Juan M Coria

Hervé Bredin
email: herve.bredin@irit.fr

Sahar Ghannay
email: sahar.ghannay@lisn.upsaclay.fr

Sophie Rosset
email: sophie.rosset@lisn.upsaclay.fr

OVERLAP-AWARE LOW-LATENCY ONLINE SPEAKER DIARIZATION BASED ON END-TO-END LOCAL SEGMENTATION

Keywords: speaker diarization, low latency, overlapped speech detection, speaker embedding

We propose to address online speaker diarization as a combination of incremental clustering and local diarization applied to a rolling buffer updated every 500ms. Every single step of the proposed pipeline is designed to take full advantage of the strong ability of a recently proposed end-to-end overlapaware segmentation to detect and separate overlapping speakers. In particular, we propose a modified version of the statistics pooling layer (initially introduced in the x-vector architecture) to give less weight to frames where the segmentation model predicts simultaneous speakers. Furthermore, we derive cannot-link constraints from the initial segmentation step to prevent two local speakers from being wrongfully merged during the incremental clustering step. Finally, we show how the latency of the proposed approach can be adjusted between 500ms and 5s to match the requirements of a particular use case, and we provide a systematic analysis of the influence of latency on the overall performance (on AMI, DIHARD and VoxConverse).

INTRODUCTION

Speaker diarization aims at answering the question "who spoke when", effectively partitioning an audio sequence into segments with a particular speaker identity. Most dependable diarization approaches consist of a cascade of several steps [START_REF] Anguera | Speaker diarization: A review of recent research[END_REF][START_REF] Diez | BUT System for DIHARD Speech Diarization Challenge 2018[END_REF]: voice activity detection to discard non-speech regions, speaker embedding [START_REF] Madikeri | Integrating online i-vector extractor with information bottleneck based speaker diarization system[END_REF][START_REF] Snyder | X-Vectors: Robust DNN Embeddings for Speaker Recognition[END_REF] to obtain discriminative speaker representations, and clustering [START_REF] Diez | BUT System for DIHARD Speech Diarization Challenge 2018[END_REF][START_REF] Lin | LSTM Based Similarity Measurement with Spectral Clustering for Speaker Diarization[END_REF][START_REF] Landini | But System for the Second Dihard Speech Diarization Challenge[END_REF] to group speech segments by speaker identity. The main limitation of this family of multi-stage approaches relates to how they handle overlapped speech (which is known to be one of the main sources of errors): either they simply ignore the problem or they address it a posteriori as a final postprocessing step based on a dedicated overlapped speech detection module [START_REF] Otterson | Efficient use of overlap information in speaker diarization[END_REF][START_REF] Bullock | Overlap-aware diarization: Resegmentation using neural end-to-end overlapped speech detection[END_REF][START_REF] Horiguchi | End-to-end speaker diarization as post-processing[END_REF][START_REF] Bredin | End-to-end speaker segmentation for overlap-aware resegmentation[END_REF]. A new family of approaches have recently emerged, rethinking speaker diarization completely. Dubbed end-to-end diarization (EEND), the main idea of this approach is to train a single neural networkin a permutation-invariant manner -that ingests the audio recording and directly outputs the overlap-aware diarization output [START_REF] Fujita | End-to-End Neural Speaker Diarization with Permutation-Free Objectives[END_REF][START_REF] Fujita | End-to-End Neural Speaker Diarization with Self-Attention[END_REF]. We propose to meet half-way between multistage and overlap-aware end-to-end diarization and design a multi-stage pipeline where overlapped speech is a first-class citizen in every single step: from segmentation to incremental clustering. In particular, our first contribution (discussed in Section 2.2.1) is a modified version of the statistics pooling layer (initially introduced in the x-vector architecture) to give less weight to frames where the intial segmentation step predicts simultaneous speakers.

Despite being competitive with multi-stage approaches, the main limitation of the overlap-aware end-to-end approaches is the strong assumption that the number of speakers is upper bounded or even known a priori. While reasonable for some particular use cases (e.g. one-to-one phone conversations), this assumption does not hold in many other situations (e.g. physical meetings or conference calls). One solution to this problem is to augment end-to-end approaches with mechanisms to automatically estimate the number of speakers. For instance, EEND-EDA [START_REF] Horiguchi | End-to-End Speaker Diarization for an Unknown Number of Speakers with Encoder-Decoder Based Attractors[END_REF] extends EEND [START_REF] Fujita | End-to-End Neural Speaker Diarization with Permutation-Free Objectives[END_REF][START_REF] Fujita | End-to-End Neural Speaker Diarization with Self-Attention[END_REF] with a recurrent Encoder-Decoder network to generate a variable number of Attractors -similar to speaker centroids. Multi-stage approaches usually do not suffer from this limitation as they rely on a clustering step for which a growing number of techniques exist to accurately estimate the number of speakers [START_REF] Tae | Auto-Tuning Spectral Clustering for Speaker Diarization Using Normalized Maximum Eigengap[END_REF]. We propose to combine the best of both worlds [START_REF] Kinoshita | Integrating end-to-end neural and clustering-based diarization: Getting the best of both worlds[END_REF] by first applying the end-to-end approach on audio chunks small enough to reasonably estimate an upper bound on the local number of speakers and, then only, apply global constrained clustering on top of the resulting local speakers. As discussed in Section 2.2.2, we say that cluster-ing is constrained because cannot-link constraints are inferred from the output of the local end-to-end diarization. The main difference between this work and [START_REF] Kinoshita | Integrating end-to-end neural and clustering-based diarization: Getting the best of both worlds[END_REF] is that we target lowlatency online speaker diarization while they address offline speaker diarization.

This work relies heavily on the speaker segmentation model introduced in [START_REF] Bredin | End-to-end speaker segmentation for overlap-aware resegmentation[END_REF] and summarized in Section 2.1 for convenience. However, they address two very different problems with radically different constraints. While [START_REF] Bredin | End-to-end speaker segmentation for overlap-aware resegmentation[END_REF] performs local offline speaker diarization of extremely short 5s chunks of audio, this work addresses online speaker diarization of (possibly infinite) audio streams. Hence, this work extends [START_REF] Bredin | End-to-end speaker segmentation for overlap-aware resegmentation[END_REF] with a mechanism to track speakers over the duration of a conversation, with a latency much lower than 5s and real-time processing.

Low-latency online speaker diarization differs from its offline counterpart in several ways. While the latter assumes that the whole audio sequence is available at once (and hence can rely on multiple passes over the whole sequence to output its final prediction), the former ingests a possibly infinite audio stream and can only afford a short delay between when it receives a buffer of audio and when it outputs the corresponding prediction (without the option to correct it afterwards). These additional constraints prevent state-of-the-art multi-stage approaches like VBx [START_REF] Landini | Bayesian HMM clustering of x-vector sequences (VBx) in speaker diarization: Theory, implementation and analysis on standard tasks[END_REF] from being used in that setting as they heavily rely on the possibility to pass several times over the audio sequence. EEND-like approaches are not suitable either because they expect large chunks of audio (30 seconds or more), leading to prohibitively high latency. One notable exception is FlexSTB [START_REF] Xue | Online Streaming End-to-End Neural Diarization Handling Overlapping Speech and Flexible Numbers of Speakers[END_REF] that astutely relies on an adaptive internal buffer to both simulate large audio chunks and support low (1s) latency.

A comprehensive set of experiments on AMI, DIHARD II, DIHARD III and VoxConverse datasets is reported and discussed in Section 4 -where FlexSTB and a state-of-the-art offline approach based on VBx [START_REF] Landini | Bayesian HMM clustering of x-vector sequences (VBx) in speaker diarization: Theory, implementation and analysis on standard tasks[END_REF] respectively serve as baseline and topline. In particular, we show how the latency of the proposed approach can easily be adjusted (without retraining) between 500ms and 5s to match the requirements of a particular use case.

OVERLAP-AWARE ONLINE DIARIZATION

As depicted in Figure 1, we propose to address online speaker diarization as the iterative interplay between two main steps: segmentation and incremental clustering. Every few hundred milliseconds (500ms in our case), the segmentation module first performs a fine-grained overlap-aware diarization of a 5s rolling buffer. This local diarization is then ingested by the incremental clustering module that relies on speaker embeddings to map local speakers to the appropriate global speakers (or create new ones), before updating its own internal state.

Segmentation

The segmentation step is the direct application of the end-toend speaker segmentation neural network introduced in [START_REF] Bredin | End-to-end speaker segmentation for overlap-aware resegmentation[END_REF], used to obtain a fine-grained local speaker diarization. As depicted in Figure 1, it ingests the 5s audio rolling buffer and outputs speaker activity probabilities {s 1 , . . . , s F } where F is the number of output frames and s f ∈ [0, 1] Kmax , with K max the estimated maximum number of different speakers that a 5s chunk may contain (K max = 4 in our case). Speakers whose activity probability exceeds a tunable threshold τ active at least once during the chunk constitute the set of local speakers. Inactive speakers are simply discarded.

Active speaker probabilities are then passed unchanged (i.e. with continuous values between 0 and 1) to the incremental clustering step. In particular, it means that overlapping speech (i.e. when two or more speakers have high probabilities simultaneously) is handled from the very beginning of the pipeline. This is in contrast with most dependable speaker diarization approaches that handle overlapping speech as a post-processing step [START_REF] Landini | But System for the Second Dihard Speech Diarization Challenge[END_REF][START_REF] Bredin | End-to-end speaker segmentation for overlap-aware resegmentation[END_REF]. This early detection of overlapping speech will prove very useful for the incremental clustering.

Incremental clustering

Because the segmentation model is trained in a permutationinvariant manner and applied locally to the rolling buffer, one cannot guarantee that one particular speaker consistently activates the same index over time. Figure 2 illustrates this lim-itation for two states of the rolling buffer: despite being only 500ms apart from each other and therefore having most of their audio content in common, notice how both active speakers are swapped. This section describes how we use incremental clustering to circumvent this limitation by tracking speakers (and detecting new ones) over the whole duration of the audio stream.

Segmentation-driven speaker embedding

Like most recent speaker diarization systems, we rely on neural speaker embeddings to represent and compare speakers. Our model is based on the canonical x-vector TDNN-based architecture, with the difference that the statistics pooling layer [START_REF] Snyder | X-Vectors: Robust DNN Embeddings for Speaker Recognition[END_REF] is modified to return the concatenation of weighted mean µ k and weighted standard deviation σ k for each active speaker k -instead of the regular mean µ and standard deviation σ:

µ = f x f F -→ µ k = f w f k • x f f w f k (1)
σ 2 = f (x f -µ) 2 F -1 -→ σ 2 k = f w f k • (x f -µ k ) 2 f w f k -f w f k 2 f w f k
where x f is the output of frame f of the last TDNN layer. One straightforward option is to derive w f k from the speaker activity probability and use w f k = s f k directly, so that the final (pooled) speaker embedding mostly relies on frames where the segmentation model is confident that speaker k is active. This generates exactly one embedding per active speaker in the current buffer, even when split into multiple speech turns (e.g. the red speaker in the lower row of Figure 2). Furthermore, as summarized in [START_REF] Bredin | End-to-end speaker segmentation for overlap-aware resegmentation[END_REF], the segmentation model is also very good at detecting overlapped speech regions (where two or more speakers are active simultaneously). Therefore, another option is to make the speaker embedding focus on frames where it is confident that speaker k is the only active speaker:

w f = s f • softmax k (β • s f ) γ (2)
where the effect of this transformation is illustrated in Figure 3. The use of softmax k weighs down frames where two or more speakers are active, and the exponent γ > 1 weighs down frames where the segmentation model is not quite confident about the activity of a speaker. Embeddings extracted with this weighing scheme are called overlap-aware speaker embeddings in the rest of the paper.

Constrained incremental clustering

Given the initial content of the rolling buffer, the segmentation and embedding steps are combined to extract one embedding for each active speaker in the first 5s of the audio stream. These speaker embeddings {c 1 , . . . , c K } are stacked to form the initial centroid matrix C with shape K × D where K is the number of active speakers so far, and D is the dimension of the speaker embedding.

Every few hundred milliseconds (e.g. 500ms), the rolling buffer is updated, and the segmentation and speaker embedding steps are combined to extract one embedding for each of the K buffer ≤ K max locally active speakers. Those K buffer speaker embeddings are then compared to the current state of the centroid matrix C to find the optimal mapping m * between local and global speakers. Denoting d (c, e) the distance between centroid c and local speaker embedding e, one option is to assign the kth local speaker to the closest centroid:

m * (k) = argmin c∈C d(c, e k ) (3)
Yet, this simple option does not take full advantage of the output of the segmentation model, as two local speakers might end up being assigned to the same centroid. This would be in contradiction to the output of the segmentation model that already chose to discriminate local speakers. Therefore, we add the constraint that any two local speakers cannot be assigned to the same centroid, while keeping the objective of minimizing the overall distance between local speakers and their assigned centroids:

m * = argmin m∈M k d(m(k), e k ) ( 4
)
where M is the set of mapping functions between local speakers and centroids with the following property:

k = k =⇒ m(k) = m(k ) (5)
In practice, this optimal mapping is obtained by applying the Hungarian algorithm on the speaker-to-centroid distance matrix, and can be seen as an incremental clustering step with cannot-link constraints.

Detecting new speakers and updating centroids

Once the optimal mapping m * is determined, for any given local speaker k and their local embedding e k

• if d(m * (k), e k ) > δ new , they are marked as new speaker (i.e. it is the first time they are active since the beginning of the audio stream) and their embedding e k is appended to the pool of centroids:

C ← C ∪ e k
• otherwise, they are marked as returning speaker, and their embedding e k is used to update the corresponding centroid.

Because of the weighing scheme described in Section 2.2.1, the quality of a speaker embedding e k is expected to be positively correlated with the estimated duration during which local speaker k is active: ∆ k = f s f k . Therefore, we propose to only update a centroid when this duration is long enough:

c m * (k) ← c m * (k) + e k if ∆ k > ρ update c m * (k) otherwise ( 6
)
where ρ update is the minimum duration below which a speaker embedding e k is considered to be too noisy to help refine the centroid. Equation 6 assumes that speaker embeddings e k are unit-normalized and optimized for cosine similarity.

Adjusting the latency

Even though the whole [t -5s, t] buffer is used to extract embeddings and assign local speakers to an existing (or new) cluster, only the (active) speaker activity probabilities s f k at its rightmost part [t -λ, t] are output: λ effectively controls the latency of the whole system. The lowest possible value for λ corresponds to the period between two consecutive updates of the rolling buffer (500ms in our case). In this configuration, the rightmost parts of two consecutive buffer states [t -5s, t] and [t + λ -5s, t + λ] do not overlap: [t -λ, t] and [t, t + λ]. Therefore, they are simply concatenated and frame-level speaker activity probabilities are passed through a final thresholding step. Local speaker k is marked as active at frame f if s f k > τ active .

The careful reader might have noticed that, at the very beginning of the audio stream, the initial buffer must be filled entirely before a first output can be provided -effectively leading to a much larger latency of 5s, an order of magnitude larger than the promised λ = 500ms. However, once this initial 5s warm-up period has passed, the latency is indeed λ = 500ms. If having a low latency from the very beginning of the stream is critical, one can simply left-pad the [0, λ] initial incomplete buffer with zeros. Fig. 4. Depending on the allowed latency λ (between 500ms and 5s), multiple positions of the rolling buffer can be aggregated to obtain a (hopefully better) prediction. Speakers are colored according to the cluster they were found to belong to.

Figure 4 shows that, for cases where longer latency λ is permitted, several positions of the rolling buffer can be combined in an ensemble-like manner to obtain a more robust output. In practice, for a given frame f , the final speaker activity probabilities are computed as the average of the speaker activity probabilities obtained from each buffer position.

EXPERIMENTS

Datasets

We ran experiments on three different datasets covering a wide range of domains and number of speakers.

DIHARD III [START_REF] Ryant | The Third DIHARD Diarization Challenge[END_REF][START_REF] Ryant | Third DIHARD Challenge Evaluation Plan[END_REF] does not provide a training set. Therefore, we split its development set into two parts: 192 files used as training set, and the remaining 62 files used as a smaller development set. The latter is simply referred to as development sets in the rest of the paper. When defining this split (shared at huggingface.co/pyannote/segmentation), we made sure that the 11 domains were equally distributed between both subsets. The test set is kept unchanged. We AMI provides an official {training, development, test} partition of the Mix-Headset audio files [START_REF] Carletta | Unleashing the killer corpus: experiences in creating the multi-everything AMI Meeting Corpus[END_REF] that is further described as the Full partitioning in [START_REF] Landini | Bayesian HMM clustering of x-vector sequences (VBx) in speaker diarization: Theory, implementation and analysis on standard tasks[END_REF].

Implementation details

We use the pretrained segmentation model available at hf.co/pyannote/segmentation, which was trained on the composite training set made of the union of AMI, DI-HARD III, and VoxConverse respective training sets. It ingests 5 second audio chunks and outputs one prediction every 16ms with K max = 4 speakers. More details about the training process can be found in [START_REF] Bredin | End-to-end speaker segmentation for overlap-aware resegmentation[END_REF]. The speaker embedding model is based on the canonical x-vector TDNN-based architecture [START_REF] Snyder | X-Vectors: Robust DNN Embeddings for Speaker Recognition[END_REF], but with filter banks replaced by trainable SincNet features [START_REF] Ravanelli | Speaker Recognition from Raw Waveform with SincNet[END_REF]. It was trained with additive angular margin loss [START_REF] Deng | Ar-cFace: Additive Angular Margin Loss for Deep Face Recognition[END_REF] using chunks of variable duration (from 2 to 5 seconds) drawn from VoxCeleb [START_REF] Nagrani | VoxCeleb: A Large-Scale Speaker Identification Dataset[END_REF][START_REF] Joon | VoxCeleb2: Deep Speaker Recognition[END_REF], augmented with reverberation based on impulse responses from EchoThief and [START_REF] Traer | Statistics of natural reverberation enable perceptual separation of sound and space[END_REF], and additive background noise from MUSAN [START_REF] Snyder | MU-SAN: A Music, Speech, and Noise Corpus[END_REF]. It reaches an equal error rate of 2.8% on VoxCeleb 1 test set using cosine distance only. We share the pretrained model and more details about the training process at hf.co/pyannote/embedding. Weights used in the statistics pooling layer of the overlap-aware speaker embeddings were obtained with γ = 3 and β = 10. Those values were not optimized with the rest of the hyper-parameters. Instead, we handpicked them based on examples like the one in Figure 3.

Experimental protocol

While the same pretrained segmentation and embedding models were used for all three datasets, we rely on their respective development sets to optimize hyper-parameters (τ active , ρ update and δ new ) specifically for each dataset. More precisely, we use the pyannote.pipeline optimization toolkit that relies on a tree-structured Parzen estimator algorithm [START_REF] Bergstra | Algorithms for Hyper-Parameter Optimization[END_REF] to minimize the overall diarization error rate (DER) -computed with pyannote.metrics [START_REF] Bredin | pyannote.metrics: A Toolkit for Reproducible Evaluation, Diagnostic, and Error Analysis of Speaker Diarization Systems[END_REF] without any forgiveness collar and including overlapped speech regions. To ensure a fair comparison between different approaches, the optimization process is applied for all of them independently. In other words, it means that every row × dataset entry in Table 1 results from one dedicated optimization process. This includes the offline topline, the proposed online approach and its ablative variants, but excludes both FlexSTB (as we unfortunately did not have access to its implementation) and experiments on DIHARD II (where we use respective hyper-parameters tuned for DIHARD III).

RESULTS AND DISCUSSION

Table 1 summarizes the whole set of experiments.

Offline vs. online. We start by reporting the performance of a strong offline topline that consists of VBx [START_REF] Landini | Bayesian HMM clustering of x-vector sequences (VBx) in speaker diarization: Theory, implementation and analysis on standard tasks[END_REF] followed by the overlap-aware resegmentation step introduced in [START_REF] Bredin | End-to-end speaker segmentation for overlap-aware resegmentation[END_REF]. Because this latter resegmentation step relies on the exact same pretrained segmentation model as our proposed approach, most of the reported decrease in performance with λ = 5s is caused by speaker confusion errors (relative +100% for DIHARD III, +160% for AMI, +130% for VoxConverse).

Incremental clustering still has a long way to go to be on par with offline multi-pass clustering.

Overlap-aware speaker embedding. The first ablative experiment shows that the overlap-aware weighing scheme introduced in Equation 2 brings a relative performance improvement of 10% on AMI, 18% on VoxConverse and 1% on DIHARD III. Given that they respectively contain 17%, 3%, and 11% of overlapped speech, there is still room for improvement on this particular aspect. In particular, while we handcrafted this weighing scheme, it should be possible to train the segmentation and speaker embedding models jointly for the latter to fully take advantage of the former's capability at detecting and separating simultaneous speakers.

Overlap-aware speaker segmentation. In a second ablative experiment, we replace the segmentation model by an oracle that provides perfect binary (i.e. s f k ∈ {0, 1}) overlap-aware segmentation. As expected, missed detection is where most of the difference occurs (caused by overlapped speech), while speaker confusion only marginally improves. The community has yet to solve the problem of overlapped speech detection.

Adjustable latency. Figure 5 shows how the performance of our online approach evolves as we decrease the allowed latency from λ = 5s to λ = 500ms. Speaker confusion error rate consistently increases as the latency decreaseswhile false alarm and missed detection remain constant. This can be explained by the ensemble-like aggregation process described in Section 2.3 that combines more views of the same problem as the allowed latency increases. Note that we kept the hyper-parameters (τ active , ρ update , δ new ) optimized for latency λ = 5s and still get reasonable performance for lower latencies. However, it is also possible to re-optimize the hyper-parameters for a specific latency. This is what we did for the λ = 1s setting marked with • in Table 1 for comparison with FlexSTB [START_REF] Xue | Online Streaming End-to-End Neural Diarization Handling Overlapping Speech and Flexible Numbers of Speakers[END_REF]. Not only do we get better overall performance, but our approach also has the advantage of a lower memory footprint, as it never ingests nor runs inference on more than 5s of audio at a time (compared to 100s of FlexSTB) and keeps a single vector per speaker in memory (compared to 100s of acoustic features and per-speaker scores in FlexSTB). Furthermore, our approach with λ = 3s reaches the same performance as the official offline baseline [START_REF] Ryant | The Third DIHARD Diarization Challenge[END_REF] of the DIHARD III challenge (25.5% vs 25.4%).

Continual learning. Figure 6 compares the performance over time of our online system against the offline topline [10].

While the performance of the latter remains somewhat constant, the former gets better as conversations unfold, almost bridging the gap after 5 minutes of conversation. As new information becomes available, our system learns better speaker centroids, hence decreasing speaker confusion error. While very long conversations can become rather expensive (if not impossible) to process with most offline models, our system can handle daylong audio streams at a practically constant memory cost, while getting better and better.

Reproducible research.

We share an open-source implementation of this work, as well as expected outputs in RTTM format at this address to facilitate future comparisons:

github.com/juanmc2005/StreamingSpeakerDiarization.

Real time. Computation time for one step of the rolling buffer is 165ms on a CPU Intel Cascade Lake 6248 (20 cores at 2.5Ghz) or 50ms on a GPU Nvidia Tesla V100 SXM2. This is suitable for real time applications, as the rolling buffer can be processed before its next update (every 500ms).

CONCLUSION

We have proposed an overlap-aware online speaker diarization system combining the end-to-end local segmentation of a 5 second long rolling buffer with incremental clustering. Apart from handling overlapping speech at every stage, our system benefits from an adjustable latency between 500ms and 5s. We show that our system outperforms FlexSTB [START_REF] Xue | Online Streaming End-to-End Neural Diarization Handling Overlapping Speech and Flexible Numbers of Speakers[END_REF] with a lower memory consumption, and that it is capable of bridging the gap to offline performance as conversations unfold. This last advantage may make it preferable to an offline system when recordings are long and resources low.

Fig. 1 .

 1 Fig. 1. Block diagram of the proposed approach.

Fig. 2 .

 2 Fig. 2. Actual output s f of the segmentation model on two consecutive positions of the 5s rolling buffer in file DH DEV 0001 from DIHARD III. Because of permutationinvariant training, green and red speakers are swapped.

Fig. 3 .

 3 Fig. 3. Effect of Equation 2 with γ = 3 and β = 10. Top row is the actual output s f of the segmentation model on a 5s excerpt of file DH DEV 0007 from DIHARD III. Bottom row depicts weights w f used for statistics pooling. Both low confidence and overlapped speech regions are weighed down.

Fig. 5 .

 5 Fig. 5. Impact of latency on the overall performance.

Fig. 6 .

 6 Fig. 6. Evolution of performance as conversations unfold. Left: proposed online approach with 5s latency. Right: offline topline [10]. Local diarization error rate computed on the 223 DIHARD III (test) conversations that are longer than 300s.

Table 1 .

 1 Conf. DER FA Miss. Conf. DER FA Miss. Conf. DER FA Miss. Conf. DER • 14.4 • 14.9 • 35.1 • Experimental results on test sets. FA, Miss. and Conf. stand for false alarm, missed detection and speaker confusion rates respectively. ( • = hyper-parameters optimized with latency λ = 1s for fair comparison) also report performance on DIHARD II [21] for comparison with FlexSTB [17].VoxConverse does not provide a proper training set either[START_REF] Joon | Spot the Conversation: Speaker Diarisation in the Wild[END_REF]. Therefore, we also split its development set into two parts: the first 144 files (abjxc to qouur, in alphabetical order) constitute the training set, leaving the remaining 72 files (qppll to zyffh) for the actual development set. Furthermore, multiple versions of VoxConverse test set have been circulating: we rely on version 0.0.2 available at

	System FA Miss. VBx [16] DIHARD III [18] Latency ∞ 3.6 12.5 6.2	AMI [19, 16] 22.3 3.1 17.2 3.8	24.1 3.1	VoxConverse [20] 4.6 3.4	11.1	5.0	DIHARD II [21] 15.3 7.4	27.7
	→ w/ overlap-aware segmentation [10]	∞	4.7	9.7	4.9	19.3 4.3 10.9	4.7	19.9 4.6	3.0	3.5	11.1	5.6	13.5	7.1	26.3
	Ours	5s	5.3 10.0	9.7	25.0 5.0 10.0	12.4	27.5 3.8	4.9	8.2	16.8	5.7	14.0	14.4	34.1
	→ w/o overlap-aware embedding	5s	4.6 11.3	9.3	25.3 3.0 16.0	11.6	30.5 4.1	5.1	11.2	20.4	5.1	15.5	13.6	34.3
	→ w/ oracle segmentation	5s	2.1	1.4	6.9	10.4 1.0	1.1	15.5	17.7 0.5	0.7	9.1	10.3	2.2	1.6	12.0	15.8
	FlexSTB [17]	1s	NA	NA	NA	NA	NA	NA	NA	NA	NA	NA	NA	NA	NA	NA	NA	36.0
	Ours	1s	6.2	9.7	11.8	27.6 6.6	9.4	14.4	30.4 5.1	3.3	11.7	20.1 5.8

github.com/joonson/voxconverse.

This work was granted access to the HPC resources of IDRIS under the allocation AD011012177 made by GENCI, and was partly funded by the French National Research Agency (ANR) through the PLUMCOT project (ANR-16-CE92-0025). Thanks to Antoine Laurent for running and sharing the VBx offline speaker diarization topline.


Héloïse Beaugendre

Alexandre Ern

Thomas Esclaffer

Eric Gaume

Irina Ginzburg

Cyril Kao

A seepage face model for the interaction of shallow water tables with the ground surface: Application of the obstacle-type method

Keywords: Finite elements, Richards equation, Seepage face, Obstacle-type model, Water table ground interaction, Numerical solutions

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The mechanisms leading to surface runoff in hillslopes exposed to heavy rainfall episodes have received significant attention over the last decades. Historically, the first approach considers overland flow to be generated by runoff over those portions of the watershed where the rainfall intensity exceeds the soil infiltration capacity; see [START_REF] Horton | The role of infiltration in the hydrologic cycle[END_REF]. This concept has been refined by the introduction of the partial area contribution concept initiated by [START_REF] Cappus | Etude des lois de l'écoulement. Application au calcul et à la prévision de débits. Bassin expérimental[END_REF] and [START_REF] Betson | What is watershed runoff?[END_REF], whose work has been further developed by [START_REF] Dunne | Partial area contributions to storm runoff in a small New England watershed[END_REF]. In this approach, the actual surface contributing to overland flow is restricted to saturated surfaces, i.e., where the water table reaches the ground level. This leads to a segmentation of the watershed into infiltration, exfiltration and runoff dominated zones. Since the contributing area represents only some portion of the watershed, a first important issue in hillslope hydrology is to determine the spatial extension of the contributing area and its temporal evolution resulting from the dynamics of subsurface flows. A further important issue is to determine the relative importance of surface and subsurface water in storm hydrographs and to compare these different contributions with results obtained by flow hydrograph separations [START_REF] Loye-Pilot | Décomposition chimique et isotopique d'un hydrogramme de crue d'un torrent méditerranéen montagnard. Réflexions méthodologiques[END_REF]; [START_REF] Iorgulescu | Analyse du comportement hydrologique par une approche intégrée à l'échelle du versant, application au bassin de la Haute Mentue[END_REF]; [START_REF] Marc | Investigation of the hydrological processes using chemical and isotopic tracers in a small Mediterranean forested catchment during autumn recharge[END_REF]; McGlynn and To tackle these issues, several field and laboratory studies have been conducted [START_REF] Abdul | Laboratory studies of the effects of the capillary fringe on streamflow generation[END_REF]; [START_REF] Barros | Runoff in shallow soils under laboratory conditions[END_REF]). However, field experiments are often strongly site-dependent, and laboratory tests are often complex and time-consuming. This has prompted the development of numerical tools offering more flexibility and faster results in the simulation of hillslopes subjected to various operating conditions [START_REF] Ogden | Saturated area formation on nonconvergent hillslope topography with shallow soils: A numerical investigation[END_REF]; [START_REF] Cloke | The effect of model configuration on modelled hillslope-riparian interactions[END_REF]; [START_REF] Weiler | Virtual experiments: a new approach for improving process conceptualization in hillslope hydrology[END_REF]). These so-called virtual hillslope experiments have emerged as a valuable complement to field and laboratory measurements for improving process conceptualization in hillslope hydrology.

Variably saturated flows in porous media can be modeled with the Richards equation [START_REF] Richards | Capillary conduction of liquids through porous mediums[END_REF]). This equation is based on two assumptions that, in some circumstances, can preclude its use. On the one hand, that the gasphase remains connected so that a single air pressure can be defined. On the other hand, that there exist an algebraic relation linking the volumetric water content to the hydraulic head and that the water flow velocity can be expressed using a generalized Darcy law involving a relative hydraulic conductivity. Despite these limitations, the Richards equation is now a well-established approach to simulate water table dynamics; see, for instance, [START_REF] Celia | A general mass-conservative numerical solution for the unsaturated flow equation[END_REF] for a thorough discussion of numerical aspects including choice of the main unknown, of the nonlinear iterative solver and of the space and time discretization schemes. Using the Richards equation presents two advantages, namely the use of a single (nonlinear) partial differential equation that can be discretized by fairly standard or more advanced finite element or finite volume techniques [START_REF] Woodward | Analysis of expanded mixed finite element methods for a nonlinear parabolic equation modeling flow into variably saturated porous media[END_REF]; [START_REF] Knabner | Adaptive hybrid mixed finite element discretization of instationary variably saturated flow in porous media[END_REF]; [START_REF] Bastian | Higher order discontinuous Galerkin methods for flow and trans-port in porous media, Challenges in scientific computing -CISC 2002[END_REF]; Bause and Knaber (2004); [START_REF] Fagherazzi | Application of the discontinuous spectral Galerkin method to groundwater flow[END_REF]; Manzini and the soil can be treated simultaneously as a single computational domain. Models based on the Richards equation can be used to simulate the response of the water table to infiltration caused by rainfall.

A difficulty appears when the water table rises so much that some portion of the ground is saturated. In this case, the intersection of the water table with the ground surface produces seepage faces where infiltration, exfiltration, overland flow and re-infiltration processes may occur, [START_REF] Esteves | Overland flow and infiltration modelling for small plots during unsteady rain: numerical results versus observed values[END_REF].

Simpler approaches neglecting flood routing and re-infiltration processes can still improve the current understanding of hillslope hydrology; see, for instance, [START_REF] Ogden | Saturated area formation on nonconvergent hillslope topography with shallow soils: A numerical investigation[END_REF]; [START_REF] Cloke | The effect of model configuration on modelled hillslope-riparian interactions[END_REF][START_REF] Cloke | Using numerical modelling to evaluate the capillary fringe groundwater ridging hypthesis of streamflow generation[END_REF] where the mechanisms leading to surface runoff in hillslopes are investigated. In particular, these studies have emphasized the need for an accurate description of the seepage face area [START_REF] Rulon | The development of multiple seepage faces on layered slopes[END_REF]). A mathematical way to determine the extent of the seepage face is to consider that on the ground surface the boundary condition for the Richards equation changes nature, e.g., a Dirichlet condition imposing the head (saturated area) or a Neumann condition specifying the water flux (unsaturated area). Since the water table position is a priori unknown, its intersection with the ground surface leads to an unsteady obstacle-type problem, in which the partition of the ground surface into saturated and unsaturated zones and the water table dynamics are solved simultaneously.

The goal of this paper is threefold. First, to formalize the mathematical setting for the obstacle-type model that can be used to describe the water table dynamics in a variably saturated soil where saturation conditions are met in an a priori unknown part of the ground surface. Second, to elaborate on previous work [START_REF] Ogden | Saturated area formation on nonconvergent hillslope topography with shallow soils: A numerical investigation[END_REF]; [START_REF] Cloke | The effect of model configuration on modelled hillslope-riparian interactions[END_REF]) to further investigate the impact of soil hydraulic parameters, hillslope geometry, and boundary conditions on model predictions, and thus to assess the suitability of the simulation tool to carry on virtual hillslope experiments accurately and efficiently. Third, to compare the obstacle-based approach to a more detailed model where the overland flow, which is assumed to be governed by the shallow water equations in the diffusive wave regime, is coupled to the Richards equation for the subsurface flow through the boundary conditions. Similar situations where these couplings occur between the shallow water equations and the Navier-Stokes equations or the Darcy equations have been recently investigated in [START_REF] Discacciati | Mathematical and numerical models for coupling surface and groundwater flows[END_REF]; [START_REF] Miglio | Coupling of free surface and groundwater flows[END_REF].

This paper is organized as follows. Section 2 presents the governing equations in the framework of the obstacle-type model and describes the numerical methods. Section 3 discusses the results obtained with the obstacle-type formulation on two-dimensional test cases at the metric and hectometric scales. The impact of hillslope geometry, boundary conditions, and soil hydraulic parameters on model predictions is studied. Section 4 investigates the more detailed model where the shallow water type equation describing the propagation of the overland flow is coupled with the water table dynamics. Section 5 draws the conclusions.

The Obstacle-Type Model (OTM)

The goal of this section is to describe the OTM and its numerical approximation. The OTM is based on the Richards equation to predict the water table dynamics and on the assumption that the height of the overland flow as well as re-infiltration processes have a negligible impact on the subsurface flow.

The numerical approximation is based on a finite element discretization and a Newton scheme embedded into a fixed-point iteration to determine those points at the ground where saturation conditions are met.

The Richards equation

The soil is modeled as an isotropic, variably saturated, porous medium in which air entrapment does not affect the flow. The movement of liquid water can be described by the Richards equation

∂ ∂t θ(ψ) + ∇•v(ψ) = 0 , ( 1
)
where θ is the volumetric water content (dimensionless), ψ the soil water pressure head (L) and v(ψ) the Darcy flow velocity (LT -1 ) defined by

v(ψ) = -k(ψ)∇H , (2)
where k is the hydraulic conductivity (LT -1 ), H = ψ + z the total hydraulic head (L) and z the vertical coordinate (upwards). The Richards equation can be rewritten as

∂ ∂t θ(ψ) = ∇• k(ψ)∇ψ + k(ψ)e z , (3)
where e z is the upward unit vector. The hydraulic conductivity is written in

the form k(ψ) = k s k r (ψ)
where k s is the hydraulic conductivity at saturation and k r (ψ) is the relative hydraulic conductivity.

Eq. (3) requires the knowledge of the soil water retention curve, ψ → θ(ψ),

and the relative hydraulic conductivity function, ψ → k r (ψ). Accurate mea-surement of the soil properties is generally tedious and costly. Consequently, more or less empirical equations have been used to describe the relations between water content, pressure and hydraulic conductivity [START_REF] Brooks | Hydraulic properties of porous media[END_REF]; [START_REF] Mualem | A new model for predicting the hydraulic conductivity of unsaturated porous media[END_REF][START_REF] Van Genuchten | A closed-form equation for predicting the hydraulic conductivity of unsaturated soils[END_REF]). The choice of one of these models can have a sizeable impact on the flow dynamics especially near saturation and, thus, substantially modify the position of the water table [START_REF] Cloke | The effect of model configuration on modelled hillslope-riparian interactions[END_REF]

Boundary conditions

The pressure head ψ is chosen as the main unknown in the Richards equation.

To specify suitable boundary conditions, we consider the two-dimensional domain Ω with boundary ∂Ω sketched on the left panel of Figure 1. Let n be the outward normal vector. A constant rainfall intensity i (LT -1 ) is considered;

the rainfall velocity is v r = -ie z .
The upper surface of the computational domain, ∂Ω t , allows for infiltration and exfiltration. The top boundary ∂Ω t is split into ∂Ω - t ∪ ∂Ω + t where ∂Ω - t corresponds to the non-saturated surface and ∂Ω + t to the saturated surface.

The key assumption made in the OTM is that the height of the overland flow as well as re-infiltration processes have a negligible impact on the water table dynamics. As a result, the pressure head can be set to the atmospheric pressure on ∂Ω + t , i.e ψ = 0. Following our notation, the flux through the top surface, v(ψ)•n, is locally negative if infiltration occurs or positive if exfiltration occurs. The rainfall flux, v r • n, is then negative. On the unsaturated area, the infiltration flux is equal to the rainfall flux, and in the saturated area the infiltration flux is smaller than or equal to the rainfall flux.

Hence, the following holds:

ψ ≤ 0 and v(ψ)•n = v r •n on ∂Ω - t , ψ = 0 and v(ψ)•n ≥ v r •n on ∂Ω + t . (4)
The saturated surface ∂Ω + t can be further divided into:

• the partial infiltration zone in which v r •n ≤ v(ψ)•n ≤ 0;

• the exfiltration zone in which v(ψ)•n > 0.

This splitting is illustrated in the right panel of Figure 1. The saturated zone ∂Ω + t extends from point B down to the toe of the slope (point D). The exfiltration zone extends from point C down to D.

For the bottom and left boundaries, an impermeable layer is assumed, i.e., the condition v(ψ)•n = 0 is imposed. On the right boundary, ∂Ω r , two boundary conditions can be considered: either a no-flow condition representing a symmetric configuration at the toe of the slope (henceforth referred to as BC1)

or a specified total hydraulic head accounting for the presence of a stream (henceforth referred to as BC2).

Numerical schemes for the OTM

Since the water table position is a priori unknown, its intersection with the ground surface leads to an unsteady obstacle-type problem in which the partition of ∂Ω t into ∂Ω - t ∪ ∂Ω + t and the water table dynamics must be solved simultaneously. Early work based on a similar approach includes that of [START_REF] Neuman | Saturated-unsaturated seepage by finite elements[END_REF] and [START_REF] Rulon | The development of multiple seepage faces on layered slopes[END_REF].

Stationary OTM

To formalize the OTM, consider first a stationary setting and, for the sake of simplicity, assume a BC1 boundary condition on ∂Ω r . Recall that H 1 (Ω) is the space of square integrable functions in Ω whose distributional derivative is also square integrable in Ω. The weak formulation corresponding to the steady problem is the following: given ∂Ω + t ⊂ ∂Ω t , let

V ∂Ω + t = { φ ∈ H 1 (Ω); φ = 0 on ∂Ω + t } , (5)
and let a be the form (non-linear in ψ, linear in φ)

a ∂Ω + t (ψ, φ) = Ω k(ψ)•( ∂ ∂x ψ + e z ) • ∂ ∂x φ + ∂Ω - t (v r •n) φ . (6)
Then, we seek ∂Ω + t ⊂ ∂Ω t and ψ ∈ V ∂Ω + t such that

                               (i) a ∂Ω + t (ψ, φ) = 0 ∀φ ∈ V ∂Ω + t , (ii) ψ ≤ 0 on ∂Ω - t , (iii) v(ψ)•n ≥ v r •n on ∂Ω + t . (7)
Note that working with the space V ∂Ω + t implies ψ = 0 on the saturated area ∂Ω + t . The well posedness of the stationary flow problem (i) requires that

∂Ω + t = ∅, i.e.
, that the water table reaches the top boundary. In addition, one can prove that a solution to (i)-(iii) exists whenever i ≤ k s , i.e., whenever the rainfall intensity does not exceed the soil infiltration capacity.

An approximate solution {∂Ω + t , ψ} of Eq. ( 7) is sought using a fixed-point iterative scheme as follows:

(1) choose an initial ∂Ω + t ;

(2) solve problem (i);

(3) check whether (ii) and (iii) are satisfied;

(4) if (ii) is satisfied and (iii) is not, move ∂Ω + t one mesh cell (or more) downhill; go to step 2;

(5) if (iii) is satisfied and (ii) is not, move ∂Ω + t one mesh cell (or more) uphill; go to step 2;

(6) if both (ii) and (iii) are satisfied, then the current pair {∂Ω + t , ψ} is the desired solution.

In this algorithm, the set of points lying on the soil surface where saturation conditions are met is determined iteratively and the boundary between ∂Ω + t and ∂Ω - t always coincides with a grid node. Nothing prevents the use of the OTM and the previous algorithm in situations with multiple seepage faces, i.e when there are multiple intersections of the water table with the ground surface.

Several numerical methods can be employed to solve problem (i) in step 2.

In this work, the Richards equation is discretized using linear and continuous finite elements on an unstructured mesh covering the whole computational domain. The discrete equations are nonlinear owing to the soil hydraulic functions. These equations are solved approximately using Newton's method. The convergence criterion is based on a weighted L 2 -norm of the solution update.

In the results presented in the following sections, a tolerance of 10 -7 is used for Newton's method and a tolerance of 10 -8 is used for the linear solver within each step of Newton's method. Since the Newton solver is called several times within the fixed-point iterative scheme, it is efficient to use the previous pair {∂Ω + t , ψ} as a starting estimate once the saturated region ∂Ω + t has been updated. Once the fixed-point iterative scheme has converged, it is possible to refine the mesh and go back to step 2 to improve the quality of the solution.

Owing to the maximum principle, both conditions (ii) and (iii) in Eq. ( 7) cannot be violated simultaneously. However, in numerical approximations, this can happen. In this case, we still consider that the water table has been correctly positioned. With this "loosened" convergence criterion, the final position of the water table depends on whether the converged position ∂Ω + t has been approached from below or above. The two resulting values yield lower and upper bounds for the water table position that typically differ from one or two mesh cells at the most.

Unstationary OTM

The unstationary OTM is solved using an implicit Euler scheme with an adaptive time step. For a time step k ≥ 0, given (∂Ω + t ) k and ψ k , we seek

(∂Ω + t ) k+1 ⊂ ∂Ω t and ψ k+1 ∈ V (∂Ω + t ) k+1 such that                                (i) a ′ (∂Ω + t ) k+1 (ψ k+1 , φ) = 0 ∀φ ∈ V (∂Ω + t ) k+1 , (ii) ψ k+1 ≤ 0 on (∂Ω - t ) k+1 , (iii) v(ψ k+1 )•n ≥ v r •n on (∂Ω + t ) k+1 , (8)
where

a ′ (∂Ω + t ) k+1 (ψ k+1 , φ) = 1 δt Ω (θ(ψ k+1 ) -θ(ψ k )) φ + a (∂Ω + t ) k+1 (ψ k+1 , φ) . ( 9
)
This problem is solved using the same iterative algorithm as that employed for the stationary OTM. In step 1, the initial choice is (∂Ω

+ t ) k+1 = (∂Ω + t ) k .
Note that in the unstationary case, problem (i) is well-posed even if the water table has not reached the top boundary.

The finite element code with Newton's method to approximate the Richards equation has been tested against analytical solutions for steady and unsteady infiltration columns. The test cases are described in [START_REF] Ginzburg | Lattice Boltzmann approach to Richards' equation[END_REF].

Virtual Hillslope Simulations

In this section, the OTM is used to perform virtual hillslope simulations at the metric and hectometric scales. The metric-scale problem is used to investigate the influence of rainfall intensity and soil hydraulic parameters within the VGM model on the dynamic response of the water table. The metric scale for these simulations allows for a very fine representation of the soil. The hectometric-scale problem focuses on the effects of geometrical properties and initial conditions since at large scales the choice of the initial condition may engender difficulties, such as downward infiltration front, depending on the soil texture.

Preliminary definitions

The saturated ground fraction

Let L be the length of the slope and let L s be the portion of the hillslope that is saturated. Henceforth, the ratio L s /L is termed the saturated ground fraction.

Infiltration, exfiltration, and runoff fluxes

Let Q rain = iL(e z •n) be the rainfall rate. We define Q in to be the infiltration flux and Q not in to be the "direct runon" flux, i.e., the water that never infiltrates. Hence,

Q rain = Q in + Q not in . ( 10
)
The exfiltration flux Q exf corresponds to the top surface exfiltration Q exf|∂Ωt plus the exfiltration into the stream Q exf|∂Ωr (subsurface flow through the right surface) if any. Hence,

Q exf = Q exf|∂Ωt + Q exf|∂Ωr . (11)
The quantity Q runoff is defined to be the exfiltration flux plus the direct runon flux. Hence,

Q runoff = Q exf + Q not in = Q exf + Q rain -Q in . ( 12
)
Note that Q runoff is not the instantaneous water flux at the toe of the slope and into the stream but the instantaneous water flux into the overland flow and into the stream.

To evaluate the infiltration and exfiltration fluxes associated with the discrete solution to the Richards equation at a given time, we use the following approach based on test functions. This method is in fact the natural way of expressing mass conservation when continuous finite elements are used. For simplicity, the method is described when no exfiltration occurs through the right boundary. The top boundary ∂Ω t is divided into two regions, an exfiltration region, ∂Ω ex , where v(ψ)•n > 0, and an infiltration region, ∂Ω in , where v(ψ)•n ≤ 0. Let ϕ 1 and ϕ 2 be the continuous piecewise linear functions defined on ∂Ω t such that ϕ 1 = 1 (respectively ϕ 2 = 1) at all the mesh nodes on ∂Ω ex (respectively on ∂Ω in ). Then, there exist two continuous piecewise linear functions defined on the whole computational domain, say φ 1 and φ 2 , such that:

φ 1 | ∂Ωt = ϕ 1 , φ 2 | ∂Ωt = ϕ 2 , and φ 1 + φ 2 = 1 on Ω. For a time step k ≥ 1, the infiltration flux Q in (t k
) and the exfiltration flux Q exf (t k ) are defined as follows:

Q exf (t k ) := ∂Ωt (v(ψ k )•n)ϕ 1 = Ω ∇φ 1 •∇ψ k + Ω φ 1 θ k -θ k-1 δt , Q in (t k ) := ∂Ωt (v(ψ k )•n)ϕ 2 = Ω ∇φ 2 •∇ψ k + Ω φ 2 θ k -θ k-1 δt , (13)
so that the following mass balance equation holds:

Q exf (t k ) + Q in (t k ) = Ω θ k -θ k-1 δt . ( 14
)
At steady-state, Q in = Q exf and, hence, Q runoff = Q rain . Finally, the time to reach equilibrium, T e , is defined as the lowest time for which

|Q in -Q exf | ≤ 5 × 10 -3 Q in .
3.1.3 The VGM and the modified VGM model

Let n > 1 and α be the van Genuchten soil parameters, and set m = 1 -1 n . Using the additional parameter ψ s < 0, the modified VGM model discussed in [START_REF] Vogel | Effect of the shape of the soil hydraulic functions near saturation on variably-saturated flow predictions[END_REF] is

θ(ψ) =                  β (1 + (-αψ) n ) -m , ψ < ψ s , 1 , ψ ≥ ψ s , (15)
where θ(ψ) is the effective saturation defined by Eq. (??) and

β = (1 + (-αψ s ) n ) m so that θ(ψ s ) = 1. We set k(ψ) = k s k r (ψ)
where the relative hydraulic conductivity is specified as a function of θ(ψ) as follows:

k r ( θ) =                    θ 1/2 1-1-(β -1 θ) 1 m m 2 1-1-β -1 m m 2 , ψ < ψ s , 1 , ψ ≥ ψ s . ( 16
)
The parameter ψ s is interpreted in [START_REF] Vogel | Effect of the shape of the soil hydraulic functions near saturation on variably-saturated flow predictions[END_REF] as a minimum capillary height. Eqs. ( 15) and ( 16) reduce to the original VGM model when ψ s = 0.

Five soils, whose hydraulic parameters are collected in [START_REF] Ogden | Saturated area formation on nonconvergent hillslope topography with shallow soils: A numerical investigation[END_REF].

Metric-scale problem

The two-dimensional domain selected to perform the study is sketched in Figure 2; it corresponds to the configuration considered by [START_REF] Abdul | Laboratory studies of the effects of the capillary fringe on streamflow generation[END_REF]. The domain dimensions are 1.4 m in length and 1 m to 0.8 m in height.

For the left boundary we assume an impermeable layer (BC1). The initial condition is a horizontal water table located at 0.7 m and the hydrostatic pressure profile. Simulations are run using a constant rainfall intensity for a duration longer than the time necessary to reach equilibrium. After assessing grid refinement issues in the next section, we investigate the influence of rainfall intensity on model predictions for YLC. Then we study the influence of the soil hydraulic parameters for SCL, YLC, and Sand 2. Particular attention is given to the shape of both the soil water retention and the relative hydraulic conductivity functions, especially near saturation.

Mesh density assessment

We consider YLC subjected to a constant rainfall intensity such that i/k s = 10%. Results obtained on two meshes are compared. The reference mesh, Mesh 1, is an unstructured triangulation consisting of 4083 nodes and 7903 elements: the node spacing is uniform and 1 cm on the top surface (142 nodes on the top surface) and 5 cm on the bottom surface.

Influence of rainfall intensity

We consider YLC subjected to four different rainfall intensities such that i/k s = 2, 5, 10, and 15%. Figure 3a plots the time evolution of the saturated ground fraction for the four rainfall intensities. As expected, the saturated ground fraction increases with the rainfall intensity. The nonlinear temporal response of the soil under different rainfall intensities is related to the soil hydraulic functions and, in particular, to the fact that for YLC, the relative hydraulic conductivity increases rapidly with pressure head near saturation.

Figure 3b plots the time evolution of the relative exfiltration flux, i.e., the exfiltration flux normalized by the rainfall rate. We observe that it is not a priori possible to rank at all times the exfiltration fluxes monotonically in terms of the rainfall intensity. The same phenomenon is observed with SCL and Sand 2.

Influence of the soil hydraulic parameters

The time evolution of the water table location is studied for SCL, YLC, and Sand 2. The corresponding relative hydraulic conductivity and soil water capacity, C(ψ) = dθ dψ , are plotted in Figure 4 as a function of pressure head for the original VGM model. Simulations are run with a constant rainfall intensity such that i/k s = 10% for each soil. Table 3 collects the predicted equilibrium times for the three soils. Sand 2 exhibits the fastest response and SCL the slowest. The parameter that exerts the major influence on the equilibrium time appears to be the hydraulic conductivity at saturation. Since the ratio i/k s is kept fixed, the rainfall intensity increases as the hydraulic conductivity at saturation does, and less time is then needed to fill the domain. Other parameters (such as (θ sθ r ), n, and α) also influence the equilibrium time, but to a smaller extent. Further studies should include a sensitivity analysis to assess quantitatively the relative importance of each parameter.

Figure 5a presents the time evolution of the saturated ground fraction. The shape of the three curves is a consequence of the respective shapes of the hydraulic functions for the three soils. At steady-state, Figure 5b shows strongly different distributions of the effective saturation along the unsaturated ground surface, although the saturated ground fraction is essentially identical for the three soils as it is controlled by the rainfall intensity for mass conservation reasons. The value of the parameter n appears to condition the curvature of the plots. For instance, a sole modification of this parameter (by setting it to 5 for YLC) yields a similar response to that observed for Sand 2. For the three soils, the time evolution of the saturated ground fraction is plotted in Figure 5a.

As sketched in Figure 1b (not in scale), only a part of the saturated ground (from C to D) contributes to the exfiltration, the saturated zone from B to C still allowing for some infiltration. In this case, at steady-state, the exfiltration part (CD) is independent of the soil texture and represents about 70% of the saturated length L s . The figure showing the time evolution of the fraction of the exfiltration part (CD/L) for the three soils is similar to Figure 5a and is therefore not presented. Figure 5c presents the time evolution of the relative exfiltration and runoff fluxes for the three soils. For each soil, the dynamics of the exfiltration are similar to that of the water table position. Furthermore, at equilibrium, the exfiltration represents 45.9% of runoff and direct runon represents 54.1%. Finally, Figure 5d presents the time evolution of the ratio of exfiltration to runoff fluxes for the three soils. The value of this ratio is roughly identical for the three soils: as soon as runoff begins, the ratio reaches a value between 40% and 50% and does not vary much until steady-state is reached. The differences between the curves result from an early occurrence of runoff (and hence exfiltration) for coarser-textured soils. To a lesser extent, fine-textured soils reach a maximum value of the ratio much faster after the beginning of the runoff; this value then slowly decreases to the steady-state value. This differs from the results for Sand 2 where the exfiltration ratio slowly increases to a maximum value that is reached just before equilibrium. This type of behavior is linked to the shape of the soil hydraulic functions.

To conclude this section, we consider the modified VGM model. In particular, we address the influence of the minimum capillary height on model results for SCL, YLC, and Sand 2. The effect of the modified VGM model on the relative hydraulic conductivity and water capacity functions is shown in Figure 6 for each soil. The same value, ψ s = -2 cm, has been chosen for the three soils.

Simulations are run with a constant rainfall intensity such that i/k s = 10%.

The time evolution of the water table is shown in Figure 7. The modified impacts its evolution in time: saturation appears earlier and, hence, exfiltration and runoff occur faster; see Figure 8. As discussed by [START_REF] Vogel | Effect of the shape of the soil hydraulic functions near saturation on variably-saturated flow predictions[END_REF], the modification of the VGM model affects more significantly those soils for which the parameter n takes values close to 1. For Sand 2, the hydraulic functions obtained with the modified VGM model are very similar to those obtained with the original VGM model. This fact is reflected in the results presented in Figures 7 and8.

Hectometric-scale problem

The geometry studied by [START_REF] Ogden | Saturated area formation on nonconvergent hillslope topography with shallow soils: A numerical investigation[END_REF] is sketched in Figure 9. A no-flow boundary condition is imposed at the bottom and left surfaces, simulating an impermeable layer. The initial condition corresponds to a horizontal water table located at the toe of the slope and a hydrostatic pressure profile. The soil consists of Sand OW to facilitate comparisons with the results reported in [START_REF] Ogden | Saturated area formation on nonconvergent hillslope topography with shallow soils: A numerical investigation[END_REF]. Simulations are run with a constant rainfall intensity of i = 30 mm/h (i/k s = 0.6%).

The effects of hillslope geometry on model predictions is investigated first. The land surface slope So is always set to 10%, while the depth to impermeable layer, D, and the slope length, L, are varied. Three geometries are studied:

Geometry 1 (L = 50 m, D = 1 m), Geometry 2 (L = 50 m, D = 2 m),
and Geometry 3 (L = 30 m, D = 1 m). A constant head boundary condition representing a stream is imposed at the right surface (BC2). Figure 10 presents the dynamic response of the hillslope for the three geometries. The mesh consists of 11996 elements for Geometry 1, of 7196 elements for Geometry 2, and of 14877 elements for Geometry 3. At equilibrium, Ogden and Watts have derived a simple expression to evaluate the saturated ground fraction based on the assumption that the hydraulic gradient is equal to the land surface slope So wherever the water table intersects the ground surface; the resulting formula is [START_REF] Ogden | Saturated area formation on nonconvergent hillslope topography with shallow soils: A numerical investigation[END_REF])

Ls L = 1 - Dk s So iL . ( 17
)
Our results, see Table 4, agree well with the above formula, indicating that at equilibrium, the velocity field is mainly tangential to the top surface. Since BC2 is imposed, the exfiltration flux essentially occurs through the right surface and is proportional to D; for instance, Q exf is found to be twice as large when D is doubled. The time to reach equilibrium naturally increases with increasing soil depth and slope length.

Figure 11 compares the two boundary conditions (BC1 and BC2) for Geometry 1. At this space scale, the impact of the boundary condition is negligible.

The differences between the two solutions are localized at the toe of the slope (in the area near the stream) and locally change the nature of the exfiltration process: through the top surface for BC1 and directly into the stream for BC2.

As can be seen from Figure 12a, the relative hydraulic conductivity of Sand OW decreases very slowly compared to that of YLC or Sand 1. With this type of soil and the initial condition considered above, a numerical solution can be obtained on a reasonable mesh size (mesh spacing of around 0.5 m). The choice of another soil, such as YLC for example, requires a much finer mesh if the same initial condition is used. Indeed, the low hydraulic conductivity of the soil on the left part of the geometry induces a downward infiltration Figures 12c and 12d present, respectively, the time evolution of the saturated ground fraction and the relative infiltration and exfiltration fluxes for YLC with the rainfall intensity i/k s = 10% and Geometry 1. The relative exfiltration flux is in agreement with the formula derived in [START_REF] Ogden | Saturated area formation on nonconvergent hillslope topography with shallow soils: A numerical investigation[END_REF],

namely Q exf Q rain = k s SoD i(e z •n)L ≈ 0.02.

Coupling with Overland Flow

The OTM examined so far is suitable to describe the coupling between the subsurface water table and surface runoff provided the height of the overland flow as well as re-infiltration processes can be neglected. A more sophisticated approach described in [START_REF] Esclaffer | Etude théorique de la formation des débits de crues à l'échelle des versants[END_REF] and [START_REF] Beaugendre | Numerical investigation of surface runoff in hillslopes with variably saturated flows[END_REF] consists of modeling the overland flow using a simplified form of the shallow water equations, the so-called diffusive wave approximation which is widely used to describe flood routing [START_REF] Moussa | Criteria for the choice of flood routing methods in natural channels[END_REF]; [START_REF] Beauquillon | Propagation des écoulements transitoires intermittents dans les systèmes hydrauliques à surfaces libres[END_REF]; [START_REF] Daluz-Vieira | Conditions governing the use of approximations for the Saint-Venant equations for shallow surface water flow[END_REF]). This section first describes briefly the diffusive wave approximation for the shallow water equations and how this model can be coupled to the Richards equation. Then, numerical results obtained with the OTM and the coupled model are compared at the metric and hectometric scales.

The Coupled Subsurface/Overland Flow Model (CSOFM)

Transient flow of shallow water can be described by the following equations [START_REF] Esteves | Overland flow and infiltration modelling for small plots during unsteady rain: numerical results versus observed values[END_REF]; [START_REF] Zhang | Modeling of two-dimensional overland flow[END_REF])

               ∂ ∂t y + ∂ ∂x (yV ) = w , ∂ ∂t V + V ∂ ∂x V + g ∂ ∂x y + g(S f -S) = 0 , ( 18
)
were y is the water depth (L), V the x-component of the flow velocity (LT -1 ),

w the mass source term, g the gravity (LT -2 ), S the river bed slope, and S f the energy line slope (S and S f are dimensionless).

We assume that the first and second terms in the momentum equation can be neglected in comparison with the third and fourth terms. This yields the diffusive wave approximation in the form

               (i) ∂ ∂t y + ∂ ∂x (yV ) = w , (ii) ∂ ∂x y + S f -S = 0 . ( 19
)
The Manning-Strickler uniform flow formula (Viollet et al. (1998)) is chosen to describe the energy line slope S f [START_REF] Panday | A fully coupled physically-based spatiallydistributed model for evaluating surface/subsurface flow[END_REF]; VanderKwaak (1999); [START_REF] Liu | A diffusive transport approach for flow routing in GIS-based flood modeling[END_REF])

V = K S R 2 3 sgn(S f )|S f | 1 2 , ( 20
)
where K S is the Strickler coefficient of roughness (L -1 3 T -1 ) and R the hydraulic radius (L) defined as the ratio between the cross-sectional flow area A (L 2 ) and the wet perimeter χ (L). Assuming that the overland flow occurs as a thin layer with a wide rectangular section, the relation R ≈ y holds. The Manning-Strickler uniform flow formula is also considered in the InHM model, [START_REF] Vanderkwaak | Hydrologic-response simulations for the R-5 catchment with comprehensive physics-based model[END_REF].

Letting q(y) = yV , Eqs. ( 19)(ii) and ( 20) readily yield

q(y) = K S y 5 3 sgn(S f )|S - ∂ ∂x y| 1 2 . ( 21
)
This is conveniently rewritten as follows:

q(y) = K S y 5 3 S -∂ ∂x y |S -∂ ∂x y| 1 2 . ( 22
)
The continuity equation then becomes

∂ ∂t y + ∂ ∂x   K S y 5 3 S -∂ ∂x y |S -∂ ∂x y| 1 2   = w . ( 23
)
One advantage of the diffusive wave approximation is to reduce the shallow water equations to a single partial differential equation. However, by doing so, the differential order of the equation is increased, and it is now necessary to supply two boundary conditions for Eq. ( 23). The condition y = 0 is imposed on ∂Ω - t and expresses the fact that the overland flow only exists on ∂Ω + t where it flows downhill. At the hillslope outlet, another simplification, the so-called kinematic wave approximation, i.e., ∂ ∂x y = 0, is made. In other models like MODHMS [START_REF] Panday | A fully coupled physically-based spatiallydistributed model for evaluating surface/subsurface flow[END_REF]) and HydroGeoSphere, based on earlier work of [START_REF] Therrien | Three-dimensional analysis of variablysaturated flow and solute transport in discretely-fracturated porous media[END_REF], it is possible to choose between the Chezy and the Manning-Strickler formula.

The CSOFM is formulated as follows. On ∂Ω + t , the boundary condition for the Richards equation becomes y = ψ. This condition expresses the fact that the vertical pressure profile in the overland flow is hydrostatic. Moreover, the source term w in Eq. ( 23) is evaluated from the rainfall and exfiltration fluxes in the form w = v r •nv(ψ)•n. More details on the computational method used to approximate the CSOFM are given in the appendix.

Since the CSOFM assumes the relation y = ψ, this model does not account for changes in water storage and flow area due to micro-topographic depressions and/or obstructions as proposed in more general models such as MODHMS, HydroGeoSphere, and InHM. Since the coupling between the overland and subsurface flows relies on the continuity of the hydraulic head, only one pressure value is defined at each node of the ground surface. On the contrary, MODHMS, HydroGeoSphere, and InHM allow for a discontinuity of the pressure field at the ground surface nodes. Then, an evaluation of the exchanged flux from the overland flow to the subsurface flow is needed [START_REF] Gerke | A dual-porosity model for simulating the preferential flow movement of water and solutes in structured porous media[END_REF]).

Numerical results

This section compares the numerical results obtained with the OTM and the CSOFM at the metric and hectometric scales.

Metric-scale problem

The dynamics of the water table are studied for Sand 1 using both BC1 and BC2. Here and in subsequent results, the Strickler coefficient of roughness K S is set to 10 m -1 3 s -1 . The geometry is that described in Section 3.2. The initial condition is a horizontal water table located at 0.8 m (the toe of the slope) and a hydrostatic pressure profile. Figure 13 and 14 present the time evolution of the saturated ground fraction and of the relative infiltration and exfiltration fluxes, respectively. The OTM and the CSOFM yield similar results for both boundary conditions. Both models predict that the final water table position is lower with BC2 than with BC1. This is related to the fact that BC1 and BC2 yield different flow velocity fields near the right boundary: with BC2, the velocity vectors on the saturated zone are nearly tangent to the surface so that most of the exfiltration occurs through the right surface. Figure 14c presents the time evolution of the relative runoff flux for the two models and the two boundary conditions. At this space scale, the boundary condition clearly affects the dynamics of runoff. Again, both models yield similar results.

Hectometric-scale problem

We consider the geometry of Ogden and Watts described in Section 3.3. The initial condition is a horizontal water table located at the toe of the slope and a hydrostatic pressure profile. Simulations are run with a constant rainfall intensity such that i = 30 mm/h, and BC2 is imposed at the right surface.

Figure 15 compares the results obtained with the two models on Geometry 1.

Under the present conditions, both models yield similar results with the exception of a slight difference in the steady-state values predicted for the exfiltration flux. This is related to the fact that because the pressure takes slightly larger values along the top boundary when predicted with the CSOFM, its normal derivative is larger, and, hence, the exfiltration flux is also larger.

Conclusions

The Richards equation combined with different models for hydraulic functions (BCM model [START_REF] Brooks | Hydraulic properties of porous media[END_REF]); original [START_REF] Van Genuchten | A closed-form equation for predicting the hydraulic conductivity of unsaturated soils[END_REF]) and modified [START_REF] Vogel | Effect of the shape of the soil hydraulic functions near saturation on variably-saturated flow predictions[END_REF]) VGM model) has been discretized by finite elements in space. The nonlinearities induced by the soil hydraulic functions can be efficiently handled by Newton's method. From a numerical viewpoint, the soil labelled SCL is the most difficult to deal with because of its strong nonlinear behavior especially near saturation. A second nonlinearity stems from the fact that the movement of the water table induces a changing character of the surface boundary condition for the Richards equation. This yields a unsteady obstacle-type problem for which a robust solution procedure based on a fixed-point iteration has been proposed.

The obstacle-type model has been used to investigate the hydraulic behavior of virtual hillslopes under constant rainfall conditions. Results on the smallscale geometry have shown that the rainfall intensity has a significant influence on the spatial extension of the saturated area of the hillslope and on the exfiltration flux. For a fixed value of the ratio i/k s , the structure of the soil, characterized by the van Genuchten hydraulic parameters, strongly affects the dynamic response of the system, whereas the steady-state values remain fairly identical. For the three soils addressed in the metric-scale problem, exfiltration occurs as soon as runoff does; this means that at any time, water exiting the system will have roughly the same origin, either from direct runon or through subsurface flow. In addition, a driving parameter for the time evolution of the saturated ground fraction and of the relative exfiltration flux appears to be the exponent n in the VGM model. Further investigations should be performed to determine whether its influence expresses itself through the relative hydraulic conductivity or the soil water capacity or both functions. Furthermore, results similar to those of [START_REF] Vogel | Effect of the shape of the soil hydraulic functions near saturation on variably-saturated flow predictions[END_REF] for infiltration in very dry soils are observed with the modified VGM model, especially for fine-textured soils where the introduction of a minimum capillary height has a significant influence on runoff and exfiltration genesis. Based on the above findings, a sensitivity analysis can be conducted as the next step to achieve a better understanding of the parameters conditioning surface runoff in hillslopes.

On the hectometric-scale problem, the numerical results are in very good agreement with the simple closed formulas derived by Ogden and Watts.

Moreover, at this space scale, boundary conditions on the right surface have a marginal impact on the solution. This medium-scale problem also underlines the importance of initial conditions in virtual hillslope simulations. This appendix briefly describes the computational method used to approximate the CSOFM. After time-discretization with the implicit Euler scheme, the subsurface flow problem consists of seeking ψ k+1 ∈ H 1 (Ω) such that for all φ ∈ H 1 (Ω),

1 δt Ω (θ(ψ k+1 ) -θ(ψ k )) φ + a (∂Ω + t ) k+1 (ψ k+1 , φ) = R(ψ k , ψ k+1 , φ) . (24)
Eq. ( 23) and the boundary condition y = ψ on ∂Ω + t are used to approximate the right-hand side as follows:

R(ψ k , ψ k+1 , φ) = - ∂Ω + t ψ k+1 -ψ k δt φ + ∂ ∂x q(ψ k+1 )φ . ( 25
)
Mass lumping is used for the unsteady term, i.e., for the vertex N i located on

∂Ω + t , ∂Ω + t ψ k+1 -ψ k δt φ ≈ ψ k+1 (N i ) -ψ k (N i ) δt L i , (26)
where L i is the distance separating the midpoints of the boundary cells on the left and on the right of the vertex N i . An upwind scheme is used to approximate the spatial derivative in the right-hand side of Eq. ( 25). Assuming that the nodes on ∂Ω + t are numbered downwards and defining the numerical flux

Q k+1 i-1,i = K s (ψ k+1 (N i-1 )) 5 3 S -ψ k+1 (N i )-ψ k+1 (N i-1 ) L i-1/2 S -ψ k+1 (N i )-ψ k+1 (N i-1 ) L i-1 2 1 2 , ( 27
)
denotes the distance between N i-1 and N i , we set

∂Ω + t ∂ ∂x q(y)φ ≈ Q k+1 i,i+1 -Q k+1 i-1,i . (28)
Eq. ( 28) is used for all nodes in the interior of ∂Ω + t . At the outlet node, say N i 0 , we set

Q k+1 i 0 ,i 0 +1 = K S (ψ k+1 (N i 0 )) 5 3 S 1 2 .
With this choice, the boundary condition ∂ ∂x y = 0 is weakly enforced at the hillslope outlet.

Considering BC2 amounts to enforce a constant total hydraulic head along the right boundary. At each time t, this value can be evaluated using the node located at the toe of the slope.

0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 saturated zone A B D water table unsaturated zone C ∂Ω - t = [AB] ∂Ω + t = [BD] Ω ∂Ω r 0 Unsaturated zone Saturated zone X A B X B C X C X D a b

Finally, we have

  compared the obstacle-type model with a coupled subsurface/overland flow model in which the dynamics of the overland flow are governed by the shallow water equations in the diffusive wave approximation. Both models yield similar results for the present test cases, both at the metric and hectometric scales, because the mechanisms leading to saturation are mainly controlled by subsurface flows and, more specifically, by the hydraulic properties of the soil. In other words, neglecting the feedback of surface runoff on the water table dynamics leads altogether to reliable results for the present test cases. Subsequent work should also consider models including reinfiltration processes under different rainfall conditions to further investigate the impact of overland flow on the genesis of surface runoff. Some preliminary test cases involving multiple seepage faces have been run. More realistic situations involving heterogeneous soil conductivity are being investigated to study the performance of the OTM code when dealing with perched water tables.

Fig. 1 .

 1 Fig. 1. (a) Schematic representation of the domain Ω. (b) Distribution along the top surface of typical pressure head, ψ (m) (solid line), and normal velocity, v•n (m/h) (dashed line).

Fig. 2 .

 2 Fig. 2. Abdul and Gillham geometry.

Fig. 3 .

 3 Fig. 3. (a) Time evolution of the saturated ground fraction for four different rainfall intensities with YLC. (b) Time evolution of the relative exfiltration flux for four different rainfall intensities with YLC.

Fig. 5 .Fig. 8 .Fig. 9 .

 589 Fig. 5. Original VGM model. (a) Time evolution of the saturated ground fraction for SCL, YLC, and Sand 2. (b) Distribution of the effective saturation along the top surface for the three soils. (c) Time evolution of the relative exfiltration and runoff fluxes for the three soils. (d) Time evolution of the ratio between exfiltration and runoff fluxes for the three soils.

Fig. 10 .Fig. 11 .Fig. 13 .Fig. 14 .Fig. 15 .

 1011131415 Fig. 10. (a) Time evolution of the saturated ground fraction and (b) of the relative exfiltration flux for the three geometries.

  Table 1, are selected for our virtual hillslope simulations: Silty Clay-Loam (SCL), Yolo Clay-Loam (YLC), Sand 2 from Kao et al. (2001), Sand 1 from Vogel et al. (2001), and Sand OW from

  To test local mesh re-finement, a new mesh, Mesh 1*, is produced from Mesh 1 by imposing a node spacing of 0.5 cm at the equilibrium water table position that results from the computation on Mesh 1. Mesh 1* consists of 7429 nodes and 14474 elements (240 nodes on the top surface). The agreement between the steady-state results produced by the two meshes is very satisfactory with relative errors lower than 1%; see Table2. Since local mesh refinement improves the results only marginally in the present simulations, quasi-uniform meshes are used in the rest of this work. Furthermore, to test mesh coarsening, a series of meshes (Mesh 2, Mesh 3, and Mesh 4) is produced by diminishing the number of nodes at the top surface. Results, which are collected in Table2, indicate that the ratio L s /L is only mildly affected by mesh coarsening while the equilibrium time is somewhat more sensitive to it.

Table 3

 3 Mesh influence on model predictions at steady state.Equilibrium time for three soils using the original VGM model.

	Table 1
	VGM soil hydraulic parameters.
	Texture	θ r (-) θ s (-) α (1/m) n (-) k s (m/h)
	Sand OW 0.069 0.435 0.326	3.9	5.0
	Sand 1	0.045 0.430 14.5	2.68 0.297
	Sand 2	0.05	0.5	3.7	5	0.1
	YLC	0.23	0.55	3.6	1.9	0.018
	SCL	0.1	0.41	1.9	1.31 0.0026

Table 4

 4 Equilibrium time and saturated ground fraction predicted numerically or using the empirical formula of Ogden and Watts.

			Geometry 1 Geometry 2 Geometry 3
	T e (h)		5.98	11.97	1.43
	L s /L (%)	67.3	34.7	45.6
	1 -DksSo iL	(%)	66.7	33.3	44.4

Acknowledgements

This work has been supported by INRIA under the Cooperative Research Project DYNAS (http://www-rocq.inria.fr/estime/DYNAS). We thank our colleagues from the Estime project at INRIA for stimulating discussions. We also thank the referees for their careful reading of the manuscript and their suggestions.


Catherine Garnier
email: catherine.garnier@inrae.fr

T Loiseleux

P Gelebart

M Chevalier

Marie-Hélène H Famelart

Thomas Croguennec

M Anton

A Riaublanc

Monitoring dairy protein aggregate structure allows to control the texture of emulsions

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


M Hennetier
email: marie.hennetier@purpan.fr

D De-Guibert

Y Gu

V Solé

Catherine Garnier

G Delaplace

Romain Jeantet

A Riaublanc

Asymetrical Flow Field-Flow Fractionation coupled with Multi-Angle Light Scattering and Differential Refractometer (AF4-MALS-DRi): An analysis tool to characterize milk protein aggregates produced in semi-industrial way

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.