# Remote Sensing Change Detection using Denoising Diffusion Probabilistic
Models

Wele Gedara Chaminda Bandara, Nithin Gopalakrishnan Nair, Vishal M. Patel
Johns Hopkins University
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Project page: $\tt https://github.com/wgcban/ddpm-cd$

###### Abstract

Human civilization has an increasingly powerful influence on the earth system,
and earth observations are an invaluable tool for assessing and mitigating the
negative impacts. To this end, observing precisely defined changes on Earth’s
surface is essential, and we propose an effective way to achieve this goal.
Notably, our change detection (CD) method proposes a novel way to incorporate
the millions of off-the-shelf, unlabeled, remote sensing images available
through different earth observation programs into the training process through
denoising diffusion probabilistic models. We first leverage the information
from these off-the-shelf, uncurated, and unlabeled remote sensing images by
using a pre-trained denoising diffusion probabilistic model and then employ
the multi-scale feature representations from the diffusion model decoder to
train a lightweight CD classifier to detect precise changes. The experiments
performed on four publically available CD datasets show that the proposed
approach achieves remarkably better results than the state-of-the-art methods
in F1, IoU and overall accuracy. Code and pre-trained models are available at:
$\tt https://github.com/wgcban/ddpm-cd$

Figure 1: Images generated from the diffusion model trained on off-the-shelf
remote sensing images. The generated images contain objects that we commonly
see in real remote sensing images, such as buildings, trees, roads,
vegetation, water surfaces, etc., demonstrating the powerful ability of the
diffusion models to extract key semantics that can be further used in remote
sensing change detection.

##  1 Introduction

The results of a recent poll published in The Washington Post shows that
people around the world are increasingly concerned about climate change and
are aware of human activities related to global climate than ever before
servey . With global climate measures global_climate_mes such as atmospheric
carbon dioxide concentration, global average surface temperature, and sea
level rise, it is now possible to observe changes at global scale due to
climate change. On the other hand, accurate predictions of local land cover
changes such as changes due to deforestation, floods, forest fires,
urbanization, and use of land for agriculture are also of critical importance
as an essential climate variable bojinski2014concept and for climate change
mitigation programs such as UN-REDD+.

Thanks to various satellite programs of the National Aeronautics and Space
Administration (NASA), the European Space Agency (ESA), the United States
Geological Survey (USGS), and many other government/private entities around
the world, we now witnesses of more than 24 million publically available
satellite images from the last 37 years as of 2021, making remote sensing
images as the effective and reliable data source for monitoring local changes
in the Earth’s surface.

Even though we have access to millions of remote sensing images to monitor
local changes, predicting changes of interest is a challenging task to tackle
through computer vision due to the fact that the high diversity of data,
seasonal variations of Earth’s surface, different levels of solar
illumination, different imaging styles and sensors, spectral variability, etc
wen2021change ; wang2021loveda . In addition, annotating changes in remote
sensing images requires comparison of two or more images in the time series
data by the human experts, which is indeed a time consuming and expensive
process, making the available CD datasets to have less annotated training
images (mostly around a couple thousand images) available to train the CD
model.

Therefore, an effective and robust CD model should not only focus on
exploiting the information from limited annotated data available for training
the model, but also focus on exploiting as much information as possible from
millions of freely available, unlabeled and uncurated remote sensing images to
improve the accuracy and robustness of the CD. How to effectively harness the
information from these large collections of uncurated remote sensing images is
a long-standing question, and much effort has been devoted to answering this
question in the past.

Remote sensing pre-training wang2022empirical is one of the most widely
adopted techniques for this purpose. With the emergence of large-scale areal-
scene recognition datasets such as MillionAID long2021creating , fMoW
christie2018functional and BigEarthNet sumbul2019bigearthnet , one can pre-
train the encoder of the CD model to capture the key feature representations
(i.e., semantics) of remote sensing images. However, the need for image-level
class labels hinders its ability to train on on millions of unlabeled and
uncurated remote sensing images, which has motivated the researchers to
explore self-supervised pre-training methods chen2020simple ; chen2020improved
; xu2021regioncl such as SeCo manas2021seasonal , where the presence of
seasonal changes in aerial scenes is used to ensure consistency between
positive samples.

Figure 2: The diffusion model as a directed graphical model ho2020denoising .

Different from the above pre-training techniques, in this paper, we show that
the diffusion probabilistic models sohl2015deep ; ho2020denoising ;
nichol2021improved can also be used to extract key semantics of remote sensing
images. Diffusion models are directed graphical models parameterized by a
Markov chain (see figure 2). We train a diffusion model using a variational
inference procedure on a large collection of off-the-shelf remote sensing
images to generate images that are closer to real images over the finite time.
Figure 1 shows some sample images generated from the diffusion model. We can
see that the generated images are very closer to the actual remote sensing
images and contains key objects find in remote sensing images, such as
buildings, trees, vegetation, roads, water surfaces, etc. Therefore, diffusion
model is powerful enough to extract key features of a given image. While any
generative model (such as VAEs kingma2013auto ; kingma2019introduction or GANs
goodfellow2014generative ; souly2017semi ) can be trained to learn key
semantics in a similar way, based on our experiments, diffusion models have
been shown to produce better feature representations. From the experiments, we
observe that the representational power oF VAE and GANs are limited compared
to diffusion models, resulting in lower CD performance (see the supplementary
material for more details). Since diffusion models can be used to extract
better semantics of given a image, we fine-tune a light-weight CD classifier
with the limited pixel-level change labels by utilizing multi-scale semantics
(i.e., deep feature representations) from the diffusion model as its inputs.
Experiments conducted on multiple CD datasets show that the proposed method
achieves significant performance improvement over the state-of-the-art (SOTA)
CD methods, demonstrating that diffusion probabilistic models can be used not
only to sample images closer to the training data distribution, but also to
extract deep semantics that are useful for low-level vision tasks.

##  2 Background

###  2.1 Diffusion Models

Diffusion probabilistic models ho2020denoising ; sohl2015deep ;
saharia2021image ; nair2022image ; perera2022sar belong to the class of
generative models where, once trained, we can use the model to generate high-
quality images that are closer to the training data distribution. Unlike other
generative models such as Generative Adversarial Networks (GANs), Variational
Autoencoders (VAEs), autoregressive models and flows, diffusion models are
efficient to train and have been widely adopted in several applications,
including super-resolution saharia2021image , debluring whang2021deblurring ,
segmentation wolleb2021diffusion ; jiang2018difnet ; baranchuk2021label ,
image colorization song2020score , inpainting song2020score , and semantic
editing rombach2021high . However, there is still no demonstration of the
application of diffusion models in remote sensing applications despite its
high potential and great success in other machine vision applications. In this
work, we demonstrate that remote sensing CD can also benefit from diffusion
models, opening a new direction to explore remote sensing applications in the
future. In the following sections, we provide a brief overview of the
diffusion framework.

#### Gaussian Diffusion Process.

Diffusion models learns the training data distribution $p(x_{0})$ by
performing variational inference on a Markovian process with $T$ number of
timesteps. The diffusion process consists of a forward process as well as a
backward process. In the forward process, we gradually add Gaussian noise to
the clean image $x_{0}\sim p(x_{0})$ until the image is completely destroyed
resulting in an isotropic gaussian distribution
$\mathcal{N}(\boldsymbol{0},\boldsymbol{I})$. The noising operation at each
time step $t$ in the forward process is defined as follows:

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

where $(x_{0},x_{1},\cdots,x_{T})$ denotes the $T$-step Markov chain (see Fig.
2), $\alpha_{1:T}=(\alpha_{1},\cdots,\alpha_{T})$ is the noise schedule that
controls the variance of noise added at each step. In the reverse process a
sequence of small denoising operations is performed using a neural network to
obtain back the original image. During each step of the reverse process, we
perform a denoising operation to the image $x_{t}$ to obtain back $x_{t-1}$.
For this, a neural network $f_{\theta}$ is used to model the parameters of the
reverse distribution
$p(x_{t-1}|x_{t}):=\mathcal{N}(x_{t-1}:\mu_{\theta}(x_{t},t),\Sigma_{\theta}(x_{t},t))$.
The parameters for this network $\theta$ is obtained by minimizing the KL
divergence between the forward and reverse distributions over all timesteps.
During optimization, sampling from the distribution $q(x_{t}|x_{t-1})$
requires knowledge of $x_{t-1}$ which requires a sequence of operations to be
computed as per equation 2. The marginal distribution of $x_{t}$ given $x_{0}$
can be derived by marginalizing out the intermediate latent variables as:

$q(x_{t}|x_{0})=\mathcal{N}(x_{t};\sqrt{\gamma_{t}}x_{0},(1-\gamma_{t})\boldsymbol{I}),$
(2)

where $\gamma_{t}=\prod_{i=1}^{t}\alpha_{i}$. Hence simpliying the calculation
procedure. Furthermore, Sohl et al.sohl2015deep has showed minimizing the KL
divergence between the forward and reverse distributions could be simplified
by using the posterior distribution $q(x_{t}|x_{t-1},x_{0})$ instead of the
forward distribution $q(x_{t}|q{x_{t}]-1]})$. Under the Markovian assumptions,
Using equations 1 and 2, the posterior distribution can be derived as:

$q(x_{t-1}|x_{t},x_{0})=\mathcal{N}(x_{t-1},\mu,\sigma^{2}\boldsymbol{I}),$
(3)

where

$\mu(x_{t},x_{0})=\frac{\sqrt{\gamma_{t-1}}(1-\alpha_{t})}{1-\gamma_{t}}x_{0}+\frac{\sqrt{\alpha_{t}}(1-\gamma_{t-1})}{1-\gamma_{t}}x_{t}\text{
and }\sigma^{2}=\frac{(1-\gamma_{t-1})(1-\alpha_{t})}{1-\gamma_{t}}.$ (4)

This posterior distribution is further utilized when parameterizing the
reverse Markov chain and formulating a variational lower bound on the log-
likelihood of the reverse chain. While optimizing, the covariance matrix for
both the distributions $q(x_{t-1}|x_{t},x_{0})$ and $p(x_{t-1}|x_{t})$ is
usually considered the same and the network $f_{\theta}$ predicts the mean of
the distribution.

#### Optimizing the Diffusion Model.

The denoising model $f_{\theta}$ takes as input the noisy image ${x_{t}}$:

$x_{t}=\sqrt{\gamma}x_{0}+\sqrt{1-\gamma}\epsilon,\text{ where
}\epsilon=\mathcal{N}(\boldsymbol{0},\boldsymbol{I})$ (5)

and the timestep $t$ and aims to recover the mean of the posterior
distribution. Recent work by Ho et al.ho2020denoising has simplified this
training objective for optimizing the parameters of the network and proposed a
simplified objective defined by:

$\mathbb{E}_{x_{0},\epsilon}\left\|f_{\theta}(\tilde{x},t)-\epsilon\right\|_{2}^{2},\text{
where }\epsilon=\mathcal{N}(\boldsymbol{0},\boldsymbol{I}).$ (6)

#### Inference (i.e., sampling).

The inference is defined as the reverse Markovian process which goes in the
reverse direction of the forward diffusion process. The inference starts from
Gaussian noise $x_{T}$, and $x_{t}$ at each timestep $t$ is iteratively
denoised to get back $x_{t-1}$ according to:

$x_{t-1}\leftarrow{}\frac{1}{\sqrt{\alpha_{t}}}\left(x_{t}-\frac{1-\alpha_{t}}{\sqrt{1-\gamma_{t}}}f_{\theta}(x_{t},t)\right)+\gamma_{t}\boldsymbol{z},\text{
where }\boldsymbol{z}=\mathcal{N}(\boldsymbol{0},\boldsymbol{I}),$ (7)

for $t=T,\cdots,1$.

###  2.2 Remote sensing CD

Although there exists a wide-variety of classical methods to detect the
changes from multi-temporal images such as ImageDiff, ImageRatio ImageRatio ,
ImageRegr ImageRegr , CVA cva , DPCA dpca , MAD MAD , IRMAD IRMAD and PCDA
PCDA , the current research on remote sensing CD has largely reshaped by deep
learning due to its powerful discriminative ability asokan2019change . The
existing deep learning based remote sensing CD frameworks are mainly based on
convolutional networks (such as FC-EF daudt2018fully , FC-Siam-conc
daudt2018fully , and FC-Siam-diff daudt2018fully ; bandara2022revisiting , and
DS-IFN zhang2020deeply , DASNet 9259045 ), attention modules (such as ADS-Net
WANG2021102348 , and DDCNN 9254128 ), transformer networks (such as BIT
chen2021remote and ChangeFormer bandara2022transformer ), or the combination
of them (such as zhang2022swinsunet ). During the early stage of CD research,
the backbones of these models (i.e., the feature extractor) are initialized
with ImageNet krizhevsky2012imagenet pre-trained weights, due to the limited
availability of large-scale aerial scene classification datasets. However, the
large domain gap between real and aerial images wang2022empirical , the
emergence of large-scale aerial scene classification datasets (like MillionAID
long2021creating , fMoW christie2018functional and BigEarthNet
sumbul2019bigearthnet ), and the new findings on unsupervised pre-training
techniques such as contrastive pre-training manas2021seasonal motivated and
made it possible to pre-train the backbone on aerial images which resulted in
faster convergence in the training and significant boost in CD performance
than before wang2022empirical . However, we see that there is still a quite
large room for improvements because current pre-training methods either
require aerial scene classification datasets for supervised pre-training (like
BigEarthNet sumbul2019bigearthnet ) or paired multi-temporal images for self-
supervised pre-training (like SeCo manas2021seasonal ), which limits their
ability to harness information from millions off-the-shelf unlabeled remote
sensing images.

In this study, for the first time, we show that diffusion probabilistic models
are extremely useful in overcoming the above limitations in current remote
sensing pre-training methods. More specifically, we train a diffusion model on
millions of off-the-shelf remote sensing images to learn the key semantics of
aerial images, and then use the multi-scale features from the diffusion
decoder as the inputs to train a CD classifier under the limited availability
of pixel-wise labels.

Figure 3: Multi-scale feature representations from the diffusion decoder for
different noise levels.

##  3 Proposed approach for CD

The proposed CD approach consists of two steps: (1) Training the diffusion
model on a large collection of off-the-shelf remote sensing images to learn
the key semantics in aerial images, and (2) Training a CD classifier on
limited pixel-wise change labels, that takes the deep feature representations
from the diffusion decoder as the inputs and outputs the change prediction
maps.

###  3.1 Pre-training the diffusion model on off-the-shelf remote sensing data

In order to train the diffusion model, we first collect a large amount of
3-channel (i.e., RGB) remote sensing images ($\sim$ 1M) of Sentinel-2
drusch2012sentinel patches without any human supervision. More specifically,
we use Google Earth Engine gorelick2017google to process and download imagery
patches from around 200K locations around the world manas2021seasonal , where
each patch covers an earth resolution of approximately $2.65\times 2.65$ km.
Next, we train the SOTA unconditional U-Net like saharia2021image diffusion
model (see Fig. 4) on the collected dataset until it can generate images
closer to real aerial images, as shown in Fig. 1.

###  3.2 Extracting representations of pre-change and post-change images from
diffusion model

The pre-trained diffusion model is then utilized to extract the feature
representations of pre-change ($I_{a}$) and post-change ($I_{b}$) images.
Specifically, we first create the noisy version of the given input image
following Eqn. (5), and then processes it through the diffusion model to
obtain the feature representations (see Fig. 4) of size $h\times w$,
$h/2\times w/2$, $h/4\times w/4$, $h/8\times w/8$, and $h/16\times w/16$.
Figure 3 shows such multi-level feature representation for a given input image
for different amount of noise levels added to the input (i.e., $t$). From the
figure, we can see that the diffusion models are also capable of extracting
the hierarchical feature representations (i.e., semantics) of the input
images, where the features at $h\times w$ and $h/2\times w/2$ scales
represents simple patterns like edges and gradients, while features at
$w/4\times w/4$, $w/8\times w/8$, and $w/16\times w/16$ scales represent more
abstract features. Furthermore, unlike other deep learning architectures, we
can also obtain different augmented versions of the feature representations by
changing the amount of noise added to the input that can be further utilized
to train a robust and generalized CD head under limited pixel-level change
labels. In an ablation study, we show that incorporating multi-level features
corresponds to different levels of noisy images can significantly improve the
CD performance.

Figure 4: Proposed CD approach. We fine-tune a light-weight CD head which
takes multi-level feature representations from the pre-trained diffusion model
as inputs and outputs change prediction map.

###  3.3 Training the lightweight CD network

Next, we train a lightweight CD head that takes multi-scale deep feature
representations of pre- and post-change images obtained from the diffusion
model as the inputs. Specifically, the network first computes the difference
between the multi-level features of pre- and post-change images, and then pass
them through a channel-spatial attention module roy2018concurrent , followed
by a convolutional classifier to obtain the final change prediction map as
depicted in Fig. 4 . See the supplementary material for more information on
the CD network architecture.

##  4 Experimental setup

###  4.1 Detests and training Details

#### Pre-training the diffusion model:

As we discussed in Sec. 3.1, we train a SOTA unconditional U-Net like
diffusion model saharia2021image on off-the-shelf remote sensing images
collected from Google Earth Engine without any human supervision. Consistent
with saharia2021image ; ho2020denoising , we use the Adam optimizer with a
linear warmup schedule over $10$k training steps, followed by a fixed learning
rate of $1e-5$.

#### Training CD network:

We perform CD experiments on four publically available CD datasets, namely
LEVIR-CD LEVIR , WHU-CD WHU-CD , DSIFN-CD DSIFN , and CDD CDD . Among these,
LEVIR-CD and WHU-CD are building CD datasets, and DSIFN-CD and CDD are general
CD datasets. Following the previous works chen2021remote ;
bandara2022transformer , we utilize the patches of size $256\times 256$ for
training. Given a image, we first extract the multi-level feature
representations from the pre-trained diffusion model and feed these features
as the inputs to the CD network. We optimize the parameters of CD head with
the cross-entropy (CE) loss using the AdamW optimizer, while keeping the
parameters of the pre-trained diffusion model freezed. We use the initial
learning rate of $1e-5$ and linearly decay it to zero over 120 epochs. We
optimize the CD head on val-set of the datasets and report the results on the
test-set. We experiment with different sampling locations (i.e., $t$) in the
noise-schedule to find out the optimal feature representations for CD such as
$t=5,50,100,400,$ and $650$, and we discuss how the CD performance varies with
$t$ in the ablation study.

###  4.2 Performance metrics

In order to compare the CD performance of our model with SOTA methods, we
report F1 and Intersection over Union (IoU) scores with regard to the change-
class as the primary quantitative indices bandara2022revisiting ;
bandara2022transformer ; chen2021remote . Additionally, we report overall
accuracy (OA) to get a global quality of predictions.

##  5 Results and discussion

In this section, we compare the results of our proposed CD method with several
SOTA methods, namely fully-convolutional early-fusion (FC-EF) method
daudt2018fully , fully-convolutional siamese-difference (FC-SD) method
daudt2018fully , fully-convolutional siamese-concatenation (FC-SD) method
daudt2018fully , daul-task constrained siamese network (DT-SCN)
liu2020building , spatial-temporal attention network (STA-Net) chen2020spatial
, densely connected siamese network (SNU-Net) fang2021snunet , bi-temporal
image transformer (BIT) chen2021remote , and transformer-based siamese network
(ChangeFormer) bandara2022transformer . Out of these methods, BIT and
ChangeFormer are latest available SOTA methods that use non-local-attention
networks (i.e., transformers) for remote sensing CD.

###  5.1 Quantitative CD Results

Table 1: The average quantitative results on LEVIR-CD, WHU-CD, DSIFN-CD, and CDD datasets∗. Method |  LEVIR-CD LEVIR |  |  WHU-CD WHU-CD |  |  DSIFN-CD DSIFN |  |  CDD CDD
---|---|---|---|---|---|---|---
F1 |  IoU |  OA |  |  F1 |  IoU |  OA |  |  F1 |  IoU |  OA |  |  F1 |  IoU |  OA
FC-EF daudt2018fully |  83.40 |  71.53 |  98.39 |  |  76.73 |  62.24 |  98.31 |  |  72.17 |  56.45 |  83.37 |  |  66.93 |  50.30 |  93.28
FC-SD daudt2018fully |  86.31 |  75.92 |  98.67 |  |  78.95 |  65.23 |  98.48 |  |  70.55 |  54.49 |  84.13 |  |  70.61 |  54.57 |  94.33
FC-SC daudt2018fully |  83.69 |  71.96 |  98.49 |  |  79.85 |  66.46 |  98.50 |  |  59.71 |  42.56 |  87.57 |  |  75.11 |  60.14 |  94.95
DT-SCN liu2020building |  87.67 |  78.05 |  98.77 |  |  91.43 |  84.21 |  99.35 |  |  70.58 |  54.53 |  82.87 |  |  92.09 |  85.34 |  98.16
STANet chen2020spatial |  87.26 |  77.40 |  98.66 |  |  82.32 |  69.95 |  98.52 |  |  64.56 |  47.66 |  88.49 |  |  84.12 |  72.22 |  96.13
IFNet zhang2020deeply |  88.13 |  78.77 |  98.87 |  |  83.40 |  71.52 |  98.83 |  |  60.10 |  42.96 |  87.83 |  |  84.00 |  71.91 |  96.03
SNUNet fang2021snunet |  88.16 |  78.83 |  98.82 |  |  83.50 |  71.67 |  98.71 |  |  66.18 |  49.45 |  87.34 |  |  83.89 |  72.11 |  96.23
BIT chen2021remote |  89.31 |  80.68 |  98.92 |  |  90.53 |  83.39 |  99.34 |  |  87.61 |  77.96 |  92.30 |  |  88.90 |  80.01 |  97.47
ChangeFormer bandara2022transformer |  90.40 |  82.48 |  99.04 |  |  88.57 |  79.49 |  99.12 |  |  94.67 |  88.71 |  93.23 |  |  94.63 |  89.80 |  98.74
ddpm-CD |  90.91 |  83.35 |  99.09 |  |  92.65 |  86.31 |  99.42 |  |  96.65 |  91.28 |  97.09 |  |  95.62 |  91.62 |  98.98
∗All the values are in %. Higher values of CF1, IoU, and OA indicate good CD
performance. Color convention: best, 2nd-best, and 3rd-best.

Table 1 compares the CD performance of the proposed method with SOTA methods
in terms of F1, IoU, and OA scores. We can see that the proposed method
outperforms existing SOTA methods by a significant margin on all metrics
across all four datasets, empirically validating that diffusion models can
extract key semantics that will be useful for CD.

###  5.2 Qualitative CD Results

In addition to the quantitative results, we also visualize some qualitative
examples to further highlight the effectiveness of the proposed method over
the SOTA methods in Fig. 5, 6, 7 and 8 for LEVIR-CD, WHU-CD, DSIFN-CD and CDD
datasets, respectively. From the visual results also we can clearly see that
the proposed method is able to detect much complex changes accurately than the
existing methods, further demonstrating diffusion model’s ability to extract
key semantics from the images.

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Figure 5: Comparision of different SOTA CD methods on LEVIR-CD dataset: (a)
Pre-change image, (b) Post-change image, (c) FC-EF, (d) FC-Siam-Di, (e) FC-
Siam-Conc, (f) DTCDSCN, (g) BIT, (h) ChangeFormer, (i) ddpm-CD (ours), and (j)
Ground-truth.

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Figure 6: Comparision of different SOTA CD methods on WHU-CD dataset: (a) Pre-
change image, (b) Post-change image, (c) FC-EF, (d) FC-Siam-Di, (e) FC-Siam-
Conc, (f) DTCDSCN, (g) BIT, (h) ChangeFormer, (i) ddpm-CD (ours), and (j)
Ground-truth.

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Figure 7: Comparison of different SOTA CD methods on DSIFN-CD dataset: (a)
Pre-change image, (b) Post-change image, (c) FC-EF, (d) FC-Siam-Di, (e) FC-
Siam-Conc, (f) DTCDSCN, (g) BIT, (h) ChangeFormer, (i) ddpm-CD (ours), and (j)
Ground-truth.

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Figure 8: Comparison of different SOTA CD methods on CDD dataset: (a) Pre-
change image, (b) Post-change image, (c) FC-EF, (d) FC-Siam-Di, (e) FC-Siam-
Conc, (f) DTCDSCN, (g) BIT, (h) ChangeFormer, (i) ddpm-CD (ours), and (j)
Ground-truth.

Table 2: How the CD performance varies with the time step $t$ that we used to extract features from the diffusion model on val-set of LEVIR-CD, WHU-CD, DSIFN-CD, and CDD datasets. Time step $t$ |  LEVIR-CD LEVIR |  |  WHU-CD WHU-CD |  |  DSIFN-CD DSIFN |  |  CDD CDD
---|---|---|---|---|---|---|---
F1 |  IoU |  OA |  |  F1 |  IoU |  OA |  |  F1 |  IoU |  OA |  |  F1 |  IoU |  OA
5 |  89.71 |  81.35 |  99.15 |  |  91.57 |  84.46 |  99.19 |  |  93.87 |  88.39 |  96.09 |  |  91.24 |  83.89 |  91.24
50 |  90.66 |  82.90 |  99.23 |  |  92.74 |  86.47 |  99.31 |  |  94.17 |  88.99 |  96.29 |  |  93.78 |  88.28 |  98.60
100 |  90.50 |  82.65 |  99.21 |  |  92.78 |  86.54 |  99.31 |  |  94.95 |  90.39 |  96.77 |  |  94.32 |  89.25 |  98.72
150 |  90.08 |  81.95 |  99.18 |  |  92.34 |  85.77 |  99.27 |  |  94.59 |  89.74 |  96.54 |  |  94.34 |  89.29 |  98.75
50, 100 |  91.02 |  83.52 |  99.26 |  |  93.09 |  87.07 |  99.34 |  |  94.51 |  89.61 |  96.51 |  |  94.91 |  90.31 |  98.85
50, 100, 400 |  91.26 |  83.92 |  99.28 |  |  93.50 |  87.80 |  99.38 |  |  95.38 |  91.18 |  97.05 |  |  95.64 |  91.64 |  99.00
50, 100, 650 |  91.10 |  83.67 |  99.26 |  |  93.02 |  86.95 |  99.33 |  |  95.07 |  90.62 |  96.87 |  |  95.24 |  90.90 |  98.92
∗All the values are in %. Higher values of F1, IoU, and OA indicate good CD
performance. Color convention: best, 2nd-best, and 3rd-best.

##  6 Ablation study

In this ablation study, we discuss how the CD performance varies with time-
step $t\in[0,T]$ that we used to extract the feature representation from the
diffusion model. To this end, we train the CD classifiers with features
sampled from different time-steps $t$ of the diffusion model to find out best
time-step range that provide best semantics required for the CD task. Table 2
shows how the CD performance varies with the features obtained at
$t=5,50,100,(50\text{ and }100),(50,100\text{ and }400)$ and $(50,100,\text{
and }650)$ used as inputs to train the CD classifier. We observed the best CD
performance across all the datasets when we utilize the feature
representations sampled from the range $t\in[100,400]$. Furthermore, we have
noticed that, concatenating feature representations of multiple time samples
such as $t=50,100$ and $400$ results in further improvement in CD. Therefore,
our final CD classifier takes feature representations sampled at $t=$50, 100,
and 400 as inputs.

##  7 Conclusion

In this paper, we proposed a novel remote sensing CD method based on denoising
diffusion probabilistic models. The proposed method first pre-train an
unconditional denoising diffusion probabilistic model on millions of off-the-
shelf remote sensing images available to the public to learn the key semantics
in the remote sensing images. The pre-trained diffusion model is then utilized
to extract multi-scale feature representations from a given satellite image to
train a light-weight CD classifier. The proposed method differs from existing
deep CD methods mainly in two aspects: (1) The training of diffusion
probabilistic model which serves as the feature extractor in our case does not
require any type of labeled data which is indeed valuable specially for remote
sensing applications due to the fact that most of the publically available
millions of remote sensing data are unlabeled or difficult to annotate. (2)
Unlike existing deep CD algorithms we keep the weights of the trained
diffusion model freezed and only optimize the parameters of lightweight CD
classifier, that helps efficient and fast deployment on new CD applications or
datasets. Above all these advantages of the proposed approach, it also
achieves significantly better performance compared to SOTA CD methods making
it as the new SOTA method for remote sensing CD.

##  8 Additional qualitative results

###  8.1 LEVIR-CD dataset

Figure 8, 8, 8, and 8 show additional qualitative results on LEVIR-CD dataset.

((a))

((b))

((c))

((d))

((e))

((f))

((g))

((h))

((i))

((j))

Figure 8: Comparison of different state-of-the-art CD methods on LEVIR-CD
dataset: (a) Pre-change image, (b) Post-change image, (c) FC-EF daudt2018fully
, (d) FC-Siam-Di daudt2018fully , (e) FC-Siam-Conc daudt2018fully , (f) DT-SCN
liu2020building , (g) BIT chen2021remote , (h) ChangeFormer
bandara2022transformer , (i) ddpm-CD (ours), and (j) Ground-truth.

((a))

((b))

((c))

((d))

((e))

((f))

((g))

((h))

((i))

((j))

Figure 8: Comparison of different state-of-the-art CD methods on LEVIR-CD
dataset: (a) Pre-change image, (b) Post-change image, (c) FC-EF daudt2018fully
, (d) FC-Siam-Di daudt2018fully , (e) FC-Siam-Conc daudt2018fully , (f) DT-SCN
liu2020building , (g) BIT chen2021remote , (h) ChangeFormer
bandara2022transformer , (i) ddpm-CD (ours), and (j) Ground-truth.

((a))

((b))

((c))

((d))

((e))

((f))

((g))

((h))

((i))

((j))

Figure 8: Comparision of different state-of-the-art CD methods on LEVIR-CD
dataset: (a) Pre-change image, (b) Post-change image, (c) FC-EF daudt2018fully
, (d) FC-Siam-Di daudt2018fully , (e) FC-Siam-Conc daudt2018fully , (f) DT-SCN
liu2020building , (g) BIT chen2021remote , (h) ChangeFormer
bandara2022transformer , (i) ddpm-CD (ours), and (j) Ground-truth.

((a))

((b))

((c))

((d))

((e))

((f))

((g))

((h))

((i))

((j))

Figure 8: Comparision of different state-of-the-art CD methods on LEVIR-CD
dataset: (a) Pre-change image, (b) Post-change image, (c) FC-EF daudt2018fully
, (d) FC-Siam-Di daudt2018fully , (e) FC-Siam-Conc daudt2018fully , (f) DT-SCN
liu2020building , (g) BIT chen2021remote , (h) ChangeFormer
bandara2022transformer , (i) ddpm-CD (ours), and (j) Ground-truth.

###  8.2 WHU-CD dataset

Figure 8, 8, 8, 8 and 8 show additional qualitative results on WHU-CD dataset.

((a))

((b))

((c))

((d))

((e))

((f))

((g))

((h))

((i))

((j))

Figure 8: Comparision of different state-of-the-art CD methods on WHU-CD
dataset: (a) Pre-change image, (b) Post-change image, (c) FC-EF daudt2018fully
, (d) FC-Siam-Di daudt2018fully , (e) FC-Siam-Conc daudt2018fully , (f) DT-SCN
liu2020building , (g) BIT chen2021remote , (h) ChangeFormer
bandara2022transformer , (i) ddpm-CD (ours), and (j) Ground-truth.

((a))

((b))

((c))

((d))

((e))

((f))

((g))

((h))

((i))

((j))

Figure 8: Comparision of different state-of-the-art CD methods on WHU-CD
dataset: (a) Pre-change image, (b) Post-change image, (c) FC-EF daudt2018fully
, (d) FC-Siam-Di daudt2018fully , (e) FC-Siam-Conc daudt2018fully , (f) DT-SCN
liu2020building , (g) BIT chen2021remote , (h) ChangeFormer
bandara2022transformer , (i) ddpm-CD (ours), and (j) Ground-truth.

((a))

((b))

((c))

((d))

((e))

((f))

((g))

((h))

((i))

((j))

Figure 8: Comparision of different state-of-the-art CD methods on WHU-CD
dataset: (a) Pre-change image, (b) Post-change image, (c) FC-EF daudt2018fully
, (d) FC-Siam-Di daudt2018fully , (e) FC-Siam-Conc daudt2018fully , (f) DT-SCN
liu2020building , (g) BIT chen2021remote , (h) ChangeFormer
bandara2022transformer , (i) ddpm-CD (ours), and (j) Ground-truth.

((a))

((b))

((c))

((d))

((e))

((f))

((g))

((h))

((i))

((j))

Figure 8: Comparision of different state-of-the-art CD methods on WHU-CD
dataset: (a) Pre-change image, (b) Post-change image, (c) FC-EF daudt2018fully
, (d) FC-Siam-Di daudt2018fully , (e) FC-Siam-Conc daudt2018fully , (f) DT-SCN
liu2020building , (g) BIT chen2021remote , (h) ChangeFormer
bandara2022transformer , (i) ddpm-CD (ours), and (j) Ground-truth.

((a))

((b))

((c))

((d))

((e))

((f))

((g))

((h))

((i))

((j))

Figure 8: Comparision of different state-of-the-art CD methods on WHU-CD
dataset: (a) Pre-change image, (b) Post-change image, (c) FC-EF daudt2018fully
, (d) FC-Siam-Di daudt2018fully , (e) FC-Siam-Conc daudt2018fully , (f) DT-SCN
liu2020building , (g) BIT chen2021remote , (h) ChangeFormer
bandara2022transformer , (i) ddpm-CD (ours), and (j) Ground-truth.

###  8.3 DSIFN-CD dataset

Figure 8, 8, 8 and 8 show additional qualitative results on LEVIR-CD dataset.

((a))

((b))

((c))

((d))

((e))

((f))

((g))

((h))

((i))

((j))

Figure 8: Comparison of different state-of-the-art CD methods on DSIFN-CD
dataset: (a) Pre-change image, (b) Post-change image, (c) FC-EF daudt2018fully
, (d) FC-Siam-Di daudt2018fully , (e) FC-Siam-Conc daudt2018fully , (f) DT-SCN
liu2020building , (g) BIT chen2021remote , (h) ChangeFormer
bandara2022transformer , (i) ddpm-CD (ours), and (j) Ground-truth.

((a))

((b))

((c))

((d))

((e))

((f))

((g))

((h))

((i))

((j))

Figure 8: Comparison of different state-of-the-art CD methods on DSIFN-CD
dataset: (a) Pre-change image, (b) Post-change image, (c) FC-EF daudt2018fully
, (d) FC-Siam-Di daudt2018fully , (e) FC-Siam-Conc daudt2018fully , (f) DT-SCN
liu2020building , (g) BIT chen2021remote , (h) ChangeFormer
bandara2022transformer , (i) ddpm-CD (ours), and (j) Ground-truth.

((a))

((b))

((c))

((d))

((e))

((f))

((g))

((h))

((i))

((j))

Figure 8: Comparison of different state-of-the-art CD methods on DSIFN-CD
dataset: (a) Pre-change image, (b) Post-change image, (c) FC-EF daudt2018fully
, (d) FC-Siam-Di daudt2018fully , (e) FC-Siam-Conc daudt2018fully , (f) DT-SCN
liu2020building , (g) BIT chen2021remote , (h) ChangeFormer
bandara2022transformer , (i) ddpm-CD (ours), and (j) Ground-truth.

((a))

((b))

((c))

((d))

((e))

((f))

((g))

((h))

((i))

((j))

Figure 8: Comparison of different state-of-the-art CD methods on DSIFN-CD
dataset: (a) Pre-change image, (b) Post-change image, (c) FC-EF daudt2018fully
, (d) FC-Siam-Di daudt2018fully , (e) FC-Siam-Conc daudt2018fully , (f) DT-SCN
liu2020building , (g) BIT chen2021remote , (h) ChangeFormer
bandara2022transformer , (i) ddpm-CD (ours), and (j) Ground-truth.

###  8.4 CDD dataset

Figure 8, 8, 8, 8, 8, 8, 8, 8, 8 and 8 show additional qualitative results on
LEVIR-CD dataset.

((a))

((b))

((c))

((d))

((e))

((f))

((g))

((h))

((i))

((j))

Figure 8: Comparison of different state-of-the-art CD methods on CDD dataset:
(a) Pre-change image, (b) Post-change image, (c) FC-EF daudt2018fully , (d)
FC-Siam-Di daudt2018fully , (e) FC-Siam-Conc daudt2018fully , (f) DT-SCN
liu2020building , (g) BIT chen2021remote , (h) ChangeFormer
bandara2022transformer , (i) ddpm-CD (ours), and (j) Ground-truth.

((a))

((b))

((c))

((d))

((e))

((f))

((g))

((h))

((i))

((j))

Figure 8: Comparison of different state-of-the-art CD methods on CDD dataset:
(a) Pre-change image, (b) Post-change image, (c) FC-EF daudt2018fully , (d)
FC-Siam-Di daudt2018fully , (e) FC-Siam-Conc daudt2018fully , (f) DT-SCN
liu2020building , (g) BIT chen2021remote , (h) ChangeFormer
bandara2022transformer , (i) ddpm-CD (ours), and (j) Ground-truth.

((a))

((b))

((c))

((d))

((e))

((f))

((g))

((h))

((i))

((j))

Figure 8: Comparison of different state-of-the-art CD methods on CDD dataset:
(a) Pre-change image, (b) Post-change image, (c) FC-EF daudt2018fully , (d)
FC-Siam-Di daudt2018fully , (e) FC-Siam-Conc daudt2018fully , (f) DT-SCN
liu2020building , (g) BIT chen2021remote , (h) ChangeFormer
bandara2022transformer , (i) ddpm-CD (ours), and (j) Ground-truth.

((a))

((b))

((c))

((d))

((e))

((f))

((g))

((h))

((i))

((j))

Figure 8: Comparison of different state-of-the-art CD methods on CDD dataset:
(a) Pre-change image, (b) Post-change image, (c) FC-EF daudt2018fully , (d)
FC-Siam-Di daudt2018fully , (e) FC-Siam-Conc daudt2018fully , (f) DT-SCN
liu2020building , (g) BIT chen2021remote , (h) ChangeFormer
bandara2022transformer , (i) ddpm-CD (ours), and (j) Ground-truth.

((a))

((b))

((c))

((d))

((e))

((f))

((g))

((h))

((i))

((j))

Figure 8: Comparison of different state-of-the-art CD methods on CDD dataset:
(a) Pre-change image, (b) Post-change image, (c) FC-EF daudt2018fully , (d)
FC-Siam-Di daudt2018fully , (e) FC-Siam-Conc daudt2018fully , (f) DT-SCN
liu2020building , (g) BIT chen2021remote , (h) ChangeFormer
bandara2022transformer , (i) ddpm-CD (ours), and (j) Ground-truth.

((a))

((b))

((c))

((d))

((e))

((f))

((g))

((h))

((i))

((j))

Figure 8: Comparison of different state-of-the-art CD methods on CDD dataset:
(a) Pre-change image, (b) Post-change image, (c) FC-EF daudt2018fully , (d)
FC-Siam-Di daudt2018fully , (e) FC-Siam-Conc daudt2018fully , (f) DT-SCN
liu2020building , (g) BIT chen2021remote , (h) ChangeFormer
bandara2022transformer , (i) ddpm-CD (ours), and (j) Ground-truth.

((a))

((b))

((c))

((d))

((e))

((f))

((g))

((h))

((i))

((j))

Figure 8: Comparison of different state-of-the-art CD methods on CDD dataset:
(a) Pre-change image, (b) Post-change image, (c) FC-EF daudt2018fully , (d)
FC-Siam-Di daudt2018fully , (e) FC-Siam-Conc daudt2018fully , (f) DT-SCN
liu2020building , (g) BIT chen2021remote , (h) ChangeFormer
bandara2022transformer , (i) ddpm-CD (ours), and (j) Ground-truth.

((a))

((b))

((c))

((d))

((e))

((f))

((g))

((h))

((i))

((j))

Figure 8: Comparison of different state-of-the-art CD methods on CDD dataset:
(a) Pre-change image, (b) Post-change image, (c) FC-EF daudt2018fully , (d)
FC-Siam-Di daudt2018fully , (e) FC-Siam-Conc daudt2018fully , (f) DT-SCN
liu2020building , (g) BIT chen2021remote , (h) ChangeFormer
bandara2022transformer , (i) ddpm-CD (ours), and (j) Ground-truth.

((a))

((b))

((c))

((d))

((e))

((f))

((g))

((h))

((i))

((j))

Figure 8: Comparison of different state-of-the-art CD methods on CDD dataset:
(a) Pre-change image, (b) Post-change image, (c) FC-EF daudt2018fully , (d)
FC-Siam-Di daudt2018fully , (e) FC-Siam-Conc daudt2018fully , (f) DT-SCN
liu2020building , (g) BIT chen2021remote , (h) ChangeFormer
bandara2022transformer , (i) ddpm-CD (ours), and (j) Ground-truth.

((a))

((b))

((c))

((d))

((e))

((f))

((g))

((h))

((i))

((j))

Figure 8: Comparison of different state-of-the-art CD methods on CDD dataset:
(a) Pre-change image, (b) Post-change image, (c) FC-EF daudt2018fully , (d)
FC-Siam-Di daudt2018fully , (e) FC-Siam-Conc daudt2018fully , (f) DT-SCN
liu2020building , (g) BIT chen2021remote , (h) ChangeFormer
bandara2022transformer , (i) ddpm-CD (ours), and (j) Ground-truth.

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# Stagnation points control chaotic fluctuations in viscoelastic porous media
flow

Simon J. Haward  Cameron C. Hopkins  Amy Q. Shen  Okinawa Institute of Science
and Technology Graduate University, Onna-son, Okinawa, 904-0495, Japan

###### Abstract

Viscoelastic flows through porous media become unstable and chaotic beyond
critical flow conditions, impacting industrial and biological processes.
Recently, Walkama et al. [Phys. Rev. Lett. 124, 164501 (2020)] have shown that
geometric disorder greatly suppresses such chaotic dynamics. We demonstrate
experimentally that geometric disorder per se is not the reason for this
suppression, and that disorder can also promote choatic fluctuations, given a
slightly modified initial condition. The results are explained by the effect
of disorder on the occurrence of stagnation points exposed to the flow field,
which depends on the initially ordered geometric configuration.

††preprint: APS/123-QED

Unlike viscous Newtonian liquids (e.g., water), many fluids exhibit an elastic
resonse to an applied strain. Such “viscoelastic” fluids are widespread in
biology (blood, mucus, synovial fluid) and industry (paints, coatings,
fracking fluids). The elasticity is imparted by a mesoscopic structure (formed
by e.g., polymers, proteins, or self-assemblies of lipids or surfactants),
that relaxes after deformation [1]. The strength of the elastic response is
quantified by the Weissenberg number $\text{Wi}=\tau\dot{\gamma}$, with $\tau$
the fluid relaxation time and $\dot{\gamma}$ the rate of strain. While flows
of Newtonian fluids become unstable and turbulent due to the onset of inertial
effects at high Reynolds number, $\text{Re}\gg 1$, viscoelastic flows can
become unstable and “elastically turbulent” even for $\text{Re}\ll 1$, purely
due to elastic effects at high Wi [2, 3, 4, 5, 6, 7].

Viscoelastic porous media flow occurs in diverse processes from enhanced oil
recovery (EOR) and filtration to drug delivery [8, 9]. Porous media flow
subjects a fluid to a complex cycle of deformation with high shear rates
through the pore-throats or between obstacles, and high elongational rates at
points of constriction or at stagnation points. Stagnation points are regions
where particularly high fluid strains develop, resulting in strong stretching
of the microstructure if $\text{Wi}\gtrsim 1$ [10, 11, 12, 13, 14, 15].
Elastic tensile stresses due to stretching on curvilinear streamlines (as
through porous media) are conditions well-established to lead to linear
instabilities in viscoelastic fluids [16, 17, 18, 19], which can be precursors
to elastic turbulence [20, 21, 22, 23, 7]. Resulting chaotic fluctuations are
expected to greatly enhance the pressure loss and the dispersion in porous
media, with positive impacts for e.g., removing oil ganglia from the pore
space in EOR [24, 25, 26].

Figure 1: Unit cell representations of two contrasting ordered hexagonal
arrays of posts used in the flow experiments. In (a) the posts are “staggered”
along the $x$-direction in which the flow is imposed. The post radius is $R$,
lattice spacing $S$. Rotating the array by 30∘ aligns the posts in the flow
direction (b). Disordered “aligned” arrays are generated by the random
displacement of each post within a hexagon of circumradius $\beta S$, as
described in Ref. 27.

Recently, in microfluidic experiments with polymer solutions in porous media
modeled by hexagonal arrangements of posts, Walkama et al. demonstrated that
introducing random disorder to the array can significantly suppress chaotic
fluctuations for $\text{Wi}\gtrsim 1$ [27]. Disorder resulted in the opening
of clear flow paths for the viscoelastic fluid, which became more dominated by
shear and less by extensional kinematics as disorder increased, thus reducing
the stretching of the dissolved polymer, and suppressing fluctuations.
However, only one initially ordered configuration of posts [similar to Fig.
1(a)] was considered. Other works show that instabilities and fluctuations in
viscoelastic flows through ordered post arrays depend on the orientation of
the array relative to the flow direction [28]. Different behavior might be
expected from an array with posts staggered along the flow direction [Fig.
1(a)] than from the same array rotated by 30∘ such that the posts are aligned
[Fig. 1(b)].

Figure 2: Evolution of the retardation fields with increasing Wi for the WLM
solution in post arrays with various $\beta$: (a) normalized, time-averaged
retardation fields $\bar{\delta}/\max(\bar{\delta})$; (b) local retardation
fluctuations $\delta_{rms}$. Field of view is centered on the coordinate
origin at the center of each array (54% of full field of view shown). Scale
bar: $250~{}\upmu$m.

In this Letter, we present microfluidic experiments of viscoelastic flow
through two ordered arrays of posts oriented as shown in Fig. 1. First, we
demonstrate that the chaotic fluctuations at $\text{Wi}\gtrsim 1$ can be
strongly supressed in a perfectly ordered array simply by aligning the posts
in the flow direction [Fig. 1(b)]. Subsequently disordering the aligned array
does not suppress, but _promotes_ chaotic fluctuations over a wide range of
Wi. Our results are explained (in context with those of Walkama et al. [27])
by the presence and role of stagnation points in the flow field, as opposed to
the degree of disorder per se.

Microfluidic channels ($W=2.4$ mm wide, $H=1$ mm high, and 25 mm long)
containing arrays of $\approx 300$ circular posts (radius $R=50~{}\upmu$m)
were fabricated in fused silica by selective laser-induced etching [29]. Two
of the channels contained ordered hexagonal arrays (disorder $\beta=0$,
lattice spacing $S=240~{}\upmu$m, porosity $\phi\approx 0.84$) in either
“staggered” or “aligned” orientations [Fig. 1]. Five arrays were generated by
disordering the aligned configuration with
$\beta=[0.05,~{}0.1,~{}0.2,~{}0.3,~{}0.4]$ (Fig. 1(b) [27]).

The model viscoelastic test fluid is an aqueous wormlike micellar (WLM)
solution of 100 mM cetylpyridinium chloride and 60 mM sodium salicylate [30].
The fluid is shear-thinning with a zero shear viscosity $\eta_{0}\approx 48$
Pa s and a single-mode Maxwell relaxation time $\tau=1.5$ s. Flow through the
post arrays is driven by a syringe pump (Cetoni GmbH) infusing at controlled
volumetric flow rate $Q$, hence average flow velocity ${U=Q/\phi WH}$, and
characteristic deformation rate $\dot{\gamma}=U/R$. The maximum Reynolds
number, $\text{Re}=\rho UR/\eta(\dot{\gamma})<10^{-5}$ (density $\rho$,
viscosity $\eta(\dot{\gamma})$), means that inertia can be neglected.

A high speed polarizing camera (Photron CRYSTA PI-1P, see Ref. 31) is used to
visualize the regions of high micelle orientation and elastic stress in the
post arrays over a range of imposed $\text{Wi}=\tau\dot{\gamma}$, via the
flow-induced optical retardation, $\delta$. Time averaged fields
$\bar{\delta}=\langle\delta\rangle_{t}$, captured at 125 Hz are shown for a
few of the arrays and several values of Wi, alongside the corresponding local
rms fluctuations
$\delta_{rms}=\sqrt{\bigl{\langle}(\delta-\bar{\delta})^{2}\bigr{\rangle}_{t}}$
[Fig. 2]. For the ordered staggered geometry ($\beta=0$, first row of Fig. 2),
at low $\text{Wi}\approx 0.75$ each post has an associated downstream wake of
high retardation, of similar appearance to that seen downstream of isolated
cylinders [15]. This indicates that the downstream stagnation point of each
post is effective at orienting and stretching the micellar microstructure. At
$\text{Wi}\approx 0.75$, the flow is steady and the rms fluctuations are low.
For increasing Wi, the flow becomes time-dependent and at $\text{Wi}\approx
7.5$ strong fluctuations are observed. The fluctuation appears as the
transverse wagging motion of each downstream wake (Movie 1 [32]). The growth
of fluctuations with Wi is consistent with the experiments of Walkama et al.
[27] using a similar staggered hexagonal array and range of Wi. We further
extend the range of Wi and observe that fluctuations remain, but apparently
become less intense for $\text{Wi}\gtrsim 7.5$. At $\text{Wi}\approx 75$, the
fluctuations appear more uniformly throughout the field and are of increased
frequency compared with $\text{Wi}\approx 7.5$ (Movie 2 [32]).

In the ordered aligned geometry ($\beta=0$, second row of Fig. 2), the
retardation field at $\text{Wi}\approx 0.75$ is qualitatively different from
the staggered array. Here, there is no significant downstream wake and the
regions of high stress are concentrated upstream and to the sides of each
post. As Wi is increased, the retardation becomes concentrated in the gaps
between the rows of aligned posts. Fluctuations in the ordered aligned array
remain low relative to the staggered array until for $\text{Wi}\approx 75$ a
similar level of time-dependence is observed throughout the field. Movie 3 and
Movie 4 show the time-resolved retardation at $\text{Wi}\approx 7.5$ and
$\text{Wi}\approx 75$, respectively [32].

For increasing disorder of the aligned array ($\beta=0.1$ and $\beta=0.3$,
third and fourth rows of Fig. 2, respectively), at low $\text{Wi}\approx 0.75$
a few posts appear to acquire high stress in downstream wake regions, although
unlike the staggered array, these are not necessarily aligned in the primary
flow ($x$) direction owing to the disorder. At intermediate $\text{Wi}\approx
7.5$ and $\text{Wi}\approx 12.5$, the flow becomes unsteady, and although less
intense than for the staggered array, fluctuations are clearly greater than
for the aligned array with $\beta=0$. For the highest Wi shown
($\text{Wi}\approx 75$), fluctuations are roughly similar in all arrays.

Figure 3: Spatially-averaged retardation fluctuations
$\langle\delta_{rms}\rangle_{i}$ _vs_ Wi in various post arrays. The shaded
gray region about the data for the aligned array with $\beta=0$ indicates the
typical variability over repeated test runs. Insert compares fluctuations
between the ordered and disordered aligned arrays.

Spatially-averaged rms retardation fluctuations
$\langle\delta_{rms}\rangle_{i}$ in the staggered array show a local peak at
intermediate $\text{Wi}\approx 7.5$, followed by a reduction and a subsequent
growth towards an apparent high-Wi plateau [Fig. 3]. In the aligned geometry
($\beta=0$), the peak at intermediate Wi is greatly diminished, although at
higher Wi fluctuations reach a similar plateau value as for the staggered
array. With increasing disorder of the aligned geometry, the fluctuations at
intermediate Wi increase. At $\beta=0.4$, a peak emerges at a similar Wi as
the peak seen in the staggered array. At higher Wi, most of the geometries
tend to a similar limiting plateau in the level of fluctuation, however at
$\beta=0.1$, fluctuations remain higher than for those $\beta=0$.

Spatially-averaged fluctuations from the disordered arrays are compared
against those from the ordered aligned array as
$(\langle\delta_{rms}\rangle_{i}-\langle\delta_{rms,\beta=0}\rangle_{i})/\langle\delta_{rms,\beta=0}\rangle_{i}$
[Fig. 3 (insert)]. Within experimental error, the data for $\beta=0.05$ does
not deviate significantly from zero, but for $\beta>0.05$, all of the
disordered arrays show a relative peak in the fluctuations around
$\text{Wi}\approx 10$ where $\langle\delta_{rms}\rangle_{i}\approx
2\times\langle\delta_{rms,\beta=0}\rangle_{i}$. Perhaps surprisingly, the
increase in fluctuations is saturated (or maximal) for a rather low disorder
of $\beta=0.1$. It is noted that Walkama et al. showed that small amounts of
disorder were also sufficient to strongly _suppress_ the fluctuations occuring
in a staggered post array configuration [27].

Figure 4: Analysis of retardation fluctuations about the local mean
$\delta(t)-\bar{\delta}$ in the ordered staggered and aligned arrays
($\beta=0$). (left) Kymographs of the fluctuations along $x=0$ for: (a,b)
$\text{Wi}\approx 0.75$; (c,d) $\text{Wi}\approx 7.5$; (e,f) $\text{Wi}\approx
75$. (right) Power spectral density (PSD) distributions of the fluctuations
for: (g) $\text{Wi}\approx 0.75$; (h) $\text{Wi}\approx 7.5$; (i)
$\text{Wi}\approx 75$, taken along the correspondingly-colored horizontal
dashed lines in (a,b).

Analysis of retardation fluctuations about the local mean
$\delta(t)-\bar{\delta}$ in the ordered staggered and aligned arrays
($\beta=0$) at three representative values of Wi is provided in Fig. 4.
Kymographs of the fluctuations [Fig. 4(a-f)] are extracted along the line
$x=0$ between the post located at the origin and its neighbor to positive $y$.
For low $\text{Wi}\approx 0.75$, the fluctuations in both the staggered [Fig.
4(a)] and ordered aligned [Fig. 4(b)] arrays are weak and the kymographs
indicate the noise level of the measurement. At this Wi, the power spectral
density, PSD, of the fluctuations with time along the four colored dashed
lines all show a uniformly flat frequency response [Fig. 4(g)]. At
intermediate $\text{Wi}\approx 7.5$, the strong fluctuations in the staggered
array [Fig. 4(c)] translate to a PSD with high power at low frequencies, that
decays steeply into the noise with a power-law exponent $-3.7$ [Fig. 4(h)]. In
contrast, in the ordered aligned array at $\text{Wi}\approx 7.5$ [Fig.
4(d,h)], the fluctuations are an order-of-magnitude weaker at low frequencies
and decay with an exponent $-2$. At high $\text{Wi}\approx 75$, both the
staggered and the ordered aligned array show fluctuations of similar intensity
[Fig. 4(e,f)] and with similar frequency content in the PSD [Fig. 4(i)]. The
PSD is shifted to higher frequencies than at $\text{Wi}\approx 7.5$ and decays
with an exponent of $\approx-2.2$. The power-law decays in the power spectra
of the fluctuating signals indicate that the fluctuations are aperiodic and
the slopes are consistent with values reported in studies of elastic
turbulence [21, 27, 25, 26, 7].

Figure 5: Flow velocimetry in various post arrays at $\textnormal{Wi}\approx
0.75$: (left) Normalized flow speed
$\lvert\textbf{u}\rvert/\max(\lvert\textbf{u}\rvert)$; (middle) flow type
parameter $\xi$; (right) normalized principal strain rate $\lambda_{1}/(U/R)$.
Disordering the aligned array introduces stagnation points, where extension-
dominated regions also have high strain rates (examples indicated by gray
arrow-heads). Scale bar: $250~{}\upmu$m.

Our results reveal a broad consistency with those of Walkama et al. [27] for
viscoelastic flow through an ordered staggered array of posts. However,
introducing disorder to an aligned array has a diametrically opposite effect
from that shown by Walkama et al. for introducing disorder to a staggered
array. The results can be understood in context with each other by examining
the flow fields through the arrays at low Wi prior to the onset of
instability. Flow velocimetry was performed on a FlowMaster volume
illumination micro-particle image velocimetry system (LaVision GmbH [33]) by
seeding the WLM test fluid with $2~{}\upmu$m diameter fluorescent tracers (PS
Fluored, Microparticles GmbH). From the measured velocity fields
$\textbf{u}=(u,v)$, the deformation rate D and vorticity $\pmb{\Omega}$
tensors are computed. Fig. 5 (left column), shows the normalized magnitude of
the velocity fields $\lvert\textbf{u}\rvert/\max(\lvert\textbf{u}\rvert)$. To
indicate the local flow kinematics, Fig. 5 (middle column) shows the “flow
type parameter”
$\xi=(\lvert\bf{D}\rvert-\lvert\pmb{\Omega}\rvert)/(\lvert\bf{D}\rvert+\lvert\pmb{\Omega}\rvert)$,
where $\lvert\textbf{D}\rvert=\sqrt{2\textbf{D}:\textbf{D}}$ and
$\lvert\pmb{\Omega}\rvert=\sqrt{2\pmb{\Omega}:\pmb{\Omega}}$ [34]. Here,
$\xi=-1$ indicates solid body rotation, $\xi=0$ simple shear, and $\xi=1$ pure
extension. The flow strength in the extensional regions is quantified by the
principle strain rate (or eigenvector of $\bf{D}$ [35]),
$\lambda_{1}=\frac{1}{2}\sqrt{(\textbf{D}_{11}-\textbf{D}_{22})^{2}+4\textbf{D}_{12}^{2}}$
[Fig. 5 (right column)]. In the staggered array [Fig. 5 (top row)], it is
evident that streamlines diverge upstream of each post and reconverge
downstream, resulting in extension-dominated regions at the up- and downstream
stagnation points. Extensional rates are particularly high downstream of each
post, where elastic stresses are also high [Fig. 2] and time-dependence is
first manifested (Movie 1, [32]). By contrast, in the ordered aligned array
[Fig. 5 (second row)] the flow is concentrated in between the aligned rows of
posts and streamlines do not connect between successive streamwise-oriented
posts. Every stagnation point is thus effectively “screened” from the flow
field (note that although the flow type parameter shows regions of extensional
flow between the posts, this is inevitable due to the symmetry of the flow
about $y=0$, however $\lvert\lambda_{1}\rvert$ in these extensional regions is
weak). As disorder is applied to the aligned array [Fig. 5 (third and fourth
rows)], locations emerge where streamlines split and reconverge, reintroducing
stagnation points to the flow field. It becomes possible to identify posts
downstream of which the flow kinematics are extension dominated and with
relatively high streamwise oriented strain rate (examples marked by gray
arrowheads in Fig. 5).

Contrary to Walkama et al. [27], our results clearly show that disorder does
not necessarily suppress chaotic fluctuations in viscoelastic flows. Depending
on the initially ordered configuration, disorder can in fact, promote such
fluctuations over a range of Wi. In general, this can be understood by
considering the rate of occurrence of stagnation points in the flow field.
Stagnation points are locations prone to elastic instability due to the high
tensile stresses they induce and the consequent feedback on the flow [17, 36,
37, 38, 39, 40, 13, 41, 23, 15, 42, 43, 44]. In the ordered staggered
configuration considered by Walkama et al., the number of such points is
maximized (every array element contributes both an upstream and downstream
stagnation point). Thus, disordering a staggered array can only serve to
reduce the incidence of such locations: some posts will become hidden in the
wakes of others, thus screening their stagnation points. By contrast, in the
ordered aligned array every stagnation point is screened from the flow field
by the previous upstream post (i.e., stagnation points are minimized). In this
case, introducing disorder must inevitably increase the occurrence of
stagnation points where instability is most probable to initiate. An
equivalent, perhaps more intuitive, way to express this is: disordering a
staggered array tends to open free paths for the fluid flow (as shown in Ref.
27), but disordering an aligned array acts to block the free paths that
previously existed. Naturally, for high disorders, both staggered and aligned
configurations should tend towards randomness, hence homogeneity. We also note
that fluctuations become effectively geometry-independent at sufficiently high
Wi, where both staggered and aligned configurations show elastic-turbulent-
like characteristics.

This work demonstrates the crucial importance of geometry in determining the
onset and strength of chaotic fluctuations in viscoelastic flows. In
particular, a disorder of just $\beta=0.1$ applied to an aligned array can
increase rms fluctuations by up to $\approx 100\%$, highlighting the
significant effect that a relatively few stagnation points can have on the
global dynamics of viscoelastic flows.

We gratefully acknowledge the support of the Okinawa Institute of Science and
Technology Graduate University (OIST) with subsidy funding from the Cabinet
Office, Government of Japan, and also funding from the Japan Society for the
Promotion of Science (JSPS, Grant Nos. 18K03958, 18H01135, 20K14656 and
21K03884) and the Joint Research Projects (JRPs) supported by the JSPS and the
Swiss National Science Foundation (SNSF). We are grateful to Mr. Kazumi Toda-
Peters (OIST) for device fabrication, and to Dr. Stylianos Varchanis (OIST)
for helpful discussions.

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