboptimal
performance. To address this issue, we propose to determine the FP composition
based on the channel-wise data distribution. Compared to previous methods
[25], our approach optimizes performance by tailoring to the unique
characteristics of the data in each channel.

Due to their high inter-channel variance and low intra-channel variance,
quantizing the activations is well known to be challenging [26]. Previous
works, such as SmoothQuant [19], aim to migrate the difficulty the
quantization of activation to weight. However, these methods still suffer from
performance degradation at lower precision levels. In this study, we push the
limits of quantization bitwidth by using random Hadamard transforms to
eliminate outliers in the activations. Additionally, we apply a corresponding
transform to the network weights to minimize quantization errors across the
network. This approach effectively reduces the impact of outliers on
quantization while introducing only a minimal increase in computational
overhead. To this end, we introduce a hybrid FP Quantization for DiT (HQ-DiT),
a simple yet efficient PTQ method designed for low-precision DiT. HQ-DiT
achieves performance levels comparable to full-precision models while
operating with precision as low as FP4. Our contributions are summarized as
follows:

  * •

We introduce HQ-DiT, an efficient PTQ method for DiTs, capable of achieving
performance on par with full-precision models using 4-bit FP (FP4)
quantization. To the best of our knowledge, HQ-DiT represents the first
attempt to quantize the DiT using FP data format.

  * •

We propose a novel algorithm that can adaptively select the optimal FP format
based on the data distribution, effectively addressing the significant
computational overhead associated with search-based methods in the prior work
[20].

  * •

HQ-DiT quantifies both weights and activations in DiTs using FP4, leading to a
$5.09\times$ speedup and $2.13\times$ memory savings compared to the full-
precision model. Our HQ-DiT achieves state-of-the-art results in low-precision
quantization, with the FP4 model outperforming the full-precision latent
diffusion model (LDM) in both Inception Score (IS) and Frechet Inception
Distance (FID).

##  2 Background and Related Work

###  2.1 Diffusion Models

Diffusion models (DMs) [27] have recently gained significant attention for its
remarkable ability to generate diverse photorealistic images. It is a
parameterized Markov chain trained through variational inference to generate
samples that match the data distribution over a finite duration. Specifically,
during the forward process of DMs, given an input image $x_{0}\sim q(x)$, a
series of Gaussian noise is generated and added to the $x_{0}$, resulting in a
sequence of noisy samples $\\{x_{t}\\},0\leq t\leq T$.

$q(x_{t}|x_{t-1})=\mathcal{N}(x_{t};\sqrt{1-\beta_{t}}x_{t-1},\beta_{t}I)$ (1)

where $\beta_{t}\in(0,1)$ is the variance schedule that controls the strength
of the Gaussian noise in each step.

During the reverse process, given a randomly sampled Gaussian noise
$\mathcal{N}(x_{T};0,I)$. The synthetic images are generated progressively
with the following procedure:

$p_{\theta}(x_{t-1}|x_{t})=\mathcal{N}(x_{t-1};\mu_{\theta}(x_{t},t),\hat{\beta_{t}}I)$
(2)

where $\mu_{\theta}(x_{t},t)$ and $\hat{\beta_{t}}$ are defined as follows:

$\mu_{\theta}(x_{t},t)=\frac{1}{\sqrt{\alpha_{t}}}(x_{t}-\frac{1-\alpha_{t}}{\sqrt{1-\bar{\alpha}_{t}}}\epsilon_{\theta,t}),\;\hat{\beta_{t}}=\frac{1-\bar{\alpha}_{t-1}}{1-\bar{\alpha}_{t}}$
(3)

Figure 2: A DiT block.

In equation 3, ${\alpha}_{t}=1-\beta_{t}$, and
$\bar{\alpha}_{t}=\prod_{i=1}^{t}{\alpha}_{t}$. $\epsilon_{\theta,t}$ denotes
the predicted noise that is generated with a backbone DNN. The backbone DNN in
the conventional DMs utilize a convolutional-based U-Net architecture [2, 28].
In contrast, the DiT model substitutes this U-Net with a transformer and
incorporates a parallel MLP that generates scale and bias for intermediate
outputs, as shown in Figure 2. DiT is recognized for surpassing conventional
DM in terms of visual generation capability.

###  2.2 Classifier-free Guidance

In DiT, an additional class label $c$ can be provided by the user as guidance
for image generation. In this situation, the reverse process becomes:

$p_{\theta}(x_{t-1}|x_{t},c)$

Classifier-Free Guidance [29] uses an implicit classifier to replace an
explicit classifier, adjusting the guidance weight to control the realism and
balanced diversity of generated images. According to Bayes’ formula, the
gradient of the classifier can be formulated as:

$\nabla_{x_{t}}\log p(c\mid x_{t})=\nabla_{x_{t}}\log p(x_{t}\mid
c)-\nabla_{x_{t}}\log p(x_{t})$
$=\frac{-1}{\sqrt{1-\bar{\alpha}_{t}}}\left(\varepsilon_{\theta}(x_{t},t,c)-\varepsilon_{\theta}(x_{t},t)\right)$

By interpreting the output of DMs as the score function, the DDPM sampling
procedure [27, 30] can be guided to sample $x$ with high probability $p(x\mid
c)$ by:

$\bar{\varepsilon_{\theta}}(x_{t},c)=\varepsilon_{\theta}(x_{t},\emptyset)+s\cdot(\varepsilon_{\theta}(x_{t},c)-\varepsilon_{\theta}(x_{t},\emptyset))+s\cdot\nabla_{x}\log
p(c\mid x_{t})\propto\theta(x_{t},\emptyset)$

where $s>1$ indicates the scale of the guidance. Classifier guidance can
control the balance of realism and diversity of generated samples and is
widely applied in the generative models such as DALL$\cdot$ E [29, 31, 32].

Figure 3: (a) Magnitude distribution of an input activation of a DiT linear
layer before and after Hadamard transform. (b) Histogram on an input
activation matrix across different time steps.

###  2.3 Post-Training Quantization

Unlike QAT, PTQ does not require model training or incurs only minimal
training costs for calibration purposes, making it very efficient in
computation. Furthermore, when calibrating generative models such as DMs,
instead of using the original training dataset, calibration datasets can be
generated using the full-precision model. This enables the calibration process
to be implemented in a data-free manner. For example, Q-Diffusion [17] applies
advanced PTQ techniques proposed by BRECQ [13] to improve performance and
evaluate it on a wider range of datasets, while PTQD [33] further breaks down
quantization error and combines it into diffusion noise. EfficientDM [18]
enhances the performance of the quantized DM by fine-tuning the model using
QALoRA [34, 35]. However, all of the aforementioned methods primarily focus on
INT quantization in conventional DMs with U-Net architectures, while HQ-DiT
aims for quantizing DiT with low-precision FP numeric format. FPQ [25]
determines the optimal FP bias and format by exhaustively search over the
design space, leading to a FP4 LLM with significant computational cost on
quantization process. In contrast, our method calculates the optimal format
based on data distribution and employs random Hadamard transforms to mitigate
outliers, achieving superior performance with negligible cost on the
quantization process.

FP quantization has been widely utilized as an alternative to INT quantization
for improving the efficiency of deep neural networks (DNNs). Compared with INT
quantization, FP data format has a wider dynamic range, making them better
adapt to the wide-spread data distribution in DNNs. Multiple customized FP
formats have been proposed and adopted in commercial hardware, including
Google’s bfloat16 [36] and Nvidia’s TensorFloat 32 [37]. Recently, Nvidia
announced that FP4 and FP8 will also be supported in the new series of
Blackwell GPUs. FP formats have also shown to be efficient in quantizing LLM
[25, 20]. In this work, we aim for applying FP for quantizing DiT.

##  3 Methodology

In this section, we describe the HQ-DiT method in detail. We first study the
data distribution within DiT in Section 3.1. Next we describe Hadamard
transform and its application in DiT in Section 3.2, followed by the detailed
quantization strategies in Section 3.3 and Section 3.4.

###  3.1 How Activations are Distributed within DiT?

Algorithm 1 MinMax quantization

Input: FP32 array $A_{fp}$, number of bits $n$, number of exponent bits
$n_{e}$.

$n_{m}\leftarrow n-n_{e}-1$ $\triangleright$ Calculate mantissa bitwidth

$max\\_val\leftarrow 2^{(2^{n_{e}}-1)}\times(2-2^{-n_{m}})$.

$A_{sign}\leftarrow\text{sign}(A_{fp})$.

$A_{abs}\leftarrow\text{abs}(A_{fp})$.

$bias\leftarrow\left\lfloor\log_{2}(\max(A_{abs}))-\log_{2}(max\\_val)\right\rfloor$

$value\\_max\leftarrow 2^{(2^{n_{e}}+bias-1)}\times(2-2^{-n_{m}})$

$A_{abs}\leftarrow max(A_{abs},value_{max})$

$x_{log\\_scales}\leftarrow\text{clamp}(\left\lfloor(\log_{2}(A_{abs})-bias\right)\rfloor,1)$

$scale\leftarrow 2^{(x_{log\\_scales}-n_{m}+bias)}$

$A^{\prime}_{abs}\leftarrow\text{quantize and round }A_{abs}\text{ by }scale$

$A^{\prime}_{fp}\leftarrow A^{\prime}_{abs}*A_{sign}$ $\triangleright$ Apply
quantization

return $A^{\prime}_{fp}$ $\triangleright$ Return the quantized array

To understand the distribution of the input activation within DiT, we collect
the input activations of a DiT block across 50 denoising steps, the histogram
is highlighted in Figure 3 (b). We then analyze the distribution of the
activation matrix at a specific time step, as shown in the left part of Figure
3 (a). Similar to the pattern observed in DMs [17], outliers exist at the
level of entire channels. Specifically, the scale of outliers in activations
is approximately $100\times$ larger than rest of the activation values. Thus,
per-token quantization would inevitably introduce substantial errors. Given
that activations exhibit high variance across different channels but low
variance within channels, with outliers confined to a limited number of
channels, implementing per-channel quantization and minimizing the impact of
outliers can hopefully result in lower quantization error.

###  3.2 Hadamard Transform for Activation Quantization

Previous work [38] has demonstrated that Hadamard transform can be applied to
eliminate outliers present in data. Similarly, we introduce random Hadamard
transforms to eliminate outliers present in the input activations $X$ of DiT
by multiplying it with an orthogonal Hadamard matrix $H\in\mathbb{R}^{n\times
n}$, where $n$ is the embedding dimension of DiT and $H$ satisfies
$HH^{\top}=H^{\top}H=I$. The resultant matrix $XH$ will exhibit a much
smoother distribution, with most outliers being eliminated, as depicted in the
right part of Figure 3 (a). To maintain the mathematical equivalence of the
linear layer within the DiT block, we need to apply the Hadamard matrix to the
corresponding weight matrices within the self-attention (SA) layers and
pointwise feedforward (FFN) layers of each DiT block. Next we will describe
these modifications in detail.

#### Hadamard Transform within Self-attention Module

To achieve computational invariance, each of the query, key, and value weight
matrices $W_{q}$, $W_{k}$, and $W_{v}$ within the SA block are multiplied by
$H^{\top}$, producing new matrices $W^{\prime}_{q}=H^{\top}W_{q}$,
$W^{\prime}_{k}=H^{\top}W_{k}$, and $W^{\prime}_{v}=H^{\top}W_{v}$, which can
be performed prior to the DiT execution and incurs no additional online
computational cost. This ensures that the Hadamard transform applied to the
input activations will not alter the output, as
$(XH)(W^{\prime})=(XH)(H^{\top}W)=XW$. This will produce queries, keys, and
values that are mathematically equivalent to the original outputs. The
resultant input activations $XH$ can then be quantized with much smaller
quantization error. This process is highlighted in Figure 4 (a).

After that, the attention matrix of each head $A_{i}$, which is the output of
the softmax layer, is multiplied by the corresponding value matrix $V_{i}$.
The results are then multiplied by the $W_{out}$ matrix, as shown in Figure 4
(a). This process produces the SA block output $Y=VW_{out}$, where
$V=\text{concat}[A_{1}V_{1},\ldots,A_{h}V_{h}]$ and $h$ denotes the number of
heads. However, outliers will also be present within $V$, necessitating the
application of the Hadamard transform to $V$. In order to apply Hadamard
transform over $V$ without any additional multiplication computation, we need
to fuse the Hadamard matrix $H_{d}$ on each head, where $H_{d}$ denotes a
$d\times d$ Hadamard matrix, and $d$ is the embedding size of each head. Since
each component of $V$ are first produced at each head and then concatenated
together to form $V$, which are then multiplied against $W_{out}$, we can
apply Hadamard transform on $W_{v}$ and $W_{out}$ with $H_{head}$ by
leveraging the properties of the Kronecker product.

$W_{v}\leftarrow W_{v}H_{head},\quad W_{\text{out}}\leftarrow
H_{head}W_{\text{out}},$

where $H_{head}=((I\otimes H_{d})(H_{h}\otimes I))$, $I$ is the identity
matrix, $H_{d}$ denotes a $d\times d$ Hadamard matrix, $H_{h}$ denotes a
$h\times h$ Hadamard matrix, $h$ is the number of heads. $\otimes$ represents
the Kronecker product. This effectively mitigates the outliers within $V$
without any additional computational overhead.

Figure 4: Quantization workflow in (a) SA block and (b) FFN block for DiT.

#### Hadamard Transform within FFN Module

The FFN module within DiT consists of two linear layers and one GELU
activation function (Figure 4 (b)). Denote $W_{fc1}$ and $W_{fc2}$ the weight
matrices associated with these two linear layers. To better quantize the
intermediate activations, we also fuse the Hadamard transform matrices into
$W_{fc1}$ by executing $W_{fc1}\leftarrow H^{\top}W_{fc1}$. Similarly, this
operation can be performed offline without any additional cost during the
inference process. The output of the first linear layer $W_{fc1}$ also
contains outliers and an additional round Hadamard transform are required to
remove the outliers. Due to the presence of the GELU activation function, the
Hadamard transform need to be performed online. To mitigate this cost, we have
designed an efficient Hadamard transform with the following computational
complexity:

Lemma 1 If the dimension $n$ of $X$ is a power of 2, the Walsh-Hadamard
transform allows us to compute $XH$ in $O(n^{2}\log n)$ time complexity. If
$n$ is not a power of 2, by leveraging the properties of the Kronecker
product, we can complete the computation in $O(n^{2}q\log p)$ time complexity,
where $p$ is the largest power of 2 such that we can construct a new Hadamard
matrix by the Kronecker approach.

We will provide a detailed description of Lemma 1 in the appendix. Applying
the Hadamard transform to activations and weights can effectively eliminate
outliers. Next, we discuss the method used for quantizing weights and
activations.

###  3.3 FP Format Selection for Weight Quantization

In this section, we describe the approach for quantizing the weight matrices
within the DiT blocks. Our objective of post-training quantization is to find
a quantized weight matrix $\hat{W}$ which minimizes the squared error,
relative to the full precision layer output. Formally, this can be formulated
as:

$\arg\min_{\hat{W}}\|\hat{W}X-WX\|_{2}^{2}$

Figure 5: HQ-DiT quantization scheme within a DiT block. Operations in FP4 and
FP32 are highlighted in blud and red, respectively.

Selecting an appropriate FP composition for weight quantization is crucial, as
an improper choice on exponent and mantissa bitwidth can lead to significant
quantization errors. A naive approach to selecting an FP data format involves
exhaustively searching through each possible combination, which results in
high computational overhead. In this work, we propose a simple yet effective
method for FP format selection. Our approach is based on the simple fact that
for FP format with a fixed total bitwidth, a greater number of exponent bits
allows for a larger range of representable values. As shown in Figure 6 (a), a
large exponent bitwidth (e.g., E2M1) will also result in an uneven
distribution of the represented data, making the long-tail phenomenon more
pronounced. For the weight matrices in the DiT model, we can determine the
optimal FP format by analyzing their data distribution via the following
indicator:

$s_{w}=\frac{\max(|W|)}{\text{Quantile}(|W|,\alpha)}$

where $\alpha$ is a hyperparameter that return the bottom $\alpha$ percentile
of the weight values. By setting $\alpha$ to be a small number (i.e., 10), we
can approximate the ratio of the maximum and minimum element within $W$, while
eliminating the impact of the outliers. Our goal is to represent all the
elements within this interval with low error. On the other hand, for a given
FP format with exponent and mantissa bitwidths of $n_{e}$ and $n_{m}$,
respectively, the ratio between the maximum and minimum values can be computed
as follows:

$r=2\times\frac{max\\_val}{min\\_val}=2^{(2^{n_{e}})}\times\frac{(2-2^{-n_{m}})}{(1+2^{-n_{m}})}$

where $max\\_val=2^{(2^{n_{e}}-1)}\times(2-2^{-n_{m}})$ and
$min\\_val=(1+2^{-n_{m}})$ are the maximum and minimum values this FP format
can represent. To maximize the representation of elements within an interval,
we aim to find the optimal FP number format to bring $r$ in close
approximation to $s_{w}$. This is easily achievable due to the limited number
of FP format combinations available for a given total bitwidth.

Based on the evaluation results of DiT models, we observe that $\alpha=25$
yields the best performance. The effect of $\alpha$ selection over the image
quality can be seen in Figure 6 (b). We then apply the FP composition
searching methods described earlier to find the optimal FP composition for
each DiT block. This results in a hybrid FP data format that significantly
enhances performance, as demonstrated by the evaluation results in Section
4.2. After an appropriate composition is found, we then use the GPTQ [21]
approach to perform the FP quantization. GPTQ is originally designed for
linear quantization, and we modify it to support FP quantization, the
implementation details can be found in the appendix.

Figure 6: (a) Distribution of FP format E2M1 and E1M2, with the bias set to 0.
(b) Effect of $\alpha$ on FP format selection. If $\alpha$ is too small, a
larger $n_{e}$ will be chosen, leading to severe data distribution imbalances.
If $\alpha$ is too big, less number can be represented. Samples generated on
W4A4 ImageNet 256 $\times$ 256 (cfg=1.5).

###  3.4 Activation Quantization

Given the need to quantize activations in real-time, employing a complex
quantization scheme would incur significant computational overhead. Therefore,
following the application of the Hadamard transform to the input activation,
we utilize the straightforward MinMax quantization method for activation, as
described in Algorithm 1. MinMax quantization sets the bias bit based on the
maximum value in each channel of activation, then maps full-precision data to
the quantized data range by dividing the scale factor. Evaluation results
demonstrate that MinMax quantization can deliver excellent performance on
activation quantization with minimal computational overhead.

As depicted in Figure 5, HQ-DiT is applied to all the fully-connected layers
within each DiT block. Computations conducted in low-precision FP
representations are highlighted in blue, while the remaining operations,
highlighted in red, are performed with FP32.

##  4 Experiments

In this section, we evaluate the performance of HQ-DiT in generating both
unconditional and conditional images on ImageNet at different resolutions,
including $256\times 256$ and $512\times 512$. We obtain the pretrained DiTs
from the official Github repository [39], apply the quantization techniques
detailed in Section 3, and evaluate the performance of the quantized DiT. To
compare the performance of HQ-DiT, we adopt the Minmax algorithm described in
Algorithm 1 as a baseline, which performs channel-wise FP quantization on both
weight and activation. Furthermore, we also implement other sophisticated
quantization approaches over DiT including FPQ [25], SmoothQuant [19] and GPTQ
[21], details can be found in appendix. We evaluate different approaches using
the criteria including Inception score (IS), FID and sFID [40] by sampling
over 50K images, subsequently using ADM’s TensorFlow Evaluation suite [28] to
produce the final results.

###  4.1 Main Results

In this section, we present results on both conditional and unconditional
image generation using HQ-DiT. In unconditional image generation [29], no
external guidance is provided during the iterative DiT denoising process. For
conditional image generation [41], specific class labels are provided as an
additional input to guide the DiT denoising process.

#### Evaluation on unconditional image generation:

The evaluation results for unconditional image generation are shown in Table
1. While SmoothQuant and FPQ are effective for 8-bit weight and 8-bit
activation (W8A8), their performance deteriorates significantly with 4-bit
weight and 4-bit activation. In contrast, HQ-DiT demonstrates superior
performance, achieving the highest IS and FID at W8A8 precision. Furthermore,
it shows only a slight increase in FID compared to the full-precision (FP32)
results. Additionally, our approach maintains the capability to perform
unconditional generation tasks even at W4A4 precision, whereas the other two
methods struggle to produce images that are recognizable to humans. This
significant improvement highlights the notable capability of HQ-DiT to
mitigate the impact of outliers and select the optimal data format for each
layer. This results in low-precision quantization with minimal quantization
error. Additional generation results can be found in the appendix.

Table 1: Quantization results for unconditional image generation with input
size of $256\times 256$.

Method Bit-width (W/A) IS $\uparrow$ FID $\downarrow$ sFID $\downarrow$ FP32
32/32 112.73 12.54 9.38 MinMax (FP) 8/8 72.3 40.06 47.23 FPQ 8/8 105.76 13.52
9.62 SmoothQuant (FP) 8/8 110.17 13.23 9.40 HQ-DiT 8/8 111.62 12.74 9.42
MinMax (FP) 4/4 1.52 300.64 423.50 FPQ 4/4 8.69 231.83 168.72 SmoothQuant (FP)
4/4 4.82 289.21 204.37 HQ-DiT 4/4 61.34 23.91 17.02

#### Evaluation on conditional image generation:

We also evaluate the performance of HQ-DiT for class-conditional image
generation, as detailed in Table 2. Experiments on ImageNet $256\times 256$
are conducted with two different Classifier-Free Guidance Scale (cfg), 1.5 and
4.0. Similarly, we provide comprehensive evaluation results with three metrics
including IS, FID and sFID. We observe that for a cfg of 4.0, compared with
the FP32 DiT, HQ-DiT achieves an average reduction of 6.98 in FID and 2.22 in
sFID for W4A8. HQ-DiT also achieves the optimal performance compared with
other baseline algorithms. For W4A4, other methods struggle to mitigate the
substantial quantization noise caused by low-bit quantization. For example,
FPQ at W4A4 bitwidth experiences a decrease in IS by 248.73 and an increase in
sFID by 9.96. In contrast, HQ-DiT can still achieving a high IS of 437.13 and
a low sFID of 9.94. Meanwhile, our model gets a 66.44 increase in IS and 0.19
decrease compared with full-precision LDM model [2].

Table 2: Results for conditional image generation on ImageNet $256\times 256$
and ImageNet $512\times 512$.

Model Method Bit-width (W/A) IS $\uparrow$ FID $\downarrow$ sFID $\downarrow$
DiT-XL/2 $256\times 256$ (steps = 100 cfg = 1.5) FP32 32/32 266.57 2.55 111The
original DiT paper [1] reports a FID of 2.27, but our replication efforts
yield an FID of 2.55. 5.34 SmoothQuant 4/8 12.84 137.60 81.22 SmoothQuant 4/4
6.93 252.34 192.87 FPQ 4/8 108.81 20.33 17.11 FPQ 4/4 12.62 127.99 69.10 GPTQ
4/8 15.09 107.63 75.33 GPTQ 4/4 3.14 200.85 329.18 HQ-DiT 4/8 145.48 12.59
11.41 HQ-DiT 4/4 136 13.12 11.3 DiT-XL/2 $256\times 256$ (steps = 100 cfg =
4.0) FP32 32/32 481.65 16.75 9.82 SmoothQuant 4/8 181.67 17.92 25.61
SmoothQuant 4/4 13.92 160.71 91.36 FPQ 4/8 449.64 10.88 9.60 FPQ 4/4 232.92
14.69 19.78 GPTQ 4/8 174.26 18.24 23.41 GPTQ 4/4 16.99 97.37 132.07 HQ-DiT 4/8
445.12 9.77 7.62 HQ-DiT 4/4 437.13 9.25 9.94 DiT-XL/2 $512\times 512$ (steps =
20 cfg = 4.0) FP32 4/4 430.59 16.26 10.47 FPQ 4/4 140.35 23.61 27.82 GPTQ 4/4
28.12 68.34 107.29 HQ-DiT 4/4 370.69 9.44 15.91 LDM-4[18] FP32 4/4 379.19
11.71 6.08

We further evaluate the performance of DiT on ImageNet $512\times 512$. The
total number of time steps is set to 20, while the remaining settings remain
the same as those for ImageNet $256\times 256$. According to the evaluation
results presented in Table 2, our W4A4 model achieves an IS of 370.69, with a
6.82 increase in FID compared to the full-precision model, while obtaining the
best performance among the baseline algorithms.

###  4.2 Ablation Study

Figure 7: Efficiency Analysis of the Quantized Model.

#### Impact of Hadamard Transform on Image Quality

We first investigate the impact of the Hadamard transform on the performance
of the quantized model under different settings. Specifically, we evaluate the
quality of images generated by HQ-DiT with and without applying the Hadamard
transform to the activations. Without Hadamard transform, HQ-DiT fails to
generate high-quality images at W4A4 and can just get an IS of 3.14, FID of
200.85. Conversely, by introducing the Hadamard transform, our method achieves
an IS of 437.13 and an sFID of 9.94, performance levels comparable to full-
precision models.

#### Impact of FP Composition

Additionally, we conduct experiments to analyze the effect of the FP format
selection described in Section 3.3. Specifically, we compare HQ-DiT with other
baselines by fixing the FP composition across the entire DiT layers, resulting
three baselines including E3M0, E2M1 and E1M2. The results presented in Table
3 illustrate that our hybrid FP format achieves an IS of 437.13, outperforming
all other FP4 formats, and a FID of 9.25, which is close to the FID result of
E1M2 (9.20). However, given the great improvement in the IS score, a
degradation of 0.05 in FID is considered insignificant. This demonstrates that
our FP composition selection and hybrid FP quantization scheme can effectively
enhance the DiT performance.

Table 3: The effect of FP format selection. Experiment conducted on ImageNet
$256{\times}\text{ 256}$

Model FP-Format Bit-width (W/A) IS $\uparrow$ FID $\downarrow$ DiT-XL/2 (steps
= 100 cfg = 4.0) E3M0 4/4 408.88 13.32 E2M1 4/4 430.33 11.65 E1M2 4/4 422.77
9.20 HQ-DiT 4/4 437.13 9.25

###  4.3 Deployment Efficiency

Finally, we evaluate the execution cost of the quantized model generated by
HQ-DiT by comparing it to the full-precision (FP32) model and the INT
quantized model. We evaluate the execution cost in terms of two aspects: model
size and theoretical computation time per DiT block. Our computation follows
the workflow depicted in Figure 5. In the evaluation of theoretical
computation time, due to the lack of hardware support, we calculated the
GFLOPs for each module and estimated the computation time of the quantized
model based on the theoretical acceleration performance. As shown in Figure 7,
quantization effectively reduces the model size. Although the introduction of
online Hadamard transforms increases computational overhead, the overall
computational cost is still significantly lower compared to INT8
implementation. More implementation details are provided in the appendix.

##  5 Conclusion

In this paper, we propose HQ-DiT, an efficient data-free PTQ method for low-
precision DiT execution. To address the difficulty in quantizing activations,
we eliminate outliers by introducing the Hadamard transform with minimal
execution cost. To select the appropriate FP format, we propose a selection
method based on data statistics. Our experiments demonstrate the superior
performance of HQ-DiT compared to other quantization methods. Notably, our
4-bit model achieves higher IS and lower FID compared to the full-precision
LDM model.

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##  Appendix A Implement Details

In our experiment, we adopt the E2M1 format for quantizating activation. For
the quantization of weight matrices, we compare different floating-point
formats with our proposed floating-point format selection method. We use a
block size of 64 during the quantization process with a calibration dataset of
512. All experiments are conducted on Nvidia A100 GPUs. Furthermore, the rest
baseline algorithms, including SmoothQuant, GPTQ, and FPQ, we adopt the same
calibration dataset as HQ-DiT. All the rest settings, including time
information at each timestep, category labels, random noise, and classifier-
free guidance are also kept the same.

In the implementation of SmoothQuant, we collect the maximum values of the
activation and weight matrices from the first layers of the attention module
and the FFN module in the forward propagation backbone network (specifically,
the query, key, and value calculations in the attention module, and fc1 in the
FFN). We set the hyperparameter $\alpha=0.5$ to determine the correction
factor and conduct related experiments based on this setup.

For the GPTQ method, we perform experiments on the ImageNet dataset with
resolutions of $256\times 256$ and $512\times 512$. In the $256\times 256$
experiments, we select a block size of 64, while for the $512\times 512$
experiments, we use a block size of 16. We quantize and update the weights of
the matrices in each linear layer of the forward propagation backbone network
according to the GPTQ method. Though GPTQ method is originally designed for
Linear quantization, we find it suitable for float point quantization. The
experimental results, focusing solely on quantizing weight matrices using the
GPTQ method, are provided in Table 4.

To implement the FPQ method, we consider all floating-point formats except
those with an exponent bit of 0 as the search range, which is equivalent to
the linear quantization. For each linear layer, we search for the optimal
floating-point format and the optimal pre-shifted bias according to the method
proposed in the FPQ paper.

To generate the evaluation results, we adopt the same sampling process across
all the baseline algorithms. We randomly sampled 50K images and calculated
evaluation metrics involving IS, FID, and sFID using the evaluation suite.

Table 4: GPTQ results for conditional image generation. Experiment conducted
on ImageNet

Model Method Bit-width (W/A) IS $\uparrow$ FID $\downarrow$ sFID $\downarrow$
$256\times 256$ cfg=1.5 FP32 32/32 266.57 2.55 5.34 GPTQ 4/32 210.8 5.20 8.98
$256\times 256$ cfg=4.0 FP32 32/32 481.65 16.75 9.82 GPTQ 4/32 447.12 11.70
7.83 $512\times 512$ cfg=4.0 FP32 32/32 430.59 16.26 10.47 GPTQ 4/32 384.46
11.64 14.38

##  Appendix B Activation Distribution in DiT

In this section, we present the range on of DiT activations of across
different blocks on the ImageNet $256\times 256$. The distribution of
activation can be found Figure 8. It is noticeable that the original DiTs’
activations exhibit high variance between different blocks. By applying the
Hadamard transform, outliers in the activations can be effectively eliminated,
leading to an accurate per-channel activation quantization.

Figure 8: Magnitude of the activations in different blocks

##  Appendix C Fast Hadamard Transform

A Hadamard matrix is an orthogonal matrix whose entries are proportional to
$\\{+1,-1\\}$. For matrices of dimension $2^{n}$, a Hadamard matrix can be
constructed. Specifically, the matrix $H_{2^{n}}$ is constructed using the
Kronecker product:

$H_{2}=\begin{bmatrix}1&1\\\ 1&-1\end{bmatrix},and\
H_{2^{n}}=\begin{bmatrix}H_{2^{n-1}}&H_{2^{n-1}}\\\
H_{2^{n-1}}&-H_{2^{n-1}}\end{bmatrix}$

For any matrix $X\in\mathbb{R}^{m\times n}$ and orthogonal Hadamard matrix
$H\in\mathbb{R}^{n\times n}$, their matrix multiplication will cause a
computing complexity of $\Theta(mn^{2})$. When $n$ is a power of 2, we can
apply the fast Walsh-Hadamard transform to compute $XH$. Utilizing a divide-
and-conquer strategy, we recursively decompose the Hadamard matrix of size $n$
into two matrices of size $\frac{n}{2}$. This allows the computation to be
completed with $O(mn\log n)$ additions and subtractions.

For dimension $n$ that is not powers of 2, we can factorize $n$ as $n=pq$,
where $p$ is the largest power of 2 such that there exists a known Hadamard
matrix of size $p$. Then, the Hadamard matrix $H$ of order $n$ can be
constructed as:

$H_{n}=H_{p}\otimes H_{q}$

where $H_{p}$ and $H_{q}$ are Hadamard matrices of order $p$ and $q$,
respectively. Then, we can compute $XH$ using a divide-and-conquer strategy
similar to the fast Walsh-Hadamard transform. This involves recursively
processing $H_{p}$, allowing the computation to be executed with $O(mnq\log
p)$ additions and subtractions. For instance, for a dimension of
$28672=1024\times 28$, this method can significantly reduce the computational
overhead.

##  Appendix D Additional Sample Visualization

In this section, we provide random samples from W4A4 quantized model based on
FPQ and our method HQ-DiT. Results are shown in figures below.

(a) FPQ

(b) Ours

Figure 9: Samples are generated by W4A4 DiT model on ImageNet $256\times 256$,
cfg is set to 1.5.

(a) FPQ

(b) Ours

Figure 10: Samples generated by W4A4 DiT model on ImageNet $256\times 256$
(cfg=4.0)

(a) FPQ

(b) Ours

Figure 11: Samples generated by W4A4 DiT model on ImageNet $512\times 512$
(cfg=4.0)


# The BELSAR dataset: Mono- and bistatic full-pol L-band SAR for agriculture
and hydrology

Jean Bouchat  Earth and Life Insitute, Université catholique de Louvain,
Louvain-la-Neuve, 1348, Belgium  corresponding author: Jean Bouchat
<EMAIL_ADDRESS> Emma Tronquo  Hydro-Climate Extremes Lab, Ghent
University, Ghent, 9000, Belgium  Anne Orban  Centre Spatial de Liège,
Université de Liège, Angleur, 4031, Belgium  Karlus A.C. de Macedo
MetaSensing B.V., DK Noordwijk, 2201, The Netherlands  Niko E. C. Verhoest
Hydro-Climate Extremes Lab, Ghent University, Ghent, 9000, Belgium  Pierre
Defourny  Earth and Life Insitute, Université catholique de Louvain, Louvain-
la-Neuve, 1348, Belgium

###### Abstract

The BELSAR dataset is a unique collection of high-resolution airborne mono-
and bistatic fully-polarimetric synthetic aperture radar (SAR) data in L-band,
alongside concurrent measurements of vegetation and soil bio-geophysical
variables measured in maize and winter wheat fields during the summer of 2018
in Belgium. This innovative dataset, the collection of which was funded by the
European Space Agency (ESA), helps addressing the lack of publicly-accessible
experimental datasets combining multistatic SAR and in situ measurements. As
such, it offers an opportunity to advance the development of SAR remote
sensing science and applications for agricultural monitoring and hydrology.
This paper aims to facilitate its adoption and exploration by offering
comprehensive documentation and integrating its multiple data sources into a
unified, analysis-ready dataset.

## Background & Summary

Agriculture and soil moisture monitoring are essential for sustainable food
production, water resource management and mitigating the impact of climate
change on crop yield and ecosystem health. Today, the best-established large-
scale operational agricultural monitoring systems are mainly based on optical
remote sensing [1]. However, they are often hampered by the presence of clouds
[2], which obscure the view of the sensors. In contrast, Synthetic Aperture
Radars (SAR) are active sensors, largely independent of weather conditions,
capable of interacting with and capturing valuable information about both
vegetation canopies and underlying soils [3, 4]. The recent European Space
Agency (ESA) Sentinel-1 (S1) mission, providing systematic and global coverage
of dense SAR time series, paved the way for the development of operational
applications like the Copernicus Emergency Services and the open-source
Sen4CAP system to implement the European Union’s Common Agricultural Policy.
The success of this operational mission triggered the intent to launch S1
companion satellite to enhance this all-weather earth-observation capacity in
a bistatic mode.

Bistatic SARs are radar systems in which the transmitter and receiver are
physically separated, unlike monostatic systems in which they share the same
location. Multistatic systems, meanwhile, have several receivers for a common
transmitter. The most simple multistatic system comprises of both an active
monostatic sensor and a passive bistatic one. These systems allow the
acquisition of images using different geometries and configurations, thereby
capturing multi-dimensional scattering effects that could not be recorded by
monostatic systems alone. These flexible systems offer the advantage of
helping to differentiate between the often intertwined relative contributions
of vegetation and soil, thus improving the retrieval performances of soil
moisture and crop biophysical variables retrieval [5]. Despite its
considerable potential, bi- or multistatic SAR applications for vegetation and
soil monitoring have remained limited. It seems that this scarcity can mostly
be attributed to a lack of comprehensive, well-documented experimental
datasets combining mono- and bistatic SAR acquisitions and in situ
measurements of vegetation and soil biogeophysical variables [6]. As a result,
the potential of bistatic SAR for these applications has mainly been
investigated via radiative transfer models [6, 7, 8, 5, 9].

In this context, the ESA funded the BELSAR-Campaign project, an airborne and
in situ measurement campaign that took place during the 2018 growing season in
Belgium. The result is the BELSAR dataset [10], a collection of data
containing high-resolution fully-polarimetric mono- and bistatic synthetic
aperture radar (SAR) times series in L-band and concurrent field measurements
of vegetation and soil bio-geophysical variables. The SAR data were acquired
with an airborne multistatic SAR system operated by MetaSensing BV. The field
measurements were collected both during and after the crop growing season in
ten maize and ten winter wheat fields simultaneously with the SAR
acquisitions.

Several studies exploiting the BELSAR dataset have already been published.
Bouchat et al. [11] have used the dataset to assess the potential of
simultaneous mono- and bistatic SAR acquisitions for agriculture and soil
moisture monitoring applications, as well as the impact of maize row structure
on the SAR backscatter. Tronquo et al. [12] have presented a semi-empirical
method based on effective roughness modeling to retrieve soil moisture in bare
agricultural fields and have shown an increase in the accuracy of soil
moisture estimation by using several polarizations at the same time. Finally,
Bouchat et al. [13] have obtained promising results for green area index (GAI)
retrieval in maize fields using the Water Cloud Model [14] and dual-polarized
SAR data in L-band. Yet, the potential of the BELSAR dataset is still far from
having been fully investigated and, given the vast possibilities offered by
such an innovative and unique collection of data, other users will want to
exploit it in their research in agriculture, hydrology, change detection, or
other SAR remote sensing techniques and applications. Therefore, this paper
aims to facilitate its uptake through its thorough documentation, as well as
through the integration of the different sources of data from BELSAR-Campaign
into a single so-called integrated dataset provided with it.

## Methods

### Airborne SAR acquisitions

The design and implementation of the airborne SAR campaign was carried out by
the Centre Spatial de Liège and MetaSensing BV. Multi-temporal mono- and
bistatic fully-polarimetric (HH, HV, VH, and VV) airborne L-band SAR data were
acquired with the MetaSAR-L systems over the BELSAR area of interest, in fig.
1, and processed with the MetaSAR-Pro software, MetaSensing BV’s proprietary
airborne SAR processor.

Mono- and bistatic airborne radar data were acquired simultaneously by two
left-looking L-band SAR operated by MetaSensing BV on-board two CESNA Gran
Caravan airplanes specifically adapted for the mission. The planes and radar
system are shown in fig. 2. The radar operated in frequency modulated
continuous wave mode (FMCW) at a central frequency of
$1.3\text{\,}\mathrm{GHz}$, with a pulse repetition frequency (PRF) at
$1004\text{\,}\mathrm{H}\mathrm{z}$, and a sampling frequency of
$50\text{\,}\mathrm{MHz}$. The authorized signal bandwidth, allocated by the
Belgian Institute for Post and Telecommunications (BIPT), was limited to
$50\text{\,}\mathrm{MHz}$. The sensors, capable of providing imaging with
spatial resolution up to $1\text{\,}\mathrm{m}$, were equipped with two flat
antennas: a squared one and a rectangular one, with a nominal antenna look-
angle of $45\text{\,}\mathrm{\SIUnitSymbolDegree}$, and a beamwidth of
$40\text{\,}\mathrm{\SIUnitSymbolDegree}$ in elevation and respectively
$40\text{\,}\mathrm{\SIUnitSymbolDegree}$ and
$20\text{\,}\mathrm{\SIUnitSymbolDegree}$ in azimuth. They were synchronized
through special techniques based on a dedicated high-accuracy GPS-disciplined
oscillator (GPSDO) system. Additionally, high-end Global Navigation Satellite
System (GNSS)/Inertial Navigation System (INS) devices were installed on the
sensors to be able to track their navigation and attitude precisely. Each
sensor transmitted and received in a fully-polarimetric ping-pong mode on a
chirp-to-chirp basis, allowing both sensors to simultaneously collect mono-
and bistatic SAR images depending on the preferred configuration.

Radar data were collected during a series of five flight missions, labeled F1
to F5, between 31 May (F1) and 10 September 2018 (F5), with a temporal
baseline of approximately one month. Table 1 lists the dates of these flights.

On each flight mission, mono- and bistatic data were acquired by flying the
airplanes in close formation in two different bistatic geometries, i.e., in
across-track (XTI) and in along-track (ATI) configuration. One aircraft flew
steadily at $2500\text{\,}\mathrm{m}$ above mean sea level, while the other
positioned itself in two distinct locations behind the first. Figure 3 depicts
the nominal XTI and ATI flight configurations. The actual XTI and ATI
baselines were about $35\text{\,}\mathrm{m}$ and $450\text{\,}\mathrm{m}$,
respectively, for all flight missions except for the first one (F1) when the
along-track baseline was about $900\text{\,}\mathrm{m}$ and the horizontal
separation was in a $6080\text{\,}\mathrm{m}$ range in the ATI configuration.
The actual baselines are shown on fig. 4 for the first two flight missions, F1
and F2. These acquisition configurations were established on the basis of
ESA’s requirement to reproduce the configuration planned for the SAOCOM-CS
satellite mission [15].

Three partially overlapping passes were necessary to cover the area of
interest. These passes divided the $4.5\text{\,}\mathrm{km}$ wide area
depicted in fig. 5 into three main tracks labeled Alpha (A), Zulu (Z), and
Bravo (B). Four trihedral corner reflectors with a side-length of
$75\text{\,}\mathrm{cm}$, one of which is shown in fig. 2, were deployed in a
fourth, shorter track labeled Zulu short (Zs). They were placed in a row in
the across-track direction, and used to produce a geometrical reference as
well as the point spread function (PSF) for monostatic acquisitions. Their
elevation angle was adjusted for each flight mission, depending on its
altitude. Their positions were measured with a precision of
$1\text{\,}\mathrm{m}$ using a GNSS receiver. The complete set of tables
describing the tracks and corner reflector installation can be found in the
final report of the BELSAR-Campaign project [10].

Radar data were processed using the Global Back-Projection (GBP) algorithm,
which also handles motion compensation. They were delivered as
$\sigma$-calibrated single look complex (SLC) focused SAR data in ground range
geometry, with a ground resolution of $1\text{\,}\mathrm{m}$ in azimuth and
$4\text{\,}\mathrm{m}$ in slant range, after Hann windowing with a
$50\text{\,}\mathrm{MHz}$ transmitted pulse. The SAR images were geo-
referenced and co-registered on a sub-pixel level based on the absolute
position accuracy of the navigation data, i.e., $0.75\text{\,}\mathrm{m}$. The
implemented SAR processing chain is shown in fig. 6.

Geometric calibration was performed using the corner reflectors. For bistatic
data, two range delays had to be defined, one for each system. One of the
range delays was adjusted relatively to the absolute calibrated one according
to a global offset obtained from a coherence-based fine sub-pixel co-
registration.

The same radiometric calibration was applied to both mono- and bistatic data.
The absolute calibration constant, $K$, was evaluated for each flight mission
using the four corner reflector responses. Antenna pointing direction and
incidence angles were computed using post-processed navigation data and an
external digital elevation model (DEM). In the XTI configuration, the corner
reflectors were assumed to behave in the same way for both mono- and bistatic
sensors. In the ATI configuration, corner reflectors were not visible on
bistatic images. The radiometric offset for bistatic ATI data was therefore
based on monostatic ATI and bistatic XTI data.

Finally, a polarimetric calibration was also applied following the procedure
described in Fore et al. [16]. Data were calibrated for co-pol and cross-pol
channel imbalances and phase bias, i.e., amplitude and relative phase
differences between co-polarization channels, at both transmission and
reception using the corner reflectors. Cross-pol imbalance and phase bias were
estimated using the ratio of the averaged cross-pol responses from a large
number of pixels.

### In situ measurements

Vegetation and soil bio-geophysical variables were measured in the ten maize
and ten winter wheat fields by two teams from the Earth and Life Institute at
UCLouvain and the Hydro-Climate Extremes Lab (H-CEL) at Ghent University
quasi-simultaneously with the airborne SAR acquisitions. The in situ
measurements were performed with a delay of maximum six days with respect to
the airborne SAR missions. Table 1 contains the dates of the different
measurements.

The sampled fields—labeled with a letter, M for maize and W for wheat, and a
number ranging from 1 to 10—are located at the BELAIR Hebania test site, in
the Hesbaye region of Belgium. This site belongs to the global Joint
Experiment for Crop Assessment and Monitoring (JECAM) network. The location of
the site and the fields is shown in fig. 1. The area corresponds to a typical
landscape of intensive agriculture in Belgium. The fields are relatively
large, flat, and homogeneous, with a uniform topsoil texture of silt loam. The
major crops in the area are wheat, potatoes, beets, and maize.

At the time of the first acquisition, winter wheat and maize crops were
already in place. Harvesting took place between the second (F2) and third (F3)
acquisitions for winter wheat, and between the third (F3) and last (F5) for
maize. Some SAR acquisitions were therefore conducted over bare fields. It
should be noted that maize stalks were still present on the fields after
harvest (see Table 1).

Finally, due to the overlap in the flight tracks, certain fields were imaged
several times on the same date in both flight configurations, i.e., ATI and
XTI. Other fields, however, were imaged only during certain flight missions
due to the length of the tracks varying between flight missions, and two maize
fields namely M9 and M10, were never imaged by the airborne SAR system.

#### Vegetation

Sowing density, canopy cover fraction and GAI, plant height, plant development
stage, wet biomass and vegetation dry matter content were measured by the team
from UCLouvain to characterize the vegetation canopy. The measurement
protocols for maize and winter wheat are similar, but adaptations were applied
in consideration of the specificities of both crops. Measurements were
recorded starting in May (F1) until the harvest, which occurred at different
dates in the sampled fields, i.e., between June (F2) and end of July (F3) in
winter wheat and between end of July (F3) and September (F5) in maize.

In each field, three homogeneous and representative plots were determined
before the campaign based on high-resolution Pleiades images and normalized
difference vegetation index (NDVI) profiles derived from them. The plots were
marked with flags and geolocalized at the beginning of the growing season.
They were named after the field they belong to with the letter a, b, or c
attached, e.g., M2a for the first plot of the maize field M2. Field
measurements of vegetation were made in the center of these plots around each
airborne SAR acquisition. They were chosen to be at least
$30\text{\,}\mathrm{m}$ from the edges of the field and from each other. The
plots were squares with a side length of $15\text{\,}\mathrm{m}$ in maize
fields and for winter wheat the side length was $25\text{\,}\mathrm{m}$, i.e.,
the distance between two tractor lines. Correct use of the values provided in
the vegetation dataset involves averaging the values of the variables in the
three plots to obtain a value representative of the field.

Plant density was measured once at the beginning of the growing season by
measuring both interline and interplant distances. To obtain the interline
distance in winter wheat fields, the distance between six rows was measured
three times per plot for different rows. It was then divided by five, the
number of intervals between six rows and averaged. In maize fields, row
spacing was measured directly between two rows three times per plot across
different rows. The interplant distance was derived in the same way in both
crops by counting the number of plants in a one-meter length of sowing row
three times per plot.

The canopy cover fraction (FCover), i.e., the fraction of ground surface
covered by green plant material, and the GAI, i.e., half of the total area of
green plant in the canopy per unit of horizontal ground surface area, were
measured by means of digital hemispherical photography (DHP) processed with
the CAN-EYE software [17]. Note that the GAI is the biophysical variable
actually estimated in the remote sensing of the leaf area index (LAI) [18]. In
each plot, ten photos were taken with a nadir view approximately
$1\text{\,}\mathrm{m}$ above the canopy using a system consisting of a camera
equipped with a hemispherical lens mounted on a $3\text{\,}\mathrm{m}$
telescopic L-shaped pole. Between each photo, the operator walked five steps,
or approximately $3.75\text{\,}\mathrm{m}$. The photos were taken each time
before the operators entered the plot to avoid altering the measurement.

The phenological development of the crops was reported on the BBCH scale [19].
Only one value was recorded per plot–in case plants in the same plot were at
different stages of development, the BBCH stage of the majority of the plants
was selected.

The height of the plants was measured with a ruler from the ground level to
its highest part, i.e., leaf, flower, ear or panicle depending on the crop and
the stage of development, without extending it manually. For each run, nine
plants were measured in each plot to derive a field average.

The wet biomass and the dry matter content of the crop vegetation were
measured by destructive sampling. In each plot, all the plants along three
$1\text{\,}\mathrm{m}$ rows were cut and then weighed in the field to
determine their total fresh weight. The three harvested rows were randomly
selected diagonally from each other. A subsample of the cut plants was then
randomly selected, weighed and stored in a micro-perforated plastic bag which
was then transported to UCLouvain for oven drying at
$60\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for $72\text{\,}\mathrm{h}$. The
dry weight of the subsample was then measured to obtain the dry matter content
of the plants. There were no weeds in the plots. Pictures of the subsamples in
the oven and of the crop cutting are shown in fig. 8.

#### Soil

The team from Ghent University monitored three soil variables: bulk density,
surface soil moisture and surface roughness. Samples were recorded over the
entire surface of the studied fields.

Bulk density was measured in all fields alongside the first airborne SAR
acquisition (F1) using Kopecky rings. Five samples were taken per field, in
order to compute field average values and within-field standard deviations.
Additional samples were taken if the bulk density changed due to tillage
operations in later flight missions.

Volumetric soil moisture samples were taken in all ten winter wheat and ten
maize fields concurrently with each flight airborne SAR acquisition. During
the first acquisition, in May (F1), no samples could be taken in fields W8, W9
and W10, because tillage operations took place during the acquisition time.
Soil moisture was monitored using Time Domain Reflectometry (TDR) sensors with
$11\text{\,}\mathrm{cm}$ rods. At least ten locations per field were
monitored, with three repetitions per location. Field average volumetric soil
moisture was then calculated, by averaging all measurements within each
reference plot.

A pin profilometer, shown in fig. 8, of $1\text{\,}\mathrm{m}$ length with a
spacing of $1\text{\,}\mathrm{cm}$, was used to measure soil surface
roughness. Roughness samples were taken in all ten winter wheat fields
concurrently with every flight campaign, except for field W7 on F4, in August,
when tillage operations were taking place at the time of sampling. Surface
roughness samples in maize fields could only be taken on F1 and F2. On F3, the
maize vegetation was too dense to take samples, resulting in missing values.
Between July (F3) and August (F4) campaigns, M1, M6 and M8 were harvested,
which made it possible to acquire roughness measurements on F4 and F5. Between
F4 and F5, three additional maize plots were harvested (M3, M4 and M9). This
way, six maize plots could be sampled on F5. The roughness measures were taken
in two directions, i.e., along and across the direction of tillage. Pictures
of the pin profilometer in both directions were taken and then digitized to
correct for tilted pictures and to allow determination of correlation lengths
($l$) and root-mean-square (RMS) heights ($s$). Both were calculated according
to the procedure described by Davidson et. al[20]. $s$ was calculated as

$s=\sqrt{\frac{\sum_{i=1}^{N}z^{2}_{i}}{N-1}},$ (1)

with $N$ the number of profile points and $z_{i}$ the surface height of the
profile points. To estimate $l$, the normalized autocorrelation function
$A(\tau)$ was first defined as

$A(\tau)=\frac{\sum_{i=1}^{N-k}z_{i}z_{i+k}}{\sum_{i=1}^{N}z_{i}^{2}},$ (2)

then $l$ was obtained by linearly interpolating between two lags, $\tau_{1}$
and $\tau_{2}$, that bracket $A(\tau)=e^{-1}$ as

$l=\tau_{1}+(e^{-1}-A(\tau_{1}))\frac{\tau_{2}-\tau_{1}}{A(\tau_{2})-A(\tau_{1})}.$
(3)

During each acquisition, at least five profiles per field and along each
direction were taken, and for each field that was sampled, the average and
standard deviation of the correlation length and root-mean-square height were
determined.

## Data Records

The BELSAR dataset is available upon request at https://doi.org/10.5270/ESA-
bccf2d9. In it, the SAR, vegetation and soil data are supplied in several
files that have been combined into an integrated dataset available at
https://figshare.com/s/e6e480924a1021a42028.

### SAR data

The airborne SAR images can be found in the RadarData folder. The SAR data are
stored in $320$ NetCDF files—$2$ sensors (SAR, i.e., monostatic, and BISAR,
i.e., bistatic) $\times$ $5$ flights missions (F1, F2, F3, F4, and F5)
$\times$ $4$ flight tracks (A, Z, B, and Zs) $\times$ $2$ bistatic
configurations (ATI and XTI) $\times$ $4$ polarizations (HH, HV, VH, and
VV)—and delivered with all ancillary metadata that are necessary for further
processing and analysis, including antenna pattern, navigation data, digital
elevation model, and position of the focused pixels in geographical
coordinates (WGS84) [21].

The flight track corresponding to each SAR acquisition is provided in
BELSAR_airborne_acquisitions_map_track_datetime.csv.

### In situ data

The vegetation and soil datasets can be found in the Insitu folder, in which
the main files are:

  * •

BELSAR_agriculture_database.xlsx

    * –

BelSAR_maize tab – Vegetation biophysical variables recorded in each plot of
each maize field

    * –

BelSAR_wheat tab – Vegetation biophysical variables recorded in each plot of
each winter wheat field

  * •

BELSAR_soil_bulkdensity_database.xlsx

    * –

Field_average_bulkdensity tab – Mean bulk density for each field
[$\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$]

    * –

Field_std_dev_bulkdensity tab – Standard deviation of the bulk density for
each field [$\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$]

  * •

BELSAR_soil_moisture_database.xlsx

    * –

Field_average tab – Mean volumetric soil moisture for each field
[$\mathrm{\char 37\relax}$]

    * –

Field_std_dev tab – Standard deviation of the volumetric soil moisture for
each field [$\mathrm{\char 37\relax}$]

    * –

Raw_data tab – Overview of the raw data: latitude, longitude (WGS84),
volumetric soil moisture [$\mathrm{\char 37\relax}$], period
[$\mathrm{\SIUnitSymbolMicro s}$], attenuation and permittivity of all TDR
samples. For F3, F4 and F5, the latter three can be missing due to sensor
failure.

  * •

BELSAR_soil_roughness_database.xlsx

    * –

Field_average_corr_length tab – Mean correlation length for each field
[$\mathrm{cm}$]

    * –

Field_average_RMSheight tab – Mean root-mean-square height for each field
[$\mathrm{cm}$]

    * –

Field_std_dev_corr_length tab – Standard deviation of the correlation length
for each field [$\mathrm{cm}$]

    * –

Field_std_dev_RMSheight tab – Standard deviation of the root-mean-square
height for each field [$\mathrm{cm}$]

### Integrated dataset

To facilitate their exploitation, the SAR and in situ datasets have been
integrated into a single dataset, BELSAR_fields_integrated_db.csv. The
integrated dataset includes zonal statistics of the radar measurements as well
as the bio- and geophysical variables of soil and vegetation for each maize
and winter wheat field. Each entry in the table corresponds to a given field
and image in the SAR dataset. The codes used to generate the integrated
dataset are provided in the same folder.

## Technical Validation

### SAR data calibration

The SAR data were geometrically, radiometrically, and polarimetrically
calibrated based on the response of the four corner reflectors deployed in
track Zs. Figure 7 shows the polarimetric RGB composite image in the vicinity
of the corner reflectors. Their responses show that the resolution of the
images are within $1\text{\,}\mathrm{m}$ resolution in azimuth and
$4\text{\,}\mathrm{m}$ in slant range, as intended, and that the corners are
correctly geolocated, with an accuracy of $0.75\text{\,}\mathrm{m}$. Their
polarimetric signatures and impulse response function also indicated a good
isolation between the polarization channels at antenna level. Cross-talk was
therefore considered negligible, as also attested by their appearance as
yellow spots on the RGB composite, i.e., only HH and VV response with no
significant imbalances. As for the radiometric calibration of the bistatic
data in the ATI flight configuration, however, given that the corner
reflectors were not visible on the images, their radiometric offset was
determined using monostatic ATI and bistatic XTI data. This may have led to an
imbalance between bistatic and monostatic data, or from one flight to the
next, in the bistatic ATI data. A complete quality assessment can be found in
the BELSAR-Campaign project final report [10].

### Interferometric SAR

The potential of the airborne data for interferometric SAR (InSAR) was
affected by very low interferometric coherence. Strong temporal coherence
losses were observed in double-pass monostatic pairs, i.e., between the
different flight missions, together with a significant geometric decorrelation
due to large orthogonal baselines as compared to the critical baseline.
Coherence was also low for single-pass bistatic interferometric pairs, despite
the short perpendicular baselines. The short signal bandwidth, restricted to
$50\text{\,}\mathrm{MHz}$ by the BIPT, has certainly played a significant role
on geometric decorrelation. This effect was strengthened by the relatively low
flight altitude. Furthermore, in addition to temporal and geometric
decorrelation, coherence might have been affected, in bistatic configuration,
by a lack of synchronization resulting from using two different clocks for the
transmitter and receiver systems, compared with a monostatic configuration
where the unique transmitter-receiver system operates with absolute time and
phase references. The consequences of mis-synchronization errors are
positioning and phase errors in the output SLC images, giving rise to
additional azimuthal fringes in the interferogram with a detrimental effect on
coherence. A solution to this last issue would be to implement a
synchronization algorithm that minimizes the impact of residual plane motion
and mitigates the fringe mis-synchronization by a phase calibration. A
multisquint based correction algorithm is proposed by de Macedo et al. [22] to
solve this.

### Vegetation and soil measurements

#### Vegetation

The vegetation measurements were performed in accordance with the guidelines
laid down by JECAM [23]. Violin plots of in situ measured vegetation
biophysical variables, i.e., phenological development stage (BBCH), plant
height, GAI, FCover, wet biomass, and dry matter content, are shown in Figure
9.

Maize was observed between BBCH stages $15$ and $89$, i.e. from the leaf
development stage (five leaves) to the fully ripe stage, and winter wheat
between stages $59$ and $76$, i.e. from the end of the heading to medium milk
stage. Few data, in a narrow range of values, are available on winter wheat
because the campaign started late in an unusually warm and dry growing season,
leading to an early maturation and harvest.

The mean plant height observed in the different fields ranges from
$0.42\text{\,}\mathrm{m}$ (with a standard deviation of
$0.08\text{\,}\mathrm{m}$) to $2.89\text{\,}\mathrm{m}$
($0.09\text{\,}\mathrm{m}$ standard deviation) in maize. These values are in
accordance with maize plant height measurements over a loamy test site in
Belgium [24]. For winter wheat plant height ranges from
$0.77\text{\,}\mathrm{m}$ ($0.09\text{\,}\mathrm{m}$ standard deviation) to
$0.78\text{\,}\mathrm{m}$ ($0.09\text{\,}\mathrm{m}$ standard deviation).

The mean FCover and GAI over all maize fields range respectively from
$0.14\text{\,}\mathrm{m}^{2}\mathrm{m}^{-2}$
($0.71\text{\,}\mathrm{m}^{2}\mathrm{m}^{-2}$ standard deviation) on F1 to
$0.55\text{\,}\mathrm{m}^{2}\mathrm{m}^{-2}$
($0.30\text{\,}\mathrm{m}^{2}\mathrm{m}^{-2}$ standard deviation) on F4 with a
maximum at $0.65\text{\,}\mathrm{m}^{2}\mathrm{m}^{-2}$
($0.10\text{\,}\mathrm{m}^{2}\mathrm{m}^{-2}$ standard deviation) on F3 and
from $0.36\text{\,}\mathrm{m}^{2}\mathrm{m}^{-2}$
($0.14\text{\,}\mathrm{m}^{2}\mathrm{m}^{-2}$ standard deviation) on F1 to
$3.0\text{\,}\mathrm{m}^{2}\mathrm{m}^{-2}$
($0.24\text{\,}\mathrm{m}^{2}\mathrm{m}^{-2}$ standard deviation) on F4, the
latter also being the maximum mean GAI value. In winter wheat, both ranges are
very limited, from $0.78\text{\,}\mathrm{m}^{2}\mathrm{m}^{-2}$
($0.06\text{\,}\mathrm{m}^{2}\mathrm{m}^{-2}$ standard deviation) to
$0.79\text{\,}\mathrm{m}^{2}\mathrm{m}^{-2}$
($0.04\text{\,}\mathrm{m}^{2}\mathrm{m}^{-2}$ standard deviation) for the
FCover and from $4.41\text{\,}\mathrm{m}^{2}\mathrm{m}^{-2}$
($0.75\text{\,}\mathrm{m}^{2}\mathrm{m}^{-2}$ standard deviation) to
$4.49\text{\,}\mathrm{m}^{2}\mathrm{m}^{-2}$
($0.42\text{\,}\mathrm{m}^{2}\mathrm{m}^{-2}$ standard deviation) for the GAI.
The small range of values for winter wheat were expected given the late start
of the campaign, which implies that winter wheat was already close to
maturation on F1. These GAI values for winter wheat at maturation stage, are
in line with field-observed GAI over a test site in Wallonia (southern part of
Belgium) [25].

The mean of dry matter content over all fields ranges from
$11.18\text{\,}\mathrm{\char 37\relax}$ ($1.68\text{\,}\mathrm{\char
37\relax}$ standard deviation) to $40.57\text{\,}\mathrm{\char 37\relax}$
($2.18\text{\,}\mathrm{\char 37\relax}$ standard deviation) in maize and from
$27.01\text{\,}\mathrm{\char 37\relax}$ ($1.96\text{\,}\mathrm{\char
37\relax}$ standard deviation) to $32.42\text{\,}\mathrm{\char 37\relax}$
($6.51\text{\,}\mathrm{\char 37\relax}$ standard deviation) in winter wheat.
The maize was not let to dry further because it was intended for silage.

The mean wet biomass over all maize fields ranges from
$0.24\text{\,}\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{m}^{2}$
($0.14\text{\,}\mathrm{k}\mathrm{g}\mathrm{m}^{-2}$ standard deviation) on F1
to a maximum mean of $9.01\text{\,}\mathrm{k}\mathrm{g}\mathrm{m}^{-2}$
($0.56\text{\,}\mathrm{k}\mathrm{g}\mathrm{m}^{-2}$ standard deviation) on F4,
in line with values found in the study of Blaes et al. [24] over a maize field
in a loamy test site. For winter wheat fields, the mean wheat biomass ranges
from $4.53\text{\,}\mathrm{k}\mathrm{g}\mathrm{m}^{-2}$
($0.77\text{\,}\mathrm{k}\mathrm{g}\mathrm{m}^{-2}$ standard deviation) on F1
to a maximum mean of $4.61\text{\,}\mathrm{k}\mathrm{g}\mathrm{m}^{-2}$
($0.72\text{\,}\mathrm{k}\mathrm{g}\mathrm{m}^{-2}$ standard deviation) on F2.

#### Soil

Figure 10 shows violin plots for the soil geophysical variables, i.e., soil
moisture, bulk density, correlation length across, correlation length along,
root-mean-square height across, and root-mean-square height along, for maize
and winter wheat fields.

Bulk density was measured in all winter wheat and maize fields during the
first flight campaign (F1), with a mean bulk density of
$1.27\text{\,}\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$ (with an average within
field standard deviation of
$0.079\text{\,}\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$) for the winter wheat
fields and $1.24\text{\,}\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$ (with an
average within field standard deviation of
$0.065\text{\,}\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$) for the maize fields.
Similar bulk density values have been reported in the study of van der Bolt et
al. [26] where agricultural fields in Flanders (Belgium) were sampled,
including sites with a texture of silt loam. Due to tillage operations that
took place between F3 and F4 on the winter wheat fields (except W7),
additional soil samples were taken during the field campaign coincident with
F4 to determine bulk density values after tillage operations. On F5,
additional samples were taken in winter wheat fields W2 and W7 and in maize
fields M1, M3 and M4. The plots show that bulk density generally decreases
after tillage operations.

The field average volumetric soil moisture values range from
$3.0318.94\text{\,}\mathrm{vol\%}$ for winter wheat fields and from
$3.6518.76\text{\,}\mathrm{vol\%}$ for maize fields. These values are within
the range of soil moisture values reported in the study of Choker et al. [27],
where in situ soil moisture measurements over numerous agricultural plots in
Europe (mainly France, Belgium, and Italy) were acquired. Note that 2018 was
marked by an exceptional dry summer in Belgium, which is depicted in the low
soil moisture values, especially on F3. The range of within field standard
deviations for winter wheat fields is $0.684.60\text{\,}\mathrm{vol\%}$ and
for maize $1.125.36\text{\,}\mathrm{vol\%}$.

The field average root-mean-square heights (correlation length) was measured
along and across the direction of tillage. Roughness along the direction of
tillage is comparable between winter wheat and maize fields, with a range of
respectively $0.411.46\text{\,}\mathrm{cm}$ ($1.285.34\text{\,}\mathrm{cm}$)
for winter wheat and $0.321.23\text{\,}\mathrm{cm}$
($1.424.30\text{\,}\mathrm{cm}$) for maize. Across the direction of tillage,
the difference is slightly larger, with a range of
$0.552.63\text{\,}\mathrm{cm}$ ($1.469.69\text{\,}\mathrm{cm}$) for winter
wheat and $0.801.55\text{\,}\mathrm{cm}$ ($3.169.07\text{\,}\mathrm{cm}$) for
maize. Especially during crop growth stages, the root-mean-square height
across the direction of tillage is substantially higher for maize compared to
winter wheat, with an average value of $1.23\text{\,}\mathrm{cm}$
($6.25\text{\,}\mathrm{cm}$) for maize compared to $0.79\text{\,}\mathrm{cm}$
($4.38\text{\,}\mathrm{cm}$) for winter wheat.

Verhoest et al. [28] summarized possible sources of errors that affect
roughness measurements, of which the limited length of the pin profilometer
and resolution in both vertical and horizontal directions are the main
disadvantages, especially for the estimation of $l$. Roughness measures found
in this study are comparable to the ones estimated by Davidson et al. [20] for
agricultural sites over Europe. A mean $s$ of 0.6 and 1.6 cm (with a standard
deviation of 0.3 and 0.7 cm) was estimated for seedbed and harrowed field
conditions respectively, which is in line with the BELSAR study. In terms of
correlation length, mean values of 3.7 and 3.8 cm (with a standard deviation
of 2.6 and 2.9 cm) for respectively seedbed and harrowed field conditions were
found [20], which is in the range of values found here for both maize and
winter wheat fields after harvest. The correlation length for maize fields
before harvest was substantially larger, which can be explained by the rough
seedbed pattern for maize.

## Usage Notes

The integrated dataset is directly accessible on figshare, while the BELSAR-
Campaign data are available online via FTP upon submission of a data access
request to ESA’s Earth Online service.

With regard to the exploitation of these data, it is advisable to use the
integrated dataset instead of the original data as it contains in one table
both the zonal statistics of the SAR data and the corresponding vegetation and
soil variables for the maize and winter wheat fields. As such, it provides an
analysis-ready dataset, thereby greatly facilitating the handling of the
BELSAR data for, among others, the development of agricultural or hydrological
applications as well as for easy comparison with other comparable datasets.
Note that for certain purposes, it is recommended to apply a negative buffer
to the polygons to avoid edge-of-field effects on the radar signal. In this
case, the polygons delineating the fields would have to be redrawn and the
integrated dataset rebuilt.

## Code availability

The codes used to produce the integrated dataset from SAR, vegetation and soil
data have been uploaded to figshare along with it. These contain a number of
tools that can be easily adapted and reused to use the BELSAR data for other
purposes. To rebuild the integrated dataset from the BELSAR-Campaign data, the
contents of the ESA repository must first be accessed. Then, once downloaded,
running extract_mini_rasters.py will extract zonal statistics from the SAR
data and integrated_dataset.py will match these extracted data to the
corresponding in situ vegetation and soil measurements and generate the
integrated dataset. For test purposes, these two scripts can be run for one or
a series of images by adding their indices, from 0 to 319, as arguments to the
python command, e.g., python extract_mini_rasters.py 1 for the second image
and python extract_mini_rasters.py 0 2 for the first three images.

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## Acknowledgements

This research was conducted in the framework of the BELSAR-Publication project
funded by the STEREO III program of the Belgian Federal Science Policy Office
(BELSPO) under contract SR/00/409.

The authors would like to thank Quentin Vandersteen and Jeroen Claessen for
the collection of the in situ data, Leila Guerriero, Nazzareno Pierdicca, Hugh
Griffiths, and Malcolm Davidson for their participation to the BELSAR advisory
committee, and the Service public de Wallonie for providing access to the Land
Parcel Information System of Wallonia.

## Author contributions statement

J.B., E.T., A.O. and K.A.C.M. collaborated in curating, analyzing and
validating the data, as well as writing the manuscript, J.B. and E.T. wrote
the code, J.B., E.T. and K.A.C.M. plotted the figures, N.E.C.V. and P.D.
provided guidance and supervision. All authors reviewed the manuscript.

## Competing interests

The authors declare no competing interests.

## Figures & Tables

Figure 1: BELSAR area of interest, with the ten winter wheat and ten maize
fields in which the in situ measurements were recorded in blue and orange
respectively, and all other agricultural fields inventoried in the Land Parcel
Information System of Wallonia, Belgium, in grey.

Figure 2: MetaSensing BV’s SAR system on-board one of the two CESNA Gran
Caravan airplanes (left) and a picture of one airplane taken from the other
while flying in the across-track (XTI) configuration (middle), and a corner
reflector in the field (right). Figure 3: Nadir view of the theoretical XTI
(left) and ATI (right) bistatic flight configurations. Figure 4: Actual
vertical, across-track, and along-track baselines between the two SAR sensors
over Zs for the first (F1; left) and second (F2; right) flight missions in the
XTI (top) and ATI (bottom) bistatic configurations. Figure 5: Flight track
design in alternating opposite directions and corner reflectors localization
within the subset area, Zs. The arrows indicate the trajectory of the left-
looking sensors for each track.

[width=.65]Images/flowchart

Figure 6: SAR processing chain Figure 7: Polarimetric RGB composite image
around the corner reflectors (indicated by the arrows), where red, green and
blue correspond to HH, VV, and HV, respectively.

Figure 8: Oven drying of the plant subsamples (left), crop cutting along a $1\text{\,}\mathrm{m}$ row (middle) and pin profilometer (right). Figure 9: Violin plots of vegetation biophysical variables measured in situ, i.e., phenological development stage (BBCH; top-left), plant height (top-right), green area index (GAI; middle-left), canopy fraction cover (FCover; middle-right), wet biomass (bottom-left), and dry matter content (bottom-right), for maize (orange) and winter wheat (blue) for all flights (F1-F5). The number of fields that were monitored during each flight is denoted by $N$. Figure 10: Violin plots of soil geophysical variables measured in situ, i.e., soil moisture (top-left), bulk density (top-right), correlation length across (middle-left) and along (middle-right) tillage direction, root-mean-square (RMS) height across (bottom-left) and along (bottom-right) tillage direction, for maize (orange) and winter wheat (blue) for all flights (F1-F5). The number of fields that were monitored during each flight is denoted by $N$.  Flight mission | Flight date | In situ measurement date | Maize stage | Winter wheat stage
---|---|---|---|---
F1 | 30/05 | 30/05, 31/05, and 05/06 | Leaf development | End of heading
F2 | 20/06 | 21/06, 22/06, and 26/06 | Stem elongation | Medium milk
F3 | 30/07 | 01/08 and 02/08 | Dough stage | Bare soil
F4 | 28/08 | 29/08 and 30/08 | Fully ripe (only M2 and M5) | Bare soil
|  |  | and bare soil with stalks |
F5 | 10/09 | 10/09 and 11/09 | Bare soil with stalks | Bare soil
Table 1: Dates (in day-month-2018 format) of airborne SAR acquisitions and in
situ measurements of vegetation and soil variables, with the corresponding
development stages of both crops.