RELA(CS Souhila Kettouche
email: souhila.kettouche@gmail.com

Larbi Ben M'hidi

Moufida Maimour
email: moufida.maimour@univ-lorraine.fr

Lakhdar Derdouri
email: derdouril@yahoo.fr

DM-RPL : Disjoint Multipath RPL for Bandwidth Provision in the Internet of Multimedia Things

Keywords: Internet of Multimedia Things (IoMT), RPL, Multipath routing, Performance evaluation, QoE, QoS, IoT-Lab

Internet of Multimedia Things (IoMT) is one extensively current topic of the Internet of Things (IoT) due to the immersive growth of multimedia applications in several fields. In Low-power and Lossy Networks (LLNs) where sensor nodes are a key component, providing a satisfactory quality of service (QoS) as well as a user quality of experience (QoE) for such applications is a challenging task. In fact, high bandwidth and substantial computation resources are required. To provide sufficient bandwidth to handle these high data rate applications, we propose to extend RPL to enable for simultaneous use of disjoint multiple paths. This is done on top of the already maintained DODAG structure with the least induced overhead. Furthermore, we suggest applying a low-complexity encoding method on the captured images. Based on both QoS and QoE metrics, we evaluate the performance of our disjoint multipath RPL (DM-RPL) for real video clip transmission using the IoT-LAB testbed. Our results show that multipath provides more bandwidth as the PDR is increased. Video quality is further improved thanks to the adopted data reduction at the source. All of this translates into less energy being consumed.

I. INTRODUCTION

The Internet of Multimedia Things (IoMT) has undergone an unprecedented development these last years. This is confirmed by the massive use of multimedia applications in various fields such as smart homes and industrial monitoring. However, IoMT has stringent requirements compared to traditional IoT in terms of quality of service (QoS) as well as quality of experience (QoE). When it comes to handle multimedia applications characterised by their high data rate in wireless sensor networks (WSN), one of the basic building blocks of the IoT, the problem becomes more challenging because of their constrained resources.

RPL [START_REF] Alexander | RPL: IPv6 Routing Protocol for Low-Power and Lossy Networks[END_REF] is the IETF standardised IPv6 Routing Protocol for low-power and lossy networks (LLNs) where directed-oriented acyclic graphs (DODAG) rooted at the sink are maintained. An objective function is used by each node to select its preferred parent toward the root node. Two objective functions namely, ETX [De Couto et al., 2003] (default) and OF0 (hops number) are predefined. Despite the fact that RPL almost meets the requirements of LLNs to handle scalar data routing, it is still far from being able to allow real time streaming of video flows as it was mainly designed for low data traffic network [START_REF] Kettouche | QoE-based performance evaluation of video transmission using RPL in the IoMT[END_REF]. To handle multimedia applications, some researchers proposed new objective functions based on energy [START_REF] Alvi | Energy efficient green routing protocol for internet of multimedia things[END_REF], [START_REF] Mortazavi | An energy-aware rpl routing protocol for internet of multimedia things[END_REF] or free bandwidth [START_REF] Bouzebiba | Freebw-rpl: A new rpl protocol objective function for internet of multimedia things[END_REF]. The proposed objective functions allow better behaviour when compared to predefined ones. However, the performance evaluation in both [START_REF] Alvi | Energy efficient green routing protocol for internet of multimedia things[END_REF] and [START_REF] Bouzebiba | Freebw-rpl: A new rpl protocol objective function for internet of multimedia things[END_REF] does not consider real video traffic and the user QoE is not evaluated either. As for [START_REF] Mortazavi | An energy-aware rpl routing protocol for internet of multimedia things[END_REF], simulations are based on H. 264 multimedia trace [START_REF] Seeling | Network performance evaluation using frame size and quality traces of single-layer and two-layer video: A tutorial[END_REF]. Nonetheless, H. 264 compression is not adapted to low-power video sensors [START_REF] Sarif | A study on the power consumption of H.264/AVC-based video sensor network[END_REF]. Besides, the user QoE has not been assessed.

Multipath routing [START_REF] Maimour ; Maimour | Interference-aware multipath routing for WSNs: Overview and performance evaluation[END_REF] has emerged in the last decades as a promising solution to provide sufficient bandwidth to handle high data rate flows. Extending RPL to allow for multipath routing has already been considered in the literature mainly to balance traffic among bottleneck nodes [START_REF] Iova | Exploiting multiple parents in RPL to improve both the network lifetime and its stability[END_REF], [START_REF] Moghadam | High throughput load balanced multipath routing in homogeneous wireless sensor networks[END_REF], [START_REF] Zhu | Multi-path data distribution mechanism based on rpl for energy consumption and time delay[END_REF], improve data transmission reliability through replication [Lagos Jenschke et al., 2021] or mitigate congestion [START_REF] Tang | Toward improved RPL: A congestion avoidance multipath routing protocol with time factor for wireless sensor networks[END_REF], [START_REF] Lodhi | Transient multipath routing protocol for low power and lossy networks[END_REF], [START_REF] Wang | Energy balancing RPL protocol with multipath for wireless sensor networks[END_REF]. The above multipath extensions are targeted to scalar data reporting and mostly make use of braided (non-disjoint) paths. More recently, authors of [START_REF] Bouacheria | Leveraging multi-instance rpl routing protocol to enhance the video traffic delivery in iomt[END_REF] leverage the RPL multi-instance opportunity to build multiple paths to enhance video delivery in the IoMT. A priority-based strategy is implemented where high priority frames are routed via the instance maintained using link quality as a metric and the low priority ones via the instance with the shortest path. They mainly showed by simulation that the disjoint multipath is more suitable to video delivery when compared to single and non-disjoint RPL variants.

Since a wireless network capacity is limited even with multipath routing, it still remains difficult to provide sufficient bandwidth able to satisfy the requirements of data intensive applications such as IoMT ones. Low complexity in-network data reduction is more than necessary to lower transmissions and the corresponding energy expenditure. Widely used standard video encoding techniques based on motion estimation algorithms such as MPEG-4, H.263 or H.264 are not suitable for sensor nodes [START_REF] Sarif | A study on the power consumption of H.264/AVC-based video sensor network[END_REF]. In this paper, we aim to handle video delivery in the IoMT by addressing both facets of bandwidth scarcity problem. First, we propose a new algorithm to obtain disjoint multiple paths in RPL. Our extension do not leverage the multi-instance opportunity provided by RPL as in [START_REF] Bouacheria | Leveraging multi-instance rpl routing protocol to enhance the video traffic delivery in iomt[END_REF]. Instead, it uses existing RPL control messages to insure disjointness through the use of the IDs of the last hop nodes toward the DODAG root. Second, we make use of low-complexity video encoding more adapted to LLNs to reduce the amount of transmitted data. Our performance evaluation considers both QoS and QoE metrics and is carried on a real WSN testbed (IoT-LAB [START_REF] Adjih | Fit iot-lab: A large scale open experimental iot testbed[END_REF]). To the best of our knowledge, this is the first work that evaluates video transmission using multipath RPL on top of a real testbed with real video streams emulation. In this paper, we begin by presenting our disjoint multipath RPL (DM-RPL) protocol in Section II. Afterwards, the compression method is presented in Section III. The experimental scenario and the obtained results are discussed in Section IV before concluding.

II. DISJOINT MULTIPATH RPL (DM-RPL)

The IoMT is made of a large number of small low-power devices that integrate sensors with on-board processing and wireless communication capabilities. Some of these devices ideally placed at strategic positions, are equipped by video cameras to provide richer and more informative data on an area of interest. These sensor nodes are generally in charge of reporting information to a central gateway called the Sink. This delivery model is known as convergecast or multipoint to point communication.

A WSN can be modelled as a unidirected connected graph G(V, E) composed of a set of sensor nodes V and a set of links E. We consider the use of RPL to establish and maintain a DODAG structure rooted at the Sink r ∈ V according to an appropriate objective function. This is achieved through the use of ICMPv6 packets called DODAG Information Objects (DIO). Based on the DODAG structure, each node v maintains a list of parents, one of which is designated as the preferred parent noted π(v). A node willing to transmit data to the Sink, sends packets to its preferred parent. Let T (V, E T ) be the spanning tree of G, rooted at the Sink (r), obtained by considering only links to preferred parents in the RPL DODAG rooted at r. We define the depth of a node (d) as the length (number of hops) of its tree path to the root. The tree root has a depth d(r) = 0 and a non-root node v ∈ T has a depth d(v) = d(π(v)) + 1. We call a subtree each tree rooted at a node v ∈ T where π(v) = r or equivalently d(v) = 1. The subtree rooted at v is noted T v and v is designated as a subroot.

When dealing with high data rate applications, it is worth remembering the important limitations of LLNs. The available bandwidth is insufficient with respect to the requirements of these applications. In an attempt to provide additional bandwidth, we suggest to leverage simultaneous transmissions on multiple disjoint paths. To do so, we propose to extend monopath RPL to allow for simultaneous use of multiple disjoint paths without incurring much more overhead. The resulting protocol is called Disjoint Multipath RPL (DM-RPL).

Paths disjointness from two nodes u and v to r using tree T child-parent links is guaranteed if these nodes belong to two different subtrees i.e. u ∈ T tu and v ∈ T tv with t u = t v . For a given source s, the default path is the one that follows the preferred parent at each hop. To get other paths that are disjoint, the source has to choose alternate parents that belong to different subtrees. In order to enable a node to identify the subtree to which belong each of its parents, one need to propagate the ID of each subroot downward. This is achieved by piggybacking this information in a field of DIO messages intended for this purpose we call P ID (Path ID). It is initialised to zero in the initial DIO advertised by the DODAG root. Upon the reception of a DIO message from the DODAG root, a subroot inserts its ID in the P ID field before advertising the obtained DIO to its neighbours. When a regular node (other than root and subroots) receives a DIO message with non-zero P ID field, it broadcasts it as it is. Based on the DIO information, a regular node updates its parents list as suggested by default RPL except that it records the P ID of each of them. When the above process converges, each node in G will learn the ID of its subroot which is the last crossed node before last hop as well as the potential parents leading to the root via disjoint paths.

Figure 1 illustrates how DM-RPL maintains multiple disjoint paths based on the DODAG structure and the subroot ID propagation downward using DIO messages. The DODAG root r broadcasts a DIO message with a zeroed P ID field. Each of the subroots i.e. nodes 1, 2 and 3, chooses r as its preferred parent and puts its own ID in the P ID field before broadcasting the resulting DIO. The other nodes (4, 5, 6 and 7), on the reception of a DIO, update the list of their parents. In each lower part of the circle representing a node, is given the list of the P IDs maintained where the first is the one related to the preferred parent. As depicted, node 4, for instance, has two disjoint paths via 1 and 2 respectively. For node 7, there are three possible paths highlighted in yellow for the primary one (via the preferred parent) and in green for the two alternative paths. Since an LLN network capacity is limited even when multiple paths are involved, it still remains difficult to provide sufficient bandwidth able to satisfy the requirements of data intensive applications such IoMT ones. Having sensors to deliver all the captured visual data is inefficient and has to be avoided. Authors of [START_REF] Bidai | Interference-aware multipath routing protocol for video transmission over zigbee wireless sensor networks[END_REF] showed that multipath routing allows to double the overall throughput. However, an acceptable video quality was difficult to obtain even when capturing and transmitting at only 6 fps a grey-scale low resolution video clip compressed using MPEG-4. As a result, in-network data reduction is more than necessary to lower transmissions and the corresponding energy expenditure. As already stated, MPEG-4 is not suitable for sensor nodes. Even JPEG low complexity still image compression algorithm is not very beneficial in terms of power consumption. This is mainly due to the discrete cosine transform (DCT) stage which consumes at least 60% of the whole power encoder [START_REF] Taylor | Design of an adaptive architecture for energy efficient wireless image communication[END_REF], [Suseela and Asnath Victy Phamila, 2018]. In order to suite the constrained nature of processing resources of low-power motes, we suggest to use a low complexity image compression provided by SenseVid [START_REF] Maimour ; Maimour | Sensevid: A traffic trace based tool for qoe video transmission assessment dedicated to wireless video sensor networks[END_REF], a video transmission and evaluation tool that considers LLNs specific characteristics.

A captured image is either intra-coded (M-frame) or inter-coded with respect to the previous M-frame in which case it is referred to as an S-frame. Whether a given frame is M-or S-encoded depends on the GOP coefficient (g). If M SE ≤ g 2 where M SE is the mean square error with respect to the previous M-frame, then the frame is S-encoded ; otherwise, it is M-encoded. When g = 0 then all the frames are M-encoded. Increasing its value results in more S-frames and thus reduces the data rate. In what follows the process of encoding is detailed for each frame type.

A. Main Frame Encoding

Figure 2 shows the different steps of an M-frame encoding process. First, the image is decomposed into blocks of 8 × 8. Then, a fast pruned DCT called binDCT-C [START_REF] Liang | A fast multiplierless approximations of the dct with the lifting scheme[END_REF] is applied on each block with a triangular pattern [START_REF] Mammeri | Modeling and adapting jpeg to the energy requirements of VSN[END_REF]. Only coefficients located at the upper left triangle of side length (ρ ≤ 8) are considered. The resulting DCT block coefficients are quantised using the JPEG standard quantisation matrix. Trade-off between quality level and compression rate can be obtained by selecting a proper quality factor (QF) that allows adjusting the quantisation matrix values. An image visual quality ranges from the poorest (QF = 1) to the best quality (QF = 100). The obtained block is then linearised following a zigzag scan before applying a lossless entropy encoding. The default entropy coder in SenseVid consists in coding all coefficients using exponential-Golomb code [Teuhola, 1978].

B. Secondary Frame Encoding

As shown in Figure 3, a secondary frame is inter-coded with respect to the previous main frame i.e. only blocks of the difference between the current frame and the previous M-frame are considered. To further save video sensors and network resources, a filtering and a thresholding operations are consecutively applied. Difference blocks (D) with the least values that satisfy d 2 ij /64 ≤ 650 are ignored. The retained blocks with difference values less than θ are zeroed where θ is the threshold similarity parameter. The rational behind this is that d 2 ij /64 > 650 corresponds to a P SN R < 20 dB, the level below which the reconstructed block is considered of bad quality. Finally, a lossless entropy encoding is applied on the resulting non-null blocks. Note that since not all blocks are transmitted, blocks sequences have to be specified in the packet header. The sequence number of the first block of a packet is given as it is. A subsequent sequence is encoded using the difference with the respect to the previous one.

IV. EXPERIMENT RESULTS

Our experiments are made using IoT-LAB [START_REF] Adjih | Fit iot-lab: A large scale open experimental iot testbed[END_REF], a very large scale open experimental IoT testbed composed of 2728 static and mobile nodes distributed over six different sites in France (Strasbourg, Grenoble, Lille, Paris Saclay and Lyon). Three types of nodes, equipped with different micro-controllers and wireless communication chips, are provided namely, M3, A8 and WSN430 nodes. IoT-LAB can be accessed through a web portal or using provided command-line tools. Additionally, it offers a hosted environment on SSH front-ends with pre-installed command-line Tools and target architectures cross-compiler tool chains. It allows to retrieve experiments results and to access devices serial ports.

In order to assess the performances of our disjoint multipath routing, we extended the RPL implementation provided in the Contiki IPv6 network stack. The last two bytes of the IPv6 address of a node in IoT-LAB testbeds are used as an ID. Its uniqueness is ensured by the fact that only the last 16 bits are varied in the IPv6 address assignment and subnetting. In DM-RPL, this 16-bit ID is used to identify possible disjoint paths. The unused 7 th (Flags) and 8 th (Reserved) bytes of the DIO packet are chosen to constitute the P ID field.

To carry out our experiments, we used M3 nodes1 installed in the F4 corridor of Grenoble site. Figure 4b shows the 15 used nodes and represents a possible tree structure in which the links are those connecting each node to its best parent. Since IoT-LAB does not provide video sensors, we emulate the transmission of a real video clip (by the node 207 to the root 240) using a traffic trace file (st-packet) generated by SenseVid. As shown in Figure 4a, a python script (client.py) gives transmission instructions to the source node. The RPL-UDP sender and receiver modules are modified to enable the former to send packets according to the instructions and the latter to retrieve the list of received packets to be stored in a receiver trace file (rt-packet).

Based on this latter, SenseVid (see Figure 5) reconstructs the video received images and generates QoE video related metrics, namely, PSNR and SSIM. The PSNR between the sent and the received, possibly distorted video frame is computed using :

P SN R = 20 log V peak
M SE where M SE is the mean square error and V peak is the maximum possible pixel value. The SSIM metric is computed as follows :

SSIM = 2µ x µ y + C 1 µ 2 x + µ 2 y + C 1 × 2σ x σ y + C 2 σ 2 x + σ 2 y + C 2 × σ xy + C 2 /2 σ x σ y + C 2 /2
where x and y are two non negative image signals, µ x , σ x and µ y , σ y are the mean and standard deviation of x and y respectively. σ xy is the sample cross-covariance between x and y. C 1 = 6.5025 and C 2 = 58.5225.

We compare the performance of DM-RPL with default RPL and focus on the impact of increasing images frequency capture on their real time transmission. To do so and coping with LLNs constraints, we consider the transmission a given number ranging from 9 to 18 grey-scale images extracted from the original 300 ones of the hall monitor video clip [yuv, ]. Moreover, these images are down-sampled to a resolution of 128 × 128. We consider transmitting images encoded independently of each other (M-encoded) as well as the case of including some S-frames where the GOP coefficient g is set to 15 in order to reduce the data transmission rate as shown in Table I. Used parameters are regrouped in Table II. Each experiment is repeated at least 10 times to avoid fluctuation and obtained results are presented using box plots to show the distribution of obtained values of the metric of concern. A segment and a small black square inside each rectangle shows the median and the mean values respectively.

Although, our multipath extension allows to build more paths, we conducted our experiments using only two paths. The obtained results and lessons learnt using two paths can be generalised to more paths and are expected for a future work. We first consider the packet delivery ratio (PDR) obtained by RPL (1-path) and DM-RPL (2-path) with the two sequences of frames (g0 and g15). As expected, Figure 6 shows a decreasing PDR when increasing the number of images to transmit due to a higher number of packets to transmit in a limited time window. Based on the mean PDR, DM-RPL outperforms RPL regardless of the transmission rate. DM-RPL is able to provide more bandwidth thanks to splitting the data flow on the two disjoint paths. Reducing the required bandwidth by inter-coding some of the frames (g = 15) allows to achieve better performances in terms of PDR when compared to the case where all frames are intra-coded (g = 0). This observation is valid for both RPL and DM-RPL.

Transmission rate reduction thanks to the introduction of S-frames leads to lower quality of images when compared to the use of only M-frames as shown by the obtained reference PSNR and SSIM given in Table I. Hence, the quality of the received images has to be considered in the evaluation. Figure 7 shows both mean and distribution plots of the obtained PSNR and SSIM of the received images as reconstructed by SenseVid considering the incurred packet losses. Triangles in the candles curves depict the reference PSNR and SSIM. Once again, the quality is getting worse when increasing the transmission rate. The images quality is still fair (P SN R > 20) for the evaluated schemes provided that the number of frames does not exceed 12. When using two paths along with more data reduction (2-path, g15), the quality remains acceptable even for the highest rate i.e. 18 frames captured and transmitted in real time. Note that the obtained PSNRs and SSIMs are higher when g = 15 compared to g = 0 even if the corresponding reference values are better in the latter case.

Figure 8 shows samples of the frame 3 when 12 images are captured and transmitted. The rightmost image is the one transmitted by the source. The others are those received by the Sink when using the four evaluated schemes. Data reduction strategy is profitable mainly when two paths are used where both PSNR and SSIM are improved.

Figure 9 plots the average consumed power for the different evaluated schemes when varying the number of captured frames. The results show that DM-RPL exhibits lower energy consumption compares to RPL. This is due to the fact that in DM-RPL, the traffic load is balanced on two paths. Thus, experimenting a lower number of packet losses which reduces the amount of MAC level retransmissions leading to lower energy expenditure.

V. CONCLUSION

In an attempt to handle multimedia traffic in constrained LLNs, we propose to extend RPL such that multiple disjoint paths can be maintained and used by video sensors. Moreover, we leverage low complexity compression techniques to cope with limited sensors resources. We carried out experiments using real WSN testbed and real video clip transmission. Both QoS (PDR and energy) and QoE (PSNR and SSIM) were adopted to assess the added value of our proposal. We mainly showed that using two paths allows to provide more bandwidth such that the amount of successful transmissions is improved. Our used data reduction technique allows additionally for better images quality. Energy consumption is also improved.

Despite the obtained results, there is still a way to go to enhance the received images quality mainly when increasing the capture frequency. As a future work, we expect to improve our multipath strategy to increase the probability of finding more than one path with the bestquality. Moreover, advanced encoding techniques with better optimised rate-distortion have to be investigated. Transmission strategies involving interleaving/concealement techniques are under study. Finally, advanced methods for image reconstruction with denoising and deblurring are to be explored.

Acknowledgement. Authors thank FIT IoT-LAB project for providing testbed and tools to perform this paper experiments.

ACKNOWLEDGMENTS

Authors thank FIT IoT-LAB project for providing testbed and tools to perform this paper experiments.

θ

  Fig. 1: Illustrative example

  ia l c o m . ( T C P ) c li e n t .p y s t -p a c k e t s e r ia l c o m . ( T C P ) n c 2 0 0 0 0 > r t -p a c k e t (a) IoT-LAB interface with SenseVid.

Fig. 4 :

 4 Fig. 4: IoT-lab Experiment Setup

Fig. 5 :

 5 Fig. 5: SenseVid Architecture (redrawn from [Maimour, 2018b])

  Fig. 6: PDR (a) distribution and (b) mean values

FigFig. 8 :

 8 Fig. 7: Obtained Images Quality

TABLE I :

 I Transmitted video characteristics

	Images	ref. PSNR (dB)	ref. SSIM	compression ratio (bpp)	transmission rate (Kbps)	packets to send	Frames sequence
	number	g = 0	g = 15	g = 0	g = 15	g = 0	g = 15	g = 0	g = 15	g = 0	g = 15	when g = 15
	9	29.50	28.52	0.9081	0.8980	0.66	0.57	7.8	6.73	99	90	MMMMSMSMS
	12	29.50	28.81	0.9010	0.8997	0.66	0.57	10.40	8.98	132	116	MMMMMSMMSMSM
	15	29.50	28.05	0.9077	0.8959	0.66	0.52	13.01	10.32	165	140	MSMMMMSMSMSMSSM
	18	29.51	28.31	0.9078	0.8960	0.66	0.51	15.6	11.94	198	162	MSMMMSMSSMSMSMSSMM

TABLE II :

 II Experimental Setup

	Number of sensors	15 (including the root)
	Testbed site	INRIA Grenoble
	Experiment duration	10 min
	operating system	Contiki OS
	Sensors type	M3 open nodes
	Packet maximum payload	128 Bytes
	Transport protocol	UDP
	Routing protocol	(IPv6) RPL ,DM-RPL
	RPL objective function	ETX
	DIO maximum interval	17.5 min
	DIO minimum interval	4 sec
	MAC	CSMA / link-layer bursts
	Radio Duty Cycling	ContikiMAC
	Channel check rates (Hz)	128
	transmission power	0 dbm
	reception sensitivity	101 dBm
	Video duration	12 seconds
	Frame resolution	128x128
	Number of captured images	9, 12, 15, 18
	GOP Coefficient (g)	0, 15
	Quality Factor	5
	DCT	Triangular BIN DCT
	DCT triangle side length ρ	8
	Entropy coder	Exponential-Golomb (EG)

https://www.iot-lab.info/docs/boards/iot-lab-m3/


Olivier Ramaré

Julien Cassaigne

Sary Aurélien Drappeau

Notes on the domain of exponent pairs

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

Introduction and results

Exponent pairs in the large E. Phillipps developped in [8] a theory of exponent pairs by furthering and simplifying the notion of exponent system introduced by J.G. van der Corput in [12]. The reader will find a modern account of this theory in the reference book [6] by S.W. Graham and G. Kolesnik. Roughly speaking a couple pκ, λq P r0, 1 2 s ˆr 1 2 , 1s is said to be an exponent pair when, given a regular function φ that is 'monomial-like' and whose first derivative on the interval rN, 2N s is of size F , the upper bound

S " ÿ N ănď2N e 2iπφpnq ! ε F κ`ε N λ`ε .
holds for any ε ą 0. The following exponent pairs are known:

p0, 1q, ´1 2

An example

In [2], Dekking and Mendès-France propose a geometrical approach to exponential sums. The reader will find there several examples and some compelling drawings. A most classical example is φpnq " tplog nq{p2πq where t is some large parameter and for instance N " t 1{3 . The sum S is then often called a zeta-sum. A first trivial bound for S is N `1. Since φ 1 pnq " t{p2πnq, we see that t 2{3 ă 4πφ 1 pnq ď 2t 2{3 , so that we may select F " N 2 . We thus get the bound S ! ε N 2κ`λ`ε , and with the pairs given in (1) above, this gives the exponents of which 13{14 is the best one for our problem.

The domain of exponent pairs

It follows from the theory that, given an exponent pair pκ, λq, we may build another one by the two formulas

´κ 2κ `2 , κ `λ `1 2κ `2 ¯, ´λ ´1 2 , κ `1 2 ¯. ( 3
)
{processi} {processi}

The first five pairs above are obtained by using these two processi while the last two have been obtained respectively by M.N. Huxley and N. Watt in [7] and by N. Watt in [13]. Furthermore, any convex combination of exponent pairs is again an exponent pair. When using the pair p0, 1q and the processi described in (3), we call the convex hull of the domain obtained the van der Corput Domain denoted by D.

Theorem 1.1. The domain D is the convex hull of the curve C defined in [START_REF] Rankin | Van der Corput's method and the theory of exponent pairs[END_REF] and which is the graph of a continuous non-increasing function.

When we add the point `89 560 , 369 560 ˘, the domain will be called the Watt Domain in the sequel and denoted by D ˚.

In [START_REF] Rankin | Van der Corput's method and the theory of exponent pairs[END_REF], R.A. Rankin started to describe the set of accessible exponent pairs, a study furthered by S.W. Graham in [START_REF] Graham | An algorithm for computing optimal exponent pairs[END_REF]. The viewpoint taken in both papers is to compute optimal values in a specific problem. The aim of the present note is to continue this work from a more geometric viewpoint. Nonetheless it is fair to say that a large part of the material we present here can be found in the previous two papers in some form or some other.

Since we also strive to describe the situation with pictures, it may be better to provide the readers with the means to play themselves with these pictures. We shall be using the SageMath, see [START_REF]SageMath, the Sage Mathematics Software System[END_REF]; the script we use is available on the web at: https://ramare-olivier.github.io/Maths/ExpPairsNote-01.sage

Copy this code is a file named, say, ExpPairs.sage, without forgetting the sage suffix, start SageMath and load this via the command load("ExpPairs.sage"). We give in the text some pointers on to how to code in SageMath, as well as commands that we write in the form [7] M. N. Huxley and N. Watt, 1988, "Exponential sums and the Riemann zeta function".

[13] N. Watt, 1989, "Exponential sums and the Riemann zeta-function. II".. [9] R. A. [START_REF] Rankin | Van der Corput's method and the theory of exponent pairs[END_REF], "Van der Corput's method and the theory of exponent pairs".

[5] S. W. [START_REF] Graham | An algorithm for computing optimal exponent pairs[END_REF], "An algorithm for computing optimal exponent pairs". [START_REF]SageMath, the Sage Mathematics Software System[END_REF] The Sage Developers, 2022, SageMath, the Sage Mathematics Software System (Version 9.5).

ExpPairs.sage/plotC (12,6) [1] to mean that the reader should type the command plotC (12,6) [1] in Sage-Math, once the main file ExpPairs.sage has been duly loaded.

A simple continuous bound

A consequence of our study is the next flexible estimate. {continuousEP} Theorem 1.2. Let S be an exponential sum of monomial type and parameters N and F . Then, for every κ P r0, 1 2 s and every ε ą 0, we have

S ! ε F κ`ε N ϑ0pκq`ε where ϑ 0 pκq " 1 ´κ log 2 log 2κ `1 2κ .
We have ϑ 0 p0q " 1, ϑ 0 p1{2q " 1{2 and ϑ 0 p1{6q " 2{3. This is proved in Lemma 2.8 below. The upper bound S ! ε F φpλ`1 2 q´1 2 `εN λ`ε also holds true for every λ P r 1 2 , 1s and every ε ą 0, and this one is better than the above one when λ ď 2{3. Theorem 5.1 belows offers a generalization of this result to higher dimensional exponential sums. Contrarily to S.W. Graham's approach that leads to optimal values at some specific points, Theorem 1.2 allows real-valued optimization at a small numerical loss in the exponent. The approximation is however very tight as shown by Figure 1. Difference between ϑ 0 and the optimal exponent pair from the van der Corput domain, drawing obtained via ExpPairs.sage/compareModelConvHull(12).

On our example, we get S ! ε N 2κ`ϑ0pκq`ε , which is minimal when κ " 0.0566 865 ¨¨¨with value N 0.926¨¨¨, whence ÿ

t 1{3 ănď2t 1{3 n it ! t 0.926{3 (

Preparing for the proofs: a change of variables

We prefer to change of variables and to use pu, vq " p2κ, 2λ ´1q.

(5) {defuv} {defuv}

The pairs above become in p0, 1q, p1, 0q, p 1 3 , 1 3 q, p 1 7 , 4 7 q, p 9 28 , 9 28 q, p 89 280 , 89 280 q, while the transformations read

f pu, vq " ˆu u `2 , v `1 u `2 ˙, cpu, vq " pv, uq. (6)
{deffandg} {deffandg}

As c is an involution, it is better to consider the transform g " c ˝f and to consider iterations of f and g. It is noteworthy that f and g preserve segments.

One can consider these transforms as restrictions of linear transforms on the projective plane P 2 :

C "

¨1 0 0 0 1 1 1 0 2 ', D " ¨0 1 1 1 0 0 1 0 2 ' ( 7
)
{eq:1} {eq:1} so that f pu, vq can be read on the first two coordinates of C ˇˇˇˇˇu v 1

(when we divide by the third one); a similar link holds between g and D.

A first player: the curve C

There are two ways to describe the van der Corput Domain. Both rest on a curve C that we now build: either by taking a limit from above, or by taking the closure of the set of points obtained by iterating f and g when starting from tp0, 1q, p1, 0qu.

Getting to C from outside

The construction we now describe will for instance make clear that we reach a connected curve.

In this section, we consider the transformation of the unit square r0, 1s 2 under the two transforms f and g. Let us first notice that these transforms are contracting.

Lemma 2.1. When P, Q P r0, 1s 2 , we have }f pP q ´f pQq} ď ρ}P ´Q} where ρ " b 3`? 5 8 ď 13{16. The same holds true for g.

Proof. The Jacobian reads, with U " u `2 and V " v `1,

J " ˆ2 U 2 ´V U 2 0 1 U , ˙so that U 4 JJ ˚" ˆ4 `V 2 ´U V ´U V U 2 ˙.
The largest eigenvalue of JJ ˚is

λ " 4 `U 2 `V 2 `aU 4 `p2V 2 ´8qU 2 `p4 `V 2 q 2 2U 4 .
It is largest when V " 2, so we are left with finding the maximum over W P r4, 9s of the quantity 8 `W `?64 `W 2 2W 2 .

As this function of W is non-increasing, the worst case is W " 4. The lemma follows readily.

Let Kpr0, 1s 2 q be the compact space of the compact subsets of r0, 1s 2 , equipped with the usual Hausdorff distance (see for instance Exercise 3 of Section 16, Chapter 3 of the reference book [START_REF] Dieudonné | Foundations of modern analysis[END_REF] by J. Dieudonné) , i.e. where notation pf Y gq ˝n means that we compose pf Y gq iteratively n times with itself. This set C corresponds to the set P of S.W. Graham in [START_REF] Graham | An algorithm for computing optimal exponent pairs[END_REF]. Plotting C is not difficult.

dpK 1 , K 2 q " max ´max k1PK1 dpk 1 , K 2 q, max k2PK2 dpk 2 , K 1 q ¯. ( 8
[3] J. [START_REF] Dieudonné | Foundations of modern analysis[END_REF], Foundations of modern analysis.

[5] S. W. [START_REF] Graham | An algorithm for computing optimal exponent pairs[END_REF], "An algorithm for computing optimal exponent pairs". if nbsteps > 0:

Getting to C from inside, I

We may get points that are on the final curve by two processi. Here is a first one.

The construction above shows also that each point of C may be reached in a unique manner either from any point with a infinite sequence of f and g, giving an adapted 'binary' writing for these points. For instance the point p1, 0q is f f f ¨¨¨, while p0, 1q is ggg ¨¨¨: indeed, the first f reduces the unit square r0, 1s 2 to a smaller parallelepiped that lies inside r0, 1{3s ˆr1{3, 1s. On applying f again, we get an even smaller parallelepiped that still contains the point p0, 1q. The intersection č ně1 f ˝npr0, 1s 2 q then reduces to the point p0, 1q. Please notice that when we associate the sequence f f f ¨¨¨to this points, the order is reverse to the one we use for the composition of functions. The point p 1 3 , 1 3 q is gf f f f ¨¨¨" f gggg ¨¨¨. The same construction shows that any finite combination of f and g applied to p0, 1q or p1, 0q belongs to C. We get rational points by considering a finite sequence, say f f gf g, and completing it on the right either by f f f ¨¨¨if we want to refer to the upper left point of the parallelepiped gf gf f pr0, 1s 2 q, or by ggg ¨¨¨if we want to refer to the lower right point of the same parallelepiped. We get in this manner rational points that are on the curve C. The reader will readily see that we get a dense family of such points. Indeed, specifying a prefix, like f f gf g, localizes the point inside gf gf f pr0, 1s 2 q and any continuation, say f f gf gf f ggf f gg, leads to points that are inside this set.

Getting to C from inside, II

We now describe a second processus to get points that are on the final curve.

A point of the curve may be attained by some sequence, say f f gf g ¨¨¨. Rather than considering the transforms f and g, we could equivalently make the corresponding product of matrices C and D. These and the resulting product is a non-singular matrix with (integer) non-negative coefficients. As such it has a single dominant eigenvalue, the so-called Perron-Frobenius (see Chapter XIII of the book [START_REF] Gantmacher | The theory of matrices[END_REF] by Gantmacher) eigenvalue and a corresponding eigenvector. So, if we iterate the transform g ˝f ˝g ˝f ˝f , or equivalently DCDCC, the image in P 2 of the cone corresponding to the square r0, 1s 2 accumulates around the line containing this eigenvector. This line is indeed a point of C; it corresponds to the code f f gf gf f gf gf f gf g ¨¨¨where we repeat the pattern f f gf g. The points we now obtain are cubic, since so is the eigenvalue as a root of a cubic polynomial, namely the characteristic polynomial (of DCDCC in our example).

Here again, we can localize these points by choosing a proper prefix. Proof. Indeed, let us consider the region R " tpu, vq, v ď θ 0 puqu. If a point P belongs to the region R " tpu, vq, v ď θ 0 puqu, then f pP q also belongs to this At this level, the point p 1 3 , 1 3 q is becoming useless, and in later steps, it will even become an interior point. So have reached the points p0, 1q, p1{15, 11{15q, p1{7, 4{7q, p2{9, 4{9q, p4{9, 2{9q, p4{7, 1{7q, p11{15, 1{15q, p1, 0q

Here is the situation when we reached the step 6, and which shows that finding a pattern to determine which points to keep and which to discard may be intricate. The next figure (in pκ, λq) shows that the adequation of our model θ 0 to this case is not as good as before but still within an acceptable magin.

  {CompareCH}

Figure 1 :

 1 Figure 1:Difference between ϑ 0 and the optimal exponent pair from the van der Corput domain, drawing obtained via ExpPairs.sage/compareModelConvHull(12).

1 6

 1 4) {eq:4} {eq:4}improving on (2), though still far from the expected t `ε.

Figure 2 :

 2 Figure 2: The first three iterations, drawing obtained via ExpPairs.sage/transformSquare([[0,0],[0,1],[1,1],[1,0]],3,2,False).

  {fig2}

Figure 3 :

 3 Figure 3: Approximation of the curve C with words of length 12, drawing obtained through ExpPairs.sage/plotC(12, 6)[1].

Figure 4 :

 4 Figure 4: The curve C in green (below) and the graph of θ 0 in red (above), drawing obtained through ExpPairs.sage/plotC(12, 6)[1]+plotUpper(6).

  {Compare}

Figure 5 :

 5 Figure 5: Difference between θ 0 and C, drawing obtained by using ExpPairs.sage/compareModelCurve(12).

Figure 6 :

 6 Figure 6: Approximate domain D with words of length 3, drawing obtained via ExpPairs.sage/plotDomainC(1)+plotDomainC(2)+plotDomainC(3).

Figure 8 :

 8 Figure 8: Domain D ˚. In blue, the part added by Watt's exponent pair, drawing obtained through ExpPairs.sage/plotSimpleDomainCS([[0,1], [89/280,89/280]],12, 6, 'blue') + plotSimpleDomainCS([[0,1]], 12).

  {CompareCHW}

Figure 9 :

 9 Figure 9:Difference between ϑ 0 and the optimal exponent pair from the Watt (resp. van der Corput) domain in red [top] (resp.in blue [below]),, drawing obtained via ExpPairs.sage/compareModelConvHull(8)+compareModelConvHull(8).

Theorem 5 . 1 .

 51 leads to the rules(6), and the geometrical problem is thus unchanged! We then infer the next result from Theorem 1.2 and Theorem 4.1.{continuousEPp} For any κ P r0, 1 2pp`1q s, the couple `κ, ϑ0ppκq`p´1 p ˘is a p-dimensional exponent pair. The same is true of `κ, ϑ ˚ppκq`p´1 p ˘, where ϑ ˚is defined in Theorem 4.1.

ToPlot = actonD(DomfC, nbsteps-1, myz+1, True, shade) ToPlot += actonD(DomgD, nbsteps-1, myz+1, True, shade) return(ToPlot) MyInitialDomain = [[0,0], [0,1], [1,1], [1,0]] actonD(MyInitialDomain, 6, 2, False, 110, figsize = 15, gridlines = "automatic", xmin = 0, xmax = 1, ymin = 0, ymax = 1, aspect_ratio = 1)

Copy this code is a file CCurve.sage, without forgetting the sage suffix, start SageMath and run the file by using the command load("CCurve.sage"). It is then possible to increase nbsteps, say to 10, and to zoom on a particular region by changing the quadruple (xmin, xmax, ymin, ymax). Let us take the opportunity of this note to explain part of the help system of Sagemath. If we set P = actonD(MyInitialDomain, 1, 1, False, 0), then we may use P.<tab> to get access to all the methods associated with the object P (a plot). And to see all the tons of options associated with the show method, enter P.show?.

The 2D-plotting reference guide is available there: https://doc.sagemath.org/pdf/en/reference/plotting/plotting.pdf

Let us comment on this picture. Introducing an adhoc definition will simplyfy our task. Definition 2.2. A tile is the convex hull of four points.

Since the functions f and g transform segments into segments, tiles are transformed into tiles by any composition-product of these two.

We starts from the tile r0, 1s 2 , which we transform by f , getting the new tile Convpp0, 1q, p 1 3 , 2 3 q, p 1 3 , 1 3 q, p0, 1 2 qq and then similarly by g, getting the tile Convpp 1 3 , 1 3 q, p 2 3 , 1 3 q, p1, 0q, p 1 2 , 0qqq. These resulting tiles join in p 1 3 , 1 3 q. On applying repeatedly the transforms f and g, we get a connected necklace of tiles.

Here is a lemma that helps structure the situation.

Lemma 2.3. Let K be a connected compact subset of r0, 1s 2 that contains the points p0, 1q and p1, 0q. The set f pKq Y gpKq is again a connected compact subset of r0, 1s 2 that contains the points p0, 1q and p1, 0q.

Proof. Indeed f pKq and gpKq are both connected and compact. Both sets contain the point f pp1, 0qq " p 1 3 , 1 3 q " gpp0, 1qq.

At each step, we get a succession of tiles hp0, 1q ´hp1, 1q ´hp1, 0q ´hp0, 0q; the distance between hp1, 1q and hp0, 0q is at most ρ n ? 2, when h is a product of n terms from tf, gu. This shows an exponential rate, and the actual rate is faster (meaning the practical 'ρ' is smaller). One shows readily that we end up with a curve. To plot it, we may only consider the transforms of the lower part of the initial square. Let us state formally a theoretical consequence of this discussion. {Ccaliscontinuous} Lemma 2.4. The curve C is the graph of a continuous non-increasing function.

We shall now see that this seemingly regular curve C contains a dense subset of rational points (i.e. points whose coordinates are rational numbers).

Additional properties

Here are three additional properties of C. {inj} Lemma 2.5. If f pP q belongs to C, then P belongs to C.

Proof. We may assume that f pP q ‰ p 1 3 , 1 3 q. Indeed, if f pP q is in C, then f pP q is a limit point of a sequence f a1 g a1 f a2 g a2 ¨¨¨. As f pP q ‰ p 1 3 , 1 3 q and the image of r0, 1s 2 by f and g only intersect on this point, we deduce that a 1 ą 0. By injectivity, we see that P " f a1´1 g a1 f a2 g a2 ¨¨¨, completing the proof. We note for verification that ϑ 0 pκq " pθ 0 p2κq `1q{2. {separation} Lemma 2.6. The three areas that are (1) the points that are strictly above C, (2) the points that are on C and (3) the points that are strictly below C are stable under the action of f and g.

Proof. Indeed f and g are injective maps. Let P " pu, vq be above C. This means that the segment rP, P 0 s where P 0 " pu, 1q does not cross C. The segment rp0, 1q, p1, 1qs on which P 0 lies remains above our curve by construction after applying f or g. The segment rf pP q, f pP 0 qs may not cross C, as a crossing point would be a f pQq, and by Lemma 2.5, Q would belong to C and to rP, P 0 s, a contradiction. {split} Lemma 2.7. If a point P is below (resp. up of ) the curve C, then f pP q and gpP q are also there.

A simple continuous bound

We now present a readily exploited upper bound for C. {upperbd} Lemma 2.8. The graph of the function θ 0 : x Þ Ñ 1 ´x logp1 `1{xq{ log 2 is stable under f and remains above C. It crosses C in three points: p0, 1q, p 1 3 , 1 3 q and p1, 0q.

[4] F. R. [START_REF] Gantmacher | The theory of matrices[END_REF], The theory of matrices. Vols. 1, 2.

region. Indeed, we compute that

This proof also shows that the graph of θ 0 is invariant under f . It is easy to show that this graph is above C when u P r1{2, 1s. The other parts of this graph are obtained by applying f , and since the region upper to C is stable under f , the graph of θ 0 remains there.

Remark 2.9. When P belongs to this region then gpP q also belongs to it. Indeed, we have

The function w Þ Ñ w logp1 `U {wq is non-decreasing when w P r1, 2s and U P r2, 3s (its derivative is plog zq `1 z ´1 for z " pw `U q{w P r2, 4s). We thus only have to prove our assertion when v " θ 0 puq and a simple plot is enough.

Proof of Theorem 1.2. Translating Lemma 2.8 in terms of pκ, λq and the definition of exponent pairs are all that is required to complete this proof.

Convex hull

Given a symmetric subset S Ă r0, 1s 2 that contains p0, 1q, we consider the smallest closed convex set C pSq that contains all the images of S under f and g. Since our set contains p0, 1q it contains p1, 0q and p1{3, 1{3q. These are the vertices of the image of r0, 1s 2 by f and g. On iterating, we find all the (opposite) vertices of the small parallepipedes that we used to build C, from which we conclude that the convex hull of C is indeed in C ptp0, 1quq which we denote by D. Note that, since f and g preserve segments, it is enough to first iterate f and g and, in a second step, to take the convex hull of the final set.

As we see in Figure 2, the curve C has a singular point at p 1 3 , 1 3 q. But since any other location on the curve in a smooth image of the full curve, the set of points where this phenomenom occurs is in fact dense on C.

Let us describe an algorithmical way of computing D. We start from p0, 1q and p1, 0q, apply f and g, get the convex hull and repeat on the set of vertices obtained. Here is a plot of the first three steps. In blue round shape, the points of C that are on the border, and in red star shape, the ones that are not needed anymore; zoom in (b) on the central part, drawing obtained via ExpPairs.sage/plotDomainC(5, False) and adding .show(xmin=0.2,xmax=0.5,ymin=0.2,ymax=0.5, aspect ratio=1) for the second one.

Adding the Huxley and Watt point

In [1], E. Bombieri and H. Iwaniec improved the Lindelöf exponent beyond what was accessible through the exponent pair method. Their work was extended to yield an exponent pair by M.N. Huxley and N. Watt in [7] and by N. Watt in [13]. This gives us the two pu, vq-points Since the points given in [START_REF]SageMath, the Sage Mathematics Software System[END_REF] are below the curve C, and further point obtained by applying f and or g still stays there. Moreover, as we saw previously, a start like f f gf g localizes the image.

The question then is to find some equivalent form to Theorem 1.2. The function θ 0 (resp. ϑ 0 if we express it in the variables pκ, λq) has (rather strickingly) a simple form, but we may as well replace it by θ ˚(resp. ϑ ˚) which parametrizes the (lower) border of D ˚. {continuousEPstar} Theorem 4.1. The θ ˚be the continuous decreasing convex function on r0, 1s that parametrizes the lower border of D ˚and ϑ ˚be the continuous decreasing convex function on r0, 1{2s defined by ϑ ˚pκq " p2θ ˚p2κq `1q{2. Let S be an exponential sum of monomial type and parameters N and F . Then, for every κ P r0, 1 2 s and every ε ą 0, we have S ! ε F κ`ε N ϑ ˚pκq`ε . The functional equation θ ˚˝θ ˚" θ ˚holds, as a consequence of the symetry pu, vq Þ Ñ pv, uq.

An upper bound for θ ˚is provided by the step-function θ 1 (or ϑ 1 in the variables pκ, λq) that links the following points: This leads to a rather decent approximation of our border, as shown by the next plots.

A remark concerning p-dimensional exponent pairs

In [START_REF] Srinivasan | The lattice point problem of many dimensional hyperboloids. III[END_REF], B.R. Srinivasan developped a multi-dimensional theory of exponent pairs. Roughly speaking, let φpx 1 , x 2 , . . . , x p q be a smooth 'polynomial-like' function whose partial derivative with respect to x i remains of size F i when x i is about N i . Then a pair pκ, λq P r0, 1 2pp`1q s ˆr 2p´1 2p , 1s is said to be a p-dimensional exponent pair when, given the above data and ε ą 0, we have


Jean-Marie Mariotti Center

Alain Chelli

Gilles Duvert

Daniel Bonneau

Guy Perrin

Eric Thiébaut

Denis Mourard

Romain Petrov

Pierre Cruzalèbes

Bruno Lopez

Fabien Malbet

Gérard Daigne

Marc Ollivier

Jean-Marie Mariotti Center for Interferometry

Keywords:

The Jean-Marie Mariotti Center is a network of 11 French Institutes, Laboratories or Observatories, appointed by CNRS in 2000. It coordinates the efforts of the member institutes to offer all the potential users of interferometric facilities the best operational environment, providing software, academic formation and stimulating the prospective on new interferometric developments. At present, besides academic formation, the major effort is focused on the development of the software to prepare the observations, to reduce the data and tointerpret the results in terms of models or reconstructed images. In this contribution, we describe the achievements and the future plans of the Mariotti Center.

INTRODUCTION

Since the 90's, the French interferometric community has promoted the creation of an interferometric center. In September 2000, the Jean-Marie Mariotti Center (JMMC, also known as Mariotti Center) was formally appointed by the French CNRS ("Centre National de la Recherche Scientifique"). The JMMC is a network of 11 French Institutes, Laboratories or Observatories which have interferometric expertise: The Mariotti Center is meant to be a delocalized laboratory, formed by a network of experts and provided with a Center of Coordination located at the Laboratoire d'Astrophysique de Grenoble. It coordinates the efforts of the member institutes to offer all the potential users of interferometric facilities the best operational environment. The mission of the JMMC is threefold and consists on:

• to develop, produce, document and maintain the software necessary for the exploitation and the follow-up of new interferometric equipments, especially the VLTI;

• to stimulate and coordinate the academic formation of non specialists;

• to participate to the prospective around new interferometric instruments.

The main activity of the Mariotti Center, besides prospective and training, is software oriented system analysis and software development. These are meant to be a structural part of the instrument and, ideally, conceived right from the beginning of the instrument development. At present, the JMMC is formed by 25 researchers, who operate in working groups and via actions of formation (equivalent to 4 researchers full time), 5 PhD students and 2 engineers.

WHY A CENTER FOR INTERFEROMETRY?

An historical review will help to answer this question. Modern interferometry in the optical domain was born in France in the 70's 1 . However, two decades have been necessary for it to become perceived as a major observational technique by the astrophysical community. Besides the experimental complexity and the novelty of the technique for the optical community, this was likely due to the difficulty of controlling and hence of calibrating the signal, giving the impression of a not much reliable technique. The situation has radically changed in the last decade, thanks to two reasons: 1. the better reliability of adaptative optics techniques has allowed to concentrate a large part of the flux within the Airy disc and hence to boost upwards the limiting magnitudes of optical interferometers, making possible observations in sensitive astrophysical domains, like star formation, extra-solar planets or nucleus of active galaxies; 2. the introduction of single mode fibers has enabled to control the interferometric signal and hence to calibrate it (at least to the percent level) with automatic procedures transparent to the user, hence opening us the use of interferometric facilities to non specialists and to a larger number of investigators. These two developments have been decisive in the choice of constructing today's large interferometric facilities: the Very Large Telescope Interferometer 2 and the Keck Interferometer 3 . However, the VLTI has a unique specificity: it is conceived as a service instrument [START_REF] Schoeller | Commissioning the VLT Interferometer: from first fringes towards a genral user facility[END_REF] . Any astronomer will be able to ask for telescope time with the VLTI and to observe as a visitor astronomer, going back home with the software, developed by the consortia building the instruments (AMBER and MIDI), to reduce his/her data. Nevertheless, this is not sufficient to make the VLTI a real service instrument. Indeed, it is necessary to develop a bunch of software:

1. to prepare the observations (observing proposal), that is to say to investigate the feasibility with the prediction of errors on the pertinent astrophysical parameters;

2. to select calibrators, that is to say point sources (not resolved by the interferometer, unit visibility sources) or sources with known visibility;

3. to interpret the astrophysical observables in terms of astrophysical models;

4. to reconstruct the object spatial brightness distribution if the UV plane coverage allows it.

These developments together with the organization of interferometric schools for the future users correspond to the minimum requirement necessary for the optimum use of the VLTI. They will need to be complemented on the mid and long term by the follow-up of the instruments in order to optimize the data reduction software provided by the consortia. All these are the goals of the JMMC.

In the following, we describe the development plans of the Mariotti Center. They can be divided into three types of actions: software developments, prospective, training and assistance to users.

RESEARCH AND ALGORITHMIC DEVELOPMENTS

The first goal of the Mariotti Center has been to fulfill the software needs described above with the creation of four groups of research and developments. Most of these software developments are meant to be transparent to the use of a given interferometer or instrument.

Preparation of observations: the ASPRO software

This software tool called ASPRO 5 allows to prepare interferometric observations with the instruments VINCI 6 , AMBER [START_REF] Petrov | Using AMBER: the near infrared VLTI focal instrument[END_REF] and MIDI 8 of the VLTI, and also with the GI2T [START_REF] Mourard | Status report on the GI2T interferometer[END_REF] . It is provided with a modular structure which allows to incorporate any other interferometer (IOTA 10 , CHARA/FLUOR 11 12 , PTI [START_REF] Lane | PTI status report[END_REF] , OHANA [START_REF] Perrin | OHANA: the Optical Hawaiian Array for nanoradian astronomy on top of Mauna Kea[END_REF] , KI, etc...) by updating 2 modules: the geometrical configuration of the interferometer and the signal to noise ratio calculator of the instrument. The other modules are transparent.

At present, ASPRO contains a set of pre-defined geometrical models (collection of point sources, Gaussian sources with or without envelopes, etc...) covering most of the needs. By fixing the observation and selecting a model, it can predict the error on the model parameters with visibility and/or phase closure constraints. In the near future, we will implement simple differential models (rotating stars and multiple sources with or without envelopes) using differential phase constraints. At that stage, ASPRO will be able to combine visibility, closure phase and differential phase information. If necessary, the user will be able to use in the future its own analytical model, and -why not -its own numerical model.

The first version of ASPRO has been delivered to the community in September 2001. The second version, provided with a Web interface and simplified procedures, will be delivered at the beginning of 2003. The next three modules described below will be interfaced with ASPRO.

Search of calibrators

The goal is to provide a tool that allows to select the reference stars required to calibrate the interferometric observations. This tool has to be as general as possible so that it can be used for observations with the VLTI, but also with any other interferometer. For this, it is necessary to develop an expert system which, from the physical parameters characterizing a given object, allows to conclude if it can be used or not as a calibrator. This means that the expert system has to be able to predict the visibility of the object at a given spatial frequency.

Given the difficulty to create a catalog of stars containing the useful informations to define a calibrator, it has been decided to develop a tool allowing the user to create, from an on line consultation of the CDS catalogs, a list of stars from which he/she will be able to select the reference stars most adapted to his/her scientific program. This approach is complementary to the ESO approach which consists to build a catalog of interferometric results: the CHARM catalog [START_REF] Richichi | [END_REF] . The list of possible calibrators provided by the expert system will be cross-checked with the CHARM catalog.

The first version of this calibrator selection tool will be delivered to the community at the beginning of 2003.

Modeling the interferometric observables

It is well known that the sparse UV plane coverage of an interferometer working with a small number of telescopes (let us say up to 5) does not allow in general to reconstruct the image of the observed object. In this case, it is necessary to model the interferometric observables. The analysis of the modeling problem can be summarized into two questions:

-what is the exact relationship between the interferometric observables and the object?

-what is the quantity (χ 2 ) to be minimized to estimate the astrophysical parameters?

The answer to the first question is instrument dependent. However in the case of the VLTI, for compact objects (smaller than the Airy disc) and even at low spectral resolution (a few tens), the problem of the exact relationship between the interferometric observables and the object arises only in the case of chromatic objects when looking for very high dynamics as for extra-solar planets detection. In most cases, we can consider that the interferometric observables are instrument independent. With respect to the second point, the main question is to know if we need to constrain the phases or complex quantities. If the visibilities are well calibrated (no systematic errors), it is preferable to work with the complex bispectrum. If this is not the case, we need to work with the closure phase in order to keep the advantage of the self-calibration. For differential measurements the use of the complex cross-spectrum in a well suited estimator is appropriate.

We are developing a modeling software to ajust the interferometric observables with the geometrical/differential models defined in the ASPRO software and if necessary with user models. The first version of this modeling software will be delivered to the community during the second semester of 2003.

Image reconstruction

The possibility of reconstructing the images from interferometric measurements is of prime importance in order to identify and to interpret the environment of the observed objects with milliarcsecond spatial resolutions. However, the irregular and sparse coverage of the UV plane makes the reconstruction difficult. In radioastronomy, this problem has been largely covered and image reconstruction algorithms have been developed (CLEAN, Maximum of Entropy, WIPE). Nonetheless, whereas radio interferometers measure the complex visibility, optical interferometers measure the modulus of the complex visibility and not directly the phase, but phase closure relations. Due to these differences, the image reconstruction methods developed in radioastronomy are not directly applicable to the case of optical interferometry. Hence, it is necessary to develop image reconstruction methods dedicated and adapted to the nature of the observables produced in optical interferometry. Our objectives are the following:

-to develop and provide algorithms of image reconstruction from the observables of optical interferometers.

These algorithms will produce an image, but also an estimate of the image quality (error bar on the intensity distribution, effective spatial resolution, reconstructed field, etc...);

-to develop an expertise on the optimization of the telescopes configuration in the phase of preparation of the observations. This expertise will be acquired by processing simulated data or real data provided by the interferometers. In due time, this expertise will allow to optimize the telescope time required to perform the desired observations.

A first version of the image reconstruction software from optical interferometric observables has already be written [START_REF] Thiebaut | Imaging with the AMBER-VLTI: the case of microjets, WS VLTI -The Very Large Telescope Interferometer Challenges for the Future[END_REF] . A user friendly version will be delivered to the community in 2004.

Expert tool for instrumental follow-up

In optical interferometry, there is a long way between the acquisition of raw data and the production of astrophysical observables. The calibration problems are highly relaxed when using single mode spatial filters and the interferometric instruments (like AMBER for example) are likely to achieve their specifications. But how close they can achieve their most ambitious goals will be known after quite some actual observations. Hence, the software packages delivered to the users together with the instruments might not be perfectly optimized.

The achievement of the ultimate performances of the VLTI will require a long term follow-up and analysis of the instrument behavior in correlation with its environment in order to identify and to quantify the long list of instrumental, atmospheric and astrophysical biases. In this context, some of the software tools to model marginal instrument errors from environmental or technical information still have to be developed. This is unavoidable to be able to observe very faint sources and to reach the high image dynamics required for example for the detection of extra-solar planets.

Besides the development of standards algorithmic tools, the fine analysis of the interferometric signal in a long term perspective to reach the ultimate performances of the instruments is one of the goals of the Mariotti Center.

PROSPECTIVE

In this section, we make some prospective analysis of the software needs which will be required to optimize future interferometric developments.

Nulling and high dynamics interferometry: DARWIN/GENIE

Nulling interferometry is a technique which combines high angular resolution and high dynamics [START_REF] Serabyn | Nulling interferometry[END_REF] . Its consists to cancel with destructive interferences the light coming from a star to enhance with constructive interferences an object (stellar or planetary companion, disc) in its vicinity. This technique allows to study the source in a large spectral domain, and combining the informations at each wavelength, to produce image maps of the observed region. To improve the performances and the sensitivity of the technique, it is necessary to modulate the signal of the source. This implies a complex processing of the data resulting from the observations in order to extract the pertinent spectral and spatial scientific information. The DARWIN spatial mission of ESA [START_REF] Gondoin | Darwin infrared space interferometer[END_REF] , to be launched in 2014, is based on the nulling concept.

To date, only very simple algorithmic approaches have been developed to estimate the performances of the instrument. In order to prepare the DARWIN mission, it will be necessary:

-to propose optimized algorithms for processing the data from nulling interferometers, which take into account the spectral information and all possible sources of modulation (internal modulation of the interferometer response, full rotation of the interferometer, etc...). These algorithms will have to be capable to extract the signal of the object to observe from the residual signal of the central star, the signal of the zodiacal environment of this star, and the other sources of noise related to the instrument itself;

-to study different geometrical configurations of the interferometer in order to define an optimal configuration.

The algorithms defined for DARWIN will be tested with the GENIE instrument [START_REF] Gondoin | Darwin Ground Based European Nulling Interferometer Experiment (GENIE)[END_REF] . GENIE is an ESA/ESO instrument which, if built according the present schedule, will see its first light on the VLTI in 2006. It will combine several telescopes (UT and/or AT) of the VLT, and provide a full scale demonstration of what DARWIN can perform. The extra difficulty of an observation from the ground (atmospheric turbulence, sky emission, etc...) will require to adapt the algorithms defined for DARWIN to the GENIE case.

Optical astrometry and phase reference imaging: PRIMA

PRIMA (Phase-Referenced Imaging and Micro-arcsecond Astrometry) [START_REF] Paresce | Scientific objectives of ESO's PRIMA facility[END_REF] is an instrument which will use the dual-beam capability of the VLTI by observing simultaneously two distinct fields: (i) to perform high precision differential astrometry and (ii) to observe very faint objects with phase reference. The main astrometric objectives of PRIMA concern the characterization of extra-solar planetary systems by observing the reflex motion of the central star, the study of the mass distribution towards the galactic center, the resolution of events like gravitational lensing, etc... The main goal of phase reference is the interferometric observation of very faint objects by controlling the atmospheric piston on an object of the dual field sufficiently bright for it. To prepare the PRIMA experiment, it will be necessary to model completely the instrument in order to perform a precise metrology between the two fields and to propose optimized algorithms to extract the differential signal.

Second generation of VLTI instruments

After more than ten years of developments, the VLTI has progressively reached its full capacity with the commissioning in 2003 of its main elements: the 10 µm recombiner instrument MIDI, the 3 telescopes near infrared recombiner AMBER, 3 auxiliary telescopes, adaptative optics modules for the 8m telescopes and fringe sensing units.

During the workshop in June 2001 on the second generation instruments, ESO has considered two priorities for the VLTI: the development of a 6 to 8 channels recombiner instrument and an extension to shorter wavelengths. These developments, that are often considered in terms of hardware development, call for an associated effort in software development which is fundamental to optimize the scientific return of the instruments. In the coming years, we will need:

-to optimize the geometrical configuration of the interferometer for the problem under study; -to propose optimized algorithms to extract the pertinent information.

Participation to ASHRA prospective

One of the mandates of the ASHRA ("Action Spécifique de Haute Résolution Angulaire"), french council having in charge the prospective, is to define a mid and long term development strategy for optical interferometry [START_REF] Arnold | High angular resolution in 2010-2020: a comparison between possible post VLT/VLTI instruments[END_REF] . For this purpose, a strategy group piloted by ASHRA and transverse to ASHRA and JMMC, has been formed. The working plan of this group is the following:

-definition of the interferometric field in terms of resolution, extent and dynamics;

-definition of observational strategies in terms of frequency coverage;

-definition of image quality criteria adapted to optical interferometry;

-simulation software tools.

TRAINING AND ASSISTANCE TO USERS

Interferometry opens up a new era in optical astronomy. The instruments installed on the VLTI will allow to push the limits of the present studies based on multiple systems, determination of stellar diameters and observation of circumstellar environments of late type stars, towards new fields such as stellar formation, extra-solar planets and, for the first time, the immense field of galaxies [START_REF] Malbet | Astrophysical potential of the AMBER/VLTI instrument[END_REF] .

The VLTI infrastructure has been the fruit of a collaboration between ESO and the European Institutes. The performances of the VLTI and its instruments are unique in terms of sensitivity and of angular resolution thanks to the large collecting area of the 8m telescopes and to the numerous available baselines via the auxiliary telescopes, up to 200 m. These performances open new possibilities never reached before and require the European community to be prepared to the specificities of interferometric observations. It is then necessary for the interferometrists participating to the different instruments to transmit their knowledge to the whole astrophysical community. It is with this aims that the Mariotti Center has already begun to train French and European astronomers during a workshop (Nice, October 2001) and a European school (Les Houches, February 2002). To optimize the exploitation of the VLTI and interferometers in general, it is of prime importance:

-to provide interferometers and instruments conceived to be accessible also to non specialists; -to provide simple tools allowing astronomers to investigate the "mysteries" of interferometry (simulators, signal to noise ratio calculators, etc...);

-to pursue the efforts to develop imaging techniques;

-to provide schools for training to the observations, preparation of observations, data reduction techniques, etc...;

-to organize permanent assistance teams for each VLTI instrument and the relevant software.

SUMMARY

The Jean-Marie Mariotti Center, appointed by french CNRS in 2000, is a newborn expertise center that coordinates the efforts of the member institutes to offer all future users of interferometric facilities the best operational environment, providing software, academic formation and stimulating the prospective on new interferometric developments. Its first goals have been to fulfill the software needs to prepare the observations, to select calibrators and to interpret the interferometric observables in terms of models or reconstructed images. In parallel, it has already organized two schools to educate the European astronomers to the "mysteries" of interferometry and started a network team of specialists for permanent assistance around each VLTI instrument and relevant software. In the next years, it will pursue this task and develop software for the next generation of VLTI instruments. The Jean-Marie Mariotti Center actively supports the creation of the future European Network for Interferometry.

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  Département d'Astrophysique (Université de Nice-Sophia Antipolis, UNSA) -Département Fresnel (Observatoire de la Côte d'Azur, OCA) -IAS (Institut d'Astrophysique Spatiale, Orsay) -LAOG (Laboratoire d'Astrophysique de Grenoble) -LAT (Observatoire de Midi-Pyrénées, Toulouse) -LESIA (Observatoire de Paris-Meudon) -LISE (Observatoire de Haute Provence) -IRCOM (Université de Limoges) -Observatoire de Bordeaux -Observatoire de Lyon -ONERA (Châtillon) Further author information: (Send correspondence to Alain Chelli) Alain Chelli: E-mail: Alain.Chelli@obs.ujf-grenoble.fr, Telephone: 33 4 7663 5837 Address: Laboratoire d'Astrophysique de Grenoble, UJF, BP 53, F-38041 Grenoble Cedex 9 JMMC-PUB-0000-0001 Jean-Marie Mariotti Center SPIE, Hawaii -22-28 August 2002

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  to design simulators to specify the second generation of imaging instruments; JMMC-PUB-0000-0001 Jean-Marie Mariotti Center SPIE, Hawaii -22-28 August 2002-to elaborate precise methods to calibrate the useful signal;


Fredrik T Rantakyrö
email: frantaky@eso.org

Pascal Ballester

Stephane Brillant

Gilles Duvert

Emmanuel Galliano

Christian Hummel

Andreas Kaufer

Mario Kiekebush

Etienne Le-Coarer

Tom Licha

Fabien Malbet

Pedro Mardones

Florentin Millour

Sebastien Morel

Isabelle Percheron

Monika Petr-Gotzens

Romain Petrov

Andrea Richichi

Thomas Rivinius

Markus Schöller

Stan Stefl

Martin Vannier

Markus Wittkowski

Gerard Zins

Experiences from the first AMBER open time observations

Keywords: AMBER, VLTI, VLT, Science Operations, Paranal, ESO

The VLTI has been operating for about 5 years using the VINCI instrument first, and later MIDI. In October 2005 (Period 76) the first Science Operations with the AMBER instrument started, with 14 Open Time proposals in the observing queues submitted by the astronomical community. AMBER, the near-infrared/red focal instrument of the VLTI, operates in the bands J, H, and, K (i.e. 1.0 to 2.5 micrometers) with three beams, thus enabling the use of closure phase techniques. Light was fed from the 8 m Unit Telescopes (UT). The Instrument was offered with the Low Resolution Mode (JHK) and the Medium Resolution Mode in K-band on the UTs. We will present how the AMBER VLTI Science Operations currently are performed and integrated into the general Paranal Science Operations, using the extensive experience of Service Mode operations performed by the Paranal Science operations and in particular applying the know-how learned from the two years of MIDI Science Operations. We will also be presenting the operational statistics from these first ever Open Time observations with AMBER.

INTRODUCTION

The ESO Very Large Telescope Interferometer (VLTI) located at Cerro Paranal in northern Chile is a part of the ESO Paranal Observatory. The VLTI is a general purpose interferometric facility that offers various instruments to the general astronomical community: VINCI 1 (no longer in use in science observations), MIDI 2 (a two beam mid infrared instrument, with spectroscopic capabilities from R=30 to 230), and AMBER 3 (a three beam JHK band instrument with spectroscopic capabilities from R=30 to R=12000). This paper focuses on the experiences from the Science Operations with AMBER in observing Period 76 (P76) covering October 2005 to March 2006. In Section 2 we present the VLTI and in Section 3 we give a brief introduction to the AMBER instrument. In Section 4 a brief presentation of the general Science Operations is given, in Section 5 details the specifics of Science Operations with AMBER, Section 6 is dedicated to the operation and statistics for the P76 and finally Section 7 presents the conclusions. A schematic view of the VLTI where the telescope pads for Auxiliary Telescopes (ATs) and siderostats are visible, as well as the four UTs. In the center are also shown the delay line tunnels, VLTI control building and the interferometric lab. The AMBER observations are currently carried out with three UTs simultaneously. Two ATs are in regular science operations with MIDI and one is currently being commissioned at Paranal and the remaining fourth AT will be installed in 2006.

THE VLTI

The general concept of the VLTI is to provide an interferometric focus to the instruments, like modern telescopes provide almost diffraction-limited beams to their instruments. Therefore the VLTI infrastructure works like a general facility which provides the following functions:

• Sampling of the (u, v) plane with four fixed Unit Telescopes (UTs) and four transportable Auxiliary Telescopes (ATs) with baselines ranging from 8m to 200m.

• Collection of light with four 8 m Unit Telescopes (UTs) and three 1.8 m Auxiliary Telescopes (ATs). See Figure 1 for an aerial view of the telescope positions.

• Wavefront correction at the telescopes: in the first phase adaptive optics for the UTs (MACAO) and tip-tilt correction for the ATs (STRAP).

• Transportation of the primary and secondary beams from the telescopes to the focal lab.

• Compensation by the delay lines (DLs) of the optical path difference due to the sidereal motion.

• Correction of the slow (<1 Hz) tip/tilt motion of the beams (caused by seeing effects in the tunnel) by means of a fast detector sensing the beam motions and sending corrections to the X-Y table so that the beams are kept centered on the optical axis (IRIS). IRIS uses the H-band for the guiding so if IRIS is used no H-band light goes to AMBER. All these subsystems need to work perfectly to allow successful AMBER observations, thus engineers are performing a rigorous maintenance plan of all these systems before any VLTI observations are executed with either AMBER or MIDI. AMBER, the near-infrared focal instrument of the VLTI is located in the VLTI lab. It is capable of operating in the bands J, H, and, K i.e. 1.0 to 2.5µm. AMBER can combine the light from three telescopes simultaneously, thus enabling also the use of closure phase techniques. It works with spectrally dispersed single mode fringes and thus gives spectral resolutions R = 35, LR (Low Resolution Mode using a prism -with simultaneous observation of the full J, H and K bands), R = 1500, MR (Medium Resolution Mode using a grism -observing in the J, H or K band) and R = 12000, HR (High Resolution Mode using another grism -observing in the J, H, or K bands).

THE AMBER INSTRUMENT

In Period 76 AMBER was offered in the LR and with the MR Mode in K band only. The magnitude limits of AMBER in P76 were K=7 with LR on UTs and K=4 with MR-K on the UTs. As AMBER is an instrument still under commissioning several new modes are expected to be offered to the astronomical community (e.g. in P78 the High Resolution K-band mode was opened for Open Time Science Operations), similarly the current magnitude limits are conservative estimates and expected to be increased with further developments in data reduction and improvements in VLTI subsystems.

GENERAL SCIENCE OPERATIONS

The aim of the VLTI Science Operations is that it should be working seamlessly within the standard Science Operations frame in use for the non-VLTI VLT UT instruments. The VLTI Science Operations(SciOps) at PSO 4 is quite complex and here we highlight only the issues relevant to the AMBER Science Operations in P76. The objectives for AMBER Science Operations are the same as any instrument at the PSO. These basic objectives [START_REF] Silva | Service Mode Scheduling: a primer for users[END_REF] for the Service Mode operations are:

• The scientific objectives of an observing run are not achieved unless all observations are completed.

• A run should not be scheduled unless it has a high probability of completion.

• It is better that a smaller number of runs are totally completed than that many runs are incomplete.

• If the observing conditions permit, the runs with higher scientific priority as defined by the OPC should be completed preferentially over lower-priority runs.

These objectives guides the planning and execution of the desired observations for all VLT and VLTI instruments. Apart from these objectives there is another cornerstone of the VLT/VLTI Science Operations which for the VLTI can be phrased as:

The VLTI with its selected instrument (MIDI or AMBER) should be completely transparent to the user whether in Service Mode or as a Visitor. He/she can stop worrying about the functioning of either the telescopes or the instrument, and instead just concentrate on achieving the scientific goals of his/her project. The Science Operations with MIDI have shown that this goal has been achieved.

AMBER SCIENCE OPERATIONS SPECIFICS

In addition to the standard VLT Science Operations constraints and requirements, operating with AMBER has a few additional ones:

• AMBER was offered in P76 with all possible combinations of three UTs, yielding four possible baseline configurations. Thus not only the observation has to be performed with the normal observing constraints but also using the desired baseline configuration requested by the user.

• By giving a LST constraint for the observation the user can select the projection angle of the three baselines on the sky when his/her target is observed. As many scientific targets in P76 are asymmetrical (stars with winds, disc around stars etc) the LST constraints were frequently used and were of fundamental importance to achieve the requirement of the user.

• For each science target there was a calibrator associated with it. This calibrator had to be observed either directly before or after the target of scientific interest. Without this calibrator the observation of the science target has no scientific validity. As each observation, either science or calibrator, takes one hour the minimum time slot for a successful AMBER observation is two hours.

How these constraints affected the AMBER Science Operations is discussed in the following section.

P76 OPERATIONS

The VLTI runs with the MIDI and AMBER instruments were typically scheduled on the UTs as a block of three to five nights every month. These blocks were shared between MIDI and AMBER and involved different telescope configurations, and mixed Visitor Mode and Service Mode operations. To assure that the VLTI setup including the telescopes, active optics and delay lines (see Section 2) were ready for the operation there was a scheduled countdown of checks of each of the subsystems before each observing block. These checks were finalized with technical tests of the instrument on the sky with all subsystems running to ensure that all were ready before the start of the observing run.

To allow flexibility in baseline configurations there were at least of three UTs scheduled in each block and occasionally there were 4 UTs available for use with the VLTI instruments, allowing the use of different baseline configurations with AMBER and six with MIDI. The night astronomer that performed the operations could select the proper baseline for each part of the night and the UTs not in use were used for non-VLTI Service Mode operation. Thus in a particular night, half the night could be spent on MIDI operations switching twice between two different baseline configurations and the end of the night dedicated to AMBER observations with a particular telescope triangle. As each change between instruments and/or baseline configuration required approximately 30 minutes the astronomer had to evaluate the final scientific output based on using different baseline configurations, instruments and fulfilling the LST constraints versus the time lost between configurations.

AMBER Observations in P76

In period 76 there were 49 programs with 77 observing runs submitted as proposals for AMBER. In the end 21 programs with 24 observing runs were finally accepted, of which 12 were Service Mode programs. This meant that 71 observing blocks (OBs) were to be executed in Service Mode for AMBER in period 76. For this approximately 14 nights in total were allocated for AMBER observations split between Visitor Mode programs and Service Mode programs, assuming an average of 10 hours per night this corresponds to a total of 140 hours of observations. Of these 14 nights nine were dedicated to service mode observations and five nights to Visitor mode observations. Table 1. Observing time statistics for AMBER in observing P76 covering October 1st, 2005 to March 31st,2006. Time, the total actual time used in this mode, OBs, the number of OBs executed in this mode, Successful, is the number of OBs that are classified as successfully executed (i.e. this considers that both the Science and Calibrator OBs in a pair has both to be executed successfully to be considered a success), Failed, is the number of OBs that failed, Not Executed, the number of OBs scheduled but not executed in this period and Reclassified, the number of OBs that originally were successfully executed but failed as the corresponding Science or Calibrator OB failed. Note that in the case of Visitor Mode programs no classifications were done on the quality of the observation, and thus columns Successful, Failed, Not Executed and Reclassified have no entries.

Observing Mode Time

OBs The difference between allocated times and the used times are that telescopes were handed back to non-VLTI SciOps as there were no programs in the queues for that particular time and/or the weather conditions did not allow AMBER operations. Instead of causing weather time losses, the individual telescopes were handed back to non-VLTI Science Operations that had programs that could be executed under those conditions and still be classified as successful. Even though telescopes were handed back to standard VLT operation it still meant that due to the rapid change in weather that programs were partially executed and had to be aborted and thus causing a loss.

In Table 2 we give the amount of time due to losses, classified as Technical, Weather and Operational. Operational losses were only due to accuracy of the used Optical Path Difference (OPD) models (the OPD models are used to move the delay line carriages to positions that compensate for the different lengths that the light has to travel from each telescope to reach AMBER). Weather losses do not necessarily mean clouds, but most commonly that the atmospheric conditions were such that the Adaptive Optics (MACAO) could not perform adequately due to the fast atmospheric turbulence (i.e. low τ 0 ) and/or poor seeing. Technical losses were a mixture of hardware and software problems, ranging from problems with technical CCDs, delay line metrology to software issues with computer control problems. We define the effective time used by AMBER as the time actually used by the instrument and subtracting the time lost due to Technical, Weather, and Operational losses. The "Success Time" is defined as the time needed to execute successfully one OB, either Science or Calibrator. This number only reflects the time needed for the execution of an individual OB and do not reflect the Calibrator-Science pair success rate. As eight OBs were completed within constraints, passing the pipeline quality control criteria, but "Reclassified" (see "Reclassified" in Table 1), these were counted as successes for purposes of calculating the "Success Time". We have also counted two OBs that were formally executed within constraints, but failed as they were more extended than expected by the user and thus had too big contrast loss to fulfill the pipeline quality criteria. The total number of successfully observed OBs were thus 40. In Table 3 we present these numbers with numbers of percentage completed of programs and runs (each program consists of one or several runs).

CONCLUSIONS AND THE FUTURE

Five main conclusions have been drawn from our experience with Science Operations in P76:

1. Despite being the first Open Time Science Observations with AMBER the actual time needed to obtain a successful observations was close to the estimated execution time of 1 hour. Including the failed observations the actual execution time was much shorter than predicted.

2. Although we successfully executed 30 out of 71 observations (43%), actual run completion was only 29%.

3. The quality of the OPD models (that are good enough for MIDI and AMBER in Medium Resolution) were not accurate enough to allow precise and successful AMBER Low Resolution (LR) observations, thus causing a 12% effective downtime.

4. The two hours needed to obtain a calibrated uv-point make AMBER very sensitive to abrupt changes in weather conditions. The system also has to be working without technical interruption for this time a technical problem however minor can jeopardize a successful observation.

5. A total Technical downtime of 10% sounds much but as AMBER is sensitive to any problem on any of the three telescopes this translates to a 3.5% downtime for each telescope, a very good number.

Of these conclusions items 2, 3, 4 represent problems that need to be solved. Below we present the solutions to the three major problems above:

1. The issue of the quality of the OPD models was addressed when this became apparent as a significant cause of downtime. Historically these models were created from the offsets determined during normal science observations with MIDI and thus the sky coverage was poor and sources with high proper motions were included in the fit, and increase significantly the RMS of the fitted model. The new procedure to determine an OPD model is to allow the dedicated use of the UTs in pairs with MIDI with the sole purpose of collecting OPD model data. The sources were selected to have no or very small proper motions and observed all over sky and thus allowing an even sky coverage.

2. By improving the way that the objects are acquired and light is centered on the AMBER fibers we have managed to decrease the time needed to perform one observation by 15 minutes in P77. The total time for the execution of one OB is 45 minutes in P77 and work is in progress to parallelize the fringe acquisition so that we can in the end reach the one hour for each science and calibrator pair. By reaching a total execution time of one hour we should be much less sensitive to any abrupt changes in the external weather conditions and/or technical problems.

3. Even with the best model the fringes may drift slowly with time and this decreases the sensitivity of the instrument. As a typical full set of exposures takes 15 minutes this slow drift could then affect the data quality. Thus in the end of Period 76 it was implemented a coherencing routine that recenters the fringes on all baselines after each single exposure. The results in the last runs of P76 and in the beginning of P77 have been excellent. The fringes are frequently stabilized and thus the last exposure has the same quality as the first one taken just after the fringe acquisition.

4. Even using effectively all the allocated time it is difficult to complete runs/programs that includes different baseline configurations and setups as it cannot be guaranteed that all baseline configurations are available at all times, thus we recommend the users when submitting new programs so that any scientific program is designed so that scientifically meaningful results can be achieved in a single night. With the ongoing improvement in execution times we will be able to observe more targets in a given time allocated to a given baseline configuration, and with even further improvements in baseline configuration flexibility this suggestion can be removed,

Figure 1 .

 1 Figure 1. A schematic view of the VLTI where the telescope pads for Auxiliary Telescopes (ATs) and siderostats are visible, as well as the four UTs. In the center are also shown the delay line tunnels, VLTI control building and the interferometric lab. The AMBER observations are currently carried out with three UTs simultaneously. Two ATs are in regular science operations with MIDI and one is currently being commissioned at Paranal and the remaining fourth AT will be installed in 2006.

Figure 2 .

 2 Figure 2. The AMBER instrument on its optical table in the VLTI lab.

Table 2 .

 2 Time accounting of losses due to different reasons during AMBER Science Operations in P76. Weather losses are when weather conditions were such that it interfered with the ongoing execution of an AMBER program. Technical losses are split up between Hardware (HW) and Software (SW) caused losses, Operational losses represent problems with OPD models and Total the total time lost in period 76.

	Obs Mode Weather Tech(HW):Telescopes Tech(HW):VLTI Tech:SW Operational Total
	Service	5h 33m	6h 25m	1h 37m	0h 30m	5h 10m	19h 15m
	Visitor	0h 0m	4h 55m	0h 30m	1h 30m	8h 13m	15h 08m

Table 3 .

 3 AMBER statistics for Period 76. Success Time, the total time (corrected for losses) divided by the number of successful OBs (counting Calibrator and Science OBs separately) and represents the average time needed to obtain a successfully executed OB. Execution Time, is the total corrected time divided by the total number of executed OBs, Runs, is the percentage of runs completed and Programs, the percentage of programs completed, where the last two are only relevant for Service Mode operations.

	Observing Mode Success Time Execution Time Runs Programs
	Service	1h 6m	0h 46m	29%	25%
	Visitor	N/A	0h 34m	n/a	n/a

ACKNOWLEDGMENTS

Without the hard work of the many people at the Paranal Observatory the Science Operations would not be possible. In particular we would like to thank the people directly involved with the Science Operations, i.e. the Software Support (B. Bauvir and A. Ramirez), Instrument Support Engineers (N. Haddad), TIOs (H. Alarcon, S. Cerda, C. Esparza, L. Faundez, C. Herrera, A. Lopez and A. Pino), the people at the User Support Group in Garching (S. Marteau, P. Nass, M. van den Ancker), PSO General Operations (O. Hainaut, C. Ledoux and R. Schmutzer), Dataflow (C. Ceron, A. Correa, J Parra, D. Preminger, and R. Sanhueza) and all others too many to name (group leaders, engineers, technicians, secretaries, the general support staff) that makes the Paranal Observatory a well functioning operation.