Houssem Eddine Boulahbal
email: houssem-eddine.boulahbal@renault.com

Adrian Voicila
email: adrian.voicila@renault.com

Andrew I Comport
email: andrew.comport@cnrs.fr

Forecasting of depth and ego-motion with transformers and self-supervision

This paper addresses the problem of end-to-end self-supervised forecasting of depth and ego motion. Given a sequence of raw images, the aim is to forecast both the geometry and ego-motion using a self supervised photometric loss. The architecture is designed using both convolution and transformer modules. This leverages the benefits of both modules: Inductive bias of CNN, and the multi-head attention of transformers, thus enabling a rich spatio-temporal representation that enables accurate depth forecasting. Prior work attempts to solve this problem using multi-modal input/output with supervised groundtruth data which is not practical since a large annotated dataset is required. Alternatively to prior methods, this paper forecasts depth and ego motion using only self-supervised raw images as input. The approach performs significantly well on the KITTI dataset benchmark with several performance criteria being even comparable to prior non-forecasting self-supervised monocular depth inference methods.

I. INTRODUCTION

Forecasting the future is crucial for intelligent decision making. It is a remarkable ability of human beings to effortlessly forecast what will happen next, based on the current context and prior knowledge of the scene. Forecasting sequences in real-world settings, particularly from raw sensor measurements, is a complex problem due to the the exponential timespace space dimensionality, the probabilistic nature of the future and the complex dynamics of the scene. Whilst much effort from the research community has been devoted to video forecasting [START_REF] Finn | Unsupervised learning for physical interaction through video prediction[END_REF], [START_REF] Mathieu | Deep multiscale video prediction beyond mean square error[END_REF], [START_REF] Rakhimov | Latent video transformer[END_REF], [START_REF] Kreiman | Deep Predictive Coding Networks for Video Prediction and Unsupervised Learning[END_REF] and semantic forecasting [START_REF] Bhattacharyya | Bayesian prediction of future street scenes using synthetic likelihoods[END_REF], [START_REF] Graber | Panoptic segmentation forecasting[END_REF], [START_REF] Saric | Warp to the Future: Joint Forecasting of Features and Feature Motion[END_REF], [START_REF] Adam M Terwilliger | Recurrent flow-guided semantic forecasting[END_REF], depth and ego-motion forecasting have not received the same interest despite their importance. The geometry of the scene is essential for applications such as planning the trajectory of an agent.

Anticipating is therefore important for autonomous driving auto-pilots or human/robot interaction as it is critical for the agent to quickly respond to changes in the external environment.

The first work that studied depth forecasting was carried out by Mahjourian et al. [START_REF] Mahjourian | Geometrybased next frame prediction from monocular video[END_REF], the aim of that paper was to use 1. The proposed method is trained using only raw sensor self-supervision, it is able to forecast an accurate geometry of the scene. The network only accesses the frames I t-3:t and forecast the depth D t+n and the pose t T t+n . Not only the network forecasts accurately the ego-motion (the depth of the static object is accurate) but also handles the dynamic objects.

the forecasted depth to render the next RGB image frame. They supervised the depth loss using ground-truth LiDAR scans and the warping was done using ground-truth poses. [START_REF] Qi | 3D motion decomposition for RGBD future dynamic scene synthesis[END_REF] used additional modalities for input, namely, a multi-modal RGB, depth, semantic and optical flow and forecasted the same future modalities. The supervision was carried out using the aforementioned ground-truth labels. [START_REF] Hu | Probabilistic Future Prediction for Video Scene Understanding[END_REF] developed a probabilistic approach for forecasting using only input images and generated a diverse and plausible multi-modal future including [START_REF] Godard | Digging into self-supervised monocular depth estimation[END_REF], [START_REF] Kendall | PoseNet: A convolutional network for real-time 6-dof camera relocalization[END_REF] is used to forecast the ego-motion. The depth network combines both CNN and transformers. The Resnet34 [START_REF] He | Deep residual learning for image recognition[END_REF] encoder extracts the spatial features for each context frame. The embedding projection module projects these features into R k×d model where k = 4 is the context frames. N = 3 transformer encoders are used to fuse the spatial temporal to obtain a rich spatio-temporal features. The output of the transformer module encodes the motion of the scene. The decoder uses simple transposed convolution. In order to recover the context, skip connections are pooled from the encoder. Only the last frame features are pooled for the context. The decoder outputs a disparity map that will be used along with the pose network to warp the source images onto the target.

depth, semantics and optical flow. However, it was supervised through ground-truth labels and the final loss was a weighted sum of future segmentation, depth and optical flow losses similar to [START_REF] Qi | 3D motion decomposition for RGBD future dynamic scene synthesis[END_REF]. While these methods enable forecasting the depth, they suffer from two shortcomings: [START_REF] Hu | Probabilistic Future Prediction for Video Scene Understanding[END_REF], [START_REF] Mahjourian | Geometrybased next frame prediction from monocular video[END_REF], [START_REF] Qi | 3D motion decomposition for RGBD future dynamic scene synthesis[END_REF] require the ground-truth labels for supervision during training and testing and [START_REF] Qi | 3D motion decomposition for RGBD future dynamic scene synthesis[END_REF] uses a multi-modal input for inference that requires either ground-truth labels or a separate network.

The work presented in this paper addresses the problem of depth and ego-motion forecasting using only monocular images with self-supervision. Monocular depth and ego-motion inference has been successful for self-supervised training [START_REF] Almalioglu | GANVO: Unsupervised deep monocular visual odometry and depth estimation with generative adversarial networks[END_REF], [START_REF] Chen | Single-image depth perception in the wild[END_REF], [START_REF] Chen | Selfsupervised learning with geometric constraints in monocular video: Connecting flow, depth, and camera[END_REF], [START_REF] Godard | Digging into self-supervised monocular depth estimation[END_REF], [START_REF] Godard | Unsupervised monocular depth estimation with left-right consistency[END_REF], [START_REF] Jaderberg | Spatial transformer networks[END_REF], [START_REF] Ranjan | Competitive collaboration: Joint unsupervised learning of depth, camera motion, optical flow and motion segmentation[END_REF], [START_REF] Vitor Guizilini Rares | 3D packing for self-supervised monocular depth estimation[END_REF], [START_REF] Wang | Can Scale-Consistent Monocular Depth Be Learned in a Self-Supervised Scale-Invariant Manner[END_REF], [START_REF] Yin | GeoNet: Unsupervised Learning of Dense Depth, Optical Flow and Camera Pose[END_REF]. The basic idea is to jointly learn depth and ego-motion supervised by a photometric reconstruction loss. In this paper, it is demonstrated that it is possible to extend this self-supervised training to sequence forecasting. An accurate forecasting requires a knowledge of the ego-motion, semantics, and the motion of dynamic objects. Powered by the advances of transformers [START_REF] Tom | Language models are few-shot learners[END_REF], [START_REF] Devlin | BERT: Pre-training of deep bidirectional transformers for language understanding[END_REF], [START_REF] Dosovitskiy | An image is worth 16x16 words: Transformers for image recognition at scale[END_REF], [START_REF] Liu | Swin transformer: Hierarchical vision transformer using shifted windows[END_REF], [START_REF] Vaswani | Attention is all you need[END_REF], and using only sensor input, the network learns a rich spatio-temporal representation that encodes the semantics, the ego-motion and the dynamic objects. Therefore avoiding the need for extra labels for training and testing. The results on the KITTI benchmark [START_REF] Geiger | Are we ready for autonomous driving? the kitti vision benchmark suite[END_REF] show that the proposed method is able to forecast the depth accurately and outperform even non-forecasting methods [START_REF] Eigen | Depth map prediction from a single image using a multi-scale deep network[END_REF], [START_REF] Liu | Learning Depth from Single Monocular Images Using Deep Convolutional Neural Fields[END_REF], [START_REF] Yang | Unsupervised learning of geometry from videos with edgeaware depth-normal consistency[END_REF], [START_REF] Zhou | Unsupervised learning of depth and ego-motion from video[END_REF].

II. RELATED WORK

Previous works [START_REF] Godard | Digging into self-supervised monocular depth estimation[END_REF], [START_REF] Watson | The temporal opportunist: Self-supervised multiframe monocular depth[END_REF], [START_REF] Zhou | Unsupervised learning of depth and ego-motion from video[END_REF] have used the term "prediction" for methods that infer depth from a single image irrespective of past and future events. This could easily lead to confusion since the term "prediction" could equally be employed to refer to the prediction of future events. Therefore, the term "monocular depth inference" will be used to replace this ambiguous term. As the goal of this paper is related to time-series future prediction, the term "forecasting" will be used to refer to methods [START_REF] Hu | Probabilistic Future Prediction for Video Scene Understanding[END_REF], [START_REF] Mahjourian | Geometrybased next frame prediction from monocular video[END_REF], [START_REF] Qi | 3D motion decomposition for RGBD future dynamic scene synthesis[END_REF] that use observations of the past and present to predict a future depth of the scene.

A. Ego-motion and monocular depth self-supervised learning

Monocular depth inference is an ill-posed problem as an infinite number of 3D scenes can be projected onto the same 2D scene. The earliest deep learning methods [START_REF] Eigen | Predicting depth, surface normals and semantic labels with a common multi-scale convolutional architecture[END_REF], [START_REF] Eigen | Depth map prediction from a single image using a multi-scale deep network[END_REF] utilized LiDAR data as ground truth for supervision. For inferring pose, [START_REF] Kendall | PoseNet: A convolutional network for real-time 6-dof camera relocalization[END_REF] proposed to use a simple regression network, based on a pretrained classifier [START_REF] Szegedy | Going deeper with convolutions[END_REF], supervised by an Euclidean loss. The work on spatial transformers [START_REF] Jaderberg | Spatial transformer networks[END_REF] offered an end-to-end differentiable warping function, thus, enabling several works [START_REF] Bian | Unsupervised scale-consistent depth and ego-motion learning from monocular video[END_REF], [START_REF] Chen | Selfsupervised learning with geometric constraints in monocular video: Connecting flow, depth, and camera[END_REF], [START_REF] Godard | Digging into self-supervised monocular depth estimation[END_REF], [START_REF] Gordon | Depth from videos in the wild: Unsupervised monocular depth learning from unknown cameras[END_REF], [START_REF] Klingner | Self-supervised Monocular Depth Estimation: Solving the Dynamic Object Problem by Semantic Guidance[END_REF], [START_REF] Ranjan | Competitive collaboration: Joint unsupervised learning of depth, camera motion, optical flow and motion segmentation[END_REF], [START_REF] Vitor Guizilini Rares | 3D packing for self-supervised monocular depth estimation[END_REF], [START_REF] Wang | Can Scale-Consistent Monocular Depth Be Learned in a Self-Supervised Scale-Invariant Manner[END_REF], [START_REF] Watson | The temporal opportunist: Self-supervised multiframe monocular depth[END_REF], [START_REF] Zhan | Unsupervised Learning of Monocular Depth Estimation and Visual Odometry with Deep Feature Reconstruction[END_REF], [START_REF] Zhou | Unsupervised learning of depth and ego-motion from video[END_REF] to formulate the problem as self-supervised training of depth and pose. To account for the ill-posed nature of this task, several works have addressed different challenges. [START_REF] Bian | Unsupervised scale-consistent depth and ego-motion learning from monocular video[END_REF], [START_REF] Chen | Selfsupervised learning with geometric constraints in monocular video: Connecting flow, depth, and camera[END_REF], [START_REF] Vitor Guizilini Rares | 3D packing for self-supervised monocular depth estimation[END_REF], [START_REF] Wang | Can Scale-Consistent Monocular Depth Be Learned in a Self-Supervised Scale-Invariant Manner[END_REF] proposed to enforce depth scale and structure consistency. [START_REF] Gordon | Depth from videos in the wild: Unsupervised monocular depth learning from unknown cameras[END_REF], [START_REF] Shu | Feature-Metric Loss for Self-supervised Learning of Depth and Egomotion[END_REF] proposed more robust image reconstruction losses to handle occlusion due to moving objects. [START_REF] Chen | Selfsupervised learning with geometric constraints in monocular video: Connecting flow, depth, and camera[END_REF], [START_REF] Klingner | Self-supervised Monocular Depth Estimation: Solving the Dynamic Object Problem by Semantic Guidance[END_REF], [START_REF] Ranjan | Competitive collaboration: Joint unsupervised learning of depth, camera motion, optical flow and motion segmentation[END_REF], [START_REF] Yin | GeoNet: Unsupervised Learning of Dense Depth, Optical Flow and Camera Pose[END_REF] proposed multi-modal training to better supervise the network. In this paper, it will be shown that it is not only possible to infer the current depth but also forecast future depths accurately.

B. Sequence forecasting

Anticipation of the future state of a sequence is a fundamental part of the intelligent decision making process. The forecasted sequence could be a RGB video sequence [START_REF] Babaeizadeh | Stochastic variational video prediction[END_REF], [START_REF] Finn | Unsupervised learning for physical interaction through video prediction[END_REF], [START_REF] Kumar | Videoflow: A conditional flow-based model for stochastic video generation[END_REF], [START_REF] Mathieu | Deep multiscale video prediction beyond mean square error[END_REF], [START_REF] Rakhimov | Latent video transformer[END_REF], [START_REF] Kreiman | Deep Predictive Coding Networks for Video Prediction and Unsupervised Learning[END_REF], depth image sequence [START_REF] Mahjourian | Geometrybased next frame prediction from monocular video[END_REF], [START_REF] Weng | Inverting the pose forecasting pipeline with spf2: Sequential pointcloud forecasting for sequential pose forecasting[END_REF], semantic segmentation sequence [START_REF] Bhattacharyya | Bayesian prediction of future street scenes using synthetic likelihoods[END_REF], [START_REF] Hsu | Segmenting the Future[END_REF], [START_REF] Graber | Panoptic segmentation forecasting[END_REF], [START_REF] Graber | Panoptic segmentation forecasting[END_REF], [START_REF] Hu | Probabilistic Future Prediction for Video Scene Understanding[END_REF], [START_REF] Luc | Predicting Deeper into the Future of Semantic Segmentation[END_REF], [START_REF] Saric | Warp to the Future: Joint Forecasting of Features and Feature Motion[END_REF], [START_REF] Adam M Terwilliger | Recurrent flow-guided semantic forecasting[END_REF] or even a multi-modal sequence [START_REF] Hu | Probabilistic Future Prediction for Video Scene Understanding[END_REF], [START_REF] Qi | 3D motion decomposition for RGBD future dynamic scene synthesis[END_REF]. Early deep learning models for RGB video future forecasting leveraged several techniques including: Recurrent models [START_REF] Kreiman | Deep Predictive Coding Networks for Video Prediction and Unsupervised Learning[END_REF], variational autoencoder VAE [START_REF] Babaeizadeh | Stochastic variational video prediction[END_REF], generative adversarial network [START_REF] Mathieu | Deep multiscale video prediction beyond mean square error[END_REF], autoregressive model [START_REF] Rakhimov | Latent video transformer[END_REF] and normalizing flows [START_REF] Kumar | Videoflow: A conditional flow-based model for stochastic video generation[END_REF]. These techniques have inspired subsequent sequence forecasting methods. Despite the importance of geometry for developing better decision making, depth forecasting is still in early development. [START_REF] Mahjourian | Geometrybased next frame prediction from monocular video[END_REF] used supervised forecasted depth along with supervised future pose to warp the current image and generate the future image. Instead of using images as input, [START_REF] Weng | Inverting the pose forecasting pipeline with spf2: Sequential pointcloud forecasting for sequential pose forecasting[END_REF] used LiDAR scans and forecast a sparse depth up to 3.0s in the future on the KITTI benchmark [START_REF] Geiger | Are we ready for autonomous driving? the kitti vision benchmark suite[END_REF]. [START_REF] Qi | 3D motion decomposition for RGBD future dynamic scene synthesis[END_REF] used a multi-modal input/output and forecast the depth among other modalities. [START_REF] Hu | Probabilistic Future Prediction for Video Scene Understanding[END_REF] handled the diverse future generation by utilizing a variational model to forecast a multi-modality output. The use of multi-modalities requires additional labels or pretrained networks. This makes the training more complicated. Instead, the work presented in this paper leverages only raw images and forecasts in a self-supervised manner.

C. Vision transformers

The introduction of the Transformers in 2017 [START_REF] Vaswani | Attention is all you need[END_REF] revolutionized natural language processing resulting in remarkable results [START_REF] Tom | Language models are few-shot learners[END_REF], [START_REF] Devlin | BERT: Pre-training of deep bidirectional transformers for language understanding[END_REF], [START_REF] Radford | Learning transferable visual models from natural language supervision[END_REF]. The year 2020 [START_REF] Carion | End-to-end object detection with transformers[END_REF], [START_REF] Dosovitskiy | An image is worth 16x16 words: Transformers for image recognition at scale[END_REF] marked one of the earliest pure vision transformer networks. As opposed to recurrent networks that process sequence elements recursively and can only attend to short-term context, transformers can attend to complete sequences thereby learning long and short relationships. The multi-head attention could be considered as fully connected graph of the sequence's features. It demonstrated its success by outperforming convolution based networks on several benchmarks including classification [START_REF] Dosovitskiy | An image is worth 16x16 words: Transformers for image recognition at scale[END_REF], [START_REF] Li | Improved multiscale vision transformers for classification and detection[END_REF], [START_REF] Zhai | Scaling vision transformers[END_REF], detection [START_REF] Carion | End-to-end object detection with transformers[END_REF], [START_REF] Harold | Grounded language-image pre-training[END_REF], [START_REF] Li | Improved multiscale vision transformers for classification and detection[END_REF] and segmentation [START_REF] Cheng | Masked-attention mask transformer for universal image segmentation[END_REF], [START_REF] Liu | Swin transformer: Hierarchical vision transformer using shifted windows[END_REF]. The has led to a paradigm shift [START_REF] Liu | Are we ready for a new paradigm shift? a survey on visual deep mlp[END_REF]. transformers are slowly wining "The Hardware Lottery" [START_REF] Hooker | The hardware lottery[END_REF]. However, training vision transformers is complicated as these modules are not memory efficient for images and needs a large dataset pretraining. [START_REF] Carion | End-to-end object detection with transformers[END_REF] has demonstrated that it is possible to combine convolution and transformers to learn a good representation without requiring large pertaining. This paper proposes to leverage a hybrid CNN and transformer network as in [START_REF] Carion | End-to-end object detection with transformers[END_REF] that is designed to forecast the geometry of the scene. The proposed network is simple and yet efficient. It outperforms even prior monocular depth inference methods [START_REF] Eigen | Depth map prediction from a single image using a multi-scale deep network[END_REF], [START_REF] Liu | Learning Depth from Single Monocular Images Using Deep Convolutional Neural Fields[END_REF], [START_REF] Yang | Unsupervised learning of geometry from videos with edgeaware depth-normal consistency[END_REF], [START_REF] Zhou | Unsupervised learning of depth and ego-motion from video[END_REF] that access the image of the depth frame.

III. THE METHOD

A. The problem formulation

Let I t ∈ R w×h×c be the t-th frame in a video sequence I = {I t-k:t+n }. The frames I c = {I t-k:t } are the context of I t and I f = {I t+1:t+n } is the future of I t . The goal of the future depth and ego-motion forecasting is to predict the future geometry of the scene D t+n and the ego-motion t+n T t corresponding to I t+n given only the context frames I c :

( Dt+n, t+n Tt) = f (Ic; θ) (1)
where f is a neural network with parameters θ.

In self-supervised learning depth inference, the problem is formulated as novel view synthesis by warping the source frame I s into the target frames I tar using the depth and the s T tar ∈ SE [START_REF] Bhattacharyya | Bayesian prediction of future street scenes using synthetic likelihoods[END_REF] pose target to source pose. The warping is defined as:

ps = π( s TtarH(π -1 (ptar, D(ptar))) (2)
where π is the inverse camera projection defined as:

π -1 p, D(p) = D(p) x -cx fx , y -cy fy , 1 (3)
(f x , f y , c x , c y ) are the camera intrinsic parameters. H is the homogeneous coordinates transformation and s T tar ∈ R 3×4 is the ego-motion. Similarly, a pixel p is projected to a 3D point P given its depth D(p) using the operator π defined as:

π(p) = (fx X Z + cx, fy Y Z + cy) (4)
using the spatial transformers [START_REF] Jaderberg | Spatial transformer networks[END_REF], The reverse warping defined here is fully differentiable and uses a bilinear interpolation. Similarly, the self-supervised depth and ego-motion forecasting is defined here by considering the source frames I c and the target I t+n with the corresponding poses ti T t+n : ∀i ∈ {t -k, ..., t}. The frame I t+n is obtained by reverse warping the context frames.

Reconstructing the frame I t+n using the depth and the pose from only the context by a warping could be formulated as a maximum likelihood problem:

θ = arg max θ∈Θ L(It+n|Ic; θ) ≡ arg max θ∈Θ m P model (I m t+n |I m c ) (5)
m is the number of samples. If P model is assumed to follow a Laplacian distribution P model (I t+n |I c ) ∼ Lap(I t+n ; µ = I t+n ; β = σ 2 I). I t+n is the warped image. Maximizing the Eq. 5 is equivalent to minimizing an L 1 error of I t+n and the known image frame I t+n . Similarly, if the distribution is assumed to follow a Gaussian distribution, the maximization is equivalent to minimizing an L 2 error.

B. The architecture

The architecture of the network is depicted in Fig. 2. The forecasting network is composed of two sub-networks: a pose net to forecast the future ti T t+n and a depth network that forecast D t+n (see Eq. 1) Similar to [START_REF] Kendall | PoseNet: A convolutional network for real-time 6-dof camera relocalization[END_REF], the posenet is composed of a classification network [START_REF] He | Deep residual learning for image recognition[END_REF] as a feature extractor followed by a simple pose decoder as in [START_REF] Godard | Digging into self-supervised monocular depth estimation[END_REF]. The pose network forecasts 6 parameters using the axis-angle representation. The depth network leverages a hybrid CNN and Transformer network as in [START_REF] Carion | End-to-end object detection with transformers[END_REF] that is designed to forecast the geometry of the scene. This network benefits from both modules. The convolution module is used to extract the spatial features of the frames as it is memory efficient, easy to train and does not require large pretraining. The transformer module is used for better temporal feature aggregation. The multihead attention could be considered as a fully-connected graph of the features of each frame. Therefore the information is correlated across all the frames rather than incrementally, one step at a time, as in LSTM [START_REF] Hochreiter | Long short-term memory[END_REF]. The architecture consists of three modules: an encoder, temporal aggregation module and a decoder.

1) Encoder: : ResNet [START_REF] He | Deep residual learning for image recognition[END_REF] is one of the most used foundation models [START_REF] Bommasani | On the opportunities and risks of foundation models[END_REF]. It has demonstrated its success as a task agnostic feature extractor for nearly all vision tasks. In this work, ResNet34 is used as feature extractor. It is pretrained on ImageNet [START_REF] Deng | Imagenet: A large-scale hierarchical image database[END_REF] for better convergence. Each context frame is fed-forward and a pyramid of features is extracted. These features encode the spatial relationship between each scene separately. Thus, at the output of this module, a pyramid of spatial features for each frame is constructed. These features will be correlated temporally using the Temporal aggregation module TAM.

2) Temporal aggregation module: Since its introduction, transformer have demonstrated their performance outperforming their LSTM/RNN counterparts in various sequence learning benchmarks [START_REF] Tom | Language models are few-shot learners[END_REF], [START_REF] Devlin | BERT: Pre-training of deep bidirectional transformers for language understanding[END_REF], [START_REF] Raffel | Exploring the limits of transfer learning with a unified text-to-text transformer[END_REF], [START_REF] Walker | Predicting video with vqvae[END_REF]. forecasting accurate depth requires knowledge of the static objects, accurate ego-motion and knowledge of the motion of the dynamic objects. The last layer of the encoder is assumed to encode higher abstraction features (e. g. recognizing objects). Therefore, correlating temporally these features allows the extraction of the motion features of the scene. The TAM consists of two sub-modules:

• Embedding projection: The dimensions after flattening the feature output of the last layer of the encoder is not memory efficient for the transformers. The embedding projection maps these features as: R K×C×H×W -→ R K×denc . • Transformer encoder: After projecting the features using the embedding layer, a Transformer encoder with N layer, m multi-head attention and d enc is used. It correlates the spatial features of the sequence producing a spatio-temporal fused features.

3) Depth decoder: After the spatio-temporal fusion, the decoder takes these spatio-temporal features along with the context features as input and decodes them to produce a disparity map. As depicted in Fig. 2, the context of the scene is obtained by pooling the features of last frame in the encoder. Two levels are pooled and concatenated. dec f t+n = [ enc f t , T AM ( enc f t-4:t )]. The high level features (skip connection before the TAM) enable learning the motion while the low level features (skip connection at the start of ResNet) recover the finer details lost by the down-sampling. Therefore, the decoder maps (context + motion -→ depth).

Each level of the decoder consists of a simple sequential layer of: transposed convolution with kernel of 3 × 3 with similar channels as the encoder, batch normalization and Relu activation in that order. The forecasting head consists of a convolution with a kernel of 1 × 1 and a Sigmoid activation. The output of this activation, σ, is re-scaled to obtain the depth D = 1 aσ+b , where a and b are chosen to constrain D between 0.1 and 100 units, similar to [START_REF] Godard | Digging into self-supervised monocular depth estimation[END_REF]. For training, each level has a forecasting head but only the last head is used for inference.

The batch is omitted

C. Objective functions

As formulated in Sec. III-A, learning the parameters θ involves maximizing the maximum likelihood of P model . Similar to the prior work, along with the L 1 error other additional regularization terms are used and defined as:

• Photometric loss: Following [START_REF] Godard | Digging into self-supervised monocular depth estimation[END_REF], [START_REF] Vitor Guizilini Rares | 3D packing for self-supervised monocular depth estimation[END_REF], [START_REF] Zhou | Unsupervised learning of depth and ego-motion from video[END_REF] The photometric loss seeks to reconstruct the target image by warping the source images using the forecast pose and depth. An L 1 loss is defined as follows:

Lrec(It+n, It+n) = p |It+n(p) -It+n(p)| (6)
where I t+n (p) is the reverse warped target image obtained by Eq. 2. This simple L 1 is regularized using SSIM [START_REF] Wang | Image quality assessment: from error visibility to structural similarity[END_REF] that has a similar objective to reconstruct the image. The final photometric loss is defined as:

Lpe(It+n, It+n) = p (1 -α) SSIM[It+n(p) -It+n(p)] + α|It+n(p) -It+n(p)| (7)
• Depth smoothness: An edge-aware gradient smoothness constraint is used to regularize the photometric loss. The disparity map is constrained to be locally smooth through the use an image-edge weighted L 1 penalty, as discontinuities often occur at image gradients. This regularization is defined as [START_REF] Heise | Pmhuber: Patchmatch with huber regularization for stereo matching[END_REF]:

Ls(Dt+n) = p |∂xDt+n(p)|e -|∂xI t+n (p)| + |∂yDt+n(p)|e -|∂y I t+n (p)| (8)
Training with these loss functions is subject to major challenges: gradient locality, occlusion and out of view-objects. Gradient locality is a result of bilinear interpolation [START_REF] Jaderberg | Spatial transformer networks[END_REF], [START_REF] Zhou | Unsupervised learning of depth and ego-motion from video[END_REF]. The supervision is derived from the fours neighbors of I(p s ) which could degrade training if that region is low-textured. Following [START_REF] Garg | Unsupervised cnn for single view depth estimation: Geometry to the rescue[END_REF], [START_REF] Godard | Digging into self-supervised monocular depth estimation[END_REF], [START_REF] Godard | Unsupervised monocular depth estimation with left-right consistency[END_REF], an explicit multi-scale approach is used to allow the gradient to be derived from larger spatial regions. A forecasting head is used at each level to obtain each level's disparity map during training. Eq. 2 assumes global ego-motion to calculate the disparity. Supervising directly using this objective is inaccurate when this assumption is violated (e. g. the camera is static or a dynamic object moves with same velocity as the camera). According to [START_REF] Godard | Digging into self-supervised monocular depth estimation[END_REF] this problem can manifest itself as 'holes' of infinite depth. This could be mitigated by masking the pixels that do not change the appearance from one frame to the next. A commonly used solution [START_REF] Godard | Digging into self-supervised monocular depth estimation[END_REF], [START_REF] Zhou | Unsupervised learning of depth and ego-motion from video[END_REF] is to learn a mask µ that weigh the contribution of each pixel, while [START_REF] Zhou | Unsupervised learning of depth and ego-motion from video[END_REF] uses an additional branch to learn this mask. This paper uses the auto-masking defined in [START_REF] Godard | Digging into self-supervised monocular depth estimation[END_REF] to learn a binary mask µ as follows:

µ(I t+n , I t+n , I t ) = L pe (I t+n , I t+n ) < L pe (I t+n , I t ) (9)
µ is set to only include the loss when the photometric loss of the warped image I t+n is lower than the original unwarped image I t . The final objective function is defined as:

L = l [ µ Lp + α d Ls ] (10)

Method

Forecasting Resolution Supervision Abs Rel Sq Rel RMSE log RMSE δ < 1.25 δ < 1.25 2 δ < 1.25 where l is the scale level of the forecasted depth.

IV. EXPERIMENTS

A. Setting 1) KITTI benchmark [START_REF] Geiger | Are we ready for autonomous driving? the kitti vision benchmark suite[END_REF]: : Following the prior work [START_REF] Eigen | Depth map prediction from a single image using a multi-scale deep network[END_REF], [START_REF] Godard | Digging into self-supervised monocular depth estimation[END_REF], [START_REF] Liu | Learning Depth from Single Monocular Images Using Deep Convolutional Neural Fields[END_REF], [START_REF] Mahjourian | Unsupervised Learning of Depth and Ego-Motion from Monocular Video Using 3D Geometric Constraints[END_REF], [START_REF] Wang | Can Scale-Consistent Monocular Depth Be Learned in a Self-Supervised Scale-Invariant Manner[END_REF], [START_REF] Yang | Unsupervised learning of geometry from videos with edgeaware depth-normal consistency[END_REF], [START_REF] Zhou | Unsupervised learning of depth and ego-motion from video[END_REF], the Eigen et al. [START_REF] Eigen | Depth map prediction from a single image using a multi-scale deep network[END_REF] split is used with Zhou et al. [START_REF] Zhou | Unsupervised learning of depth and ego-motion from video[END_REF] pre-processing to remove static frames. Frames without sufficient context are excluded from the training and testing. This split has become the defacto benchmark for training and evaluating depth that is used by nearly all depth methods.

2) Baselines: : As discussed above, previous work on depth forecasting has been supervised using LiDAR scans, and has used a multimodal network that provides depth. Their evaluation is not performed on the Eigen split, nor does it use the defacto self-supervised metrics. In order to fairly evaluate the proposed method, a self-supervised monocular formulation will be used to compare performance with the KITTI Eigen split benchmark. Comparisons will be made with three approaches: prior work on self-supervised depth inference [START_REF] Eigen | Depth map prediction from a single image using a multi-scale deep network[END_REF], [START_REF] Godard | Digging into self-supervised monocular depth estimation[END_REF], [START_REF] Liu | Learning Depth from Single Monocular Images Using Deep Convolutional Neural Fields[END_REF], [START_REF] Wang | Can Scale-Consistent Monocular Depth Be Learned in a Self-Supervised Scale-Invariant Manner[END_REF], [START_REF] Yang | Unsupervised learning of geometry from videos with edgeaware depth-normal consistency[END_REF], [START_REF] Zhou | Unsupervised learning of depth and ego-motion from video[END_REF]; copy of the last observed LiDAR frame as done in [START_REF] Qi | 3D motion decomposition for RGBD future dynamic scene synthesis[END_REF]; and ForecastMonodepth2, a modified version of [START_REF] Godard | Digging into self-supervised monocular depth estimation[END_REF] that is adapted for forecasting pose/depth.

3) Evaluation metrics: : For evaluation, the metrics of previous works [START_REF] Eigen | Depth map prediction from a single image using a multi-scale deep network[END_REF] are used for the depth. During the evaluation, the depth is capped to 80m. To resolve the scale ambiguity, the forecasted depth map is multiplied by median scaling where s = median(Dgt) median(D pred ) . For the pose evaluation, the Absolute Trajectory Error (ATE) defined in [START_REF] Sturm | A benchmark for the evaluation of rgb-d slam systems[END_REF] is evaluated on KITTI odometery benchmark [START_REF] Geiger | Are we ready for autonomous driving? the kitti vision benchmark suite[END_REF] sequences 09 and 10.

4) Implementation details: : PyTorch [START_REF] Paszke | Pytorch: An imperative style, high-performance deep learning library[END_REF] is used for all models. The networks are trained for 20 epochs, with a batch size of 8. The Adam optimizer [START_REF] Diederik | Adam: A method for stochastic optimization[END_REF] is used with a learning rate of lr = 10 -4 and (β 1 , β 2 ) = (0.9, 0.999). As training proceeds, the learning rate is decayed at epoch 15 to 10 -5 . The SSIM weight is set to α = 0.15 and the smoothing regularization weight to α d = 0.001. l = 4 scales are used for each output of the decoder. At each scale, the depth is upscaled to the target image size. d model = 2048, m = 16 and N = 3 for the TAM projection. The input images are resized to 192×640. Two data augmentations were performed: horizontal flips with probability p = 0.5 and color jitter with p = 1. k = 4 frames are used for context sequence and n = 5 is used for short term forecasting and n = 10 for mid-term forecasting as in [START_REF] Qi | 3D motion decomposition for RGBD future dynamic scene synthesis[END_REF] which corresponds to forecasting 0.5s and 1.0s into the future. The ForecastedMonodepth2 is the same as [START_REF] Godard | Digging into self-supervised monocular depth estimation[END_REF] with a modified input. The context images are concatenated and used as input for both depth and pose networks.

B. Depth forecasting results

Table I shows the results of the proposed method on the KITTI benchmark [START_REF] Geiger | Are we ready for autonomous driving? the kitti vision benchmark suite[END_REF]. As specified in Sec. IV-A2, the method is compared to three approaches: prior work on depth inference; copying last frame; and adapting monodepth2 [START_REF] Godard | Digging into self-supervised monocular depth estimation[END_REF] for future forecasting. The proposed method outperforms the forecasting baselines for both short and mid-term forecasting especially for short range forecating. The results are even comparable to non-forecasting methods [START_REF] Eigen | Depth map prediction from a single image using a multi-scale deep network[END_REF], [START_REF] Liu | Learning Depth from Single Monocular Images Using Deep Convolutional Neural Fields[END_REF], [START_REF] Yang | Unsupervised learning of geometry from videos with edgeaware depth-normal consistency[END_REF], [START_REF] Zhou | Unsupervised learning of depth and ego-motion from video[END_REF] that have access to I t+n . The gap between state-of-the-art depth inference and the proposed forecasting method is reasonable due to the uncertainty of the future, the unobservability of certain events such as a new object entering the scene and the complexity of natural videos that requires modeling correlations across space-time with much higher input dimensions.

Fig. 1 shows an example of depth forecasting on Eigen test split. Several observation can be made: EIGEN SPLIT [START_REF] Eigen | Depth map prediction from a single image using a multi-scale deep network[END_REF] • The network learned the correct ego-motion: The position of the static objects is accurate. These results suggest that the network is able to learn a rich spatio-temporal representation that enable learning the motion, geometry and the semantics of the scene. Thus, extending the self-supervision depth inference to perform future forecasting with comparable results. A further analysis is done to evaluate and validate the choices of the network in Sec. IV-D.

C. Ego-Motion forecasting results

Table II shows the results of the proposed network on the KITTI odometry benchmark [START_REF] Geiger | Are we ready for autonomous driving? the kitti vision benchmark suite[END_REF]. Similar to depth, the assessment is made by comparing with non-forecasting prior works. To avoid data leakage, the network is trained from scratch on the sequences 00-08 of the KITTI odometry benchmark. The network takes only the context images I c and forecasts t T t+n . The ATE results in Table II show the proposed network achieves a competitive result relative to other non-forecasting approaches. All the methods are trained in the monocular setting, and therefore scaled at test time using ground-truth. These results suggest that using the proposed architecture along with the self-supervised loss function successfully learns the future joint depth and ego-motion.

D. Ablation study

To further analyse the network, several ablations are made. Table III depicts a comparison of the proposed model with several variants. The evaluation is done for short-term forecasting n = 5 using k = 4 context frames.

1) Effect of the Temporal Aggregation Module: In order to evaluate the contribution of the multi-head attention, a variant of the proposed method is designed by replacing the TAM module by a simple concatenation of the last layer features. From Table III, the improvement induced by the TAM module is significant across all metrics. These results suggest that the performance obtained by the proposed method is achieved through the TAM module. Since the TAM aggregates the temporal information across all frames using a learned attention, the temporal features are better correlated and the final representation successfully encodes the spatio-temporal relationship between the images.

2) Effect of sharing the encoder of depth and ego-motion: Since both pose and depth networks encode the future motion and geometry of the scene, it is expected that sharing the encoders of these networks yield better results. However, as reported by Table III, the degradation is significant. Even though these tasks are collaborative, sharing the encoder will result in a set of parameters θ that are neither the best local optima for the depth nor for the pose. By alleviating this restriction and separating the encoders, the network learns better local optima for both pose and depth.

3) The benefit of using multiple scales: In order to evaluate the multi-scale extension, a variant of the proposed method that uses only one scale is trained. As illustrated in the Table III The network benefits from the multi-scale. The reverse warping uses bi-linear interpolation. As mentioned earlier, each depth point depends only on the four neighboring warped points. By using a multi-scale depth at training-time the gradient is derived from a larger spatial region directly at each scale.

4) Effect of auto-masking: Table III compares the proposed method with a variant without using the auto-masking defined is Sec. III-C. The results show that using auto-masking improves all four evaluation criteria. This demonstrates that using auto-masking, for pixels that do not change appearance, reject these outliers that inhibit the optimization. This leads to better accuracy of the forecasted depth.

E. Limitations and perspective

Even though the proposed forcasting method yields good results, there exists a gap with respect to non-forecasting methods. Several limitations contribute to this:

• A common assumption across SOTA methods is that the environment is deterministic and that there is only one possible future. However, this is not accurate since there are multiple plausible futures. Given the stochastic nature of the forecasting proposed here, the network will tend to forecast the mean of all the possible outcomes [START_REF] Babaeizadeh | Stochastic variational video prediction[END_REF]. • The network does not forecast the correct boundaries of the objects. This is due to the formulation as a maximum likelihood problem with a Laplacien distribution assumption and the deterministic nature of the architecture. As a result, the boundaries of the dynamic objects are smoothed. • Due to the problem formulation, the scale of the forecasted depth is ambiguous. • The model fails to account for the motion of distant dynamic objects due to lack of parallax. and fine objects are ignored by the network. To address these limitations future work will investigate constraining depth to be conditional on the input image structure as in [START_REF] Boulahbal | Are conditional gans explicitly conditional?[END_REF]. Concerning scale, one solution could be to infer directly from the scene as in [START_REF] Bian | Unsupervised scale-consistent depth and ego-motion learning from monocular video[END_REF], [START_REF] Chen | Selfsupervised learning with geometric constraints in monocular video: Connecting flow, depth, and camera[END_REF], [START_REF] Wang | Can Scale-Consistent Monocular Depth Be Learned in a Self-Supervised Scale-Invariant Manner[END_REF].

V. CONCLUSION

This paper proposed an approach for forecasting future depth and ego motion using only raw images as input. This problem is addressed as end-to-end self-supervised forecasting of the future depth and ego motion. Results showed significant performances on several KITTI dataset benchmarks [START_REF] Geiger | Are we ready for autonomous driving? the kitti vision benchmark suite[END_REF]. The performance criteria are even comparable with non-forecasting self-supervised monocular depth inference methods [START_REF] Eigen | Depth map prediction from a single image using a multi-scale deep network[END_REF], [START_REF] Liu | Learning Depth from Single Monocular Images Using Deep Convolutional Neural Fields[END_REF], [START_REF] Yang | Unsupervised learning of geometry from videos with edgeaware depth-normal consistency[END_REF], [START_REF] Zhou | Unsupervised learning of depth and ego-motion from video[END_REF]. The proposed architecture demonstrates the effectiveness of combining the inductive bias of the CNN as a spatial feature extractor and the multi-head attention of transformers for temporal aggregation. The proposed method learns a spatio-temporal representation that captures the context and the motion of the scene.

Fig. 3. Qualitative results of the comparison of the proposed method with the ForecastMonodepth2 baseline. This comparison shows that the proposed method performs better than the baseline, especially for nearby dynamic objects. This observation is further validated in Table IV. In addition, the baseline method is showing a lack of detection of moving objects, which leads to a degradation of the forecasted depth. The proposed method is able to detect moving objects, thus accurately forecasting the depth of the scene.

Range

VI. MORE QUALITATIVE RESULTS

In order to further analyse the depth forecasting results, an assessment based on the ground-truth LiDAR distance is done. Table IV shows the comparison of the non-forecasting method Monodepth2 [START_REF] Godard | Digging into self-supervised monocular depth estimation[END_REF], ForecastMonodepth2 and the proposed methods.

The results suggest that the proposed method outperform the adaptation of Monodepth2 for short-range with a improvement of the Abs Rel of -16.7% and medium-range with a improvement of the Abs Rel of -8.8%. These regions are the most significant regions of the forecasting as it has enough parallax for the ego-motion and dynamic object motion. Besides, this region assess several challenges including out-of-view objects and occlusion. For the long-range forecasting, the results shows that the two methods performer badly due to the lack of parallax in these region and down-sampling that ignore small objects. Moreover, this region has a high likelihood of new-object entering the scene which the forecasting is unable to handle by definition. The reported performances and the qualitative results suggest that the two forecasting networks only fit the road and completely ignore any other object. These results are shown qualitatively in Fig. 3.

  Fig.1. The proposed method is trained using only raw sensor self-supervision, it is able to forecast an accurate geometry of the scene. The network only accesses the frames I t-3:t and forecast the depth D t+n and the pose t T t+n . Not only the network forecasts accurately the ego-motion (the depth of the static object is accurate) but also handles the dynamic objects.

Fig. 2 .

 2 Fig.2. Illustration of the proposed architecture. Two sub-networks are used for training: The PoseNetwork as in network[START_REF] Godard | Digging into self-supervised monocular depth estimation[END_REF],[START_REF] Kendall | PoseNet: A convolutional network for real-time 6-dof camera relocalization[END_REF] is used to forecast the ego-motion. The depth network combines both CNN and transformers. The Resnet34[START_REF] He | Deep residual learning for image recognition[END_REF] encoder extracts the spatial features for each context frame. The embedding projection module projects these features into R k×d model where k = 4 is the context frames. N = 3 transformer encoders are used to fuse the spatial temporal to obtain a rich spatio-temporal features. The output of the transformer module encodes the motion of the scene. The decoder uses simple transposed convolution. In order to recover the context, skip connections are pooled from the encoder. Only the last frame features are pooled for the context. The decoder outputs a disparity map that will be used along with the pose network to warp the source images onto the target.

TABLE I QUANTITATIVE

 I PERFORMANCE COMPARISON OF ON THE KITTI BENCHMARK WITH EIGEN SPLIT [20] FOR DISTANCES UP TO 80M. IN THE Supervision COLUMN, D REFERS TO DEPTH SUPERVISION USING LIDAR GROUNDTRUTH AND (SS) SELF-SUPERVISION. AT TEST-TIME, ALL MONOCULAR METHODS (M) SCALE THE DEPTHS WITH MEDIAN GROUND-TRUTH LIDAR.

	Method	forecasting	Seq.09	Seq.10
	Mean Odom	-	0.032 ± 0.026	0.028 ± 0.023
	ORB-SLAM [45]	-	0.014 ± 0.008	0.012 ± 0.011
	SfMLearner [69]	-	0.021 ± 0.017	0.020 ± 0.015
	Monodepth2 [21]	-	0.017 ± 0.008	0.015 ± 0.010
	Wang et al. [60]	-	0.014 ± 0.008	0.014 ± 0.010
	Ours	0.5s	0.020 ± 0.011	0.018 ± 0.011
		TABLE II
	ATE ERROR OF THE PROPOSED METHOD AND THE PRIOR
	NON-FORECASTING METHODS ON KITTI [20]. THE PROPOSED METHOD
	IS COMPARABLE TO THESE METHODS EVEN IF IT ONLY ACCESSES PAST
		FRAMES.

TABLE III ABLATION

 III STUDY OF THE DIFFERENT COMPONENT OF THE PROPOSED METHOD. THE EVALUATION WAS DONE ON KITTI BENCHMARK USING

QUANTITATIVE

  PERFORMANCE COMPARISON ON THE KITTI BENCHMARK WITH EIGEN SPLIT[START_REF] Geiger | Are we ready for autonomous driving? the kitti vision benchmark suite[END_REF] FOR MULTIPLE DISTANCES RANGE. FOR ABS REL, SQ REL, RMSE AND RMSE LOG LOWER IS BETTER, AND FOR δ < 1.25, δ < 1.25 2 AND δ < 1.25 3 HIGHER IS BETTER. THREE RANGES ARE CONSIDERED: SHORT RANGE [0 10M] WHICH REPRESENTS 37.95%, MEDIUM-RANGE [10 30]WHICH REPRESENTS 50.74% AND LONG-RANGE[30 80] WHICH REPRESENTS 11.30%. THE RESULTS SHOWS THAT THE PROPOSED METHOD IS ABLE TO FORECAST GOOD DEPTH AND OUTPERFORM THE BASELINE AT SHORT AND MEDIUM FORECASTING RANGE.

		Monodepth2 [21]	-	0.066	0.264	0.106	1.076	0.959	0.987	0.994
		ForecastMonodepth2	0.5s	0.138	0.586	0.178	1.697	0.847	0.957	0.985
		Ours	0.5s	0.112	0.595	0.155	1.573	0.893	0.964	0.986
		Monodepth2 [21]	-	0.119	0.858	0.192	3.706	0.876	0.956	0.978
	[10m 30m]	ForecastMonodepth2	0.5s	0.192	1.673	0.258	5.169	0.725	0.906	0.963
		Ours	0.5s	0.167	1.453	0.241	4.803	0.782	0.921	0.965
		Monodepth2 [21]	-	0.188	3.094	11.115	0.288	0.709	0.897	0.950
	[30m 80m]	ForecastMonodepth2	0.5s	0.213	3.526	11.940	0.292	0.631	0.874	0.953
		Ours	0.5s	0.224	4.052	12.638	0.312	0.622	0.862	0.941
					TABLE IV

Method Forecasting Abs Rel Sq Rel RMSE log RMSE δ < 1.25 δ < 1.25 2 δ < 1.25 3 [00m 10m]

• The network handles correctly the out-of-view object (i. e.the bicycle on the scene).


Lylian Challier
email: lylian.challier@ifremer.fr

Amélie Forget

Charlène Bazin

Simon Tanniou

Justine Le Doare

Romain Davy

Hélène Bernard

Raphael Tripier

Agathe Laes-Huon

Nicolas Le Poul

An ultrasensitive and highly selective nanomolar electrochemical sensor based on an electrocatalytic peak shift analysis approach for copper trace detection in water

Keywords: Copper sensor, modified graphite electrode, electrolyte engineering, hydrogen evolution reaction, peak shift analysis

Introduction

Copper (Cu) is an essential micronutrient for the marine environment since it is involved in many oceanic biochemical processes [START_REF] Peers | Copper-containing plastocyanin used for electron transport by an oceanic diatom[END_REF][2] [START_REF] Arellano | The donor side of photosystem II as the copper-inhibitory binding site Photosynth[END_REF]. However, copper also plays a dual role as a poison in affecting the metabolism of biological organisms [START_REF] Leal | Copper pollution exacerbates the effects of ocean acidification and warming on kelp microscopic early life stages[END_REF] [START_REF] Leal | Copper pollution exacerbates the effects of ocean acidification and warming on kelp microscopic early life stages[END_REF] and the toxic concentration of copper depends on the considered biological species. For example, the acute toxicity of copper was evaluated to be equal to 9.5 µmol.L -1 for green crab, 92 nmol.L -1 for blue mullet, [START_REF]Quality criteria for water[END_REF] whereas for some microalgae, the lethal concentration of copper is close to the optimal content for their own growth [START_REF] Cairns | Parker Effects of Temperature on Aquatic Organism Sensitivity to Selected Chemicals Virginia Water Resources Research Center[END_REF]. The dissolved fraction of copper (particles size < 0.45 µm) is considered as more harmful than the particulate fraction (particles size > 0.45 µm). It can include free Cu 2+ (0.2%) but is mainly constituted (99.8%) of complexes resulting from Cu(II) chelation by natural ligands (humic substances). As a landmark, recent works have reported low concentrations of total dissolved copper in rivers, estuaries and coastal oceans (3 to 4 nmol.kg -1 ) [START_REF] Dulaquais | Links between size fractionation, chemical speciation of dissolved copper and chemical speciation of dissolved organic matter in the Loire estuary[END_REF] but these concentrations are increasing due to human industrialization [START_REF] Gledhill | Brown The toxicity of copper (II) species to marine algae, with particular reference to macroalgae[END_REF] reaching tens of nanomolar [START_REF] Abualhaija | Competition between copper and iron for humic ligands in estuarine waters[END_REF] in estuaries. Thus, total dissolved Cu(II) concentration must be sensed in the nanomolar range, and free copper in the picomolar one, to monitor local toxic events affecting biological activities.

For this purpose, electrochemical methods, such as the well-known anodic stripping voltammetry (ASV), have been intensively used [START_REF] Gibbon-Walsh | Pseudopolarography of copper complexes in seawater using a vibrating gold microwire electrode[END_REF] [START_REF] Croot | Polarographic determination of half-wave potentials for copper-organic complexes in seawater[END_REF]. These methods have even been applied in situ for copper analysis in the marine environment with the voltammetric in situ profiling technique (VIP) [START_REF] Lou Tercier | A Novel Voltammetric In-Situ Profiling System for Continuous Real-Time Monitoring of Trace Elements in Natural Waters[END_REF] [START_REF] Braungardt | Analysis of dissolved metal fractions in coastal waters: An inter-comparison of five voltammetric in situ profiling (VIP) systems[END_REF] or more recently with the ElViDor device [START_REF] Cathalot | ELVIDOR-vibrating gold micro-wire electrode: In situ high resolution measurements of Copper in marine environments[END_REF]. These instruments can achieve limit of detection (LOD) values in the nanomolar range. However, due to the high toxicity of mercury and in agreement with the 2008/105/EC and 2011/65/UE European guidelines, the mercury layer used in the VIP needs to be replaced. Moreover, the resolution for the ElViDor system can be negatively impacted by peak overlapping of other electroactive non-metallic species. To address these issues, electrode surface modification for sensing purposes can be a relevant strategy, since it selectively pre-concentrates a metal of interest [START_REF] Liu | In-situ synthesis of reduced graphene oxide/gold nanoparticles modified electrode for speciation analysis of copper in seawater[END_REF] [START_REF] Zuo | Utilization of AuNPs dotted S-doped carbon nanoflakes as electrochemical sensing platform for simultaneous determination of Cu (II) and Hg (II)[END_REF] [START_REF] March | Modified electrodes used for electrochemical detection of metal ions in environmental analysis[END_REF] [START_REF] Garciá-Miranda Ferrari | Recent advances in portable heavy metal electrochemical sensing platforms[END_REF]. In that field, cyclam [START_REF] Peers | Copper-containing plastocyanin used for electron transport by an oceanic diatom[END_REF][START_REF] Leal | Copper pollution exacerbates the effects of ocean acidification and warming on kelp microscopic early life stages[END_REF][START_REF] Dulaquais | Links between size fractionation, chemical speciation of dissolved copper and chemical speciation of dissolved organic matter in the Loire estuary[END_REF] has been recognized as an excellent ligand for copper detection by adsorptive stripping voltammetry (AdSV) [START_REF] Goubert-Renaudin | Cyclam-functionalized silica-modified electrodes for selective determination of Cu(II)[END_REF][21] [START_REF] Goubert-Renaudin | Factors affecting copper(II) binding to multiarmed cyclam-grafted mesoporous silica in aqueous solution[END_REF][23] [START_REF] Ben Mefteh | Voltammetric Detection of Copper Ions on a Gold Electrode Modified with a N-methyl-2-naphthylcyclam film[END_REF][25] [START_REF] Chergui | Uptake of copper ions by carbon fiber/polymer hybrids prepared by tandem diazonium salt chemistry and in situ atom transfer radical polymerization[END_REF]. Different processes of surface functionalization [START_REF] Goubert-Renaudin | Cyclam-functionalized silica-modified electrodes for selective determination of Cu(II)[END_REF][21] [START_REF] Feier | Influence of the electrografting method on the performances of a flow electrochemical sensor using modified electrodes for trace analysis of copper (II)[END_REF][24] [START_REF] Ben Mefteh | Gold electrodes functionalized by methyl-naphthyl substituted cyclam films for the detection of metal ions[END_REF] on various electrode types [START_REF] Goubert-Renaudin | Cyclam-functionalized silica-modified electrodes for selective determination of Cu(II)[END_REF] [START_REF] Jasmin | Straightforward grafting approach for cyclam-functionalized screen-printed electrodes for selective Cu(II) determination[END_REF][23] [START_REF] Ben Mefteh | Voltammetric Detection of Copper Ions on a Gold Electrode Modified with a N-methyl-2-naphthylcyclam film[END_REF] have been reported, leading to various cyclam surface densities and consequently to diverse analytical performances. On one hand, a drop casting method demonstrated the highest cyclam coverage associated with the largest linear range for copper detection (3 orders of magnitude) starting at 1 nmol.L -1 [START_REF] Ben Mefteh | Voltammetric Detection of Copper Ions on a Gold Electrode Modified with a N-methyl-2-naphthylcyclam film[END_REF]. However, due to the presence of an inhomogeneous and disorganized receptor layer [START_REF] Kaliyaraj Selva Kumar | A mini-review: How reliable is the drop casting technique[END_REF] [START_REF] Dong | Three-ring, branched cyclam derivatives and their interaction with nickel(II), copper(II), zinc(II) and cadmium(II)[END_REF], this sensor is also the less selective one against iron, nickel and zinc. On the other hand, cyclambased sensing layers obtained with surface modification by silane chemistry [START_REF] Goubert-Renaudin | Cyclam-functionalized silica-modified electrodes for selective determination of Cu(II)[END_REF] and electroreduction of diazonium salts [START_REF] Jasmin | Straightforward grafting approach for cyclam-functionalized screen-printed electrodes for selective Cu(II) determination[END_REF] displayed better selectivity for copper sensing. However, due to the low surface concentration of cyclam, the limit of quantification (LOQ) was one order of magnitude higher compared to drop casting process. Finally, using a high surface to volume ratio electrode such as graphite felt [START_REF] Feier | Influence of the electrografting method on the performances of a flow electrochemical sensor using modified electrodes for trace analysis of copper (II)[END_REF] was shown to be helpful for decreasing Cu(II) LOQ. So far, in these reported works, only N-functionalized cyclams were used as bifunctional chelators, while it has been demonstrated that this way of modification is deleterious for copper chelation selectivity compared to a C-functionalized one [29]. Moreover, despite their excellent conductivity, low capacitive current, good reproducibility, adjustable surface area and very low cost, pencil graphite electrodes (PGE) have never been used for these applications.

In the meantime, many copper complexes [START_REF] Liu | Water splitting based on homogeneous copper molecular catalysts[END_REF] [START_REF] Walcarius | Electrocatalysis, sensors and biosensors in analytical chemistry based on ordered mesoporous and macroporous carbon-modified electrodes[END_REF], the catalytic reaction allowing significant enhancement of the electrochemical signal [START_REF] Wu | Electrocatalytic Mechanism for Improving Sensitivity and Specificity of Electrochemical Nucleic Acid-Based Sensors with Covalent Redox Tags -Part i[END_REF]. In those cases, current intensities are exclusively converted to substrate concentration.

In this work, we propose to explore a different kind of electrochemical sensors, based on heterogeneous electrocatalysis, where the targeted species to analyse is the catalyst (Cu) and no longer the substrate. Indeed, under certain conditions ("total catalysis", see below) and according to theoretical works for electrocatalysis at a modified electrode [START_REF]Savéant Elements of Molecular and Biomolecular Electrochemistry : an Electrochemical Approach to Electron Transfer Chemistry[END_REF], the potential of the catalytic peak obtained by CV can vary with the surface concentration of the catalyst, as described by equation (1) for a Nernstian (fast and reversible electron transfer) case:

𝐸 cat 0,ap = 𝐸 P/Q 0 + 𝑅𝑇 𝑛𝐹 𝑙𝑛 ( 𝑘𝛤 P/Q 0 √ 𝐷 s 𝑛𝐹𝑣 𝑅𝑇 ) (1)
with 𝐸 cat 0,ap being the apparent standard catalytic potential, 𝐸 P/Q 0 the standard potential of the P/Q couple, 𝛤 P/Q 0 the surface concentration of the catalyst, 𝐷 s the diffusion coefficient of the substrate, 𝑘 the rate constant of the catalytic reaction and 𝑣 the scan rate.

The equation ( 1) is valid for the well-known "total catalysis" case, for which the rate of the catalytic reaction is so fast that the catalytic reaction is controlled by the diffusion of the substrate. According to this equation, it may be possible to directly determine the surface concentration of the catalyst by simply measuring the value of the potential ascribed to the catalytic peak. Thus, the namely peak shift analysis (PSA) may be considered as a new way to report copper surface concentration. For that purpose, we used cyclam under its C-functionalized version by a para-aminobenzyl pendant (p-H 2 N-Bn-cyclam). This ligand was first covalently immobilized onto PGE by a diazonium approach (see experimental section for details). In a second step, the modified PGE was dipped in a solution of copper. Finally, the Cu-modified electrode was studied in a copper-free solution thanks to AdSV and PSA approaches (Figure 1).

To validate this strategy, we have initially optimized the experimental conditions (supporting electrolyte, pH) and performed comparative studies with PGE surfaces modified with copper layer but without cyclam. From the resulting experiments, we have then explored the sensing properties of our modified cyclam PGE system in the frame of the HER reaction leading to significant enhancement of the LOD compared to the classic stripping methods. This work thus paves the way for a new eco-friendly strategy for highly sensitive detection of pre-concentrated metals at modified electrode.

Fig. 1. Schematic diagram of the experimental principle envisioned in this work.

EXPERIMENTAL

Chemicals and reagents

In order to avoid any copper contamination, reagent preparation was performed according to trace metal clean techniques using high quality products. Sulfuric acid (95-97%), sodium chloride (99.99%), potassium chloride (99.999%) were purchased from Merck. Sodium nitrite (99.999%), calcium chloride dihydrate (> 99%), perchloric acid (70%) and ruthenium(III) hexamine chloride were purchased from Sigma Aldrich.

Magnesium chloride hexahydrate, sodium sulphate anhydrous NORMAPUR and ethanol 96% were purchased from VWR. Sodium perchlorate was acquired from Alfa Aesar, ortho-phosphoric acid 85% and mono-sodium phosphate were purchased from Fluka. p-H 2 N-Bn-cyclam was synthesized as previously described [START_REF] Breton | Modification of carbon electrode with aryl groups having an aliphatic amine by electrochemical reduction of in situ generated diazonium cations[END_REF]. All solutions were prepared with purified water (18.2 MΩ.cm, pH = 6.3) from a Milli-Q system (Millipore). Prior to any use, solutions were deoxygenated by bubbling nitrogen for 5 min. To avoid metal contamination, electrochemical cells were rinsed firstly with 10 % of HCl solution and then with Milli-Q water before use. Pencil leads (Pilot ENO-G 0.3mm HB) were commercially purchased.

Electrode preparation

PGE were produced from a 6 cm pencil lead cut into 2 cm rods with a scalpel blade, sonicated for 5 min in 96 % ethanol solution, rinsed 3 times in ethanol 96 % and dried at 70 °C during 15 min. To avoid any metal contamination from the electrode surface, the electronic link between the potentiostat and PGE was ensured thanks to a homemade disposable connection with a 0.7 mm diameter pencil graphite rod and a small piece of silicon sleeve. Finally, only 1 cm of the PGE was dipped in the studied solutions (geometric area S = 0.094 cm²).

Electrochemical measurement

All electrochemical measurements were performed with a Bio-logic potentiostat (VSP), using a conventional three electrodes system, comprising a platinum counter electrode, a silver/silver chloride reference electrode (Ag/AgCl) and a pencil-graphite working electrode.

Electrochemical PGE modification with p-H 2 N-Bn-cyclam

Electrodes were modified by electrochemical reduction of in situ generated diazonium salt according to a reported procedure [START_REF] Breton | Modification of carbon electrode with aryl groups having an aliphatic amine by electrochemical reduction of in situ generated diazonium cations[END_REF]. To generate the diazonium salt, 1 mg.mL -1 solution of p-H 2 N-Bn-cyclam was added to a solution of NaNO 2 (1.1 eq.) in 0.01 mol.L -1 H 2 SO 4 (figure 1.1). The lifetime of the diazonium species was monitored thanks to its reactivity with N-(1-naphtyl)-ethylenediamine as already published [START_REF] Tarafder | Spectrophotometric determination of nitrite in water[END_REF] (Section 1a, SI). PGE were functionalized by applying a fixed potential (E = 0.7 V vs. Ag/AgCl) until a total charge of 50 µC was passed through the circuit (Section 1b, SI).

Preparation of PGE electroplated with copper

Copper electroplating on PGE was performed by chronocoulometry, applying of a reductive potential of 0.8 V vs. Ag/AgCl in a solution of 1.10 -3 mol.L -1 CuSO 4 / 1 mol.L - 1 H 2 SO 4 . Depending on the desired copper amount, various deposition times were applied.

Analytical procedure for Cu(II) measurements

Copper sulfate solutions (1-1000 nmol.L -1 ) were used as model solutions to explore the PSA method using cyclam-modified PGE. Considering this new methodology for an eco-friendly marine use, artificial seawater (450 mmol.L -1 NaCl, 10 mmol.L -1 KCl, 9

mmol.L -1 CaCl 2 , 30 mmol.L -1 MgCl 2 , 16 mmol.L -1 MgSO 4 , pH = 7.8 adjusted with NaOH) [START_REF] Aminot | Hydrologie des écosystèmes marins. Paramètres et analyses[END_REF] was spiked with known amounts of Cu 2+ in the nanomolar range, to demonstrate the feasibility of the measurement in natural sample . The electrodes were dipped for 60 minutes in copper solution to ensure that the thermodynamic equilibrium between chelated and free copper was reached (figure 1.2). After incubation, electrodes were rinsed with Milli-Q water. Voltamperometric measurements were performed at various pH values and for various supporting electrolytes by soaking the modified electrodes in different electrolytic solutions (figure 1.3).

RESULTS ANS DISCUSSION

Investigation of the catalytic proton reduction by Cu-cyclam-modified electrodes

Our first objective was to demonstrate that cyclam-modified PGEs could effectively trap Cu 2+ ions for further sensing by electrochemical methods. For that purpose, cyclam-modified PGE were soaked in a 1 µmol.L -1 concentrated Cu 2+ solution for 1 h [START_REF] Ben Mefteh | Gold electrodes functionalized by methyl-naphthyl substituted cyclam films for the detection of metal ions[END_REF], and studied by CV in sodium perchlorate at pH = 3 after rinsing [START_REF] Goubert-Renaudin | Factors affecting copper(II) binding to multiarmed cyclam-grafted mesoporous silica in aqueous solution[END_REF]. After biasing the potential to 0.6 V, the resulting voltammogram (Figure SI.3) revealed on the backscan an anodic peak ascribed to Cu(0) dissolution. The potential of this oxidation was found to vary between 0.075 and 0.015 V vs.

[20][21][22][23][24]
Ag/AgCl. As the cathodic limits was pushed towards more negative values, the Cu(0)

oxidative peak intensity increased, reaching a maximal faradic current of 0.72 ± 0.08 µA for a cathodic limit at 1.0 V. From this result, the maximal quantity of copper chelated at the electrode was evaluated from the area of the oxidation peak. A value of 12.5 ± 0.9 µC.cm -² was found, meaning that a maximum of 65 ± 5 pmol.cm -² of cupric ion could be trapped.

Knowing that copper uptake was possible with the cyclam-modified PGE, the next goal was then to evaluate how equation ( 1) could be relevant to measure copper amounts in a sensing perspective. For this purpose, experimental conditions for obtaining a well-defined and reproducible heterogeneous copper induced electrocatalytic signal were explored. Considering the abundant literature on copper complexes for the production of dihydrogen, we first investigated the possibility to perform proton reduction at cyclam-modified PGE with chelated copper.

For this, CV experiments were performed in two different electrolytes (CH 3 CO 2 NH 4 and NaClO 4 ) and for various pH values (7, 6, 5, 4 and 3), starting from the open circuit potential (E oc ) to strong cathodic potential to explore HER, followed by a reversed scan to reveal copper re-oxidation. These two electrolytes were chosen because the former was systematically used in previous papers describing copper detection by cyclam-modified electrodes [START_REF] Goubert-Renaudin | Cyclam-functionalized silica-modified electrodes for selective determination of Cu(II)[END_REF][21] [START_REF] Goubert-Renaudin | Factors affecting copper(II) binding to multiarmed cyclam-grafted mesoporous silica in aqueous solution[END_REF][23] [START_REF] Ben Mefteh | Voltammetric Detection of Copper Ions on a Gold Electrode Modified with a N-methyl-2-naphthylcyclam film[END_REF][25], whereas the latter has the advantage to be weakly coordinating, unlike acetate, and also to be redox-innocent [START_REF] Ujvari | On the stability of perchlorate ions against reductive attacks in electrochemical systems and in the environment[END_REF]. For these experiments, the cylam-modified PGE were pre-dipped in a Cu 2+ solution (100 nmol.L -1 ) for 1 h, then rinsed with water and studied in a copper-free electrolytic solution.

As shown in Figure 2A, CVs performed in ammonium acetate led to the appearance of an intense cathodic wave (onset at 1.1 V) upon negative scan, while a much less anodic peak corresponding to Cu(0) oxidation was detected at 0.1 V when scanning positively. The current for the cathodic peak was found to strongly increase inversely to the pH, in strong contrast to anodic peak which remained constant for 3 < pH < 6

(inset Figure 1B). Bubble formation was observed from 1.3 V as expected for HER.

Interestingly, when using sodium perchlorate, current densities for the reduction process were significantly lower than for acetate. Moreover, a peak current instead of a wave was observed for pH < 5 (E pc = 1.20 V at pH = 4 and 1.45 V at pH = 3, see Fig. 1B, blue and pink curves). However, changing the supporting electrolyte did not modify the anodic signature. For cyclam-modified PGE which were not soaked in any copper(II) solution, no cathodic peak current could be observed for this potential range (Fig. 4A, red curve). These results thus i) show that copper accumulated by the ligand is involved in a reduction process that depends on pH, indicating HER and ii) confirm the results obtained by Siewert et al. [START_REF] Nestke | A Copper Complex as Catalyst in Proton Reduction[END_REF], who demonstrated that the counter anion used in the electrolyte may strongly interfere during HER. In order to confirm this hypothesis, three supplementary sodium-based supporting electrolytes (acetate, sulfate and phosphate) were investigated at pH = 3. As shown in Figure 3, the anion of the electrolyte has a significant influence on the cathodic electrochemical signal, but minor effect on copper(0) oxidation. Considering acetate (Fig. 3, black curve) and phosphate (red curve), an important reduction current appeared at -1.0 V and -1.2 V respectively, decreasing almost linearly, without showing a well-defined electrochemical peak, in contrast to perchlorate (pink curve)

and sulfate anions (blue curve). For the latter, a sharp catalytic electrochemical signal was observed at -1.37 V, close to the catalytic potential obtained for perchlorate but with a peak intensity 2.5 times higher.

For the anodic signal (Fig. As the counter anion seemed to play a significant role during the reduction reaction, and considering the SO 4 2-/SO 2 couple (E 0 = 0.16 V vs. NHE), we hypothesized that sulfate ion could be involved as a co-factor into the HER catalytic process or may be a part of the faradic yield. For this purpose, the catalytic peak obtained with a Cu- Further studies were then carried out to rationalize the impact of the electrode material on the electrochemical sensing, since it is well-known that the HER overpotential is fully material-dependent [START_REF] Zeng | Recent advances in heterogeneous electrocatalysts for the hydrogen evolution reaction[END_REF]. As shown in Figure 4A, the catalytic peak height intensities reached 300 µA whatever the electrode interface. This observation is consistent with a substrate-diffusion controlled catalytic process ("total catalysis"). In strong contrast, the peak potential was highly material-dependent, varying from 1.70 (for PGE/cyclam) to 1.32 V (for PGE/Cu(0)). Here, the shift in peak potential may be ascribed to the catalytic rate, which is likely the highest for PGE/Cu(0)and PGE/cyclam/Cu(II). The Cu-free cyclammodified PGE (Fig. 4A, red curve) displayed a more negative and broader peak than the bare PGE (Fig. 4A, black curve), as probably resulting from an insulating effect due to the organic cyclam layer. The slight difference in potential between cyclammodified electrode with copper (Fig. 4A, blue curve, E = 1.34 V) and copper electroplated electrode (green curve, E = 1.28 V) may arise from the difference of copper amount onto the electrode surface, determined at 252 and 304 nC from the oxidation peak (inset Fig. 4A), respectively. To ensure that the HER electrochemical signal obtained with Cu-cyclam-modified PGE was not related to the release of free copper from ligand decomposition [START_REF] Nestke | A Copper Complex as Catalyst in Proton Reduction[END_REF], a simple test was carried out consisting in reusing the PGE after electrochemical demetallation at 0.0 V and re-metallation with Cu 2+ . As shown in Figure 4B (pink curve), the catalytic peak after stripping was shifted negatively, and approached the value found for the cyclam-modified PGE (Fig. 4B, red curve). Dipping the unmetallated electrode into a copper(II) solution led to the full recovering of the catalytic signal at closed potential (Fig. 4B, blue curve vs black curve) accompanied by a stripping signal equivalent to the previous one (206 nC and 216 nC, respectively). These results show that even after being used for HER and copper stripping, cyclam moieties at the electrode maintain their complexing properties, since both cathodic signal during HER and anodic signal during copper stripping were recovered. Thus, the cathodic signal does not seem associated with a decomposition of the cyclam ligand.

Comparative determination of nanomolar copper(II) concentration by AdSV and

PSA approaches.

After evidencing that the HER catalytic signal was optimized in sodium sulfate at pH = Since copper concentration can be calculated from the anodic peak integration, the copper quantities accumulated at E oc were plotted against the corresponding copper concentration in solution. Results were fitted with both Langmuir (equation ( 2)) and

Hill isotherms (equation ( 3)):

𝛤 Cu 2+ 𝛤 Cu 2+ ,max = 𝐾 a,surf [Cu 2+ ] 1+𝐾 a,surf [Cu 2+ ] (2)
𝛤 Cu 2+ 𝛤 Cu 2+ ,max = (𝐾 a,surf [Cu 2+ ]) 𝑞 1+(𝐾 a,surf [Cu 2+ ]) 𝑞 (3)
where 𝛤 Cu 2+ is the surface concentration in Cu(II) (mol.cm -2 ), 𝛤 Cu 2+ ,max is the maximum surface concentration, 𝐾 a,surf the association constant characterizing the coppercyclam complex at the electrode surface and 𝑞 is a cooperative coefficient [START_REF] Koopal | Analytical isotherm equations for multicomponent adsorption to heterogeneous surfaces[END_REF].

According to these nonlinear fittings, Hill isotherm offers the best results (adjr² = 0.99836 vs. 0.99502 for Langmuir, (Figure SI.5.C) with a maximal surface concentration estimated at 97 ± 5 pmol.cm -2 . The latter is similar to that found from peak integration after dipping in a 1 µmol.L -1 copper solution (65 ± 5 pmol.cm -²).

Accordingly, the value of K a,surf was found to be very low compared to the association constant K a,sol reported in solution. Indeed, log K a,surf was estimated at 6.24 ± 0.05, while log K a,sol = 26.50 [START_REF] Croot | Polarographic determination of half-wave potentials for copper-organic complexes in seawater[END_REF]. This difference may originate because it is well known that after immobilization, a receptor displays lower affinities for the targeted species mainly because of steric hindrance or vicinity interactions [52][55]. Finally, it is very interesting to note that since Hill coefficient is superior to one (q = 1.255 ± 0.049), positive cooperation between the binding sites seems to occur during the accumulation [START_REF] Al-Ghouti | Guidelines for the use and interpretation of adsorption isotherm models: A review[END_REF].

The voltammetric data were then analysed through the PSA approach. As shown in Figure 5A, a positive shift of the peak potential was detected when the copper concentration in solution was increased from 0 to 175 nmol.L -1 . Analysis of these data (Fig. 5B) showed two trends according to copper concentration. A linear variation of the peak potential was found for [Cu 2+ ] ranging between 0 and 10 nmol.L -1 , with a slope of 0.020 ± 0.001 V/[Cu 2+ ]. The LOD calculated with the 3σ method reached a remarkable value of 1.1 nmol.L -1 , which is more than one order of magnitude lower compared to the AdSV method (16 nmol.L -1 ). For higher copper concentration (10 nM < [Cu 2+ ] < 175 nM), the peak potential also shifted towards more positive values but asymptotically reached a steady value at -1.33 V from [Cu 2+ ] = 100 nM. Further analysis was carried out by plotting the cathodic peak potential against surface concentration. Since surface concentration below 10 nM could not be achieved from ASV, we used the Hill isotherm (equation ( 3)) to evaluate it for [Cu 2+ ] < 10 nM. A slope of 62.9 mV / ln (𝛤 Cu 2+ 0 ) was obtained by the non-linear regression (Figure SI.6). This value is higher than that expected (12.6 mV / ln (𝛤 Cu 2+ 0 )) for a classical "total catalysis" case (equation ( 1)), and may result from interference of the heterogeneous electron transfer in the kinetics of the whole process [START_REF]Savéant Elements of Molecular and Biomolecular Electrochemistry : an Electrochemical Approach to Electron Transfer Chemistry[END_REF]. In order to fully evaluate our cyclam-based system, comparative studies were then performed with PGE electroplated with various amounts of copper (no cyclam).

Indeed, we wanted to evaluate an "ASV-like" protocol where the electroplated copper is first used as a catalyst for HER before being stripped. (A) copper through HER) and then reversing until 0.5 V (revealing copper by its oxidation). As the amount of electrodeposited copper was increased, the catalytic peak shifted towards more positive potential, as expected (Figure SI.7). However, since HER occurred on bare PGE near 1.4 V (Figure 3), the method was found to be unsuitable for low quantities of copper because of the interference of the HER catalysed by carbon. By this way, we measured that a minimal quantity of copper of 8 pmol.cm -2 was necessary to produce a detectable signal by PSA, while the stripping method could allow lower Cu(II) detection. In other words, ASV appeared as a more sensitive method than PSA during HER with the Cu-electroplated electrode. This result emphasizes that for PSA measurements, the support material needs to be well controlled not to interfere into the catalytic signal. In order to have a full comparison between cyclam-modified and non-modified electrodes, values of peak potential were plotted against copper surface concentration (Figure 6). It can be observed that the LOD in PSA methods is limited by the electrocatalytic properties of the material, while for ASV the LOD is imposed by the S/N ratio. Moreover, since the limit of linearity of the PSA method (LOL ≈ 10 nM) proposed in this work corresponds to the LOD in AdSV (LOD = 16 nmol.L -1 ), it is interesting to note that performing AdSV in Na 2 SO 4 at pH = 3, PSA can be a complementary method to the conventional stripping analysis, confirming copper amount at the electrode surface, or helping to interpret ambiguous oxidation signal. Together, these two approaches allow detection over a 10 3 surface concentration, from 0.03 to 30 pmol.cm -2 .

Fig. 6 Influence of the surface concentration of copper on the peak potential for proton reduction, depending on the electrode : red points report the results obtained with copper electroplated on unmodified PGE, blue points report the results obtained with copper accumulated at E oc on cyclam modified electrode for whom Γ Cu are calculated from the copper oxidation peak and green points are results obtained with copper accumulated at E oc on cyclam modified electrode for which Γ Cu were evaluated from Hill isotherms.

At last, our method was applied for more realistic situation for further use, i.e. in a pseudo-marine environment or in presence of other metal ions. We thus used artificial seawater (i.e a medium with the same ionic strength and pH, but without any trace of heavy metal and organic material) to demonstrate the feasibility of the measurement in real seawater sample. Indeed, total copper concentration determination in marine water is usually performed by acidifying the sample with HCl for storage (final pH < 2). Acidification has the advantage of limiting copper adsorption onto surfaces and to digest the organic matter which traps the cupric ions.

In this work, copper sensing experiments using PSA performed after preconcentration in artificial seawater at pH = 8.2 showed no difference compared to experiments performed in Milli-Q water (Figure 5B), as expected [START_REF] Jasmin | Straightforward grafting approach for cyclam-functionalized screen-printed electrodes for selective Cu(II) determination[END_REF] Experiments were also carried out to ensure the selectivity of the ligand for copper in natural conditions. For this, copper sensing with the PSA method were performed in artificial seawater, spiked with 1 nmol.L -1 of copper and with various amounts of competitor (10, 10 2 and 10 3 higher concentration) although the large amount used here does not reflect the natural concentration of the competitors, which are naturally found in same order of magnitude than copper. As shown in Figure 7, the competing metal ions induced little variation of the HER peak potential compared to the average of the single copper signal. This was expected given the higher affinity of the cyclam moiety towards Cu 2+ than for the other cations. Furthermore, these results are very close to those obtained by Jasmin et al. [START_REF] Jasmin | Straightforward grafting approach for cyclam-functionalized screen-printed electrodes for selective Cu(II) determination[END_REF] and Walcarius et al. [START_REF] Goubert-Renaudin | Cyclam-functionalized silica-modified electrodes for selective determination of Cu(II)[END_REF], who demonstrated the very good selectivity of cyclam for copper sensing by AdSV.

Conclusions

This work mainly aimed at exploring how heterogeneous electrochemical catalysis may be an elegant manner to detect Cu 2+ at the nanomolar level in solution. Since copper is known for catalysing proton reduction, HER was employed to precisely determine copper concentration thanks to electrocatalysis. The originality of this strategy has relied on the enhancement of the detection limit by indirectly measuring Cu 2+ concentration from the shift of the electrocatalytic HER peak through the PSA approach. So far, this strategy has never been used for such a purpose. Hence, after full optimization of the experimental conditions in terms of pH and supporting electrolyte, the three-part work (HER catalytic signal ↔ Cu(II) surface concentration ↔ Cu(II) solution concentration) was shown to be very effective since a LOD of 1.1 nmol.L -1 was achieved from PSA analysis. This value is competitive to those found by the classical VIP and ElViDor [START_REF] Grand | Developing autonomous observing systems for micronutrient trace metals[END_REF] approaches (0.95 nmol.L -1 and 2.0 nmol.L -1 , respectively). Another positive aspect of our strategy is that PSA can be fully combined to AdSV within a single measurement and allows detection for higher concentration (> 10 nM), yielding a dual sensing method. A third advantage is that the chelating bifunctional cyclam ligand which was specifically chosen under its Cfunctionalized version, was shown to be highly selective towards Cu 2+ compared to other divalent cations (Ni 2+ , Co 2+ and Zn 2+ ). At last, the simplicity of preparation, the robustness of the covalent grafting, the possible miniaturization of the system, the possibility of re-using the modified electrodes by an ASV step, the low risk of eutrophication and the low consumption of reactant (only proton) make this process very attractive for field measurements in both fresh water and marine environments.

Many perspectives can be now imagined from this proof-of-concept. Indeed, this new approach based on PSA may also be extended to other metals by changing the chelating ligand to be immobilized. Other catalytic reactions than HER allowing the employment of PSA can be envisaged. For instance, several Co 2+ species have been reported as excellent electrocatalysts for O 2 reduction [59]. In the meantime, Nicyclam complexes are well-known for being able to perform catalytic CO 2 reduction in aqueous media at low overpotential [START_REF] Froehlich | Homogeneous CO 2 reduction by Ni(cyclam) at a glassy carbon electrode[END_REF]. One main challenge will be to find a metal catalyst which displays high turnover frequency combined with a high chelating strength after immobilization such that LOD will be as low as possible.

Acknowledgements

This work was supported by IFREMER.

Notes and references

Fig. 2 .

 2 Fig. 2. Cyclic voltammograms recorded at 50 mV.s -1 at pH = 3, 4, 5, 6 and 7 in

Fig. 3 .

 3 Fig. 3. Cyclic voltammogram recorded at 50 mV.s -1 of PGE/cyclam/Cu(II) in different electrolytes (0.1 mol.L -1 ) at pH = 3 in acetic acid / sodium acetate, sodium phosphate /

  Four different PGE (bare, cyclammodified, Cu-electroplated and Cu-dipped modified-cyclam) were tested in Na 2 SO 4 0.1 mol.L -1 at pH = 3. The Figure 4 displays the CV responses for proton reduction (cathodic peak) and Cu dissolution (anodic peak).

Fig. 4 .

 4 Fig. 4. (A) Forward scan of the cyclic voltammograms of PGE, copper electroplated PGE, cyclam/PGE and PGE/cyclam/Cu(II), in Na 2 SO 4 pH = 2,7 at v = 50 mV.s -1 . (B)

3 ,

 3 we explored the possibility to exploit the peak potential shift to precisely determine [Cu 2+ ] in the solution, i.e the PSA approaches. For this purpose, cyclammodified PGE were dipped in Cu 2+ solution of variable concentration (ranging from 1 to 150 nM). After 1 h, the electrodes were rinsed with ultrapure water and placed in a solution of Na 2 SO 4 0.1 mol.L -1 at pH = 3 for voltamperommetric measurements. Anodic stripping voltammetry was first used to correlate copper surface concentration to solution concentration. Well-defined anodic peaks (Figure SI.5.a) were detected at E > -0.125 V for [Cu 2+ ] > 10 nmol.L -1 . The peak integration (Figure SI.5.b) allowed us to determine the copper LOD at 16 nmol.L -1 (3σ method) in accordance with Walcarius [20] et al. (LOD = 10 nmol.L -1 ) and Jasmin [21] et al. (LOD = 13 nmol.L -1 ). For lower Cu 2+ concentration, no significative peak could be obtained by cyclic voltammetry, because of the decrease of the signal to noise ratio (S/N).

Fig. 5 .

 5 Fig. 5. (A) Forward scan of the catalytic peaks measured after incubation of PGE/Cyclam in solution containing increasing Cu 2+ concentrations (0 (black), 4 (red), 8 (blue), 10 (pink), 20 (green), 100 (dark blue) nmol.L -1 ) (B) Potential evolution of the catalytic peak represented in A: E peak (V) = f([Cu 2+ ]).

  have been developed for sustainable energy production, generating dihydrogen and dioxygen through Hydrogen and

	Oxygen	Evolution	Reaction	(HER	[31][32][33][34][35][36]	and	OER
	[37][38][39][40][41][42])	in	homogeneous	[31][32][33][34][35][36][37]	or
	heterogeneous [32][33][34][35][36][37][38][39][40][41][42] ways. For the latter,
	many copper complexes have been reported to serve as precursors for copper-based

nanomaterial electrodeposition, leading to electroactive films with excellent catalytic efficiency. Nevertheless, and to the best of our knowledge, these catalytic activities have never been exploited for the development of copper detection. Sensors based on electrocatalysis are principally used to directly measure substrate concentration in solution

  SI.4), the copper quantities that were oxidized were very close for sulfate and perchlorate anions (202 and 208 nC, respectively). Slightly lower charges quantities were measured for phosphate and acetate electrolytes (186 and 142 nC). According to these results, best conditions for Cu(II) detection by CV were obtained with Na 2 SO 4 (0.1 mol.L -1 ) at pH = 3.

  cyclam-modified PGE was first analysed from CVs at different pH values in sulfate medium. A current variation of -151 ± 4 mA.[H + ] -1 as well as a potential shift of 92 ± 1 mV/pH were found. Since the expected shift for a Nernstian (fast) reaction involving n

	electrons and m protons is (m/n)59 mV/pH, the latter value may indicate that the
	redox reaction implies n=m proton and electron, but is kinetically limited, biasing the
	experimental measurement of the m/n ratio [51][52]. By increasing sulfate
	concentration at pH = 3, current values showed a linear variation with the sulfate
	concentration with a slope of -0.221 ± 0.001 mA.[ SO 4 2-] -1 , i.e. 3 orders of magnitude
	lower than the case of proton (-151 mA.[H perchlorate concentration at pH = 3 does not increase the catalytic intensities during
	HER unlike sulfate and (iv) sulfates have less effect on the HER catalytic current on
	bare PGE compared to Cu-cyclam-modified electrodes. These results are thus in
	favour of a specific process implying copper, sulfate and protons, with a faradic yield
	exclusively attributed to H 2 production.

+ ] -1 ). Thus, considering the high consumption of proton at the vicinity of the electrode producing H 2 and the need of proton for sulfate reduction, it is reasonable to propose that sulfate acts preferably as a co-substrate during HER rather than being reduced into SO 2 . This hypothesis was verified by using perchlorates instead of sulfates, or with bare PGE instead of Cucyclam-modified PGE. Results obtained showed that (i) sodium sulfate as electrolyte increases proton reduction efficiency compared to sodium perchlorate (3-fold increase), (ii) the current variation measured in sulfates at different pH values with Cu-cyclam-modified PGE is close to that measured with bare PGE (iii) increasing

  For this objective, different amounts of copper were electroplated onto PGE by chronocoulometry, and CVs in Na 2 SO 4 at pH = 3 were performed, starting at E oc towards 1.8 V (revealing

			0	nM
		0
			4	nM
			8	nM
		-50
			10	nM
			20	nM
		-100	100	nM
	i(µA)	-150
		-200
		-250
		-300
		-1.8				-1.6	-1.4	-1.2	-1.0	-0.8
							E(V)	vs	Ag/AgCl
			Milli-Q Water
	(B)	-1.3	Artificial seawater
		-1.4					-1.45
	E peak (V) vs Ag/AgCl	-1.6 -1.5				E peak (V) vs Ag/AgCl	-1.70 -1.65 -1.60 -1.55 -1.50
							-1.75
		-1.7						0	2	4	6	8	10
								[Cu 2+ ] (nmol.L) -1
		0				50		100	150
							[Cu 2+ ] (nmol.L) -1

  [START_REF] Goubert-Renaudin | Factors affecting copper(II) binding to multiarmed cyclam-grafted mesoporous silica in aqueous solution[END_REF]. Considering the volcano curve for HER on metal electrode and the potential metal ions that can bind to cyclam[START_REF] Yan | A review on noble-metal-free bifunctional heterogeneous catalysts for overall electrochemical water splitting[END_REF], it appears that Ni 2+ , Co 2+ and Zn 2+ are natural copper competitors (log K a,sol = 22.2, 12.7 and 15.5 respetively[START_REF] Goubert-Renaudin | Factors affecting copper(II) binding to multiarmed cyclam-grafted mesoporous silica in aqueous solution[END_REF]) that can produce a catalytic signal during an analysis, similar to the copper one. The interference of these metals was therefore evaluated in artificial seawater. Accumulations were performed in 100 nmol.L -1 solutions of Ni 2+ , Co 2+ and Zn 2+ and the electrodes were cycled in sodium sulfate at pH = 3 to evaluate HER with these different metals in solution. As shown in Figure SI.8, CVs showed little variation of the peak potential for proton reduction compared to that obtained with Cu 2+ . This observation may be fully ascribed to the better cyclam selectivity for Cu 2+ .Fig. 7 Impact of high amount of competitor on the catalytic peak potential attributed to HER by chelated copper.

		1.71 1.72	1:10	1:100	1:1000	[Cu 2+ ] = 0 nM Epeak @
		1.70
		1.68 1.69				[Cu 2+ ] = 1 nM Epeak @
	E peak (V)	1.65 1.66 1.67
		1.64
		1.63
		1.62
		1.61
		1.60
			Copper + Nickel	Copper + Cobalt	Copper + Zinc