Melissa Sanabria
email: sanabria@unice.fr

Frédéric Precioso
email: precioso@unice.fr

Thomas Menguy
email: thomas@wildmoka.com

Hierarchical Multimodal Attention for Deep Video Summarization

Keywords: Event stream data, Soccer match data, Video Summarization, Multimodal data, Sports Analytics

The way people consume sports on TV has drastically evolved in the last years, particularly under the combined effects of the legalization of sport betting and the huge increase of sport analytics. Several companies are nowadays sending observers in the stadiums to collect live data of all the events happening on the field during the match. Those data contain meaningful information providing a very detailed description of all the actions occurring during the match to feed the coaches and staff, the fans, the viewers, and the gamblers. Exploiting all these data, sport broadcasters want to generate extra content such as match highlights, match summaries, players and teams analytics, etc., to appeal subscribers. This paper explores the problem of summarizing professional soccer matches as automatically as possible using both the aforementioned event-stream data collected from the field and the content broadcasted on TV. We have designed an architecture, introducing first (1) a Multiple Instance Learning method that takes into account the sequential dependency among events and then (2) a hierarchical multimodal attention layer that grasps the importance of each event in an action. We evaluate our approach on matches from two professional European soccer leagues, showing its capability to identify the best actions for automatic summarization by comparing with real summaries made by human operators.

I. INTRODUCTION

The consumption of multimedia content has drastically evolved in the last decade. It is well-known that the amount of multimedia content stored, produced, published, exchanged on internet, is continuously increasing for more than ten years now, with a particular focus on videos. However the amount is not the only parameter which increases, more and more multimedia content evolves towards being more user immersive. This evolution of multimedia content consumption is also noticeable in the field of sport broadcasting where content augmentation improves immersive user experience, providing the user with all possible modality viewpoints of the content so as to enhance user engagement in the media. For instance, in the main world cycling races, sensors provide bike speed, pedaling frequency, heart rate, power produced, by cyclists, allowing the viewers to focus on specific content targeting a specific competitor. It is the same with soccer where This work has been co-funded by Région Provence Alpes Côte d'Azur (PACA), Université Côte d'Azur (UCA) and Wildmoka Company. new analytics about displacements on the field, run speed, run distance, heart rate, provide the viewer with an augmented experience of the matches.

In addition, specifically in the sport domain, the legalization and democratization of sport betting companies (recently in Europe) have accentuated the need for augmenting even more live content on players, teams, games, etc. Sport broadcasters aim at appealing and retaining subscribers to watch always more sports on TV, providing them with an enriched user experience. All the main sport broadcasters mandate companies, like Opta, Wyscout, Instat, Sportradar, Gamebreaker, SportsCode and many more, to send human observers in the stadiums to collect live data of all the events happening on the field during the match (pass, shoot, foul, players' position, cards, substitutions...) . Those event data contain meaningful information providing a very detailed description of all the actions occurring during the match to "feed" coaches and staffs, fans, spectators, viewers, and gamblers. Sport broadcasters want also to generate extra multimedia content such as match highlights, match summaries, etc, for their subscribers. There is no fully automatic solution to produce such content augmentation, in a short time (targeting real-time), on real sport content (one soccer match to be summarized is at the very least a 90 minute long video). The current solution for sport broadcasters is thus to rely on human operators to generate in live, highlights, summaries, specific content for social networks, and any extra content that will build viewer loyalty.

In this paper, we want to exploit the event data which provide us with unique information compared to all other available modalities (audio, visual, text...) to build a deep architecture for automatic summarization. Furthermore, event data are lighter in memory than video modality and thus allow us to process faster each whole match.

Contributions:

• Proposal Generation: Event metadata are first exploited to detect relevant categories of actions (i.e. segments of matches). Based on the intra-category diversity and inter-category possible similarity, we approach this first step as a Multiple Instance Learning problem (MIL) and address it through a LSTM MIL pooling schema.

• Hierarchical multimodal attention: Actions proposed to be in the summary, are classified into belonging or not to the summary. We design for this second step a specific multimodal attention model which combines event and audio modalities, at the event level, in a hierarchical LSTM network. Fig. 1: General schema of our approach. The left part of the figure represents the first block of our approach: Proposals Generation with a LSTM MIL Pooling. It gets as input the bags of events and outputs action Proposals. The right part of the figure is the second block of our approach: Hierarchical Multimodal Attention. It gets as input the action proposals (events data and audio data) and predict the likelihood for the given action to be in the summary.

II. RELATED WORK

Event data. While our approach uses metadata acquired in live during the matches for summarization, several approaches use this same information for other tasks like recognizing teams [START_REF] Bialkowski | Identifying team style in soccer using formations learned from spatiotemporal tracking data[END_REF], analyzing advantage of playing on your home field [START_REF] Lucey | Assessing team strategy using spatiotemporal data[END_REF], automatically discovering patterns in offensive strategies [START_REF] Van Haaren | Automatically discovering offensive patterns in soccer match data[END_REF], [START_REF] Gyarmati | Automatic extraction of the passing strategies of soccer teams[END_REF], predicting passes [START_REF] Vercruyssen | Qualitative spatial reasoning for soccer pass prediction[END_REF], detecting tactics [START_REF] Decroos | Automatic discovery of tactics in spatio-temporal soccer match data[END_REF], predicting the chance to score the next goal [START_REF] Liu | Deep reinforcement learning in ice hockey for context-aware player evaluation[END_REF] and evaluating the performance or contributions of the players [START_REF] Pappalardo | Playerank: data-driven performance evaluation and player ranking in soccer via a machine learning approach[END_REF]- [START_REF] Bransen | Measuring football players' on-the-ball contributions from passes during games[END_REF].

In addition, these metadata can now more and more be found either on program websites directly managed by the companies producing them (Prozone, GeniusSports, Opta, WyScout, and others) or through other open data sources (Kaggle competition, open datasets, etc) [START_REF] Pappalardo | A public data set of spatio-temporal match events in soccer competitions[END_REF], [START_REF] Bergmann | Linked soccer data[END_REF].

Sports Video Summarization. Early works in video summarization for sports mainly relies on hand-crafted heuristics. They exploit the characteristics of the field (lines, goal mouth), cinematographic properties like the camera motions, slow motion or zooming, and also specific edition patterns like the replays [START_REF] Ekin | Automatic soccer video analysis and summarization[END_REF]- [START_REF] Tavassolipour | Event detection and summarization in soccer videos using bayesian network and copula[END_REF] to select representative frames. More recently, approaches have migrated to machine learning techniques. Liu et al. [START_REF] Liu | Soccer video event detection using 3d convolutional networks and shot boundary detection via deep feature distance[END_REF] use 3D convolutional networks to classify the different clips of soccer videos, and Agyeman et al. [START_REF] Agyeman | Soccer video summarization using deep learning[END_REF] use this type of networks as feature extractor to then train an LSTM for action classification. Javed et al. [START_REF] Javed | Replay and key-events detection for sports video summarization using confined elliptical local ternary patterns and extreme learning machine[END_REF] propose an extreme learning machine to detect key-events based on the replay information. The main limitations in the state-of-theart of video sport summarization is the lack of standardization in the evaluation process and the use of heuristics to make decisions. Many of the aforementioned papers do not evaluate their methods using summarization metrics, they usually rather focus on the detection of most important actions like goals.

Multimodal Sports Summarization. For sports summarization the video is not the only source of information. Some methods propose to use social networks like the tweet streams during the game [START_REF] Corney | Two sides to every story: Subjective event summarization of sports events using twitter[END_REF]- [START_REF] Huang | Event summarization for sports games using twitter streams[END_REF]. Mend et al. [START_REF] Mendi | Sports video summarization based on motion analysis[END_REF] summarize a match detecting the intervals with highest motion from the optical flow. Tang et al. [START_REF] Tang | Autohighlight: Automatic highlights detection and segmentation in soccer matches[END_REF] design a deep learning algorithm to classify soccer actions from the text timeline provided by several web pages. Other methods exploit audio features [START_REF] Rui | Automatically extracting highlights for tv baseball programs[END_REF], [START_REF] Baijal | Sports highlights generation bas ed on acoustic events detection: A rugby case study[END_REF] since they help to identify the excitement of the commentators and the crowd, and sometimes the ball hit like for tennis or baseball. Multiple modalities play an important role to choose the best moments of sports videos. Several methods [START_REF] Bettadapura | Leveraging contextual cues for generating basketball highlights[END_REF]- [START_REF] Shukla | Automatic cricket highlight generation using event-driven and excitement-based features[END_REF] merge different modalities like the sound energy, the score, camera motions, players' reactions, referee whistle, etc. In this work, we are also going to consider different modalities but merging them hierarchically.

Video Summarization. Most of the works on video summarization are not specifically dedicated to sports domain and its peculiarities. For more general purpose videos, approaches focus on different criteria. For instance, the observation that similar videos share similar summary structures [START_REF] Zhang | Summary transfer: Exemplar-based subset selection for video summarization[END_REF]- [START_REF] Panda | Collaborative summarization of topic-related videos[END_REF]. Taking as inspiration semantic segmentation, Rochan et al. [START_REF] Rochan | Video summarization using fully convolutional sequence networks[END_REF] use a fully convolutional network across time, where the output is a mask showing the relevant frames for the video summary. On the other hand [START_REF] Gygli | Video summarization by learning submodular mixtures of objectives[END_REF], [START_REF] Li | A general framework for edited video and raw video summarization[END_REF] use a combination of objectives like interestingness, uniformity, representativeness to identify the most appealing moments. Recent successes of Generative Adversarial Networks have led to several works based on unsupervised approaches for video summarization [START_REF] Rochan | Video summarization by learning from unpaired data[END_REF], [START_REF] He | Unsupervised video summarization with attentive conditional generative adversarial networks[END_REF]. Zhang et al. [START_REF] Zhang | Summary transfer: Exemplar-based subset selection for video summarization[END_REF] were the first ones using LSTM for video summarization, their method is a bidirectional LSTM followed by a Multi-Layer Perceptron. Although LSTM is able to model long-range structural dependencies, Zhao et al. [START_REF] Zhao | Hierarchical recurrent neural network for video summarization[END_REF] propose a hierarchical LSTM to help the model to handle particularly long sequences. Most of the methods on the summarization of general purpose videos are based on the maximization of diversity, trying to minimize the number of similar shots [START_REF] Rochan | Video summarization by learning from unpaired data[END_REF], [START_REF] Wang | Stacked memory network for video summarization[END_REF], [START_REF] Zhang | Video summarization with long short-term memory[END_REF]. However, such a criterion does not apply for sport summarization. For instance in soccer, the easiest way to create a summary is to choose the goal clips, even if they are all visually very similar. And this situation holds for many sports.

To the extent of our knowledge, our method is the first one exploiting event data to automatically generate summaries. Event data significantly reduces the amount of information to be processed per match compared with other modalities like video where the number of frames in a 90 minutes match makes video data intractable. We propose a Multiple Instance Learning approach that, unlike comparable approaches, exploits LSTM sequentiality to process time-dependent instances and generate proposals. Furthermore, this work also introduces a new multimodal attention architecture that helps to learn a multimodal representation at the event level instead of learning a representation at the action level as existing methods.

III. PROPOSAL GENERATION

The goal of the first block of our method is to identify the action proposals of the match, that is to say consecutive relevant events. For instance, an action of a goal might be corresponding to the sequence {pass, interception, pass, goal}. Such groups of events are considered as proposals if they are parts of the match that might belong to the summary.

In the last few years, progress has been made in the task of object detection and one of the common core elements of all these approaches is to split the process in two tasks, object proposal generation as a preprocessing stage that later provides candidate windows to an object classifier. Examples of this are the architectures based on Region Proposal Network (RPN) [START_REF] Ren | Faster r-cnn: Towards real-time object detection with region proposal networks[END_REF]- [START_REF] Li | High performance visual tracking with siamese region proposal network[END_REF]. This concept was later developed for videos [START_REF] Xu | R-c3d: region convolutional 3d network for temporal activity detection[END_REF]- [START_REF] Buch | Endto-end, single-stream temporal action detection in untrimmed videos[END_REF]. In this paper we follow the same idea of splitting the detection in two tasks, first the generation of proposals and second the classification of these proposals.

However, we believe that in our context, a traditional learning method is not enough. In the same match we can find identical sequences of events, some labeled as positive (i.e. in the summary) and some as negative (i.e. not in the summary). In the case of soccer for instance, the sequence pass, interception, pass can be the beginning of an action of goal which is part of the summary but the same sequence can belong to some section of the match where nothing relevant is happening.

We thus decide to tackle the similarity of inter-categorical actions with a Multiple Instance Learning (MIL) approach.

As it will be detailed further in Section V-A, our ground truth dataset contains only the parts of matches belonging to summaries which means that we do not have access to the labels of the actions. To tackle this issue, we define as candidate all the sequences of events that are identical to the ones labeled as summary. To be more specific, for instance in one of the summaries of the ground truth, there is a goal action which corresponds to the sequence of events = {out, throw-in, long ball, aerial, pass, goal}. We then look for all the instances of this exact same sequence of events in the rest of the match and in all the remaining matches in the training set to label them as candidates.

We follow the event and bag representation proposed in [START_REF] Sanabria | A deep architecture for multimodal summarization of soccer games[END_REF]. A match is a sequence of events {e 1 , e 2 , ..., e N } which are all the events occurring on the field (possibly not broadcasted on TV). X = {x e1 , x e2 , ..., x e N } represents the set of instances, where x en is the feature vector characterizing the n-th event of the match. We denote a bag b as a set of consecutive events and B as the set of bags in a match

B = {b 1 , b 2 , ..., b F }.
A bag is considered positive if at least half of the events of the bag belongs to any of the candidates.

As in test phase we do not have access to the ground truth intervals, the bags are created in a class-agnostic way, using a sliding window with overlap across the entire match.

In the classical supervised learning problem the objective is to find a model that predicts a target value y ∈ {0, 1}, for a given instance. In the case of MIL, instead of a single instance there are groups of instances called bags. There is also a single binary label Y associated with the bag. Furthermore, it assumes that individual labels exist for the instances within a bag, i.e., y 1 , ..., y k and y k ∈ {0, 1}, however there is no access to those labels and they remain unknown during training. Then in MIL, a bag is labeled as negative if all the instances inside the bag are negative and a bag is labeled as positive if at least one instance of the bag is positive:

Y = 0, iff k y k = 0, 1, otherwise (1)

A. LSTM MIL Pooling

MIL is classically regarded as a general and abstract learning paradigm and, as such, it does not require or involve any feature extraction process. However, recent works integrate the process of solving MIL problem, within the process of learning features by using a fully-connected neural network and showing result improvements [START_REF] Wang | Revisiting multiple instance neural networks[END_REF], [START_REF] Hori | Attention-based multimodal fusion for video description[END_REF]. In our approach, we follow this line of research because it allows to consider MIL into an end-to-end trained model.

MIL paradigm assumes neither ordering nor dependency of instances within a bag. However that does not apply in our problem since the selection of an action to be part of a summary is highly dependent on the sequence of its events. For instance, a penalty or a free-kick are always preceded by a foul. More importantly, the permutation of events could completely change the meaning or the interest of an action.

For this reason, we argue that fully-connected layers as proposed by previous works are not completely suitable to capture this sequentiality. Recurrent neural networks are better suited to model such dependency. At the core of the LSTMs are memory cells which encode, at every time step, the knowledge of the inputs that have been observed up to that step. Therefore, we propose an LSTM network followed by a MIL Pooling to get the bag representation.

Each input sample is a sequence of event feature vectors representing a bag. The sequence of feature vectors is then fed to an LSTM, and the hidden state of the LSTM is given by:

h t = LST M (h t-1 , x en ) (2)
where LST M (h t-1 , x en ) represents an LSTM function of hidden state h t-1 and input vector x en .

Let b f be a bag (of events) of size K. Each event is defined by its feature vector x en (the detail of the event metadata features used in this work are provided in the supplementary material). We then learn an embedding for each event feature vector using an LSTM to preserve the sequential dependency between events. Let H b f = {h 1 , ..., h k } be the K embeddings of the K events from bag b f . Each h k embedding is of size L. We propose then a MIL pooling schema to learn the final bag representation z b f , as defined in Eq.(3):

∀ l=1,...,L : z b f l = max k=1,...,K h kl (3)
where z b f is a feature vector of size L, and is obtained from getting the maximum of each position l across all the K event embeddings of the bag.

Finally, this representation z b f is input into a single sigmoid neuron which provides the score O b f , a value between 0 and 1, for the bag b f to be a proposal or not.

B. Proposal Definition

From the output of the MIL network we get a score O b providing the likelihood per bag to be proposed for the final summary, but the overlap between consecutive positions of the sliding window lead some events to belong to more than one bag and thus to get related to different scores O b . In order to rely on the importance of each event to select or not an action in the final summary, we need to define a score per event instead of per bag, merging all the possible scores associated to the given event. The score S en for the event e n is then given by the Log-Sum-Exp (LSE) used in [START_REF] Sanabria | A deep architecture for multimodal summarization of soccer games[END_REF], the function is defined in Eq. ( 4). The LSE is a smooth version and convex approximation of the max function. The hyperparameter r controls the smoothness of approximation [START_REF] Wang | Revisiting multiple instance neural networks[END_REF].

S en ≥ r -1 • log 1 |{b f | e n ∈ b f }| b f |en∈b f r • O b f (4)
After obtaining the score per event, we use a threshold to select the positive events, this threshold is defined using the validation set. Thus an action proposal a p is a set of positive consecutive events. We denote A the set of all action proposals in a match.

IV. SUMMARIZATION: HIERARCHICAL MULTIMODAL ATTENTION

One of the biggest challenges for automatic soccer video summarization is to produce summaries provoking as much emotion as the ones made by human operators. To decide which actions are added to the summary, human editors use different sources of information. For this reason, we propose a multimodal approach that use event metadata and audio.

A. Multimodality

The event data have been manually collected by human observers sitting inside the stadium during the game. Each time an event occurs on the field, the human annotates the event with: the type (e.g., pass, foul, out or card), a timestamp, the team and players involved, outcome (i.e., if the action has been successful), the location (i.e., (x,y) position) on the field. Depending on the type of event, other information is available. For example, the final position on the field of the event, and descriptors of each type of event (e.g., yellow or red for the event type card), which are called qualifiers. The detailed description of the metadata extracted from the event data is presented in the supplementary material.

Audio plays a very important role in sports, where crowd cheering and excitement in the commentators' voice are usually indicators of an important action. For this reason, our summarization approach does not only exploit the event metadata features but also the audio signal extracted from the broadcasted match video. We use 9 different audio features also detailed in the supplementary material. These features extract the energy, the spectrum, the MFCC and the variation of the audio signal.

Since each event e n has a timestamp corresponding to a time in the match, we extract its corresponding video time time en in seconds. Since the different sounds in a broadcasted video come from the reactions of humans (spectators or commentators) just after the event occurs, the audio features of the event e n are extracted from the interval [time en , time en + 2]. Inside this interval, the audio signal is first divided into short-term windows (frames) of 100 ms with 50% overlap, then for each frame all features are calculated.

B. Multimodal Attention

In Section III we have described how we use Multiple Instance Learning to obtain a score per event and how we have defined that an action proposal a p is a set of positive consecutive events. Now the goal of the second block of our approach is to define which of these proposals indeed belong to the summary.

As we stated before, the use of multiple modalities is relevant to decide which actions are important in a sport match. The audio of an action can significantly vary not only from the type of the action but also from the events occurring inside the action. For instance, it is not the same kind of goal event, if the goal is preceded by several slow passes as if it is the result of an action starting by an interception or an error from the opponent team. For this reason, instead of learning the importance of each modality per action, we propose a hierarchical multimodal attention mechanism that in the first stage learns the importance of each modality at the event level and in the second stage learns the importance of each event inside the action (see Figure 2c).

Thus, in the first stage of the hierarchy, the multimodal representation vector per event is given by a weighted average:

c i = λ M i h M i + λ A i h A i (5)
where the weights of each modality {λ M i , λ A i } are determined by an attention layer shared across time-steps, see Fig. 2.

An action might contain several events that are not considered as important in a match but they are relevant to provide a context to the fans. For instance, there are many fouls during the match, but if a card action belongs to the summary it is important to show the foul and pass events that provoked this card. However, depending on the excitement of the crowd or the type of card, the importance of the events may vary. Therefore, after obtaining a multimodal representation per event c i , we train an attention layer that learns the importance of each event inside the action, resulting in the weight β c i . Thus, the representation vector per action proposal is given by a weighted average:

d ap = Lp i=1 β c i h c i ( 6
)
where L p is the length (number of events) of action a p . Finally, each of this d ap action representation is given to a sigmoid neuron which outputs a value between 0 and 1, that indicates the likelihood of the action a p to be included in the summary.

For the sake of space, we detail in the supplementary material the methodological comparison of our hierarchical multimodal attention model Fig. 2c with a Naive baseline Fig. 2a, and with Hori et al. model [START_REF] Hori | Attention-based multimodal fusion for video description[END_REF], Fig. 2b.

V. EXPERIMENTS

We first describe the dataset, then we split the evaluation in the two main tasks of our approach, the action proposals generation using LSTM MIL Pooling and the Summarization using the multimodal attention.

A. Dataset and metrics

Our dataset consists of event data for 20 matches from the 2017-2018 season of the French Ligue 1 and 50 matches from the 2019-2020 season of the English Premier League. There are 43 different types of events, related either to the flow of the match like a yellow card or to the action on the pitch like a shot. Each match corresponds to an average of 1700 events.

The only ground truth available are the 70 video summaries created by professional broadcasters. We created a set of intervals I t , corresponding to the time location on the video match of all clips inside the summary. To obtain the ground truth in terms of events, we found all events that have a timestamp inside the intervals I t and created a new set of intervals I. All the results reported in this section are evaluated with respect to the event ground truth I. The Missing Intervals rate and the F-score are computed from the fact that an interval refers to a summary interval of the video summaries created by professional broadcasters.

In order to obtain a fair comparison we use a 10-fold-crossvalidation. Each fold has 80%, 10% and 10% of the matches for train, validation and test set respectively. The matches of the two leagues are equally distributed in each fold.

For all the experiments, we replicate the models from the literature using Keras library and choose the parameters that have shown high classification performance on our dataset.

B. Proposal Generation

We compare our LSTM MIL Pooling method with three state-of-the-art methods. For all the experiments reported on Proposal Generation, in test phase we consider that the method found an interval if at least 50% of the action is inside any interval of I.

As we mentioned previously, action proposal generation problem has been tackled by several approaches in video processing. SST [START_REF] Buch | Sst: Single-stream temporal action proposals[END_REF] was created to generate temporal action proposals in untrimmed video sequences. It uses a recurrent neural network to produce confidence scores of multiple proposal sizes at each time step. We use the same approach but instead of video features, we use our metadata features as input. We use an LSTM network with 128 neurons, the output proposals sizes are {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} and in the training phase an output interval is considered as positive if at least 50% of it belongs to a candidate.

In terms of Multiple Instance Learning approaches, we compare with two Neural Networks based approaches. MI-Net [START_REF] Wang | Revisiting multiple instance neural networks[END_REF] uses three fully connected layers to generate a representation per sample and then with a max-pooling layer over all the samples of the bag gets a score per bag. Ilse et al. [START_REF] Ilse | Attention-based deep multiple instance learning[END_REF] is a similar approach but instead of using a maxpooling layer, it adds an attention mechanism. For this reason in our experiments we call this method MI-Net Attention. The number of neurons for the three fully connected layers are 256, 128 and 64 respectively. For the attention we used 32 neurons.

For LSTM MIL Pooling we used a LSTM with 64 neurons, Adam optimizer, binary cross-entropy as loss function and a batch size of 32 bags.

For the MIL approaches, in the training phase a bag is considered positive if at least 50% of it belongs to a candidate.

As the goal of the first block of our method is to detect all the possible actions of the match, we will focus on a low Missing Intervals rate and a high Recall. This means we will pay more attention on not losing any potential summary interval of the match. For Metadata we have both categorical and real-valued features. We consider also important to analyze the impact of mixing these types of features. Thus we compare the performance of the methods using two different representations for the categorical features, by converting the features either to One-Hot Encoding vectors or to real values using Target Encoding.

In One-Hot Encoding we created a vector of the total number of types/qualifiers with zero in all the positions and one in the position corresponding to the type/qualifier of the current event. On the other hand, Target Encoding gives only one real value that represents the fraction of times an event with this feature (type/qualifier) is labeled as 1 out of all the times an event with this feature is in the training set.

Table I shows that Target Encoding outperforms One-Hot Encoding in all the methods, which might be due to the sparsity of the features. There are 43 types of events and 47 qualifiers, an event can only have one event type and very rarely more than 3 qualifiers.

In Table I we can also see that MIL methods are clearly better on identifying the different proposals of the match, since SST performs at least 20% worse than the rest of the methods. LSTM MIL Pooling outperforms state-of-the art methods, it misses at least 3% less intervals and gets a Recall at least 3% higher compared with the second-best method (MI-Net Attention).

C. Summarization

The inputs of our Summarization block are the actions generated by the LSTM MIL Poolling method. We compare our Hierarchical Multimodal Attention model (cf. Fig. 2c) with two state-of-the-art methods, Sanabria et al. [START_REF] Sanabria | A deep architecture for multimodal summarization of soccer games[END_REF] and Hori et al. [START_REF] Hori | Attention-based multimodal fusion for video description[END_REF] (see Fig. 2b), and a baseline (see Fig. 2a).

Although Sanabria et al. [START_REF] Sanabria | A deep architecture for multimodal summarization of soccer games[END_REF] method does not integrate an attention model, we consider it as a relevant approach to Fig. 3: Examples of attention in different actions of Hori et al [START_REF] Hori | Attention-based multimodal fusion for video description[END_REF]. On the left side, attention weights of audio part: The x axis is the sequence of events in the action and the y axis represents the weight values learned by the attention layer. On the right side, multimodal attention weights in the action level: the y axis is the weight values learned by the attention layer. Blue and orange represent the audio and the event metadata respectively. compare with, since their goal is to summarize soccer matches using event data. They propose to concatenate audio energy with metadata features to train a hierarchical LSTM to decide which actions belong to the summary. We used the same features and parameters as mentioned in their paper.

As an attention baseline we have created a model called Naive Fusion (Fig. 2a) that has a LSTM per modality, then the last state of each LSTM passes through an attention layer that learns the importance of each modality. Finally the weighted sum is passed to a sigmoid neuron to make the decision.

Hori et al. [START_REF] Hori | Attention-based multimodal fusion for video description[END_REF] proposed an attention-based multimodal fusion for video description (Fig. 2b). We keep the same schema and replicate it for our video summarization problem.

For the Naive Fusion and Hori et al. approaches, we used 32 neurons for the LSTM of each modality and our audio and metadata features as input. Our model has 32 neurons in h M and h A , and 16 neurons in h c

Table II shows the Missing Intervals rate and the F-score of the aforementioned methods. The big gap between the performance of Sanabria et al. method and the others shows that the concatenation of different modalities is not enough to learn a representation of the actions. In Table II we can also see that our method misses at least 5% less actions and gets an increase of 2% in F-score than the state-of-the-art method.

We believe that our method outperforms [START_REF] Hori | Attention-based multimodal fusion for video description[END_REF] because in this method the multimodal fusion is done at the action level. Indeed their method has an attention layer at event level but it is done separately per modality. Learning the importance of the event using only the audio features of a soccer match is a very difficult task (see sumpplementary material for a clearer explanation of the architectures). The left side of Figure 3 displays the attention learned in the audio part by [START_REF] Hori | Attention-based multimodal fusion for video description[END_REF] in four different actions. It seems that the attention is just learning that the last events (i.e. the end of the actions) are more important no matter the type of the events. The right side of this figure displays the importance learned by the attention for the audio and metadata modalities. This not only shows that for this model the metadata is often more important but also that the audio modality is most of the times neglected.

On the other hand, Figure 4 shows some qualitative results of our model. We can see that the multimodal attention does not follow a particular pattern, audio and metadata importance can be very different from one action to another. And the attention learned by the second stage of our model considers many important events where the audio was considered as more relevant from the previous stage.

Soccer Baselines. As we previously mentioned, the evaluation of most of the methods on video sports summarization are based on the detection of most important actions, then to perform a fair comparison, we propose three baselines:

• Only Goals: Only the goals of the match are predicted as positive. Since the easiest way to create a summary from a soccer video is to extract the goals of the match. • All Shots-on-Target: All Shots on Target actions (i.e. goals, goalkeeper saving a shot on goal, any shot on goal which goes wide or over the goal and whenever the ball hits the frame of the goal) are predicted as positive. • Random: The prediction is a random value between 0 and 1, where the samples with values below 0.5 are negatives and the ones greater or equal than 0.5 are positives. Table III depicts the performance of these baselines. Our Fscore is clearly the highest. The Precision of our approach is only outperformed by Only Goals, considering it is very likely that all the goals of the match belong to the summary, however the Recall of this baseline is the lowest since it misses many other type of actions. The Recall of our approach is only outperformed by All Shots-on-Target, since the Shots on Target actions represent a big percentage of the actions included in summaries, yet the Precision of this baseline is at least 34% lower than ours. Multimodality. In order to show the importance of merging multiple modalities, we evaluate the performance of each modality separately using a LSTM with an attention layer.

Table IV shows that our method obtains the highest F-score compared with the models using only audio and only metadata features. Although using only audio features less intervals are missing, the low F-score reveals the low precision of this method since it predicts a lot of false positives. Comparing our results with the method using only the metadata we can see that adding the audio features helps to reduce almost 5% of missing intervals.

Audio Features. We consider that it is also important to show if the use of additional audio features improves the results. Sanabria et al. [START_REF] Sanabria | A deep architecture for multimodal summarization of soccer games[END_REF] only used the energy of the audio signal, however as it was mentioned in section IV-A, there are many other audio features that have helped to improve classifications in other contexts. We created two different models that take as input either the energy features proposed by Sanabria et al. or the audio features proposed in this paper (details are in the supplementary material), in order to predict which action belong to the summary. One model (Attention in Table V) has an attention layer that learns the importance of each event and the second model is a regular LSTM (No-Attention in Table V).

Table V shows that the models using our feature set outperforms at least by 2% the models using only the Energy features. And this behavior holds for models with and without attention mechanisms.

VI. CONCLUSION

In this paper, we proposed a method to summarize soccer matches using multimodal event data. We introduce a Multiple Instance Learning approach where the instances are sequentially dependent and we have empirically shown that it is a good method to generate proposals in a context where the instance labels are not available. Different from existing multimodal attention approaches, our method focuses on the importance of each modality at event level to then, in a second stage, learn the relevance of each event at action level. Experiments in a dataset composed of two different soccer leagues show the capability of our approach to identify the best actions for automatic summarization by comparing with real summaries made by human operators and outperforming state-of-the-art methods.

Fig. 2 :

 2 Fig. 2: Definition of our hierarchical multimodal attention schema and comparison with state-of-the-art approaches. Blue indicates metadatada, orange indicates audio and green indicate the multimodal representation of the events. λ and β are attention weights.

Fig. 4 :

 4 Fig. 4: Examples of attention in different actions learned by our model. On the bottom, multimodal attention weights at the event level: The x axis is the sequence of events in the action and the y axis represents the weight values learned by the attention layer. On the top, attention weights learned from the multimodal representation of each event. Blue and orange represent the audio and the metadata respectively.

TABLE I :

 I Performance Comparison of Proposal Generation methods.

		Missing Intervals	Recall
	Method	Target	One-Hot	Target	One-Hot
		Encoding	Encoding	Encoding	Encoding
	SST [46]	39.79	45.53	60.11	54.35
	MI-Net [49]	18.62	23.34	81.33	76.6
	MI-Net Attention [51]	16.07	19.39	83.89	80.56
	LSTM MIL Pooling	13.01	22.83	86.96	77.11

TABLE II :

 II Performance comparison of Multimodal Attention methods.

	Method	Missing Intervals F-score
	Sanabria et al. [48]	47.95	64.30
	Naive Fusion	36.19	71.23
	Hori et al. [50]	32.99	72.03
	Ours	27.38	74.09

TABLE III :

 III Performance comparison of Soccer Baselines.

	Method	Precision Recall F-score
	Only Goals	99.55	28.29	44.18
	All Shots-on-Target	40.77	75.71	52.99
	Random	41.87	48.72	45.03
	Ours	75.46	72.76	74.09

TABLE IV :

 IV Performance comparison of Separete Modalities.

	Method	Missing Intervals F-score
	Only Audio	27.75	64.17
	Only Metadata	32.09	72.05
	Ours	27.38	74.09

TABLE V :

 V F1-score comparison of Audio Features.

	Audio Features No Attention Attention
	Energy Features	59.37	59.51
	Our Features set	62.78	64.17


Catherine Annen

Alain Burgisser

Modeling water exsolution from a growing and solidifying felsic magma body

Keywords: magma, volatiles, water exsolution, granite

We modelled water exsolution from a growing and crystalizing felsic magma body using a conductive thermal model with simplified physical mechanisms of volatile transport. Magma solidification leads to the release of the water initially dissolved in the melt. Solidification behaviour depends, in turn, on dissolved water content. The water is channelled where the magma is solidified enough to form a crystal network and rapidly ascends until it encounters solid rock or liquid-rich magma. The rate of water exsolution depends on the basal magma emplacement rate, the cooling rate, and the magma initial water content. Water-rich layers form and are eventually trapped in the solid rock as cooling and crystallization proceeds. We ran our model with a quasi-eutectic granite composition and a noneutectic monzogranite composition. The non-eutectic magma crystallises on a wider range of temperature than the eutectic magma, produces more mush, and is more prone to the formation of water-rich lenses. Exsolved water accumulates on the sides of the chamber for both tested compositions and also above the top of the chamber for the non-eutectic composition. The actual presence of these water-rich layers in nature and their lifetime depends on whether the water is further released by fracturing. Our results accounts for the observed decoupling of volatiles from magma and for the episodic nature of volcano deformation. It has implications for the interpretation of magma chambers tomography as water-rich lenses would be difficult to distinguish from melt-rich lenses.

Introduction

Magmas are mixtures of melt, crystals, and volatiles. At depth, they contain various dissolved volatile species, including H 2 O, CO 2 , and SO 2 . Magma decompression or crystallization induces volatiles exsolution and formation of an independent phase of supercritical fluid.

Volatiles play an important role in volcanology. Their proportions strongly affect eruptions styles as gas-rich magmas are more explosive and hazardous than their gas-poor counterpart. The presence of exsolved volatiles also affects geophysical measurements. On the one hand, exsolved volatiles increase magma compressibility, so that magma volumes may be underestimated if based on ground surface deformation [START_REF] Tait | Pressure gas content and eruption periodicity of a shallow, crystallising magma chamber[END_REF]. On the other hand, the change in volume caused by water exsolution in closed system induces an overpressure that may lead to the rupture of the chamber walls [START_REF] Blake | Volatile oversaturation during the evolution of silicic magma chambers as an eruption trigger[END_REF][START_REF] Fowler | Phase equilibria trigger for explosive volcanic eruptions[END_REF][START_REF] Fowler | Phase equilibria trigger for explosive volcanic eruptions[END_REF][START_REF] Tait | Pressure gas content and eruption periodicity of a shallow, crystallising magma chamber[END_REF]. Volatiles also affect seismic wave velocities and electrical resistivity and make magma chamber imaging difficult to interpret.

Volatiles transfer is believed to play a role in petrogenesis [START_REF] Collins | Critical role of water in the formation of continental crust[END_REF][START_REF] Weinberg | Water-fluxed melting of the continental crust: A review[END_REF]. The presence of H 2 O increases melt fractions. Fluxing of the crust by H 2 O exsolved from a magma is expected to promote crustal melting. The long-term influence of volatiles on the fate of magmatic systems is thus not restricted to eruptive processes. Fluxes and concentration of magmatic volatiles are also keys in the generation of ore deposits. Metals present in the magma are partitioned in the supercritical phase and transported by the volatiles towards upper levels above the magma chamber where they precipitate [START_REF] Richards | Magmatic to hydrothermal metal fluxes in convergent and collided margins[END_REF].

Analytical and thermodynamical models have been used to investigate the conjugate effect of crystallisation and volatiles exsolution on chamber pressure [START_REF] Blake | Volatile oversaturation during the evolution of silicic magma chambers as an eruption trigger[END_REF][START_REF] Boichu | A model for episodic degassing of an andesitic magma intrusion[END_REF][START_REF] Tait | Pressure gas content and eruption periodicity of a shallow, crystallising magma chamber[END_REF] and magma fragmentation [START_REF] Fowler | Phase equilibria trigger for explosive volcanic eruptions[END_REF] in order to evaluate the role of volatiles in triggering eruptions. Excess mass of volcanic SO 2 emission suggests that volatiles separate from the magma at depth [START_REF] Christopher | Crustal-scale degassing due to magma system destabilization and magma-gas decoupling at Soufrière Hills Volcano[END_REF][START_REF] Wallace | Volcanic SO2 emissions and the abundance and distribution of exsolved gas in magma bodies[END_REF]. Laboratory experiments and numerical simulations [START_REF] Oppenheimer | Gas migration regimes and outgassing in particle-rich suspensions[END_REF][START_REF] Parmigiani | The mechanics of shallow magma reservoir outgassing[END_REF][START_REF] Parmigiani | Bubble accumulation and its role in the evolution of magma reservoirs in the upper crust[END_REF] indicate that the rise of suspended bubbles in melt-rich magma is limited and that volatiles preferentially ascend in crystalrich material through fingering channels and capillary fracturing. For this paper, we built on these former studies, and modeled H 2 O exsolution and transport within a growing and solidifying magma body. We tested the influence of magma chamber growth rate and magma composition, including initial water content, on exsolved volatile spatial and temporal distributions.

At the pressures and temperatures inside the magma body, the exsolved volatiles are supercritical fluids; for simplicity, we still use the terms bubbles. We will also use the term water independently of its state.

2 Numerical models

Magma chamber growth

Several conceptual models of magma bodies emplacement and growth have been proposed. They include diapirism (e.g. [START_REF] He | Magmatic diapirism of the Fangshan pluton, southwest of Beijing, China[END_REF][START_REF] Miller | In defense of magmatic diapirs[END_REF], ballooning (e.g. [START_REF] Molyneux | Evidence for significant granite space creation by the ballooning mechanism: The example of the Ardara pluton, Ireland[END_REF], cauldron subsidence and stopping (e.g. [START_REF] Burchardt | The Slaufrudalur pluton, southeast Iceland-An example of shallow magma emplacement by coupled cauldron subsidence and magmatic stoping[END_REF][START_REF] Myers | Cauldron Subsidence and Fluidization: Mechanisms of Intrusion of the Coastal Batholith of Peru into Its Own Volcanic Ejecta[END_REF]Yoshinobu et al., 2003), and repeated additions of sills or dykes (e.g. [START_REF] Allibon | The tectonically controlled emplacement of a vertically sheeted gabbro-pyroxenite intrusion: Feeder-zone of an ocean-island volcano (Fuerteventura, Canary Islands)[END_REF][START_REF] Michel | Incremental growth of the Patagonian Torres del Paine laccolith over 90 k.y[END_REF][START_REF] Miller | Growth of plutons by incremental emplacement of sheets in crystal-rich host: Evidence from Miocene intrusions of the Colorado River Region, Nevada, USA[END_REF]. Here, we model the growth of a magma chamber by under-accretion of sills. This growth geometry is probably most common in the brittle upper crust and is supported by field observations [START_REF] Grocott | Fault-assisted vertical pluton growth: Coastal Cordillera, north Chilean Andes[END_REF][START_REF] Michel | Incremental growth of the Patagonian Torres del Paine laccolith over 90 k.y[END_REF], geophysics [START_REF] Cruden | Growth of plutons by floor subsidence: Implications for rates of emplacement, intrusion spacing and melt-extraction mechanisms[END_REF], and laboratory experiments [START_REF] Kavanagh | An experimental investigation of sill formation and propagation in layered elastic media[END_REF][START_REF] Menand | Physical controls and depth of emplacement of igneous bodies: A review[END_REF]. The presence and size of a magma chamber (defined here as a body of eruptible magma larger than a single sill) depends on the magma body emplacement rate [START_REF] Annen | From plutons to magma chambers: Thermal constraints on the accumulation of eruptible silicic magma in the upper crust[END_REF]. This emplacement rate (or growth rate or growth velocity) is the thickness of sills divided by the time interval between sill emplacements (rest interval). It is equivalent to the flux, i.e. the added volume by unit of time divided by footprint area, that is commonly used in the literature [START_REF] Sliwinski | Rapid Magma Generation or Shared Magmatic Reservoir? Petrology and Geochronology of the Rat Creek and Nelson Mountain Tuffs, CO, USA[END_REF]. If emplaced in a cold crust and if the emplacement rate is low enough, the first sills completely solidify during the rest interval. With each sill intrusion, heat is transferred to the surroundings and the temperatures of the whole system increase. When temperatures are high enough, melt subsists in the sills during the rest interval. We call incubation time the moment when a magma that is liquid and eruptible (i.e. with a melt fraction >~0.5) starts to accumulate. Both the incubation time and the size reached by the magma chamber depend on the emplacement rate. We did not consider mechanical effects on the stability or growth of the chamber and did not allow the country rock to partially melt [START_REF] Karlstrom | Magma chamber stability in arc and continental crust[END_REF]. More details on the magma chamber growth model can be found in [START_REF] Annen | Conditions for the growth of a long-lived shallow crustal magma chamber below Mount Pelee volcano (Martinique, Lesser Antilles Arc)[END_REF] and [START_REF] Annen | From plutons to magma chambers: Thermal constraints on the accumulation of eruptible silicic magma in the upper crust[END_REF].

For the present paper, the exsolution and transfer of water was implemented in the simulation of magma chamber growth. H 2 O fractions, melt fractions, and temperatures are calculated both during the growth and the solidification of the magma body. We modelled the magma body growth by addition of 100-m thick sills until the total thickness of intruded material reaches 2000 m (Fig. 1), after which new sill injections no longer occur. The first sill is emplaced at 5.5 km depth and the last at 7.5 km depth. After the emplacement of the last sill, the simulation continues to run until no melt remains in the system. The diameter of the sills is 2000 m. The initial temperature of the sills at the time of emplacement is their liquidus, which depends on their composition (see section 2.3). We tested sills emplacement rate of 1, 5, 10, and 100 cm/yr. We also simulated the cooling of a 2000 m thick, 2000 m wide magma body that was instantaneously emplaced. The final geometry and volume of the intrusive body is similar for all simulations.

The simulations are run in 2D on a square grid with cylindrical coordinates. The numerical domain is 10 km x 2.5 km (Fig. 1). The grid cells dimension is 25m x 25 m. The initial temperatures of the numerical domain follow a linear geothermal gradient of 20 o C/km. The boundary conditions are a constant temperature T = 0 o C at the top of numerical domain that corresponds to the Earth surface, and no heat flow at the right and left boundaries. The horizontal dimension of the domain (2.5 km)

was chosen so that the thermal anomaly induced by the sills does not reach the right boundary on the timescale of the simulation. The heat flux through the bottom boundary is calculated by first calculating the depth Z l where the temperature reaches the liquidus temperature T l of the emplaced magma (which varies as a function of the magma composition) according to the initial geothermal gradient. This depth is outside the numerical domain and is assumed to be the source of the magma.

The bottom heat flux Q B is then calculated as follows:

(1)

Where T B is the temperature at the bottom boundary, Z B is the depth of the bottom boundary (10km), and  is the thermal conductivity.

Sills have a cylindrical geometry and pile up to build a cylindrical composite intrusion, the final shape of which is axisymmetric and isometric (Fig. 1). The volume of each new sill is accommodated by moving downwards the cells below the sill. It is an approximation of magma accommodation by chamber floor subsidence [START_REF] Cruden | Growth of plutons by floor subsidence: Implications for rates of emplacement, intrusion spacing and melt-extraction mechanisms[END_REF].

The finite difference formulation of the heat equation is used to calculate temperature:

(2)

(3)

With  the density, c the specific heat capacity, T the temperature, X m the melt mass fraction, and L the latent heat of crystallisation. The list of symbols is in Table 1. Details of the discretization by finite difference of the equations can be found in [START_REF] Annen | Conditions for the growth of a long-lived shallow crustal magma chamber below Mount Pelee volcano (Martinique, Lesser Antilles Arc)[END_REF]. The thermal conductivity  depends on temperature and on the lithostatic pressure [START_REF] Chapman | Thermal state of the continental lower crust[END_REF]:

(4)
where  0 is the thermal conductivity at surface temperature and pressure, and z is depth.

The density and the specific heat capacity do not depend on temperature in our simulation in order to avoid issues with heat conservation at the system scale.

Volatile exsolution

In this paper, we specifically address the exsolution rate of water as the main volatile phase in magmas emplaced in the upper crust. We assume that exsolution due to decompression during ascent has occurred before magma emplacement. According to [START_REF] Duan | A general model for predicting the solubility behavior of H2O-CO2 fluids in silicate melts over a wide range of pressure, temperature and compositions[END_REF], in silicic melts, all CO 2 might be already lost before H 2 O starts to exsolve. The solubility of water, C w , is only weakly influenced by temperature and composition [START_REF] Duan | A general model for predicting the solubility behavior of H2O-CO2 fluids in silicate melts over a wide range of pressure, temperature and compositions[END_REF]. It is mainly a function of pressure P [START_REF] Newman | VOLATILECALC: a silicate melt-H2O-CO2 solution model written in Visual Basic for excel[END_REF][START_REF] Schöpa | Magma Emplacement Rates and Porphyry Copper Deposits: Thermal Modelling of the Yerington Batholith, Nevada, USA[END_REF][START_REF] Sparks | The dynamics of bubble formation and growth in magmas: A review and analysis[END_REF][START_REF] Tait | Pressure gas content and eruption periodicity of a shallow, crystallising magma chamber[END_REF]:

C w = sP n (5)
with s = 4.11 x 10 -6 Pa -n , and n = 0.5 [START_REF] Burnham | A method for determining the solubility of water in silicate melts[END_REF].

If a magma is undersaturated, dissolved H 2 O concentrates in the melt while the magma cools and crystallizes:

(6)

where X d is the weight fraction of H 2 O dissolved in the melt, X 0 is the initial weight fraction of water dissolved in magma, X c is the fraction of H 2 O that enters hydrous minerals, and X m is the weight fraction of melt. When X d exceeds C w , water is exsolved.

Melt fractions

We based our relationship between melt fractions, dissolved water content, and temperatures on the experimental data of [START_REF] Whitney | The Origin of Granite -the Role and Source of Water in the Evolution of Granitic Magmas[END_REF] that were obtained at a pressure of 2 kbar. At this pressure, the solubility of water C w is about 6 wt%. The relationship between melt fraction and temperature depends on the melt water content [START_REF] Whitney | The Origin of Granite -the Role and Source of Water in the Evolution of Granitic Magmas[END_REF]. If the water exsolved during crystallization is transported upwards, any reheating of the material that lost its water produces a melt that is undersaturated in water. The melt fraction would then be lower than it was originally at the same temperature [START_REF] Caricchi | Experimental petrology of monotonous intermediate magmas[END_REF]. This effect is for the first time integrated in a numerical simulation, which allows us to estimate the error induced by neglecting the effect of water loss when modelling incrementally growing magma bodies.

Figure 2 shows the relationships between melt fraction and temperature used in the numerical simulation. To build figure 2a, we used the experimental data of [START_REF] Whitney | The Origin of Granite -the Role and Source of Water in the Evolution of Granitic Magmas[END_REF] for Cape Ann granite (Massachussets). The liquidus temperatures for different water content were estimated from Whitney's figure 2 [START_REF] Whitney | The Origin of Granite -the Role and Source of Water in the Evolution of Granitic Magmas[END_REF]. For water-undersaturated compositions, the relationship between liquidus temperature T l and magma water content X 0 is close to linear, so that in the numerical simulation (Fig. 2):

(7)

The melt fraction-temperature curve for a saturated composition (6 wt% H 2 O) is obtained by linearly interpolating between experimental data points. For magmas that are undersaturated at liquidus, saturation is reached at melt fraction X ms = (X 0 -X c )/0.06. The temperature T sat corresponding to X ms is read on the saturation curve. The undersaturated curves join the saturated curve at this point, and for any lower temperatures the melt fraction is that of a saturated composition [START_REF] Caricchi | Experimental petrology of monotonous intermediate magmas[END_REF]. To fit Whitney's experiment data with H 2 O content of 1, 2, 3, and 5 wt%, we used linear curves that relies T sat with 800 o C and 800 o C with T l . The melt fraction at 800 o C depends on the water content (Fig. 2a). The numerical simulation is only valid for H 2 O content X 0 in excess of the amount of water trapped in crystal X c . The results are presented for X c = 0.4 wt%, which corresponds to the crystallization of about 10 wt% biotite. For all water contents between 0.4 wt% (=X c ) and 6 wt% (saturation), X m (T=800°C) is found by interpolation.

To test the effect of the magma composition, we simulated the injection of a water-saturated and a water under-saturated Cape Ann magma. We also ran simulations with the melt fractiontemperature relationship obtained with Rhyolite-MELTS for the Mt. Capanne monzogranite composition (Barboni et al., 2015 and fig. 2b). In this last composition, we neglect the effect of H 2 O loss on the melt fraction-temperature curve. We will show a posteriori using the Cape Ann granite curves that this effect is insignificant. In comparison with the Cape Ann granite curve, the Mt.

Capanne curve is less steep at low temperatures and the mush state, where crystals form a network and can channel water, covers a wider range of temperatures. The initial temperature of the magma at the time of emplacement is that of the liquidus and the magma is crystal-free.

Volatiles transport

When saturation is exceeded, bubbles form. Their fate depends on the crystal and volatile fractions.

In contrast to other models (e.g. [START_REF] Black | The eruptibility of magmas at Tharsis and Syrtis Major on Mars[END_REF], the bubbles in our model do not rise in the liquid magma, but in the mush only. For gas volume fraction < 0.5 and crystal volume fraction < 0.4, bubble rise velocity v is controlled by Stokes law. The presence of other bubbles and of suspended crystals hinder bubble ascent and decrease their velocity. At 2 kbar, bubble diameters are typically a few tens of m [START_REF] Lyakhovsky | Bubble growth in rhyolitic melts: experimental and numerical investigation[END_REF]. Figure 3 shows the rise velocity of 50-m-diameter bubbles suspended in a melt with a viscosity of 10 5 Pa s. The bubble velocity is on the order of mm per year.

Considering the very slow rise of the bubbles, we followed [START_REF] Parmigiani | The mechanics of shallow magma reservoir outgassing[END_REF] in assuming that the exsolved water is immobile and remains in the magma for crystal volume fractions < 0.4.

It has been shown experimentally [START_REF] Barth | Valve-like dynamics of gas flow through a packed crystal mush and cyclic strombolian explosions[END_REF][START_REF] Holtzman | Capillary Fracturing in Granular Media[END_REF][START_REF] Oppenheimer | Gas migration regimes and outgassing in particle-rich suspensions[END_REF][START_REF] Varas | Flow and fracture in water-saturated, unconstrained granular beds[END_REF] and numerically [START_REF] Parmigiani | The mechanics of shallow magma reservoir outgassing[END_REF] that mushes are favorable environments to form permeable pathways for gas migration. For crystal volume fractions in the range 0.4-0.7, crystals form a connected network with gas and melt confined in the interstitial space.

At gas volume fraction between a critical value  cr g and 0.5, gas viscous fingering occurs. Fingering channels allow the upwards migration of the gas through the interstitial melt within the crystal network.  cr g was determined through numerical simulations by [START_REF] Parmigiani | The mechanics of shallow magma reservoir outgassing[END_REF]:

(8)
Where  c is crystal volume fraction. The velocity of exsolved volatiles transported through channeling is on the order of a few mm to a few cm per seconds [START_REF] Parmigiani | The mechanics of shallow magma reservoir outgassing[END_REF]. So the time needed to cross any reasonable thickness of mush is much shorter than the numerical time step of the simulation, which is 5 yrs for a cell dimension of 25 m. At the scale of our simulation, we consider that degassing and transport through channels are instantaneous. At each time step, the water exsolved in an area of magma reservoir where the crystal volume fraction is between 0.4 and 0.7 and the volatile volume fraction is between  cr g and 0.5 is transported vertically upwards until it hits a domain where these conditions are not fulfilled.

Results

Distribution of exsolved H 2 O

In the simulation, the exsolved water is arrested at the interface between mush and melt-rich magma ( c <0.4, bubbles are suspended) and at the interface between mush and quasi-solid material ( c >0.7, bubbles are trapped by crystals). Figures 4 and5 show how the exsolved water volume fraction and the crystal weight fraction evolve during magma chamber formation and solidification.

For an igneous body growing by accretion of sills, the first sills completely solidify between two magma injections, and water can only be transported over the thickness of a single sill. Water concentrations underline the shape of these first sills (Fig. 4a-d, Fig. 5 a-d, and Fig. 6d). After an incubation period that depends on the sill emplacement rate, a magma chamber forms (Figs 4 e-h

and 5 e-h). Channeling and water accumulation happen at the chamber periphery where the magma cools down and turns into a mush. Magma chambers form later when the sill emplacement rate is low [START_REF] Annen | From plutons to magma chambers: Thermal constraints on the accumulation of eruptible silicic magma in the upper crust[END_REF]. In the limiting case where the emplacement rate is too low for a magma chamber to form at all over the duration of the simulation, the concentration of water is limited by the size of the sills (Fig. 6 g-i).

At injection rates high enough for a magma chamber to grow (>~1 cm/yr), the final water distribution is an integration of the syn-and post-emplacement evolutions. Post emplacement, as it cools down, the magma chamber is surrounded by a corona of mush propitious to gas channeling. As detailed below, the thickness of this mush envelope depends on magma composition (compare Fig. 4e-h and Fig. 5e-h). The concentration of water is maximum atop the mush column. The thicker is the mush, the more water is channeled upwards. The mush is thin at the chamber bottom because of a steep temperature gradient. Thus, the accumulation of water at the bottom of the chamber is limited, especially for the Cape Ann composition (Fig. 4). The thickest mush columns are at the outer periphery of the axisymmetric magma chamber. So exsolved water preferentially accumulates on the upper side of the chamber. The locus of maximum water concentration moves inwards as the magma chamber retracts by cooling, which leaves behind trails of high water concentration (Fig. 6a,d). As the chamber solidifies, water-rich pockets and layers get trapped in the solidified granite while the mush is water depleted.

Magma composition

Magma composition and volatile content affect the slopes of the melt fraction-temperature curve (fig 2). The Cape Ann granite water-saturated composition evolves along a steeper melt fractiontemperature slope than an undersaturated granite composition or the Mt. Capanne monzogranite composition. As a result, the envelope of mush around the quasi-eutectic magma chamber (Fig. 4) is narrower than that around the non-eutectic chambers (Fig. 5).

Figure 7 shows in more detail how the volume fraction of crystals, melt, and water -before any water transport -evolve with temperature. It also shows the critical gas fraction. Although granitic undersaturated compositions produce more mush than a saturated composition (Fig. 2), less volatile transport occurs through them because volatiles saturate and exsolve at lower temperature and higher crystallinity compared to saturated compositions (Fig. 7b and8). For the Cape Ann composition with 2 wt% initial water, no volatile transport and accumulation occur (Fig. 7c and8c);

The water exsolves at a melt fraction (0.27 wt%) that is below the critical value for channeling (Fig. 7c) and the bubbles remain trapped in the crystal-rich material. The Mt. Capanne monzogranite, water-saturated composition fosters the most transport and accumulation of volatiles (Fig. 7d). A channel-prone mush is present between solidus and about 770 °C and the gas proportion is above the critical fraction in this temperature range (Fig. 7d). At the end of the solidification stage (26,000 yrs on Fig. 5), a body of mush without liquid core is present for this composition. At this stage, most water concentration occurs within and atop the mush body (Fig. 5h).

Volatiles output rates

Figure 9 shows the cumulated mass of H 2 O exsolved that is assumed extractable because not trapped in the liquid magma. The water fluxes reflect the magma fluxes, i.e. they depend on the sill emplacement rate and radius (Fig. 9a). The radius is the most important parameter since it controls both the amount of magma and water available, and the cooling surface area. As expected, the exsolution rate is lowest for an undersaturated magma (Fig. 9b). The saturated, Mt. Capanne monzogranite produces exsolved volatiles at a slightly higher rate than the Cape Ann saturated granite (Fig. 9b).

Effect of water loss on melt fractions

We tested the effect of water loss on the evolution of melt fractions for the water-saturated Cape Ann granite. We ran simulations where the melting curve was readjusted to the new conditions in cells that previously lost water and the magma became under-saturated upon remelting, and simulations where water loss or gain has no effect on the melting curves. In the second case, only the saturation curve (blue upper curve in Fig. 2a) was used in the computation of melt fractions independently of the actual melt water content.

Water loss has no effect on melt fraction for an instantaneously emplaced body that is only cooling because there is no reheating and remelting of partially solidified magma. Since only the water that is in excess of saturation is exsolved, the melt stays saturated and remains on the saturation curve.

For an incrementally growing body, the water loss effect on melt fractions is not null but is very weak. Reheating and remelting of former sills by new ones is limited. Figure 10 shows the difference in melt wt% between the two simulations. In the case shown here, i. e. for an emplacement rate of one 100 m-thick sill every 1000 yrs, neglecting the effect of water results in a maximum melt fraction overestimation of 3 wt% at the outer boundary of the mush and an underestimation of less than 1 wt% melt at the bottom of the chamber. Overall, the difference does not exceed 4% of the total melt volume in the system.

Discussion

The simulations presented in this paper provide new insights on what might happen in a growing and solidifying magma chamber. A series of processes, however, have been neglected and can be the focus of further studies.

Volatiles channeling

When considering volatile transport through channeling, we neglected the capillary effect, which causes a small fraction of gas to remain within the channels. The amount of gas retained depends on grain sizes and interfacial tensions between crystals and melt, which cannot easily be constrained [START_REF] Edmonds | Exsolved volatiles in magma reservoirs[END_REF]) more precisely than stating that it is much smaller than  cr g (6-10 vol.%).

In our simple model, volatiles only move upwards and not sideways (e.g. [START_REF] Belien | Gas accumulation in particle-rich suspensions and implications for bubble populations in crystal-rich magma[END_REF], which constrains their level of accumulation to the top of the mush columns they are issued from. The accumulation of volatiles is stopped when their proportion reach 50 vol%, which results in volatile layering. In the real world, we expect further volatile transfer by capillary fracturing, which would result in volatile layers connection and further volatile transport toward the surface.

Since the volatiles accumulate atop the mush, the actual distribution of the volatiles depends on the geometry of the magma body. We adopted a simple geometry where the magma chamber regularly grows by under-accretion of sills of similar dimensions. We expect real intrusions to be more irregular and to cause different patterns of volatiles depletion and accumulation.

Magma convection

We neglected the possibility that the volatiles that accumulate at the base of the liquid magma chamber might be entrained in the chamber by magma convection. Magma convection is not included in our model and the cooling rates might be underestimated. Scaling of magma convection usually assumes that the magma is heated from below or cooled from above (Worster et al., 1990), whereas in our model and in nature, an intrusion of magma is cooled from both surfaces. In such case, scaling relationships linking thermal gradient, fluid viscosity, and the effects of thermal expansion to a characteristic convection velocity are generally limited to the boundary layer region (e.g. [START_REF] Huber | Homogenization processes in silicic magma chambers by stirring and mushification (latent heat buffering)[END_REF]. Estimating the rate of bubble redistribution across the convecting magma without solving for convection itself is thus a complex task. Our results suggest that neglecting volatile entrainment by magma convection is of little consequence because volatile accumulation at the base of the magma that would be subject to convective motions, although present, is limited compared to the accumulation at the sides (granite and monzogranite compositions) and atop the chamber (monzogranite composition).

Pressure of exsolved volatile

We do not take into account the effect of pressure change due to volatile exsolution on the phase relationship. Indeed, the increase in pressure due to water exsolution is expected to modify water solubility [START_REF] Blake | Volatile oversaturation during the evolution of silicic magma chambers as an eruption trigger[END_REF] until the system reaches a new pressure equilibrium [START_REF] Tait | Pressure gas content and eruption periodicity of a shallow, crystallising magma chamber[END_REF]. Taking these feedback effects into account significantly increases the complexity of the simulations and is beyond the scope of this paper.

Implications for natural systems

Our simulations produce a decoupling between volatiles and magma, a process that had been inferred from the imbalance between the excess mass of SO 2 released by volcanoes and the mass of erupted melt [START_REF] Christopher | Crustal-scale degassing due to magma system destabilization and magma-gas decoupling at Soufrière Hills Volcano[END_REF][START_REF] Wallace | Magmatic Vapor Source for Sulfur Dioxide Released During Volcanic Eruptions: Evidence from Mount Pinatubo[END_REF]. However, in contrast with most conceptual models (e.g. [START_REF] Richards | Magmatic to hydrothermal metal fluxes in convergent and collided margins[END_REF][START_REF] Wallace | Volcanic SO2 emissions and the abundance and distribution of exsolved gas in magma bodies[END_REF], in our simulations bubbles are channeled by the mush instead of ascending in the liquid magma. The consequence is that accumulation of water occurs at the top of the mush and in the solidified magma rather than atop the liquid-rich magma chamber.

Pegmatites, hydrothermal veins, and drusy cavities testify to the presence of water in granites [START_REF] Bartley | The Granite Aqueduct and Autometamorphism of Plutons[END_REF]. Exhumed granites have lost their water and it remains to be determined if this loss is continuous, i.e. if capillary fracturing and water transport in the solid rock are contemporaneous with exsolution and channeling through the mush. Indeed, any capillary fracturing and further water loss must be delayed for the water-rich layers generated in our numerical simulation to actually form in real intrusive bodies.

The possible presence of water-rich layers in magmatic bodies has a series of implications. Water and melt both reduce rock electrical resistivity, density, and seismic wave velocity. Thus, the presence of water in the solidified part of a magma body would affect the interpretation of tomography based on those physical properties and, if not taken into account, would lead to an overestimation of magma chamber dimensions.

High proportion of water results in the fragmentation and fluidization of the rock even at low melt fraction [START_REF] Fowler | Phase equilibria trigger for explosive volcanic eruptions[END_REF]. We speculate that it might play a role in the formation of schlieren structures [START_REF] Ardill | Schlieren-bound magmatic structures record crystal flow-sorting in dynamic upper-crustal magma-mush chambers[END_REF]. In case of chamber failure and eruption, the entrainment of those fragmented layers from the roof could explain the common presence of older crystals and the evidence for cold storage in volcanic product [START_REF] Cooper | Rapid remobilization of magmatic crystals kept in cold storage[END_REF][START_REF] Rubin | Rapid cooling and cold storage in a silicic magma reservoir recorded in individual crystals[END_REF].

It has been argued that the overpressure due to volatile exsolution is high enough to fracture the wall rocks and trigger eruptions [START_REF] Blake | Volatile oversaturation during the evolution of silicic magma chambers as an eruption trigger[END_REF][START_REF] Fowler | Phase equilibria trigger for explosive volcanic eruptions[END_REF][START_REF] Tait | Pressure gas content and eruption periodicity of a shallow, crystallising magma chamber[END_REF][START_REF] Tramontano | Internal triggering of volcanic eruptions: tracking overpressure regimes for giant magma bodies[END_REF]. The pressure in the chamber depends on the chamber geometry and dimensions, on the magma compressibility, and on the wallrock viscoelastic properties [START_REF] Jellinek | A model for the origin of large silicic magma chambers: precursors of caldera-forming eruptions[END_REF].

In our simulations, those parameters vary in time and there are gradients in phase proportion, hence in viscosity and compressibility. Computing pressures is beyond the scope of this paper. We note however that in a viscoelastic crust, the pressure is controlled by the rate of volume increase [START_REF] Jellinek | A model for the origin of large silicic magma chambers: precursors of caldera-forming eruptions[END_REF]. This rate is slow since volatile exsolution is controlled by heat diffusion. This aspect was not taken into account in former studies, thus it remains to be shown if bubble exsolution can trigger eruption in a viscoelastic crust.

The flux of water released from our modeled magma is of the order 10 7 -10 8 kg per year. Comparing the flux of water from our simulations with field data is not straightforward. Data on water release from volcanic areas are scarce. The water flux is controlled both by the quantity of magma and by its cooling rate, hence by the cooling area. With radius of 1 to 2 km, the modeled magma bodies are relatively small. With a radius of several tens of kilometers, the cooling surface area would be multiplied by 100 and the exsolution rate would also increase by orders of magnitude. The deformation recorded in the Phlegrean Fields (Italy) between 1982 and 1984 has been linked to the crystallization of 0.83 km 3 of magma and the release of ~6.2 × 10 10 kg of H 2 O [START_REF] Bodnar | Quantitative model for magma degassing and ground deformation (bradyseism) at Campi Flegrei, Italy: Implications for future eruptions[END_REF].

According to [START_REF] Bodnar | Quantitative model for magma degassing and ground deformation (bradyseism) at Campi Flegrei, Italy: Implications for future eruptions[END_REF], this water may have been trapped beneath an impermeable layer before being suddenly released when this layer was breached. More generally, ground deformation is often episodic and the volcanoes are said to be breathing. The source of the deformation is attributed to magma [START_REF] Battaglia | Magma Intrusion Beneath Long Valley Caldera Confirmed by Temporal Changes in Gravity[END_REF][START_REF] Parks | Evolution of Santorini Volcano dominated by episodic and rapid fluxes of melt from depth[END_REF] or to volatiles transport [START_REF] Bodnar | Quantitative model for magma degassing and ground deformation (bradyseism) at Campi Flegrei, Italy: Implications for future eruptions[END_REF][START_REF] Vargas | Breathing of the Nevado del Ruiz volcano reservoir, Colombia, inferred from repeated seismic tomography[END_REF]. If volatiles are involved, the episodic aspect of deformation suggests that water is able to accumulate before being released by fracturing events [START_REF] Christopher | Crustal-scale degassing due to magma system destabilization and magma-gas decoupling at Soufrière Hills Volcano[END_REF].

Conceptual models for the generation of porphyry copper deposits involve metal transport through volatiles-rich dykes originating from the top of a magma chamber (see Fig. 7 in [START_REF] Richards | Magmatic to hydrothermal metal fluxes in convergent and collided margins[END_REF]. If, as our simulations suggest, water accumulates preferentially at the chamber sides and if it plays a role in the initiation of dykes, we expect volatiles-rich dykes to be initiated from the sides of the chamber.

Interestingly, on a reconstruction of the Yerington batholith based on field exposure, the source of the dykes associated with mineralization originates mostly from the side of the granite body (see Fig. 1 in [START_REF] Schöpa | Magma Emplacement Rates and Porphyry Copper Deposits: Thermal Modelling of the Yerington Batholith, Nevada, USA[END_REF]. Whether this match of locations is fortuitous or not warrants further investigation.

Since water-undersaturated Cape Ann granite produces less melt than when it is saturated, [START_REF] Caricchi | Experimental petrology of monotonous intermediate magmas[END_REF] argue that the effect of water loss needs to be taken into account in modeling of incremental magma emplacement. Our results shed a different light on this issue as they show that, in case of incremental emplacement with injection of a magma of Cape Ann granite composition, the effect of water loss on melt production is not significant. This is because re-melting of previously emplaced sills is limited. Indeed, if the preceding sill is already cold, reheating by the new sill is not enough for the cold sill to reach solidus. If the preceding sill is still hot, i.e. above about 700°C, it has not lost its water by the time the new sill is emplaced and its melting curve remains identical. The situation might be different if the new magma is of different composition and much hotter than the resident magma.

Conclusions

We simulated volatile exsolution and transport within a growing and solidifying magma body.

Transport of water through the mush results in an efficient decoupling of water and magma. Our simulations suggest that the geometry and size of the mush that surrounds a magma chamber controls volatiles transfer and accumulation. If the water does not escape immediately through solid rock fracturing, water-rich pockets get trapped in the roof and walls of the intrusive body upon solidification. Magma compositions that are water-saturated but non-eutectic are most susceptible to transfer, concentrate, and eventually expel water compared to eutectic compositions. In contrast, if the magma is strongly under-saturated, water exsolves when the crystal content is already high and cannot move upwards and accumulate.

The accumulation and eventual sudden release of water could explain the episodic deformation of volcanoes. The possible presence of water-rich layers in the solidified roof and wall of a magma chamber could lead to an overestimation of the magma reservoir size if not taken into account when interpreting magma chamber tomography.

If bubbles play a role in the rupture of a magma reservoir and the initiation of dykes, our result that water accumulates atop the mush rather than atop the liquid magma might explain why many erupted magmas are crystal rich. [START_REF] Whitney | The Origin of Granite -the Role and Source of Water in the Evolution of Granitic Magmas[END_REF]. The relationship varies with water content. The symbols show the positions of experiment data points from [START_REF] Whitney | The Origin of Granite -the Role and Source of Water in the Evolution of Granitic Magmas[END_REF]. (b) Mt. Capanne monzogranite composition saturated with water (modified from [START_REF] Barboni | Evaluating the construction and evolution of upper crustal magma reservoirs with coupled U/Pb zircon geochronology and thermal modeling: A case study from the Mt. Capanne pluton (Elba, Italy)[END_REF].

Figure 3: Ascent velocity of gas bubbles as a function of crystal volume fraction. The velocity is calculated with equations B4 to B7 in [START_REF] Parmigiani | The mechanics of shallow magma reservoir outgassing[END_REF]. The bubbles radius is 50 m. yrs during 20,000 yrs. After 20,000 yrs, temperatures relax and the magma chamber contracts. The composition is Cape Ann granite [START_REF] Whitney | The Origin of Granite -the Role and Source of Water in the Evolution of Granitic Magmas[END_REF] initially saturated in H 2 O.

Figure 5: As figure 4 for water-saturated Mt. Capanne monzogranite [START_REF] Barboni | Evaluating the construction and evolution of upper crustal magma reservoirs with coupled U/Pb zircon geochronology and thermal modeling: A case study from the Mt. Capanne pluton (Elba, Italy)[END_REF].

Figure 6: Exsolved water in wt% at the end of simulations. The composition is Cape Ann granite [START_REF] Whitney | The Origin of Granite -the Role and Source of Water in the Evolution of Granitic Magmas[END_REF] 100-m sills were emplaced every 1000 yrs during 20,000 yrs. The snapshot is taken at time 22,000 yrs, which is 2000 yrs after the last sill emplacement. [START_REF] Whitney | The Origin of Granite -the Role and Source of Water in the Evolution of Granitic Magmas[END_REF]. The relationship varies with water content. The symbols show the positions of experiment data points from [START_REF] Whitney | The Origin of Granite -the Role and Source of Water in the Evolution of Granitic Magmas[END_REF]. (b) Mt. Capanne monzogranite composition saturated with water (modified from [START_REF] Barboni | Evaluating the construction and evolution of upper crustal magma reservoirs with coupled U/Pb zircon geochronology and thermal modeling: A case study from the Mt. Capanne pluton (Elba, Italy)[END_REF]. After 20,000 yrs, temperatures relax and the magma chamber contracts. The composition is Cape Ann granite [START_REF] Whitney | The Origin of Granite -the Role and Source of Water in the Evolution of Granitic Magmas[END_REF] initially saturated in H2O.

Figure captions Figure 1 :

 captions1 Figure captions

Figure 2 :

 2 Figure 2: Relationships between melt fractions (wt%) and temperatures. (a) Cape Ann granite

Figure 4 :

 4 Figure 4: Snapshots of magma chamber formation and solidification. Upper row: Exsolved water

  and was initially saturated with 6 wt% water. Half-domain cross sections are shown on the left; vertical profiles of water contents close to the igneous body central axis are shown in the middle; vertical profiles of water contents closer to the igneous body outer edge are shown on the right. (a) (b) (c): The magma chamber was instantaneously emplaced. (d) (e) (f): 100-m sills were emplaced every 1000 years. (g) (h) (i): 100-m sills were emplaced every 10,000 yrs.

Figure 7 :

 7 Figure 7: Proportions of crystals, melts, and exsolved volatiles before transport as a function of

Figure 8 :

 8 Figure 8: Snapshots of exsolved water volume fraction. In these simulations, 100-m sills were

Figure 9 :

 9 Figure 9: Cumulated mass of exsolved water available as a function of time. We assume that the

Figure 10 :

 10 Figure 10: Effect of water loss on melt fractions. The colors show the difference in melt mass

Figure 1 :

 1 Figure 1: Model setup. The numerical half domain is a grid 10 km x 2.5 km with cells 25 m x 25 m. The system is axisymmetric. The radius of the magma body is 1 km and its final thickness is 2 km.

Figure 2 :

 2 Figure 2: Relationships between melt fractions (wt%) and temperatures. (a) Cape Ann granite composition[START_REF] Whitney | The Origin of Granite -the Role and Source of Water in the Evolution of Granitic Magmas[END_REF]. The relationship varies with water content. The symbols show the positions of experiment data points from[START_REF] Whitney | The Origin of Granite -the Role and Source of Water in the Evolution of Granitic Magmas[END_REF]. (b) Mt. Capanne monzogranite composition saturated with water (modified from[START_REF] Barboni | Evaluating the construction and evolution of upper crustal magma reservoirs with coupled U/Pb zircon geochronology and thermal modeling: A case study from the Mt. Capanne pluton (Elba, Italy)[END_REF].

Figure 3 :

 3 Figure 3: Ascent velocity of gas bubbles as a function of crystal volume fraction. The velocity is

Figure 6 :

 6 Figure 6: Exsolved water in wt% at the end of simulations. The composition is Cape Ann granite (Whitney, 1988) and was initially saturated with 6 wt% water. Half-domain cross sections are shown on the left; vertical profiles of water contents close to the igneous body central axis are shown in the middle; vertical profiles of water contents closer to the igneous body outer edge are shown on the right. (a) (b) (c): The magma chamber was instantaneously emplaced. (d) (e) (f): 100-m sills were emplaced every 1000 years. (g) (h) (i): 100-m sills were emplaced every 10,000 yrs.

Figure 7 :

 7 Figure 7: Proportions of crystals, melts, and exsolved volatiles before transport as a function of temperature. (a) a saturated Cape Ann granite, (b) Cape Ann granite with 4 wt% H2O, (c) Cape Ann granite with 2 wt% H 2 O, (d) a saturated Mte Capanne monzogranite. The grey area shows the temperature range over which transport through channels happens, i.e. where the volatile curve is above the critical volatile fraction and the crystal fraction is between 0.4 and 0.7. No transport occurs for the granite with 2 wt% H 2 O (c), because the proportion of exsolved volatiles never exceeds the critical value.

Figure 8 :

 8 Figure 8: Snapshots of exsolved water volume fraction. In these simulations, 100-m sills were emplaced every 1,000 yrs during 20,000 yrs. All snapshots are at time 22,000 yrs, i.e 2000 yrs after the last sill emplacement. Results are shown for a Cape Ann granite composition that initially contains (a) 6 wt% H2O, (b) 4 wt % H2O, (c) 2 wt% H2O, and (d) for the Mt. Capanne monzogranite that initially contains 6 wt% H 2 O. The red lines on the snapshots show the 0.4 and 0.7 crystal volume fraction isolines and delimit the areas where water in transported through channels. These areas are very thin in (a) and (b). In (c), there is no magma with crystal fraction below 40 vol%. (e) and (f) are enlargement of (a) and (d), respectively.

Figure 9 :

 9 Figure 9: Cumulated mass of exsolved water available as a function of time. We assume that the volatiles in suspension as bubbles in the liquid magma are unavailable and are excluded from the calculation. (a) Exsolved water for different emplacement rates and sills radii corresponding to magma fluxes of 0.3 km3/yr for 1 km radius and 0.1 m/yr emplacement, 0.03 km3/yr for 1 km radius and 0.01 m/yr emplacement, and 1.26 km 3 /yr for 2 km radius and 0.1 m/yr emplacement. The composition is Cape Ann water-saturated granite. (b) Exsolved water for different compositions of magma emplaced at the same rate of 0.01 m/yr and same sill radius of 1 km. The total and final mass of water exsolved is shown by the horizontal dashed lines.

Figure 10 :

 10 Figure10: Effect of water loss on melt fractions. The colors show the difference in melt mass fractions in wt% between a model that takes into account water loss and a model that neglects the effect of water loss on the melting curves. The composition is water-saturated Cape Ann granite. 100-m sills were emplaced every 1000 yrs during 20,000 yrs. The snapshot is taken at time 22,000 yrs, which is 2000 yrs after the last sill emplacement.

  Worster, M.G., Huppert, H.E.,Sparks, R.S.J., 1990. Convection and Crystallization in Magma Cooled from Above. Earth and Planetary Science Letters 101, 78-89. Yoshinobu, A.S., Fowler, J., Paterson, S.R.,Llambias, E., Tickyj, H., Sato, A.M., 2003. A view from the roof: magmatic stoping in the shallow crust, Chita pluton, Argentina. Journal of Structural

	Geology 25, 1037-1048.

Table 1 :

 1 list of symbols and units. Their value or the equation used to calculate them is given when relevant.

	c			specific heat capacity	1100	J K -1 kg -1
	t			time		s
	z			depth		m
	C w		H 2 O solubility	6	wt%
	H			Heat content	eq. 3	J m -3
	L			latent heat	350,000	J kg -1
	P			pressure		kbar
	Q B		bottom boundary heat flux	eq. 1	W m -2
	T			temperature		˚C
	T B			numerical domain bottom temperature		˚C
	T l			liquidus temperature	eq. 7	°C
	T sat		saturation temperature		°C
	X 0			initially dissolved H 2 O mass fraction
	X c			H 2 O mass fraction in minerals	0.4	wt%
	X d			dissolved H 2 O mass fraction
	X m		melt mass fraction
	X ms		melt mass fraction at saturation
	Z B			numerical domain bottom boundary depth	10	km
	Z l			depth of liquidus		km	
	 g			gas volume fraction
	 cr	g		critical gas volume fraction
	 c			crystal volume fraction
				thermal conductivity	eq. 4	W m -1 K -1
	  		thermal conductivity at Earth's surface 2.25	W m -1 K -1
				density	2600	kg m -3

Figure 5: As figure 4 for water-saturated Mt. Capanne monzogranite[START_REF] Barboni | Evaluating the construction and evolution of upper crustal magma reservoirs with coupled U/Pb zircon geochronology and thermal modeling: A case study from the Mt. Capanne pluton (Elba, Italy)[END_REF].

Acknowledgements

CA was supported by the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 794594 and by a grant from the Institut Rhônalpin des Systèmes Complexes. We thank three anonymous reviewers for helping us to improve and clarify the manuscript.