Anthony Trioux

Franc ¸ois-Xavier Coudoux

Patrick Corlay

Mohamed Gharbi

A REDUCED COMPLEXITY/SIDE INFORMATION PREPROCESSING METHOD FOR HIGH QUALITY SOFTCAST-BASED VIDEO DELIVERY

Keywords: Wireless Video Transmission, SoftCast, Uncoded Transmission, Preprocessing

SoftCast has been recently proposed as an original wireless video transmission scheme to deal with the issues encountered by conventional wireless video broadcasting systems (e.g. cliff effect). Lately, a preprocessing method for Soft-Cast that takes into account the asymmetric energy distribution between coefficients has been introduced to significantly improve the received video quality at the price of an increased amount of side information to be transmitted. In this paper, we propose an alternative method based on a zigzag scan that greatly reduces the amount of additional side information (metadata) to be transmitted (75%) and divides the computation time by 2.5 while keeping similar received video quality improvements (up to 5.4dB in terms of PSNR score). Preprocessing performances are compared under different channel signal-to-noise ratios using two metrics: Peak Signal-to-Noise Ratio (PSNR) and Structural SIMilarity (SSIM), showing the effectiveness of the proposed version.

INTRODUCTION

Broadcast video content constitutes a challenge because each user is subject to unreliable and different wireless channels that vary over time. Traditional approaches based on video codec such as H.264/AVC [START_REF] Richardson | The H.264 advanced video compression standard[END_REF] or HEVC [START_REF] Sullivan | Overview of the High Efficiency Video Coding (HEVC) Standard[END_REF] are not suitable for broadcasting video content to multi-users since they require a permanent adaptation of the source and channel coding parameters by the transmitter. Indeed, they are adjusted to match a bitrate available that is given under predicted or assumed channel state. Due to the heterogeneity of each user's channel, receivers whose channel conditions are degraded are subject to significant visual disturbances (e.g. freeze) while receivers experiencing a better channel than the estimated one cannot take full advantage of it.

A radical approach known as SoftCast [START_REF] Jakubczak | SoftCast: Clean-slate scalable wireless video[END_REF] has been proposed to tackle these problems. SoftCast represents the pioneer work of linear video coding systems where pixels are processed by linear operations and directly transmitted without quantization or channel coding. This allows the users to receive a video quality that varies linearly with channel quality without any feedback information, while avoiding the complex adaptation mechanisms of conventional schemes.

Following the original works [START_REF] Jakubczak | SoftCast: Clean-slate scalable wireless video[END_REF], linear video coding has gathered a significant interest from the research community [START_REF] He | Adaptive GoP dividing video coding for wireless broadcast based on power allocation optimization[END_REF][START_REF] Xiong | Analysis of decorrelation transform gain for uncoded wireless image and video communication[END_REF][START_REF] Fujihashi | High-Quality Soft Video Delivery With GMRF-Based Overhead Reduction[END_REF][START_REF] He | Structurepreserving hybrid digital-analog video delivery in wireless networks[END_REF]. The authors in [START_REF] He | Structurepreserving hybrid digital-analog video delivery in wireless networks[END_REF] proposed improvements for the scheme based on the characteristics of the Human Visual System (HVS) whereas [START_REF] Fujihashi | High-Quality Soft Video Delivery With GMRF-Based Overhead Reduction[END_REF] proposed efficient signal energy modelings to better allocate bandwidth resources and therefore improve the received video quality. Recent works [START_REF] Xiong | Analysis of decorrelation transform gain for uncoded wireless image and video communication[END_REF] showed that reducing the energy of the transmitted signal help to increase the received quality. Consequently, He et al. [START_REF] He | Adaptive GoP dividing video coding for wireless broadcast based on power allocation optimization[END_REF] proposed a preprocessing method known as Optimized Power Allocation for SoftCast scheme (OPA-SoftCast). This preprocessing method consists of selecting high-energy components and sending them as side information to reduce the energy of the analog-transmitted signal. This results in an improved power allocation that helps to increase the received video quality. However, the selection process is based on a exhaustive search over the GoP that increase the computation time and the amount of transmitted side information to recover the signal at the decoder.

In this paper, we propose an alternative preprocessing method that reduce the required bandwidth for side information transmission and the computation time. Instead of using exhaustive search, a direct zigzag scan is used while keeping similar received video quality improvements.

The rest of this paper is organized as follows: Section 2 gives an overview of the SoftCast scheme. In Section 3, we analyze the OPA-SoftCast method proposed by He et al. [START_REF] He | Adaptive GoP dividing video coding for wireless broadcast based on power allocation optimization[END_REF] and introduce our alternative method. The proposed method is compared to classical SoftCast and OPA-SoftCast schemes in Section 4. Conclusions are presented in Section 5.

SOFTCAST OVERVIEW

The block diagram of SoftCast [START_REF] Jakubczak | SoftCast: Clean-slate scalable wireless video[END_REF] is given in Fig. 1 where colored blocks represent the additional parts proposed in [START_REF] He | Adaptive GoP dividing video coding for wireless broadcast based on power allocation optimization[END_REF]. The SoftCast transmitter consists of several linear transforms and can be divided into 3 major parts: the data compression part made after 3D-DCT (Three-Dimensional Discrete Cosine Transform) by discarding micro-blocks of frequency coefficients called chunks; the error resilience block that consists of a power allocation between remaining chunks; and the packet loss resilience block performed by a Hadamard Transform. The 3D-DCT aims at compacting the information and exploits the separability of the DCT transform, i.e., the transmitter first transforms each frame with a spatial full-frame 2D-DCT and then performs a temporal 1D-DCT over each Group of Pictures (GoP). After 3D-DCT, the transformed frames are divided into small rectangular blocks called chunks and rearranged to form a new matrix X where each row contains a chunk. These chunks are sorted in decreasing energy order.

After the power allocation and Hadamard transform, the obtained coefficients from each chunk are directly mapped in pairs (I and Q planes in the Orthogonal Frequency Division Multiplexing technology) and transmitted without any coding step in a pseudo-analog manner referred as Raw-OFDM [START_REF] Xiong | Analysis of decorrelation transform gain for uncoded wireless image and video communication[END_REF].

In the Raw-OFDM, the Forward Error Correction (FEC) is bypassed and Pseudo-Analog Modulation replaces the classical modulation part of OFDM. Therefore, instead of bitrate, only symbol rate is considered hereafter. Since coefficients are sent in pairs (I and Q planes), the maximal resulting matching channel bandwidth (BW max ) in SoftCast video transmission can be described as follows

BW max = N r • N c • F r 2 , (1)
where N r , N c corresponds to the resolution of the video and F r is the frame rate expressed in frame per second (fps). For instance, a HD720p video format with 30fps represents a data volume of 1280 • 720 • 30 = 27.64 • 10 6 real values per second to be transmitted [START_REF] He | Structurepreserving hybrid digital-analog video delivery in wireless networks[END_REF]. The resulting matching channel symbol rate considering SoftCast transmission is 27.64 • 10 6 /2 = 13.82Msymbols/s. If the available bandwidth at the transmitter is less than BW max (i.e., bandwidth-constrained environments), Soft-Cast discards chunks, starting with lowest energy, until the source bandwidth matches the available bandwidth. The discarded chunks are replaced by null values at the receiver.

The SoftCast receiver consists of a Linear Least Square Error (LLSE) decoder that is used to get the best estimation of the received values. These values are then reassembled to form frames and passed through inverse processes. To retrieve video data signal at the receiver side, SoftCast needs also to send side information known as metadata (map of discarded chunks, mean and variance of each chunk to compute power allocation). They need to be strongly protected (Forward Error Correction code) and sent in a robust way (BPSK for example [START_REF] Fujihashi | High-Quality Soft Video Delivery With GMRF-Based Overhead Reduction[END_REF]) to ensure an error-free decoding process even in unreliable environments. Due to strong protection code and low channel coding used, it is of paramount importance to keep the amount of metadata small, since it causes large overhead, resulting in video quality degradation due to power and rate loss [START_REF] Fujihashi | High-Quality Soft Video Delivery With GMRF-Based Overhead Reduction[END_REF]. Jakubczak et al. [START_REF] Jakubczak | SoftCast: Clean-slate scalable wireless video[END_REF] proposed to use a reasonable size of chunk (64 chunks per frame) as a trade-off between quality received, amount of metadata and computation cost.

The end-to-end performances of SoftCast have been modeled by Xiong et al. [START_REF] Xiong | Analysis of decorrelation transform gain for uncoded wireless image and video communication[END_REF]. They showed that the theoretical received quality is directly related to the concept of activity denoted by

H = 1 N N i=1 √ λ i where λ i = E[X i 2 ]
is the energy of the i th chunk [START_REF] Jakubczak | SoftCast: Clean-slate scalable wireless video[END_REF] and N the total number of chunks per GoP. They demonstrated that this term directly affects the reconstructed PSNR at the receiver side as follows:

P SN R dB = c + CSN R dB -20 log 10 H (2)
with c = 20 log 10 (255) and CSN R dB corresponds to the Channel Signal-to-Noise Ratio i.e., the channel quality.

Please note that this model considers a Zero-Forcing (ZF) decoder instead of the LLSE one. This induces a small bias of around 1-2dB between the actual received PSNR and the theoretical one at very low CSNR.

Under the same channel characteristics, the higher the activity, the lower the received quality. Therefore, this formula underlines the importance of having a reduced activity and hence emphasizes the benefits of reducing the energy before applying the power allocation in the SoftCast scheme. For this purposes, He et al. [START_REF] He | Adaptive GoP dividing video coding for wireless broadcast based on power allocation optimization[END_REF] proposed to add a deviation optimization block at the transmitter side (colored block in Fig. 1) that selects and removes high energy DCT coefficients. These coefficients are transmitted as additional metadata to ensure a full recovery at the receiver side. In the next section, we analyze their method denoted as OPA-SoftCast and propose an alternative solution that helps to reduce the amount of additional metadata as well as the computation time while keeping the quality improvements given by OPA-SoftCast.

ANALYSIS OF OPA-SOFTCAST

OPA-SoftCast operates after 3D-DCT, chunks division and selection process. The input data of the algorithm is the matrix X where each row represents a zero-mean chunk [START_REF] Jakubczak | SoftCast: Clean-slate scalable wireless video[END_REF]. OPA-SoftCast is an iterative algorithm that aims at reducing the energy of the chunks. This is done by removing and transmitting trough metadata channel N d high energy DCT coefficients, called Special Frequency Coefficients (SFC). To find one SFC, the energy of each chunk is first computed and the highest value among all chunks selected. The SFC is then obtained by taking the highest energy coefficient within the selected chunk. This SFC is then removed and transmitted through metadata channel. The mean of the selected chunk is adjusted to keep a zero-mean value. After the end of the loop, the new energy of the selected chunk is recomputed. The loop is performed N d times corresponding to N d selected SFC. Due to the unknown positions of the highest energy chunk and coefficients, the values of the SFC and adjusted means, OPA-SoftCast needs to transmit these four values for each SFC to be able to reconstruct the DCT frames at the receiver side.

The authors assumed that each of these four values is quantized with 20 bits on average. With a BPSK modulation and a redundancy FEC code of 1/2, the total amount of bits that needs to be transmitted as additional metadata is 4 [START_REF] He | Adaptive GoP dividing video coding for wireless broadcast based on power allocation optimization[END_REF]. Thus, the necessary bandwidth for the transmission of this additional metadata is

• 20 • 2 • N d = 160 • N d for each GoP
4•20•2•N d •Fr
GoP where F r and GoP denote the number of frames per second and the GoP-size, respectively. This results in a non-negligible increase of the required bandwidth for metadata transmission. It is not desirable to have such amount of information to be transmitted since allowing more bandwidth to the metadata leads to a huge amount of chunks that cannot be transmitted in analog way [START_REF] Fujihashi | High-Quality Soft Video Delivery With GMRF-Based Overhead Reduction[END_REF]. To alleviate this drawback, the authors proposed a trade-off between quality improvements and additional amount of metadata. They selected an equivalent threshold of 2 SFC per frame (16 SFC for a GoP-size of 8 frames). We show by analyzing the algorithm that the amount of metadata can be further reduced while achieving similar performances.

SI = max time {std space [Sobel(F k (i, j))]}, (3)
TI = max time {std space [F k (i, j) -F k-1 (i, j)]}, (4)
where F k (i, j) represents the k th frame, (i, j) the corresponding spatial coordinates and Sobel() the Sobel filtering operation, respectively. However, as mentioned in [START_REF] Brandao | Quality assessment of H.264/AVC encoded video[END_REF] due to the current definition of these indexes that selects the highest value along the time axis, performing the TI computation for a video with slow motions that contains cut results in a high TI value. In order to have more representative (SI, TI) values, we choose to average the results over the entire sequence. Fig. 2 shows the resulting average (SI, TI) values for each sequence.

As shown in Fig. 3 and Fig. 4, the selected SFC (small white squares) are logically located in the first chunk (big black square) containing the low frequencies. For most of the video sequences, these SFC are located on the top left corner of the first frame of the GoP. Based on extensive simulations on each video sequence in Fig. 2, we see that, for almost all sequences, the position of the selected chunk does not change if less than 80 SFC per GoP are removed. Indeed, after 3D-DCT, the energy is mostly located on low frequency coefficients (i.e., the top left corner of the first frame). Furthermore, He et al. [START_REF] He | Adaptive GoP dividing video coding for wireless broadcast based on power allocation optimization[END_REF] showed that the PSNR gain slows down after one SFC per frame. Besides, transmitting more than 80 SFC is not a realistic case due to large increase of metadata, as a consequence, there is no need to transmit the position of the selected chunk as it is usually the first. We note some exceptions for very high TI values sequences, e.g., ParkJoy, where a few SFC have been selected in the upper left chunk of the second frame of the GoP. However, we show in Section 4 that performances remain similar with the proposed method.

Moreover, the placement of most of the selected SFC by OPA-SoftCast in Fig. 3 and Fig. 4 can be approximated by a simple zigzag scan (starting from the DC value in the top left corner). Therefore, we propose to use an alternative solution with low computation cost that does not require an iterative process to remove frequency coefficients. Indeed, we select the first chunk representing the 3D low-frequency values and use a zigzag scan to select N d frequency coefficients as OPA-SoftCast. However, since the proposed method is not an iterative process the adjusted zero-mean does not need to be transmitted since it is computed once and sent as classical metadata in the SoftCast scheme. Furthermore, since the positions of the selected chunk and frequency coefficients are always the same in our approach, the proposed method decreases the additional metadata from 4 values per frequency coefficients to only 1 saving 75% of the additional needed bandwidth compared to OPA-SoftCast (reduction from 9600Hz to 2400Hz when N d =16). Even if the placement of the SFC does not always follow a zigzag rule (e.g. Shields) and/or are not located in the first chunk, we show in the next section that the performance of the proposed method remains competitive.

PERFORMANCE ANALYSIS OF THE PREPROCESSING METHODS

The proposed method (OPA2) is evaluated with the OPA-SoftCast algorithm (OPA) and the original SoftCast scheme (SC) through extensive simulations described below.

Transmissions through AWGN channels are here considered in the range of [0∼30dB] as in [START_REF] He | Adaptive GoP dividing video coding for wireless broadcast based on power allocation optimization[END_REF][START_REF] Xiong | Analysis of decorrelation transform gain for uncoded wireless image and video communication[END_REF][START_REF] Fujihashi | High-Quality Soft Video Delivery With GMRF-Based Overhead Reduction[END_REF][START_REF] Cagnazzo | Shannon-Kotelnikov mappings for softcast-based joint source-channel video coding[END_REF].

Four available channel bandwidth cases are evaluated: full, three-quarter, half and quarter bandwidth. As explained in Section 2, the coefficients are directly mapped in pairs on I and Q planes in OFDM, therefore, the four resulting matching channel bandwidths correspond to 13.82, 10.36, 6.91 and 3.45MHz. We choose to show results for Full and quarter bandwidth cases, which correspond to 100% i.e., no compression applied (denoted by CR=1) and 25% of transmitted coefficients (CR=0.25), respectively. Results for others cases are similar since the first chunk is always sent regardless of the available bandwidth at transmitter side.

The simulations are performed on the selected HD720p sequences (1280 × 720 pixels, 30fps) and CIF sequences (352 × 288 pixels, 30fps) as shown in Fig. 2. The Mixed CIF and Mixed HD video sequences described in Section 3 are also used in this paper to evaluate the proposed method.

The PSNR is used as a purely objective metric to assess performances in [START_REF] He | Adaptive GoP dividing video coding for wireless broadcast based on power allocation optimization[END_REF]. In this paper, we also add the SSIM index, which provides a quality index more correlated with the Human Visual System (HVS) [START_REF] He | Structurepreserving hybrid digital-analog video delivery in wireless networks[END_REF]. As a first performance indicator, we give the resulting Data Activity H expressed in dB from eq. ( 2) for the sequences that do not comply with the key assumption of this paper (i.e., the fact that the frequency coefficients are selected in the first chunk and/or the fact that the selected coefficients do not follow a zigzag rule). Results given in Table 1 show that, even if these videos do not follow the above rules, the reduction of activity remains close to the optimal performances given by the OPA-SoftCast algorithm (recalling that a lower activity implies a better received quality). This is due to the fact that even if the selected frequencies with the proposed method are not the highest energy coefficients, they are still high energy DCT coefficients since they are located on the low-frequency band after 3D-DCT. As a consequence, the received quality remains almost the same as shown with simulations in Fig. 5. As explained in the Section 2, we note a difference between the theoretical PSNR given by eq. ( 2) and the displayed one in Fig. 5 due to the fact that eq. ( 2) considers a ZF decoder instead of a LLSE decoder as used in the SoftCast scheme. Even if there exists a small bias of about 1-2dB in terms of PSNR scores at very low CSNR (0dB typically), the Data Activity H represents a quick way to evaluate the preprocessing methods. The average PSNR loss between the proposed method and OPA-SoftCast for these sequences is about 0.2dB, which is insignificant.

An example of a visual comparison is given in Fig. 6, the CSNR is set to 0dB to accentuate the noise during transmission. The selected frame corresponds to a case where the zigzag rule is not verified, i.e., the Shields sequence as shown in Fig. 4. We can clearly observe that the original SoftCast scheme gives the lowest video quality received. By contrast, the preprocessing methods achieve better reconstructed quality under the same channel characteristics. The loss between the two methods is imperceptible (≤0.1dB) even if the coefficients selected by OPA-SoftCast do not follow a zigzag rule.

The proposed method increases the received quality between 1.3dB (ParkJoy sequence) and up to 5.4dB (Johnny sequence) as shown in Fig. 5. The difference between the ob- tained gains is due to the characteristics of the videos. Indeed, the Johnny sequence that has low (SI, TI) values is easy to decorrelate and most of the energy is therefore concentrated on low frequency coefficients after 3D-DCT. Protecting the most important low frequency coefficients allows to greatly improve the received quality. On the contrary, due to strong motions and textures in the ParkJoy sequence, the signal is difficult to decorrelate and the energy is spread across the entire GoP. As a result, the received video quality is not as good as the Johnny sequence. We verified that similar conclusions were obtained with the CIF sequences. In addition, the computation time and the required additional bandwidth to perform the preprocessing methods are also indicated in Table 2. The computation time is defined as the total time required to do the preprocessing at the transmitter and the receiver part over the full Mixed HD sequence composed of 1280 frames. The different times were obtained with Matlab R2018b on a computer equipped with an Intel processor (R) Core (TM) i7-4510U CPU, 2GHz, 12G RAM. As we can see, the proposed method divides the computation time by 2.5 compared to the OPA-SoftCast method. This is due to the fact that the proposed method is not an iterative process. Furthermore, since only the value of each frequency coefficient is sent as additional metadata, the required bandwidth for the proposed method is reduced by 75% compared to the one for OPA-SoftCast. We note that similar results were obtained for the Mixed CIF sequence.

Finally, we show in Fig. 7, the average quality scores for the Mixed HD and Mixed CIF sequences. Regardless of the total available bandwidth for transmission, preprocessing methods brings significant improvements to the received quality compared to the classical SoftCast scheme. The average gain is about 2.24dB for the PSNR score and 0.031 for the SSIM index, respectively. The loss of quality between the proposed method and OPA-SoftCast is only about 0.12dB for the PSNR score and 0.004 for the SSIM index in average, whereas the additional needed bandwidth for metadata is reduced by 75% and the computation time is divided by 2.5. The loss of quality is due to the fact the proposed method does not ensure to always select the highest frequency coefficients compared to OPA-SoftCast. However, as mentioned above, most of the energy is located on low frequency coefficients and as verified above, even when the selected frequencies are not well described by a zigzag scan (see Fig. 3 and Fig. 4), the PSNR loss remains under 0.5dB (see the zoom in Fig. 5) showing the effectiveness of the proposed version.

Fig. 1 :

 1 Fig. 1: Block diagram of the SoftCast scheme.

Fig. 2 :

 2 Fig. 2: Illustration of the average spatio-temporal indexes for the selected HD and CIF video sequences.

Fig. 3 :

 3 Fig. 3: Visual representation of the position of the 16 selected SFC (small white squares) by OPA-SoftCast in the upper left chunk (44 × 36 coefficients) of the first frame of the GoP for the selected CIF video sequences.

Fig. 4 :

 4 Fig. 4: Visual representation of the position of the 16 selected SFC (small white squares) by OPA-SoftCast in the upper left chunk (160 × 90 coefficients) of the first frame of the GoP for the selected HD720p video sequences. Note that only the left upper part (40 × 22) of each chunk is showed for display purposes. Please enlarge the figure to observe details.

Fig. 5 :

 5 Fig. 5: PSNR dB per frame for the Mixed HD sequence, CSNR=0dB, CR=1 (no compression applied).

Fig. 6 :

 6 Fig. 6: Visual quality comparison at a CSNR = 0dB and full available bandwidth for Shields sequence (first frame). From left to right, top to down: (a) Original image, (b) Classic Soft-Cast, (c) OPA-SoftCast, (d) The proposed method.

Fig. 7 :

 7 Fig. 7: Average quality scores vs CSNR. First column: PSNR results. Second column: SSIM results.

Table 1 :

 1 Data activity H in dB for the SoftCast scheme, OPA-SoftCast method and the proposed one

	Sequences H (SC)	H (OPA) H (OPA2)	PSNR loss
	Soccer	23.88dB 20.11dB	20.27dB	0.16dB
	Into Tree	21.52dB 17.94dB	18.10dB	0.14dB
	ParkJoy	26.53dB 25.42dB	25.47dB	0.05dB
	Shields	23.58dB 21.99dB	22.18dB	0.19dB

Table 2 :

 2 Comparison between the OPA-SoftCast method and the proposed one

	Method	Average Gain	Comput. Time	Required BW
	OPA-SoftCast [4]	PSNR = 2.36dB SSIM = 0.032	12.43s	9600Hz
	Proposed method	PSNR = 2.24dB SSIM = 0.031	5.05s	2400Hz

CONCLUSION

In this paper, we propose an alternative preprocessing method based on a zigzag scan for the SoftCast scheme. Compared to the original scheme, this helps to improve the received quality in terms of PSNR scores up to 5.4dB and 2.24dB in average and in terms of SSIM index up to 0.105 and 0.031 in average. In comparison to the OPA-SoftCast method, the proposed one reduces by 75% the additional needed bandwidth for metadata and divides by 2.5 the computation time while keeping similar received video quality improvements. The saved bandwidth can be used to transmit more frequency coefficients in analog way and is useful for hardware-constrained or low available bandwidth applications.


Anthony Trioux

Mohamed Gharbi

François-Xavier Coudoux

Patrick Corlay

An Analytical Model of the End-to-End Performance for Linear Video Delivery Under Bandwidth Constraints

Keywords: Wireless Video Transmission, SoftCast, Distortion Model, Bandwidth constraints, Linear Video Delivery

Recently, Linear Video Coding and Transmission (LVCT) schemes have been proposed as an alternative to traditional video transmission schemes, the latter experiencing cliff-effect [3] in wireless error-prone environments. In this paper, we propose an analytical model to estimate/predict the end-to-end performance of Soft-Cast [2] video transmission, pioneer of the LVCT schemes. This model, based on the PSNR metric, accounts for channel conditions (CSNR) as well as data compression due to bandwidth constraints. We show that regardless of the available channel bandwidth, the end-to-end video quality can be accurately modeled and predicted according to the characteristics of the video and the channel conditions.

Introduction

The basic scheme of SoftCast [START_REF] Jakubczak | SoftCast: Cleanslate scalable wireless video[END_REF] is introduced in Fig. 1. SoftCast first takes a Group of Pictures (GoP) and uses a 3D full-frame DCT as a decorrelation transform. These DCT frames are then divided into N small rectangular blocks of transformed coefficients called chunks. In the SoftCast scheme, the data compression can be done after the decorrelation transform. Specifically, when the available channel bandwidth for the transmission is less than the signal bandwidth, SoftCast discards chunks with less energy. This is generally the case especially for the transmission of High Definition (HD) content as mentioned in [START_REF] Liang | Superimposed Modulation for Soft Video Delivery with Hidden Resources[END_REF][START_REF] Trioux | Temporal information based GoP adaptation for linear video delivery schemes[END_REF]. At the receiver side, these discarded chunks are replaced by null values [START_REF] Jakubczak | SoftCast: Cleanslate scalable wireless video[END_REF].

For ease of reading, the compression level denoted hereafter CR is usually used in SoftCast and in LVCT schemes. It is defined as follows [START_REF] Li | Compressed uncoded screen content video transmission in bandwidth-constrained wireless networks[END_REF]:

CR = K N (1)
where K represents the number of transmitted chunks per GoP and N the total number of chunks within a GoP. This ratio is between 0 (no information transmitted, i.e., K = 0) and 1 (no compression, i.e., K = N ).

The third block at the transmitter level called Power Allocation or Scaling is used to provide error resilience. SoftCast scales the magnitude of the DCT coefficients to offer a better protection against transmission noise. Since the total transmission power available P is limited and fixed, it is distributed to all the chunks in a way that minimizes the Mean Square reconstruction Error (MSE) between transmitted and decoded chunks. This is a typical Lagrangian problem which leads to the following solution [START_REF] Fujihashi | High-Quality Soft Video Delivery With GMRF-Based Overhead Reduction[END_REF][START_REF] Jakubczak | SoftCast: Cleanslate scalable wireless video[END_REF] given by:

g i = λ -1/4 i • P i √ λ i , (2)
where g i , i = 1, 2, . . . , N is the scaling factor for the i th chunk, and λ i the energy of the i th transmitted coefficient (after 3D-DCT) [START_REF] Jakubczak | SoftCast: Cleanslate scalable wireless video[END_REF]. The Hadamard transform is then applied to the scaled chunks to provide packet loss resilience. This process transforms the chunks into slices. Each slice is a linear combination of all scaled-chunks.

Finally, these packets are transmitted in a pseudoanalog manner using Raw-OFDM [START_REF] Jakubczak | SoftCast: Cleanslate scalable wireless video[END_REF], i.e., classical coding (e.g., Forward Error Correction code) and modulation stages are skipped.

In parallel, the SoftCast transmitter sends an amount of data referred as metadata. These data consist of the mean and the variance of each transmitted chunk as well as a bitmap, which indicates the positions of the discarded chunks into the GoP. Metadata are strongly protected and transmitted in a robust way (e.g., BPSK [START_REF] Jakubczak | SoftCast: Cleanslate scalable wireless video[END_REF]) to ensure correct delivery and decoding process.

At the receiver side, if an estimation of the channel noise is available, a Linear Least Square Error (LLSE) decoder can be used to get the best estimation of the received values. Otherwise, a Zero-Forcing (ZF) decoder is used. Using the metadata, the decoded values are then reassembled to form DCT-frames, which are then passed through an inverse 3D-DCT process.

In a recent paper [START_REF] Xiong | Analysis of decorrelation transform gain for uncoded wireless image and video communication[END_REF], Xiong et al. modeled the endto-end performance of SoftCast for any channel Signal-to-Noise Ratio (CSNR, expressed in decibels).

They showed that the total distortion that affects the reconstructed video quality without data compression can be obtained from:

D [ZF/FB] = N i=1 D i = σ 2 n P N i=1 λ i 2 , (3)
where σ 2 n is the noise power. Based on the following definition of the CSNR and PSNR expressed in decibels: where c = 20 log 10 (255) and

CSNR = 10log 10 ( P /σ 2 n ), P = P/N, (4)
H = 1 N N i=1 λ i , (7)
refers to the data activity of the video content [START_REF] Xiong | Analysis of decorrelation transform gain for uncoded wireless image and video communication[END_REF]. The higher the data activity H, the lower the reconstructed PSNR, showing the importance of taking into account the characteristics of the transmitted video content in a SoftCast context. Note the linear characteristic of the PSNR [ZF/FB] that depends on the channel transmission conditions. We note that Xiong's model relies on two assumptions: the first is that the available bandwidth of the application allows the transmission of the N elements of x (i.e., CR=1, no compression applied). However, in practice, this is generally not the case since bandwith ressources are limited. This is especially true when considering the transmission of high resolution (HD, 4K, etc.) video formats. The second hypothesis assumes that a ZF estimator used at the receiver side. This is not valid when considering the original SoftCast scheme proposed by [START_REF] Jakubczak | SoftCast: Cleanslate scalable wireless video[END_REF] which uses an LLSE decoder.

Proposed work

The first objective of this work is to consider the more realistic and general case i.e. only the K ≤ N largest energy chunks are transmitted due to bandwidth constraints.

The challenge consists in proposing a more realistic theoretical model than Xiong's model [START_REF] Xiong | Analysis of decorrelation transform gain for uncoded wireless image and video communication[END_REF], i.e., addressing the weaknesses of the initial model (inaccurate prediction for bandwidth constrained applications).

Since N -K chunks are discarded due to bandwidth constraints, the total distortion D [ZF/CB] now consists of two parts:

• The distortion D i that affects each of the K transmitted coefficients x i , given by:

D i = E (x i -x i ) 2 .
For ease of reading, let us denote the total distortion due to the transmitted coefficients

D s = K i=1 D i .
• The distortion D j due to each of the N -K discarded coefficient x j , given by:

D j = E (0 -x j ) 2 .
Likewise, we denote the total distortion due to the discarded coefficients D d = N j=K+1 D j . Therefore, the overall distortion (3) becomes:

D [ZF/CB] = D s + D d , (8)
= σ 2 n P K i=1 λ i 2 + N j=K+1 λ j .
We note that the average transmission power in (4) becomes P = P/K as the total transmission power is here distributed over the K transmitted coefficients and in ( 5)

D = D [ZF/CB] /N .
By inserting ( 8) into ( 5), we get:

PSNR [ZF/CB] = 10 log 10 255 2 • N D s + D d , (9)
= c -10 log 10 1 +

D d D s + 10 log 10 P σ 2 n -10 log 10   1 N K K i=1 λ i 2   .
By analogy with (6), we identify the new data activity of the transmitted coefficients as:

H t = 1 √ N K K i=1 λ i . ( 10
)
For ease of reading we also define E d , the overall energy of all dropped coefficients:

E d = 1 N N j=K+1 λ j . ( 11
)
According to these new definitions, the end-to-end video quality considering bandwidth constraints for the ZF estimator is finally given by:

PSNR [ZF/CB] = c + CSNR -20 log 10 H t (12) -10 log 10 1 + CSNR lin • E d H t 2 .
where CSNR lin = P

σ 2 n .
The above equation includes a new term in comparison to [START_REF] Trioux | Temporal information based GoP adaptation for linear video delivery schemes[END_REF] that reflects the effect of the data compression applied. The PSNR now depends on three parameters: first, the CSNR which depends on the transmission conditions, and then the two other terms E d and H t that are directly related to the video data characteristics. For a given bandwidth, the higher E d , the greater degradation. However, as E d is multiplied by the CSNR lin , the degradation becomes less noticeable at low CSNR environments.

When K = N , i.e., CR=1, (12) and ( 6) are identical. In other words, the video quality scales linearly with the CSNR as stated in [START_REF] Xiong | Analysis of decorrelation transform gain for uncoded wireless image and video communication[END_REF].

Results

To evaluate the effectiveness of the proposed model, we perform full end-to-end simulations. We create a Mixed sequence by slicing the first 128 frames of ten HD 720p sequences (Ducks, Four People, In to tree, Johnny, Kristen and Sara, Old town, Parkjoy, Parkrun, Shields and Stockholm). We use GoPs of 16 frames and divide each frame into 64 chunks as it represents the original and mostly used configuration [START_REF] Jakubczak | SoftCast: Cleanslate scalable wireless video[END_REF]. We verified similar results for other GoP-sizes and chunk-sizes. We then consider four different compression ratio CR = 1, 0.75, 0.5, 0.25.

As observed in Fig. 2, the proposed model (colored lines) perfectly matches the simulation results (colored dots), regardless of the considered CR or CSNR values. When CR=1 (red color), we logically obtain the same linear characteristic as in [START_REF] Xiong | Analysis of decorrelation transform gain for uncoded wireless image and video communication[END_REF].

However, Xiong's model represented with the red line [START_REF] Xiong | Analysis of decorrelation transform gain for uncoded wireless image and video communication[END_REF] is no longer valid when the available channel bandwidth decreases (cyan, green and blue dots) as the data compression is not considered. In practice, it is mandatory to consider such loss since it drastically degrades the received video quality and implies non-linear characteristics at high CSNR values. This is the well-known levelingoff effect [START_REF] Liang | Superimposed Modulation for Soft Video Delivery with Hidden Resources[END_REF] that appears and implies huge decibel losses (e.g. ∆PSNR up to 20dB for the considered case). Unlike Xiong's model, ours (colored lines) perfectly predicts and models this leveling-off effect regardless of the amount of discarded chunks.

Conclusion and perspectives

In this paper, we present a theoretical model that can be used in the context of linear video delivery under bandwidth constraints. In contrast to Xiong's model [START_REF] Xiong | Analysis of decorrelation transform gain for uncoded wireless image and video communication[END_REF], the proposed model takes into account the losses due to data compression/bandwidth constraints. Regardless of the available channel bandwidth, results show that the model accurately represents the full end-to-end performance by predicting the leveling-off [START_REF] Liang | Superimposed Modulation for Soft Video Delivery with Hidden Resources[END_REF] effect that appears when some chunks are discarded. This model can help for parameters optimization in an LVCT transmission context subject to bandwidth limitations as in [START_REF] Trioux | Temporal information based GoP adaptation for linear video delivery schemes[END_REF]. It can also be used to quickly evaluate schemes without requiring extensive end-to-end simulations. Further works concern the extension of the model to different versions of SoftCast that bring additional PSNR gain (e.g., by taking into account the LLSE estimator) or the extension of the model to other objective metrics such as SSIM [START_REF] Zhou Wang | Image quality assessment: from error visibility to structural similarity[END_REF]. Configuration: GoP-size=16 frames, 64 chunks/frame. The colors red, cyan, green and blue represent CR=1, 0.75, 0.5 and 0.25, respectively.

PSNR

  = 10log 10 (255 2 / D), D = D [ZF/FB] /N. (5) They showed that the expected reconstructed video quality can be finally obtained from: PSNR [ZF/FB] = c + CSNR -20 log 10 H ,

Figure 1 :

 1 Figure 1: Block diagram of the SoftCast scheme.

PSNRFigure 2 :

 2 Figure 2: Average PSNR results for the proposed theoretical model (solid lines) and SoftCast simulations with ZF estimator: (dots markers) for the Mixed HD720p sequence.Configuration: GoP-size=16 frames, 64 chunks/frame. The colors red, cyan, green and blue represent CR=1, 0.75, 0.5 and 0.25, respectively.


Bénédicte Cenki
email: benedicte.cenki@umontpellier.fr

Patrice F Rey

Diane Arcay

Julian Giordani

Timing of partial melting and granulite formation during the genesis of high to ultra-high temperature terranes: insight from numerical experiments

Significance statement

Even though long-lived high to ultra-high temperature (HT-UHT) terranes which formed mostly during the Proterozoic have been widely studied, their mode of formation is still controversial. We address the intertwined timing of partial melting and granulite formation during the genesis of HT-UHT terranes in a manner that cannot be derived from the natural rocks archive. This paper is innovative in the sense that it illustrates this problem in a dynamic manner. We illustrate through 2D thermo-mechanical modelling of an orogenic cycle that i) the formation of HT-UHT terranes can be explained by an elevated crustal radiogenic heat production and ii) the timing and duration of partial melting and the formation of granulites (syn-convergence and/or syn-collapse). The numerical modelling code used is the well-tested open-source code Underworld (https://www.underworldcode.org) making the results and outcomes reproductible and transparent for the scientific community.

Introduction

High to ultra-high temperature (HT > 650°C, UHT > 900°C) metamorphism plays a key role in the stabilization of continents during orogenesis [START_REF] Harley | A matter of time: The importance of the duration of UHT metamorphism[END_REF], the formation and preservation of stable crustal roots [START_REF] Cenki-Tok | Strain and retrogression partitioning explain longterm stability of crustal roots in stable continents[END_REF] and in the formation of supercontinents (Touret et al., 2016). There is a robust record of natural examples documenting the duration of UHT events and their corresponding geodynamic setting [START_REF] Kelsey | On ultrahigh temperature crustal metamorphism: Phase equilibria, trace element thermometry, bulk composition, heat sources, timescales and tectonic settings[END_REF][START_REF] Brown | Evolution of geodynamics since the Archean: Significant change at the dawn of the Phanerozoic[END_REF]. [START_REF] Harley | A matter of time: The importance of the duration of UHT metamorphism[END_REF] proposed that "slow" (long-lived) granulites predominantly form in large hot orogens (LHO) as defined by [START_REF] Beaumont | Models of large hot orogens containing a collage of reworked and accreted terranes[END_REF] whereas "fast" (short-lived) episodes of granulitefacies metamorphism may be restricted to back-arc settings. However, the respective timing, duration and volume of migmatites and granulites is difficult to extract from the rocks archive.

Here, the first part of this study explores the equilibrium thermal state of the continental lithosphere for a broad range of crustal thicknesses (20 to 80 km), and mantle heat flow (4 to 80 mW.m -2 ), radiogenic heat production and crustal differentiation estimated by the ratio between the heat production measured at the surface of the crust and the crustal mean (Perry et al., 2006).

Conditions for partial melting and granulite facies may be established for a wide range of crustal thicknesses. However, these conditions which assume thermal equilibrium are unlikely to be achieved as the resulting hot crust would collapse under gravitational deviatoric stresses. Hence, the second part of the paper focusses on 2D thermo-mechanical experiments to constrain the respective tempo of partial melting and granulites formation as well as predict the shape of Pressure-Temperature-time (PTt) paths of particles via modelling of whole orogenic cycles, from shortening to lithospheric mantle thinning and post-orogenic extension. Whilst the establishment of the conditions for partial melting culminates during convergence, granulite facies conditions generally occurs during orogenic collapse.

Migmatite and granulite potential at thermal equilibrium

Neglecting heat advection and heat production in the mantle, the solution of the steady-state heat equation allows for the prediction of the equilibrium geotherm, and therefore the potential for the continental crust to reach HT-UHT conditions. The thermal state of the continental lithosphere at equilibrium depends on i/ the mantle basal heat flow (BHF), ii/ the total amount of radiogenic heat production (RHP), iii/ its distribution throughout the crust (H(z)), iv/ the thermal diffusivity of rocks (, and v/ the surface temperature (T o ). For conditions ii to v fixed, the graphs in Figure 1 map the Moho temperature controlled by crustal and lithospheric thicknesses modulated by the

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This article is protected by copyright. All rights reserved BHF. For a global average 40 km -thick continental crust [START_REF] Christensen | Seismic velocity structure and composition of the continental crust: A global view[END_REF], a BHF of 20 mW/m -2 , and with an average total RHP of 1.0483 W.m -3 , close to present-day values (Taylor and McLennan, 1995), the Moho temperature ranges from 650 °C when the RHP distribution is uniform (Fig. 1A) to 575 °C when the crust is differentiated with a length scale factor hc of 20 km (i.e., H(z) = Ho x exp(-z x hc), where Ho and H(z) are the heat production at the surface and at depth z, respectively Fig. 1B). At this moderate level of RHP, continental crusts must achieve a thickness > 70 km to experience granulite facies conditions (purple region in Fig. 1). For a higher average total RHP of 2.0966 W.m -3 , appropriate for Paleoproteroic to Archean times, the temperature at the Moho reaches 1020ºC for uniform RHP distribution in the crust, and 810ºC when the RHP decreases exponentially with depth (from 5 W.m -3 at the surface) with hc = 20 km. The potential for a 40-km thick crust to reach migmatite and granulite conditions is significantly enhanced (Fig. 1 C , D;Vanderhaeghe et al., 2019). However, such high temperatures are unsustainable on long time scales as such a hot crust would collapse, spreading onto adjacent ocean basins [START_REF] Bailey | Gravity-driven continental overflow and Archaean tectonics[END_REF]. This confirms that the conditions to form significant volumes of HT-UHT rocks can be controlled by the total RHP and the degree of differentiation of the continental crust (McLaren et al., 1999;[START_REF] Gerya | Thermal regime and gravitational instability of multi-layered continental crust: implications for the buoyant exhumation of high-grade metamorphic rocks[END_REF][START_REF] Clark | How does the continental crust get really hot?[END_REF][START_REF] Beucher | UWGeodynamics: A teaching and research tool for numerical geodynamic modelling[END_REF]McLaren, 2002, Kelsey and[START_REF] Kelsey | On ultrahigh temperature crustal metamorphism: Phase equilibria, trace element thermometry, bulk composition, heat sources, timescales and tectonic settings[END_REF]. In what follows, 2D thermo-mechanical experiments of a complete orogenic cycle allow to explore the additional effect of the tectonic velocities and assess the tempo and duration of migmatitic and granulite conditions.

Numerical experiments, code, and model setup

The numerical models are performed with Underworld, a well-tested open-source finite element code, to solve the equations of conservation of momentum, mass and energy for an incompressible fluid on a Cartesian Eulerian mesh (Moresi et al., 2002(Moresi et al., , 2003(Moresi et al., , 2007;;[START_REF] Beucher | UWGeodynamics: A teaching and research tool for numerical geodynamic modelling[END_REF]Mansour et al., 2020). The 2D thermo-mechanical experiments involve a geological model of dimension 480 km x 160 km discretized over a computational grid made of 240 x 80 elements.

The initial set up consists of a 35 or 40 km thick crust with 20 km air-like material above, and mantle below (Fig. 2A-B). Each model runs through three stages: i) a shortening phase during which the crust thickens to ~ 60 km with either a slow total velocity of 0.24 cm.y -1 during 70 My or a fast total velocity of 2.4 cm.y -1 during ~ 7 My (delivering a strain rate averaged over the length of the model of 1.6 x 10 -16 s -1 and 1.6 x 10 -15 s -1 respectively); ii) a rapid increase in BHF (from 0.020 W.m -2 to 0.030 W.m -2 ) over 2.5 My while the velocities imposed on the vertical

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This article is protected by copyright. All rights reserved boundaries are set to zero (v x = v y = 0 cm.y -1 ), mimicking the thermal impact of a mantle delamination phase; iii) a relaxation phase in which the crust returns to normal thickness, under slow extensional boundary conditions (total velocity of 0.10 cm.y -1 ) associated to a decrease in BHF from 0.030 W.m -2 to 0.020 W.m -2 in ~ 70 My. Details of modeling procedures, rheological and thermal parameters as well as the input python script are available as supplementary data.

The data that supports the findings of this study are available in the supplementary material of this article These experiments focus on two end-member crustal structures with average values of total RHP at ~ 1 and ~ 2 W.m -3 (Fig. 2). A value of ~ 1 W.m -3 is in line with RHP calculation predicted from the present-day composition of the bulk continental crust determined by Taylor and McLennan (1995). Models RHP1_unif mimic a Phanerozoic orogenic cycle involving a continental crust with a uniform RHP (1.0483 W.m -3 ), yielding an initial Moho temperature of 650°C at 40 km depth (Fig. 1A). However, Mareschal and Jaupart (2013), [START_REF] Artemieva | Heat production in granitic rocks: Global analysis based on a new data compilation GRANITE2017[END_REF] and [START_REF] Gard | Variations in continental heat production from 4 Ga to the present: Evidence from geochemical data[END_REF] showed that the crustal RHP may have been higher than 1 W.m -3 during the Proterozoic, having varied between 0.8 and 4 W.m -3 between 0.5 and 2.5 Ga with an average RHP close to ~ 2 W.m -3 . In addition, recent studies reveal that, in tectonically stable regions, the upper crust's RHP may be higher than in the lower crust [START_REF] Goes | Continental lithospheric temperatures: A review[END_REF][START_REF] Alessio | Thermal modelling of very long-lived (> 140 Myr) high thermal gradient metamorphism as a result of radiogenic heating in the Reynolds Range, central Australia[END_REF].

The conditions for model RHP2_diff include a total average RHP of 2.0922 W.m -3 with high RHP in the upper crust (5 W.m -3 ) that decreases exponentially with a length scale factor hc of 20 km, yielding an initial Moho temperature at 35 km depth of 650°C (Fig. 1D). Models RHP2_diff aim at approaching thermal conditions of a differentiated crust prevailing during the Proterozoic.

Results

Despite similar initial Moho temperatures (~ 650 °C), the two experiments show very different thermal evolutions. In models RHP2_diff the production of migmatites occurs at the onset of crustal thickening (Fig. 3A,C). This contrasts with models RHP1_unif, in which the conditions for migmatites appear after a few 10's of My of crustal thickening (Fig. 3B,D). Granulite conditions generally appear a few 10's of My after the onset of migmatisation but the bulk of granulite formation occurs during the post-convergence collapse phase (Fig. 3A,B,C). Model Slw_RHP2_diff (Fig. 2C-D, 3A) allows for the prograde formation of granulites and migmatites reaching a uniform 20 km-thick granulite facies lower crust and a 20 km-thick migmatitic layer on

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This article is protected by copyright. All rights reserved top at the end of the shortening phase (Fig. 2C). At the end of the experiment, the temperature at the Moho reaches UHT conditions (Fig. 3A,4A). The thickness of the HT-UHT crust is only slightly reduced after the relaxation phase (Fig. 2D). Note that the unmetamorphosed upper crust has thinned to 15 km after the relaxation phase assisted by erosion. Model Fst_RHP2_diff (Fig. 2G-H, 3C) does not show the formation of granulites during shortening and delamination and only a very low degree of partial melting in a large volume of crust (Fig. 2G). It is only at a late orogenic stage, during collapse, that the crust is hot enough to generate granulites (Fig. 2H,3C).

Model Slw_RHP1_unif (Fig. 2E-F, 3B) allows the local formation of a granulite-cored migmatitic dome but there is no pervasive formation of HT rocks in the crust at the end of the shortening phase (Fig. 2E). Through gravitational collapse, a ~10-15 km thick layer of partially molten rocks and a thin ~ 5-10 km granulitic layer form (Fig. 2F,3B). Model Fst_RHP1_unif (Fig. 2I-J, 3D) does not lead to the prograde formation of granulite and/or migmatites during shortening (Fig. 2I).

It is only when the crust is thermally relaxed, that the lower crust becomes partially molten (Fig. 2J), but granulites do not form (Fig. 3D).

Figure 4 shows that PTt paths are mainly clockwise, with prograde paths showing gentle slopes for slow shortening velocity (Fig. 4A,C), and steeper slopes for fast shortening velocity (Fig 4B ,D).

For the RHP2_diff models, the upper crust records PTt histories involving isobaric heating then cooling across the andalusite and sillimanite fields (Fig. 4A,B), whereas the middle and lower crust start in the stability field of kyanite and evolve into the sillimanite field (Fig. 4A,B). This strongly contrasts with models RHP1_unif, in which the continental crust remains in the kyanite stability field (Fig. 4C,D). Particles generally reach their peak temperature 40 to 60 My after collapse has begun and stay at HT for at least 10 to 15 My. However, PTt paths in model Fst_RHP2_diff (Fig. 4B) are more complex and record a short (a few My) near peak P isothermal heating phase before a decompression and cooling path. Interestingly, in models RHP2_diff (Fig. 4A,B), particles starting in the upper half of the crust (< 15 km) rather record near-isobaric heating followed by near-isobaric cooling.

Discussion and conclusions

In these numerical experiments, prograde granulites evolving through the sillimanite field, similar to many HT-UHT terranes (Fig. 4E), form when shortening velocities are slow and RHP is high (Slw_RHP2_diff, Fig. 3A,4A). In most cases, the formation of granulites is a late orogenic process starting ca. 8 to 20 My after collapse has begun (Fig. 3B-C, 4B-C). Granulites can be very

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This article is protected by copyright. All rights reserved localized in the deep crust as in Slw_RHP1_unif (Fig. 2E-F, 3B). This case highlights another way to form localized granulites in addition to the back-arc geodynamic settings suggested by [START_REF] Harley | A matter of time: The importance of the duration of UHT metamorphism[END_REF]. This model may be an analogue for the Variscan crust, dominated by partially molten products (Lardeaux, 2014) in which the lower crust records an episode of late orogenic granulites documented in xenoliths (310-290 Ma, [START_REF] Costa | Lower crustal rejuvenation and crustal growth during postthickening collapse: Insights from a crustal cross section through a metamorphic core complex[END_REF]Puelles et al., 2019). In the Variscan belt, km-size lenses of granulites can be found in the Alps (Gruf Complex, [START_REF] Galli | U-Pb zircon dating of the Gruf Complex: disclosing the late Variscan granulitic lower crust of Europe stranded in the Central Alps[END_REF], in the Pyrenees (Agly charnockite, Odlum et al., 2019) or in migmatitic domes of the Bohemian Massif (Lexa et al., 2011;Maierovà et al., 2014). The peak PTt conditions and PT path shapes for granulites (Fig. 4) are in agreement with the parametric study for different tectonic units of the Bohemian Massif that reflects a large variability in total RHP and shortening velocities (Maierovà et al., 2014).

Finally, the formation of long-lived regional scale HT-UHT rocks, ubiquitous through Earth history [START_REF] Harley | A matter of time: The importance of the duration of UHT metamorphism[END_REF]Fitzimmons, 2016) and generally associated with large hot orogens [START_REF] Beaumont | Models of large hot orogens containing a collage of reworked and accreted terranes[END_REF], is best explained by the model Slw_RHP2_diff. It is characterized by a differentiated shallow crust enriched in radiogenic elements and undergoing slow shortening rates.

Migmatites form throughout the orogenic cycle and granulites form mainly during the collapse phase (Fig. 2C-D, 3A, 4A). These terranes typically record HT-UHT metamorphism with retrograde near-isothermal decompression paths (Fig. 4E) as predicted in Slw_RHP2_diff (Fig. 4A). Similarly, models for the Grenville orogen [START_REF] Jamieson | The Grenville Orogen explained? Applications and limitations of integrating numerical models with geological and geophysical data[END_REF] present PTt paths for particles starting in the lower crust with a steep prograde burial path, followed by isothermal decompression, culminating at ~ 14 kbar and ~ 950 °C (Fig. 4E) that is in line with the rock archive. Other granulitic terranes present near-isothermal decompression paths culminating at 8 -13 kbar and 900 -1000°C depending on the starting depth of the particles (Fig. 4E; In Ouzzal, Algeria; Mather gneisses, Antarctica; Palni Hills, India; references in [START_REF] Harley | A matter of time: The importance of the duration of UHT metamorphism[END_REF]. Interestingly, particles starting in the upper crust for models RHP2_diff and independent of the shortening velocity (Fig. 4A,B), present isobaric heating-cooling path similar to migmatitic terranes like, for example, southern India that records UHT melt-producing and melt-consuming steep reactions [START_REF] Cenki | Melt-producing and melt-consuming reactions in the Achankovil cordierite gneisses, South India[END_REF]. Finally, the interplay between total RHP, its distribution and shortening velocities can explain PTt paths for dominantly granulitic or migmatitic terranes as well as differences in the relative tempo of formation of migmatites and granulites during the orogenic cycle (syn-convergence and/or syn-collapse) which is not accessible from the natural rock archive.

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This article is protected by copyright. All rights reserved Fast shortening

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Slow shortening

Slw_RHP2_diff

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Slow shortening

Figure showing two snapshots for each model: i) shortening-delamination and ii) collapse. Shortening velocity is either slow (0.24 cm.y -1 , C-F) or fast (2.4 cm.y -1 , G-J). This article is protected by copyright. All rights reserved

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Figure caption Figure 1 .

 caption1 Figure caption

Figure 2 .

 2 Figure 2. A-B. Model geometry, initial conditions as well as geotherm, viscosity and density profiles. The circles pattern superimposed on the continental crust represents the finite strain ellipses. White squares represent the Lagrangian particles recording the PTt paths presented in Fig. 4. A. Initial conditions for models RHP2_diff, mimicking a Proterozoic highly differentiated and highly radiogenic crust. B. Initial conditions for model RHP1_unif, simulating a Phanerozoic uniform and less radiogenic crust. C-J. Orogenic modeling results showing two snapshots for each model: i) shortening-delamination and ii) collapse. Shortening velocity is either slow (0.24 cm.y -1 , C-F) or fast (2.4 cm.y -1 , G-J).

Figure 3 .

 3 Figure 3. Depth -time profiles indicating the onset of partial melting and granulite formation through the evolution of the models presented in Fig. 2.

Figure 4 .

 4 Figure 4. A-D. PTt paths for all particles (light grey background) shown on Fig. 2. The PTt paths for the particles initially located in the center of the model are highlighted in dark grey. As a reference, the stability field for aluminosilicates and the crustal melting domains (pink) are represented. Horizontal triangles pinpoint the onset (in My) of melt (purple) and granulite (brown) formation. E. Selected representative PT path for natural analogues.

  Figure1

Figure 1 .

 1 Figure caption

Figure 2 .

 2 Figure 2. A-B. Model geometry, initial conditions as well as geotherm, viscosity and density

Figure 3 .

 3 Figure 3. Depth -time profiles indicating the onset of partial melting and granulite formation

Figure 4 .

 4 Figure 4. A-D. PTt paths for all particles (light grey background) shown on Fig. 2. The PTt

Acknowledgements

We acknowledge the funding from the European Union's Horizon 2020 research and innovation program under grant agreement No 793978. This research was undertaken with the assistance of resources from the National Computational Infrastructure (NCI), through the National Computational Merit Allocation Scheme supported by the Australian Government; the Pawsey Supercomputing Centre with funding from the Australian Government and the Government of Western Australia, and support from the Australian Research Council grants ARC-IH130200012 and ARC-LP190100146. We thank R. Beucher for his expert support with Underworld, M. Hand and F. Garel for constructive discussions, an anonymous reviewer and K. Mezger for editorial handling.

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This article is protected by copyright. All rights reserved Puelles, P., Ibarguchi, J.I.G., García de Madinabeitia, S., Sarrionandia, F., Carracedo-Sánchez, M., and Fernández-Armas, S., 2019. Granulite-facies gneisses and meta-igneous xenoliths from the Campo de Calatrava volcanic field (Spain): Implications for the tectonics of the Variscan lower crust. Lithos, 342-343, 114-134. Sandiford M., and McLaren S., 2002. Tectonic feedback and the ordering of heat producing elements within the continental lithosphere. Earth and Planetary Sciences Letters, 204 (1-2), 133-150. Taylor, S.R., and McLennan, S.M., 1995. The geochemical evolution of the continental crust. Reviews of Geophysics, 33, 241-265. Touret, J. L. R., Santosh, M., and Huizenga, J.M., 2016. High-temperature granulites and supercontinents. Geosciences Frontiers, 7, 101-113. Vanderhaeghe, O., Guergouz, C., Fabre, C., Duchêne, S., and Baratoux, D., 2019. Secular cooling and crystallization of partially molten Archaean continental crust over 1 Ga. Comptes rendus -Geoscience, 351, 562-573.

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