complex systems are ubiquitous in nature . on all scales from genes to societies , various real systems are composed of many units which collectively perform complicated tasks  . in the recent years , the framework of complex networks became recognized as an excellent formalism for studying complex systems . by modeling units as nodes and their interactions as links  , graph analysis methods entered physics , biology , engineering and even sociology  .this allowed for a variety of real and artificial complex systems to be extensively examined , typically via computational modeling  .crucial aspect of a complex network is its _ structure _ , i.e. the topology of connections among its nodes . properties of network structure dictate its global behavior , and are key to understanding the network s functioning and potentials for its control . for phase - repulsive oscillator models , profound intertwinement between network structure and network dynamicswas recently shown  .  since the structure of many natural networks is only partially known , it is of central interest to develop methods for reconstructing the network topology from the available empirical information .various experimental techniques in this directions are already in use , specially in the context of gene regulation networks  . the question of network reconstruction is entering new fields , such as climatology  and neuroscience  .recently , the topology of a social network was inferred using mobile phone data  . in the context of oscillator networks ,various theoretical  as well as experimental  results are available .in addition , a range of mathematical results is also available  .however , most interest lies in detecting the structure of real networks from experimentally measurable outputs , such as time series  .theoretical reconstruction methods usually rely on examining computational network models , and are generally divided into two classes ._ invasive _ methods involve interfering with the network dynamics via controllable perturbation , which allows for structural data to be easily extracted  .these methods generally give very good results , specially when the usage of perturbation is not limited in strength and frequency .however , it is often unpractical or even impossible to interact with the on - going network dynamics ._ non - invasive _ reconstruction methods focus on investigating the observable network outputs , such as time series of the node s dynamics , or similar measurable quantities describing the internal network states  .the relevance of the non - invasive approach is increasingly recognized , particularly due its suitability for detecting links in biological networks  . however , by not needing to interfere with the network dynamics , non - invasive approaches are typically limited to studying equilibrium network behavior , which often contains only a small amount of information on the system .alternatively , reconstruction methods also rely on other mathematical techniques , such as control theory  , compressive sensing  , adaptive random walk  , and even treat non - equilibrium scenarios  .it is also worth noting that , in somewhat different context , there is a large interest in investigating networks induces from time series  , which allow for higher - order statistical analysis of data .  in this contribution, we propose a conceptually novel non - invasive network reconstruction method .classical studies usually involve correlations among the dynamic variables , i.e. network nodes  . on the other hand , typical models of network dynamics rely on first - order differential equations ( in genetic interactions ,it is known that one protein s concentration determines the increase or decrease of another protein s concentration  ) .inspired by this , we examine the correlation between _ the variable of one node , and the derivative of another node_. our central assumption is the precise knowledge of the functional forms of the intra - network interactions . as we show , depending on the quantity of network information contained in the empirical data , our method can give very precise results even for time series of length comparable to network size .apart from being non - invasive , our method is conceptually very simple and easy to numerically implement .based on similar hypothesis , shandilya and timme recently proposed another reconstruction method  .in contrast to their result , our method avoids solving the overdetermined linear system , and in principle allows for the reconstruction error to be estimated .the paper is organized as follows : after exposing the reconstruction method in next section , we illustrate its implementation via simple example in section [ results ] . a generalized framework of our reconstruction method in presented in section [ generalized ] .the discussion of our findings and conclusions are given in section [ discussion ] .we consider a complex system composed of interacting units , which we represent as a network consisting of nodes , whose links model the node interactions .each node is assigned a dynamical state defined by the variable , where .our first assumption is the possession of the dynamical trajectories , which describe the network evolution over a certain time interval .the available data consists of sequences , each containing values .the measurements of are separated by the observation interval , which defines the resolution of the time series ( sampling frequency ) .the time interval is uniform and assumed smaller than the characteristic dynamical time scale .we further assume the evolution of the node to be governed by : where the local node dynamics is described via function , and the network ( inter - node ) coupling by the function .the network structure is encoded in the adjacency matrix , whose element specifies the strength with which the node acts on the node .the complete dynamics of the node is the cumulative effect of the local dynamics and the contributions from its neighbors that come with different strengths .our final assumption is the exact knowledge of both interaction functions and .we seek to reconstruct the network adjacency matrix under the named assumptions . by having the `` fingerprint '' of the network behavior ( time series of the node trajectories ), we attempt to reveal its structure .many natural systems are modeled using network equations such as eq.[eq-1 ] : examples include gene regulation and neural interactions , for which the interaction functions are widely investigated .modern experimental techniques allow for high - resolution measurements of quantities such as gene expression , although thus obtained time series are typically short .inter - dependence between two variables is usually quantified through correlation  , while the network models usually rely on expressing the time derivative of a node as a function of other terms , as in eq.[eq-1 ] .inspired by this , we investigate the _ derivative - variable correlation _ between the node ( variable ) and the node ( derivative ) .we define the following matrices :  denotes the time - average of a dynamical quantity ( i.e. , the average over the recorded time - evolution ) .we now re - write the eq.[eq-1 ] in the matrix form : which is our main reconstruction equation . to obtain a more stable function and derivative estimates ,we introduce a new set of time points  : so that /\delta_t ] .we rely on this calculation scheme for the computational implementation of our theory in the next section . in principle , our method is applicable to any directed or weighted network .the reconstruction is precisely correct in the limit of large dynamical data .however , since the obtainable data are not only finite , but typically very short , our method will in general yield an approximate reconstruction .we term the reconstructed adjacency matrix in order to discern from the original matrix , and quantify the matrix reconstruction error as follows : a natural test of the obtained , is to quantify how well does it reproduce the original data . to achieve this, we apply the following procedure : for all network nodes , we start the dynamics from , and run it using the reconstructed matrix for the time interval , i.e. until the time .denote thus obtained values , re - start the run from running until , accordingly obtaining , and so on . the discrepancy that the time series in comparison to is the simplest measure of the reconstruction precision .we name it trajectory error and define as follows : this way we measure point - by - point exactness of the reconstructed trajectory , which quantifies how well does it conform to the actual data .as we show in what follows , two errors are in general related , meaning that small suggests small .we test our reconstruction method using a simple illustrative example .a network with nodes is constructed by putting 17 directed links between randomly chosen pairs of nodes , while requiring the resulting network to be connected .links are weighted with positive and negative weights , uniformly selected at random from  ] , and numerically integrate eq.[eq-4 ] from time to . during the run , we store 15 values for each , equally spaced in time , starting with .the obtained time series for all nodes are shown in fig.[figure-2 ] ., obtained for the first set of initial conditions . ]we now assume that these time series are obtained from an `` external '' source , such as an experimental measurement , and employ them to reconstruct the network s adjacency matrix using the method described in the previous section . to this end, we numerically compute the derivatives and the matrices , and , in order to obtain the reconstructed adjacency matrix via eq.[eq-3 ] .the result is shown in fig.[figure-3 ] , through link - by - link comparison of the original and the reconstructed matrices and .  .link - by - link comparison of the original values of ( circles ) and the reconstructed values ( crosses ) .the resulting matrix and trajectory errors are and , respectively . ]the matrix approximates rather well , for both existing and non - existing links ( non - zero and zero weights ) .the matrix error is , and the trajectory error is , indicating a good reconstruction precision .now , we run another simulation of our dynamical system eq.[eq-4 ] with the same underlying network , but this time starting from a different set of initial conditions .the new time series of equal size and resolution is obtained and shown in fig.[figure-4 ] .we use them to reconstruct our network again , and compare two reconstructions of the same network , obtained via two sets of time series ., obtained for the second set of initial conditions . ]the new results are shown in fig.[figure-5 ] , in analogy with fig.[figure-3 ] .( done as fig.[figure-3 ] above ) . , . ]the new has the matrix error and the trajectory error of , which is considerably worse than in the previous example .the reconstruction errors are bigger on most of the links , as it can be clearly seen by comparing two figures .  despitethat both time series are originating from the same dynamical network , two reconstructions are different , one of which is closer to the actual network .this shows that besides the length and resolution of the time series , the reconstruction precision crucially depends on the `` quality '' of the time series as well , i.e. on the quantity of network information contained in them .time series showing generic patterns ( e.g. periodic / synchronized oscillations ) are more easy to reproduce , than the time series displaying more peculiar dynamics ( e.g. strong chaos ) .the former dynamical data contains less extractable network information than the latter data .in fact , both considered time series are transiental in nature , but the the first ones contain more network information than the second ones , which explains the difference in precision .this is related to the concept of momentary information transfer , recently introduced in the context of coupling analysis of time series  . on the other hand , it is generally difficult to direct the experimental measurements towards maximizing the available network information. is there a way to optimize the reconstruction precision , in the sense of extracting all usable network information , when faced with arbitrary time series ?the next section is devoted to answering this question .the proposed reconstruction method applies to any network whose internal interactions are described by the eq.[eq-1 ] , and whose interaction functions and are known. however , the final reconstruction precision depends on a number of factors : ( _ i _ ) length and resolution of the time series , also related to the precision of derivative estimates ; ( _ ii _ ) quantity of network information contained in the empirical data , which can be seen as the reproducibility of time series , or coverage of the dynamical phase space with data ; ( _ iii _ ) noise in the systems , which can manifest itself either as an error in the measurement of or imprecision in the knowledge of and ; ( _ iv _ ) invertibility of the matrix , which becomes unstable for ( very important for larger networks ) ; and finally , ( _ v _ ) properties of the network itself some networks are more reconstructable than others , due to exhibiting qualitatively different collective dynamics . in a concrete reconstruction problem , it is difficult to isolate how much each factor contributes to . instead of quantifying this , we propose a generalization of our method , done towards improving and controlling the reconstruction precision , regardless of all named factors .our method is based on calculating the correlations between the variable and other terms , as defined in eq.[eq-2 ] .more generally , we can replace by , where is an arbitrary function , without changing the main result .eq.[eq-2 ] now becomes : where notation indicates that the matrix was calculated via eq.[eq-5 ] using a pre - defined function .eq.[eq-3 ] , which now becomes : still holds for any choice of function . as before , in the limit of very large dynamical data , the reconstruction is precisely correct for any choice of . in realistic scenarios involving short time series , the reconstruction precision , except always being finite , will strongly depend on .namely , two different -s will in general yield two different -s , each with its own precision .this means that plays the role of a tunable parameter , which can be used to find the best reconstruction .considering various choices of , one can compute for each one of them , and define as the best that whose reconstructed dynamics shows minimal .this way , we can manipulate the reconstruction for any given time series towards finding the optimal , and hence , the best .such will extract maximal available network information hidden in the time series , and improve the simple reconstruction obtained for .moreover , the variations of with are related to the reconstruction precision . for a reliable reconstruction ,the obtained will not strongly depend on changes of .a bad reconstruction will be recognized by a drastic dependence of on .note also , that the functional properties of itself are irrelevant the _ only _ role of is the computation of .  to illustrate the implementation of our generalized method, we examine again the second time series shown in fig.[figure-4 ] ( `` lower quality '' ones ) . for simplicity , we choose the set of functions , where for we take integers between -20 and 20 ( except 0 ) .the network is reconstructed using eq.[eq-6 ] for each such , and the corresponding and are calculated .the results are reported in fig.[figure-6 ] : in ( a ) , we show the dependence of and on the exponent , while in ( b ) we show the scatter plot of vs .   using 40 functions , for ( except ) .( a ) : plot of and as function of the exponent .( b ) : scatter plot obtained by considering and as the two coordinates .points corresponding to , and are indicated . ]  clearly , different choices of lead to very different reconstruction precisions .the correlation between the two errors is visible in both plots , suggesting that smaller , on average , leads to a smaller .moreover , the dependence of and on the exponent appears relatively smooth .the best reconstruction , in terms of matrix error , is found for ( , ) .however , the reconstruction displaying minimal is obtained for ( , ) . despite the minimal coinciding with the minimal , their values are still relatively close .note that , as indicated in fig.[figure-6]b , both of these results are much better than what initially obtained for .in addition , the obtained precision is also better than the one found for the first ( `` better quality '' ) time series from fig.[figure-2 ] .this indicates , that despite the lower network information content in the time series from fig.[figure-4 ] , we can considerably improve the reconstruction precision by adequately tuning the function .of course , searching the -functional space beyond these 40 functions is likely to yield even better precision .we show the reconstruction for in fig.[figure-7 ] , in analogy with fig.[figure-3 ] and fig.[figure-5 ] .   using the function ( analogous to fig.[figure-5 ] ) .matrix and trajectory errors and , respectively .]  the role of is to compensate for errors in the reconstruction , occurring for the reasons discussed at the beginning of this section . as already noted , considering more ( linearly independent ) -s will improve the precision .however , the question of selecting the optimal which extracts _ all _ the network information contained in the time series remains open .most straightforwardly , one can search for such via monte carlo method , using randomly chosen functions , or through techniques such as evolutionary optimization algorithms  . on the other hand , in the actual reconstruction problem we can only measure . from fig.[figure-6] , this leads to as our `` best guess '' for . here , one can seek to estimate by examining the changes of such for small variations around .we presented a novel method of reconstructing the topology of a general dynamical network from the time series and knowledge of interaction functions . by investigating the derivative - variable correlation, our method reduces to a simple matrix equation , as often studied in linear systems theory  .notably , our method works reasonably well for time series of length comparable to the network size ( we reconstructed matrix from data points ) .  to analyze in more detail the convergence properties of our reconstruction theory, we consider the dependence of errors and on the length of the time series .we consider the first 150 time points of time series from fig.[figure-4 ] . in fig.[figure-8 ] , we show the plots of and as function of the time series length , for functions and .   and as function of the time series length , computed for the time series from fig.[figure-4 ] . functions and are considered , as indicated in the legend . ]  for small , all plots are very erratic , while for larger , they become more stable , eventually saturating at given constant values . the behavior at small confirms the importance of choosing the best , due to its major influence on the reconstruction precision .interestingly , the precision does not always improve with increasing .namely , the considered time series whose initial transient is shown in fig.[figure-4 ] , eventually become periodic for all nodes .this explains the finite precisions obtained for : they are intimately related with the network information content of the time series .note that final precisions depend less on the choice of .moreover , this accounts for the reduction of precision observed for : with increase of , examined time series include more and more periodic data , which carries less information than the transient data , in turn degrading the precision .a significant limitation of our method is the ubiquity of noise .simple estimation of derivatives is very sensitive to noise in .however , by using the appropriate data smoothing techniques  , one might still construct a reasonable approximation .noise can also be present in form of stochastic time - evolution of our system ( additive noise term in eq.[eq-1 ] ) . here ,if the noise intensity is known , one could modify eq.[eq-6 ] to include this term as well .our strongest hypothesis is the precise knowledge of the interaction functions and . lifting this assumptionwould greatly enhance the generality of our theory , and render it far more applicable to the actual empirical data .however , when the approximate functional forms are known , interaction functions can be expanded in series , facilitating their reconstruction. this would mean that for each , we obtain not just , but also and .this leads to a possibility of obtaining many different networks , all reproducing empirical data equally well , but in combination with different interaction functions .another limiting factor regards the reconstruction of large real networks with . for such networks ,the typical length of the obtainable time series is much smaller than the network size .moreover , matrix inversion can become unstable for larger , inducing additional imprecisions .future developments in this direction will necessarily involve inversion check , done for instance via singular value decomposition .the problem of network reconstruction is similar to the problem of designing a network with a prescribed dynamics .one can in principle use our method to design a network that displays given time series , by specifying the error , which here plays the role of tolerance .of course , the key difference between network reconstruction and network design , is the interpretation of the solutions . in case of many different networksthat satisfying , any of them is the solution of the design problem . in reconstruction theory however , one faces the issue of determining which one of them is behind the observed dynamics .we also note that our key assumption is the mathematical form of network interactions given by eq.[eq-1 ] . while a similar theory could be developed for any known form of eq.[eq-1 ] , the problem arises for networks whose interaction form is not ( precisely ) known , which in often the case in real ( e.g. biological ) networks .our method is in principle extendable to networks with multivariate node dynamics , where is now a vector .this will however involve additional complications , depending on which components of the node interact with which components of the node linked to .a similar situation is encountered in networks with coupling that involves two variables , such as kuramoto model  . on the other hand ,the invasive version of our theory would involve time series measured immediately after an external perturbation .such additional assumption would greatly improve our method , by providing more transient dynamics which contains more extractable network information .finally , we note that the key contribution of our theory is its conceptual novelty , coming from examining derivative - variable correlations .this calls for a comparative study of the reconstruction methods , primarily using experimental data .besides identifying the best methods depending on the reconstruction context , this comparison will also potentially improve the existing and suggest entirely new methods .this work was supported by arrs research program p1 - 0383 `` complex networks '' .many thanks to arkady for suggesting the original idea , in addition to misha , uki , bernard and matja for constructive feedback .a. s. mikhailov , v. calenbuhr , _ from cells to societies : models of complex coherent action _( springer , berlin , 2006 ) .a. barrat , m. barthlemy , a. vespignani , _ dynamical processes on complex networks _ ( cambridge university press , cambridge , 2008 ). l. f. costa _ et al _ , analyzing and modeling real - world phenomena with complex networks : a survey of applications , _ adv .phys . _ * 60*(3 ) , 329 ( 2011 ) .z. levnaji , a. pikovsky , phase resetting of collective rhythm in ensembles of oscillators , _ phys .e _ * 82 * , 056202 ( 2010 ) .z. levnaji , emergent multistability and frustration in phase - repulsive networks of oscillators , _ phys .e _ * 84 * , 016231 ( 2011 ) .f. geier , j. timmer , c. fleck , reconstructing gene - regulatory networks from time series , knock - out data , and prior knowledge , _ bmc sys .biol . _ * 1 * , 11 ( 2007 ) . m. hecker _ et al . _gene regulatory network inference : data integration in dynamic models - a review , _biosystems _ * 96*(1 ) , 86 ( 2009 ) .f. emmert - streib _ et al . _ statistical inference and reverse engineering of gene regulatory networks from observational expression data , front . genet . * 3 * , 8 ( 2012 ) .a. tsonis , p. roebber , the architecture of the climate network , _ physica a _ * 333 * , 497 ( 2004 ) .j. donges _ et al ._ , the backbone of the climate network , _ europhys .lett . _ * 87 * , 48007 ( 2009 ) .e. bullmore , o. sporns , complex brain networks : graph theoretical analysis of structural and functional systems , _ nat .neurosc . _* 10 * , 186 ( 2009 ) .n. eagle , a. pentland , d. lazer , inferring friendship network structure by using mobile phone data , _ proc .usa _ * 106*(36 ) , 15274 ( 2009 ) .z. levnaji , a. pikovsky , network reconstruction from random phase resetting , _ phys .lett . _ * 107 * , 034101 ( 2011 ) .b. kralemann , a. pikovsky , m. rosenblum , reconstructing phase dynamics of oscillator networks , _ chaos _ * 21 * , 025104 ( 2011 ) .l. prignano , a. daz - guilera , extracting topological features from dynamical measures in networks of kuramoto oscillators , _ phys .e _ * 85 * , 036112 ( 2012 ) .r. autariello , r. dzakpasu , f. sorrentino , estimating the structure of small dynamical networks from the state time evolution of one node , _ phys .e _ * 87 * , 012915 ( 2013 ) .k. blaha _ et al ._ , reconstruction of two - dimensional phase dynamics from experiments on coupled oscillators , _ phys .e _ * 84 * , 046201 ( 2011 ) .l. l , t. zhou , link prediction in complex networks : a survey , _ physica a _ * 390*(6 ) , 1150 ( 2011 ) .s. g. shandilya , m. timme , inferring network topology from complex dynamics ,_ new j. phys . _ * 13 * , 013004 ( 2011 ) .s. hempel _ et al ._ inner composition alignment for inferring directed networks from short time series , _ phys .lett . _ * 107 * , 054101 ( 2011 ) . s. deroski , l. todorovski , equation discovery for systems biology : nding the structure and dynamics of biological networks from time course data , _ curr ._ * 19 * , 360 ( 2008 ) .m. timme .revealing network connectivity from response dynamics , _ phys .lett . _ * 98 * , 224101 ( 2007 ) .q. xia _ et al ._ , inference of gene regulatory networks with the strong - inhibition boolean model ,_ new j. phys . _ * 13 * , 083002 ( 2011 ) .d. yu , m. righero , lj .kocarev , estimating topology of networks , _ phys .lett . _ * 97 * , 188701 ( 2006 ) .w. wang , network reconstruction based on evolutionary - game data via compressive sensing , _ phys .x _ * 1 * , 021021 ( 2011 ) .l. prignano , y. moreno , a. daz - guilera , exploring complex networks by means of adaptive walkers , _ phys .e _ * 86 * , 066116 ( 2012 ) .y. roudi , j. hertz , mean field theory for nonequilibrium network reconstruction , _ phys .* 106 * , 048702 ( 2011 ) .x. xu , j. zhang , m. small , superfamily phenomena and motifs of networks induced from time series , _ proc .sci . _ * 105 * , 19601 ( 2008 ) .l. lacasa _ et al ._ , from time series to complex networks : the visibility graph , _ proc .sci . _ * 105 * , 4972 ( 2008 ) .r. donner __ , recurrence - based time series analysis by means of complex network methods , _ int .. chaos _ * 21 * , 1019 ( 2011 ) .et al . _ ,noise bridges dynamical correlation and topology in coupled oscillator networks , _ phys .lett . _ * 104 * , 058701 ( 2010 ) .s. widder , j. schicho , p. schuster , dynamic patterns of gene regulation 1 : simple two gene systems , _ j. theor* 246 * , 395 ( 2007 ) .d. hansel , h. sompolinsky , solvable model of spatiotemporal chaos , _ phys .lett . _ * 71 * , 2710 ( 1993 ) .b. pompe , j. runge , momentary information transfer as a coupling measure of time series , _ phys .e _ * 83 * , 051122 ( 2011 ) .z. levnaji , evolutionary design of non - frustrated networks of phase - repulsive oscillators , preprint : arxiv:1210.2786 ( 2012 ) .j. p. hespanha , _ linear systems theory _( princeton university press , princeton , 2009 ) .j. s. simonoff , _ smoothing methods in statistics _ ( springer , new york , 1996 ) .a. arenas _ at al ._ , synchronization in complex networks , _ phys .rep . _ * 469 * , 93 ( 2008 ) .

Abstract: we propose a conceptually novel method of reconstructing the topology of dynamical networks . by examining the correlation between the variable of one node and the derivative of another node , we derive a simple matrix equation yielding the network adjacency matrix . our assumptions are the possession of time series describing the network dynamics , and the precise knowledge of the interaction functions . our method involves a tunable parameter , allowing for the reconstruction precision to be optimized within the constraints of given dynamical data . the method is illustrated on a simple example , and the dependence of the reconstruction precision on the dynamical properties of time series is discussed . our theory is in principle applicable to any weighted or directed network whose internal interaction functions are known .  * derivative - variable correlation + reveals the structure of + dynamical networks * + zoran levnaji + _ faculty of information studies in novo mesto , + 8000 novo mesto , slovenia _ +


advancements in quantum research have enabled breakthroughs in biology , chemistry , physics , engineering , and medicine including better methods to understand the structure of macromolecules used in biochemical signaling and drug delivery , to facilitate the fast and efficient storage of information , and to yield higher resolution medical images for diagnosis and treatment of early stage cancer .most , if not all , measurements and manipulations of quantum systems are achieved through the appropriate design of externally applied time - varying electromagnetic pulses , or controls .these pulses guide the system to produce a desired time - evolution or a specific terminal state .the design of such pulses is made significantly more difficult by inherent variations within the systems of interest .inhomogeneity is one of the fundamental obstacles for the practical implementation and physical realization of quantum science and quantum technology . in classical systemsthe dispersions resulting from inhomogeneity is often compensated for by feedback control .significant research effort has been employed in the area of quantum feedback control with several promising theoretical and practical discoveries in recent years .there is still a large portion of quantum systems for which state feedback is either impractical or difficult to achieve due to the short timescales and large state - space of quantum phenomena .these limitations motivate us to consider the open - loop synthesis of optimal pulses that achieve a desired goal while compensating for the inhomogeneity present in the quantum ensemble .the behavior of a bulk quantum _system _ is the aggregate behavior of a large ensemble of individual quantum _systems_. although in isolation these individual systems , e.g. atoms , spins , qubits , etc . , are fundamentally identical , in a physical system they are distinct due to different chemical and electromagnetic environments .this variation across the ensemble exhibits itself at the macroscopic level as variation in the values of parameters that characterize the dynamics of the bulk quantum system .for example , adjacent atoms within the same macromolecule shield the full strength of the applied external magnetic fields . varied levels of shielding create a dispersion in the frequency of the quantum spins , which is observed as inhomogeneity in the value of the natural frequency of the bulk sample .in addition , hardware imperfection causes attenuation in the applied electromagnetic pulse over the ensemble and can be represented as variation in a scaling factor multiplying the applied pulse . often several pulses are applied in sequence in order to achieve an intricate time - evolution of the system .each pulse is designed assuming an exact ( usually uniform ) initial system state , however , in practice , the previous pulses only prepare the system to within a neighborhood of the assumed initial state .the additive error in such a pulse sequence can cause significant performance degradation .guiding the evolution of inhomogeneous ensembles is a central idea in the design and implementation of quantum experiments . as such, there is a rich literature of methods addressing this class of challenging problems .initially these were intuitive or ad - hoc methods motivated by the symmetry of the state space , which were then augmented with various heuristic and specifically designed techniques .more recently pulse design problems have been cast as optimal control problems .here we present a methodology that addresses the difficulties of the current methods and is easily generalizable to any inhomogeneous ensemble or uncertain system .the proposed method has both theoretical , such as convergence rates and computational complexity , and practical , such as ease of implementation and computation time , advantages .  in this articlewe describe a framework to pose robust quantum pulse design problems in the language of mathematical control theory with support from new theoretical concepts in ensemble control and computational advances in multidimensional pseudospectral methods adapted for ensemble systems . in a larger context, we provide a rigorous methodology to study and control inhomogeneous ensembles or systems with parameter uncertainty from any area or application . in the following section we introduce the problem statement as well as our theoretical and computational tools . in section[ sec : examples ] , we take several examples from nuclear magnetic resonance ( nmr ) in liquids modeled by the bilinear bloch equations , including broadband excitation in the presence of inhomogeneity , a sequence of broadband pulses robust to variation in the initial conditions , and systems with a time - varying frequency .we then provide empirical and theoretical justifications that the solutions computed using the pseudospectral method converge to solution of the original optimal control problem .in this section , we review the underlying concepts involved in our approach to design robust quantum pulses as well as the broader mathematical formulations necessary to characterize and solve such design problems . inwhat follows , we present a highly general model of quantum dynamics , which demonstrates the abundance of inhomogeneity in these problems and motivates studying the control of a family of parameterized systems .we then show how the notion of ensemble control is well suited for dealing with the inherent variation and uncertainty in practical quantum systems and formulate a new type of optimal control problem based on ensemble control .the dynamics of a quantum system is given by the time - evolution of its density matrix .we consider here general dynamics in which the system may have interaction with the environment that leads to dissipation in the system state . under the markovian approximation , where the environment is modeled as an infinite thermostat which has constant state ,the evolution of the density matrix can be written in lindblad form in terms of the system hamiltonian and superoperator which model the unitary and nonunitary dynamics , respectively , -l(\rho ) , \qquad ( \hbar=1).\ ] ] the expression of the hamiltonian has components corresponding to free evolution hamiltonian , , and the control hamiltonians , where are externally applied electromagnetic pulses that can be used to manipulate , or guide , the evolution of the system state .typical pulse design problems involve designing these pulses , or controls , to bring the final state of the density matrix as close as possible to a target operator .this problem can be transformed , by taking the expectation values of the operators involved in the state transfer , to the vector - valued , bilinear control problem , and given by , ,\ ] ] where corresponds to the drift evolution representing and , corresponds to the controlled evolution representing , and  ] , which can be achieved by a simple affine transformation .[ prob : c ] where is the terminal cost ; the running cost , , where is the space of continuous functions with classical derivatives , and dynamics , , where is the space of -vector valued functions , with respect to the state , , and control , ; and are terminal and path constraints , respectively ; is the -vector valued sobolev space .the norm associated with the sobolev space with , , is given with respect to the norm ,     the pseudospectral method was originally developed to solve problems in fluid dynamics and since then has been successfully used for optimal control and applied to various areas .pseudospectral discretization methods use expansions of orthogonal polynomials to approximate the states of the system and thereby inherit the spectral accuracy characteristic of orthogonal polynomial expansions ( the coefficient of the expansion decreases faster than any inverse power of ) . through special properties , derivatives of these orthogonal polynomialscan be expressed in terms of the polynomials themselves , making it possible to accurately approximate the differential equation that describes the dynamics with an algebraic relation imposed at a small number of discretization points .an appropriate choice of these discretization points , or nodes , facilitates the approximation of the states as well as ensuring accurate numerical integration through gaussian quadrature .  as a collocation ( or interpolation ) method ,the pseudospectral method uses lagrange polynomials to approximate the states and controls of the optimal control problem , where and because the lagrange polynomials have the property , where is the kronecker delta function and are the interpolation nodes . therefore , the coefficients and are the discretized values of the original problem and become the decision variables of the subsequent discrete problem .although the interpolation with lagrange polynomials discretizes the original problem , we require a means to ensure that both the integral in the cost functional is computed accurately and the dynamics are obeyed. the integral can be approximated through gauss quadrature ; here we use legendre polynomials as the orthogonal basis for the pseudospectral method .the legendre - gauss - lobatto ( lgl ) quadrature approximation , is exact if the integrand and the nodes , where denotes the set of polynomials of degree at most and where are the lgl nodes determined by the derivative of the order legendre polynomial , , and the interval endpoints .  using the lgl nodes, we can rewrite the lagrange polynomials in terms of the orthogonal legendre polynomials , which is critical to inherit the special derivative and spectral accuracy properties of the orthogonal polynomials despite using lagrange interpolating polynomials .given , we can express the lagrange polynomials as , the derivative of ( [ eq : ix ] ) at is then , where is the constant differentiation matrix .we are now able to write the discretized optimal control problem using equations ( [ eq : ix ] ) , ( [ eq : iu ] ) , ( [ eq : gaussquad ] ) , and ( [ eq : dinterpsc_j ] ) .we transform the continuous - time problem to a constrained optimization ,    the pseudospectral method lends itself to a natural extension to consider the ensemble case , which we develop here .this is most readily apparent for a single parameter variation , i.e. , \subset\mathbb{r} ] , is the dispersion of natural frequencies ,  ] and  ] khz and duration  .pulses developed in this manner have been implemented experimentally in true protein nmr experiments to yield significant improvement in signal recovery .although designing individual pulses is of importance and benefit there are a myriad of other variations and uncertainties within typical quantum experiments , which calls for an approach that can address such new inhomogeneities and their corresponding challenges .   in most experiments , individual pulses , such as the one in figure [ fig : pattern ] ,are combined into a longer pulse sequence , which performs a more complicated manipulation of the system state with intermediate steps and goals . even in the case of highly optimized individual pulses ,as shown in the prior example , there is an error between the desired and actual final states .moreover , pulses depend upon an exact ( and usually uniform ) initial condition in order to achieve their expected levels of performance .these effects combine to create a magnified accumulated error at the termination of the pulse sequence .the variation of the initial conditions of these pulses , therefore , causes significant degradation in achievable performance .evolution of the bloch equations .the upper plot displays the concatenation of individually optimized and pulses , which achieves the dashed terminal profiles shown below , with respective average performances : 0.99 , 0.98 , 0.97 ( 0.91 minimum ) .the middle plot displays a 3-part simultaneously - optimized pulse robust to variation in the initial condition and achieves the solid terminal profiles shown below , with respective average performances : 0.99 , 0.99 , 0.99 ( 0.97 minimum ) .the noticeable enhancement in performance and uniformity is due to compensating for the inhomogeneity in the initial condition of the individual pulses . ]a representative example of such a pulse sequence is to perform a three step pulse sequence , which rotates the magnetization of the ensemble ( 1 ) from equilibrium ( ) to a point on the transverse plane ( e.g. ) ; ( 2 ) to the opposite point on the transverse plane ( e.g. ) ; ( 3 ) back to the equilibrium position ( ) .such pulses generally include `` phase locking '' pulses before and after the second pulse during which the magnetization dissipates .this dissipation is the portion of the experiment that is important to recover accurately and reflects a quantity to be measured , for example , a metabolic rate . if , in addition , there is accumulated error due to uncertainty in the initial conditions of the individual pulses , this leads directly to measurement inaccuracy . here , by removing the `` phase locking '' pulses , we can abstract this pulse sequence to a unitary process and directly address any losses due to error .the controllability of the bloch equations is shown constructing parameter - dependent ( e.g. frequency , rf inhomogeneity ) rotations of the spin vectors .this , therefore , ensures that the problem with variation in initial conditions can be solved provided that the initial conditions can be parameterized by the frequency and rf - inhomogeneity .figure [ fig : initialcond ] displays a three - stage optimized pulse designed by the multidimensional pseudospectral method which is robust to frequency dispersion and variation in the initial conditions of the three stages .this pulse was run as three concurrent optimizations , with the final states of one pulse fed in as the initial conditions of the next .this optimized pulse is compared with the combination of three separately optimized pulses ; these combined pulses were designed with equal total duration .the terminal profiles at each intermediate goal quickly show the evidence of accumulated error in the case of the individually optimized pulses ( each individual pulse has an average performance greater than 0.98 ) .most importantly , the uniformity of the inversion is lost in the additive error , with dips in performance down to 0.91 .   until now, we have considered that the dispersion and uncertainty of the system are stationary . however , addressing time - varying fluctuations in parameters is also of particular theoretical and practical importance .for example , in the formulation of quantum control problems given in ( [ eq : quantum_ensemble ] ) we noted that the hamiltonians can be time - varying , motivated by such phenomena as random telegraph noise .the first step to addressing stochastic variations in such physical systems is to demonstrate control of time - varying systems , such as given by the expectation value of the corresponding random process .  .a single - system state transfer is designed using the terminal cost and running costs ( left ) , ( middle ) , ( right ) .the terminal time was free in all cases , bounded by . ]figure [ fig : timevarying ] presents a series of optimizations designing pulses providing a state transfer to , while compensating for a time - varying frequency , .various choices of cost functional yield different results .the arbitrary control pulse profile corresponding to the terminal cost ( fig .[ fig : timevarying ] , left ) motivates studying optimal control methods that provide the capacity for hybrid objectives resulting in more physically meaningful controls , e.g. minimizing energy ( middle ) and time ( right ) .by accepting and implementing a numerical method , we implicitly assume that the transformations and discretization used to prepare the problem for computational work does not fundamentally alter the nature of the problem .it is then critically important to show that this assumption is justified .here we do so by both empirical and theoretical means . more specifically , we show that as the number of discretizations in the pseudospectral method ( and samples in the multidimensional pseudospectral method ) increases the solution of the algebraic nonlinear programming problem converges to the solution of the original continuous - time optimal control problem . for this argument, we consider a modified nonlinear programming problem statement .[ prob : d ] where is a positive constant ; we define the discrete norm , for ,  ] and  ] , $ ] , , and ( all dimensionless ) .note the factor in the time scaling is introduced to convert from units of hertz to radians / second . designing the time - varying controls and is equivalent to the original design of amplitude and phase .[ thm : rellich ] suppose is a sequence in such that                t. e. skinner , t. reiss , b. luy , n. khaneja , s. j. glaser , `` application of optimal control theory to the design of broadband excitation pulses for high resolution nmr , '' _ j. magn8 - 15 , 2003 .n. khaneja , t. reiss , c. kehlet , t. schulte - herbrggen , s. j. glaser , `` optimal control of coupled spin dynamics : design of nmr pulse sequences by gradient ascent algorithms , '' _ j. magn . reson .296 - 305 , 2005 .i. i. maximov , z. tosner , n. c. nielsen , `` optimal control design of nmr and dynamic nuclear polarization experiments using monotonically convergent algorithms , '' _ j. chem .128 , pp . 184505 , 2008 .i. ross and f. fahroo , `` legendre pseudospectral approximations of optimal control problems '' , in _ new trends in nonlinear dynamics and control _ , edited by w. kang et al .( springer , berlin , 2003 ) ; pp .327 - 342 .d. gottlieb , y. hussaini , s. orszag , `` theory and applications of spectral methods '' , in _ spectral methods for partial differential equations _, edited by r. voigt , d. gottlieb , and m. y. hussaini , siam , philadelphia , 1984 , pp . 1 - 117 .q. gong , m. ross , w. kang , f. fahroo , `` connections between the covector mapping theorem and convergence of pseudospectral methods for optimal control '' , _ comput .appl _ , vol .307 - 335 , 2008 .

Abstract: inhomogeneity , in its many forms , appears frequently in practical physical systems . readily apparent in quantum systems , inhomogeneity is caused by hardware imperfections , measurement inaccuracies , and environmental variations , and subsequently limits the performance and efficiency achievable in current experiments . in this paper , we provide a systematic methodology to mathematically characterize and optimally manipulate inhomogeneous ensembles with concepts taken from ensemble control . in particular , we develop a computational method to solve practical quantum pulse design problems cast as optimal ensemble control problems , based on multidimensional pseudospectral approximations . we motivate the utility of this method by designing pulses for both standard and novel applications . we also show the convergence of the pseudospectral method for optimal ensemble control . the concepts developed here are applicable beyond quantum control , such as to neuron systems , and furthermore to systems with by parameter uncertainty , which pervade all areas of science and engineering .


one of the most challenging goals of the future space missions is to detect life on extrasolar planets .these missions are designed to image a planet as a single dot floating beside its parent star , using high - contrast imaging techniques .photons from the unresolved planet will feed a low - resolution spectrograph and give first insights into planet s chemistry .recent works suggest that earth s vegetation is detectable in the earth integrated spectrum at visible wavelengths , assuming extraterrestrial plants , if they exist , are similar to earth s plants .our green vegetation indeed has a reflectance spectrum showing a sharp increase - the so - called vegetation red edge ( vre ) - around ( fig .[ vegetation ] ) that could be considered as a possible global biomarker .a question arising from these results is whether vegetation was also detectable when the ( early ) earth looked significantly different than today . in other words ,if an earth - like planet with a climate different from the modern earth s climate is detected one day , will we be able to detect vegetation on it ? to answer this , we investigate the vegetation spectral detectability during the last quaternary climatic extrema , for which past climate simulations have been made by general circulation models ( gcms ) .this simulated climate is used as a driver for an equilibrium vegetation model , providing biome maps of the earth at these epochs .the last glacial maximum ( lgm ) occurred about 21 kyr ( kilo - year ) ago .temperatures globally of the order of colder than today were responsible for a large extent of sea and continental ice sheets , especially in the northern hemisphere .sea level was also significantly lower ( m ) resulting in larger emerged lands .  during the holocene optimum ( 6 kyr before present , hereafter kyrbp ) ,earth s northern hemisphere was warmer than today .the sea level was rising , although still slightly lower than today , because the amount of water released from continental ice was partially compensated by an isostatic response of the continents , _i.e. _ a rise of the continents following the deglaciation of the northern hemisphere .the sahara was more vegetated than today , its desertification being the main large - scale change in land cover for the last 6000 years .  here , based on a combination of biomes and simulated cloud cover maps for the periods mentioned above ,associated with 300 to 800 nm spectra for each biome and cloud , we build earth integrated spectra , for given earth phases and observer positions .the vegetation spectral signature is then extracted from these spectra .the first part of the paper describes the model of the earth from which we derive global reflectance spectra .the second part of the paper discusses the results .note that some results showing the rotating earth and corresponding instantaneous spectrum are available on - line as animated gif files ( http://www.obs-hp.fr//results/2009_ijastrobio/vre.html ) .   at  at .note that the spectrum of a non - vegetated soil also shows a higher reflectance in the near - ir than in the visible.,width=377 ]the equilibrium terrestrial biosphere model biome3 is a process based vegetation model simulating biogeography and biogeochemistry .model inputs consist of monthly climate variables ( temperature , precipitation , and sunshine ) data , minimum annual temperature and atmospheric co2 concentration .a coupled hydrological model calculates moisture availability on the basis of a soil texture class .the model simulates the productivity and cover ( leaf area index ) of a set of plant functional types .the comparative dominance of these types is then used to estimate the dominant biome for a given grid cell .a total of 24 biomes may be simulated , including barren ground and snow / ice cover ( fig .[ biomes_legend ] and [ biomes_maps ] ) .the model was run for the three periods of interest : modern , holocene optimum and lgm earth ( fig . [ biomes_legend ] and [ biomes_maps ] ) .for the modern period , gridded climate data on a 0.5 grid was taken from , and atmospheric co2 concentration was set to a pre - industrial level of 280 ppm .for the two past periods , climate data was obtained from simulations run for under the paleoclimate modelling intercomparison project .we used data from the uk meteorological office general circulation model ( ukmo gcm ) , run under climatic forcings for these two periods .the gcm output was downscaled to the same resolution grid as the modern period by calculating anomalies for each past period .the gcm control run value was subtracted from the past value for each gcm grid point , and then these anomalies were added to the modern climatology grid points .atmospheric co2 concentration was set to 265 ppm and 180 ppm for the holocene optimum and lgm respectively .the results obtained agree well with observational evidence of past vegetation change .the mid - holocene is dominated by a greening of the sahara , and a poleward shift of the high latitude treelines .the lgm is characterised by a large expansion of steppe and grassland vegetation at mid - latitudes , with a near disappearance of temperate forest .there is also the fragmentation of tropical forest and widespread expansion of tundra vegetation at high latitudes .there are , however , some disagreements with the data , for example the expansion of grassland in australia .+  +   we used a collection of gome spectra taken the 18 , 19 and 20th of may 1999 , corresponding to springtime and the middle of the greenup period in the northern hemisphere . in maythe average ndvi ( normalized differential vegetation index ) is just half of its maximum value reached in july .sun declination is for the observing dates in may .it is fall in the southern hemisphere and the polar night is present at latitudes below .the gome spectra are reflectances obtained from the ratio of a given ground pixel radiance over solar irradiance , where is the solar zenit angle .a reference spectra for each biome is extracted from the gome data ( fig .[ gome_spectra ] ) . in order to find a given biome spectrum with minimal cloud contamination, the algorithm choses the spectrum with the lowest signal in the interval 450 to 500 nm , considering that the signal in this spectral range rapidly increases with the presence of clouds .this algorithm fails for three types of surfaces : ocean , snow / ice and cloud .these spectra are chosen by hand.  it must also be noted that this algorithm chooses the most vegetated pixel for a given biome : if a cloudless pixel is a mixture of sand and vegetation , it will be ignored in favor of a pixel with more vegetation , which will display a lowest signal in the spectral interval mentioned .this suggests that the reference pixel ( and spectrum ) for each biome is probably biased toward the most vegetated pixel of a given biome area .lastly , since it is spring in the northern hemisphere and fall in the southern hemisphere , the reference spectrum for the biomes of deciduous species will be preferentially picked up in the greener northern hemisphere .therefore , our simulations will be more relevant for reconstruction of the northern hemisphere rather of the southern hemisphere , with a probable overestimate of the vegetation signal if the southern hemisphere is mainly in view .   for the biome legend .note that all plots do not have the same vertical scale . , width=472 ]    to build a realistic earth s spectrum , we need both the cloud cover map and the reflectance map of that cloud cover , .we used cloud covers from the international satellite cloud climatology project ( isccp ) .we selected only data for the month of may from the 1999 to 2004 vis - ir observations , to be consistent with the gome spectra .since low , middle and high altitude clouds have different typical optical depth and reflectances , we build the cloud cover reflectance from the sum of the three different altitudes cloud cover maps , and , weighted by their albedo , , and respectively 0.69 , 0.48 and 0.21 , and normalized to the albedo of low altitude clouds , i.e.    the result from this formula is that the three different types of clouds contribute to a given pixel s reflectance proportionally to their cover over that pixel .the formula also can lead to a few pixels with reflectance above 1 , which are then simply clipped to 1 .the obtained map ( fig .[ cc_ref_isccp ] ) is normalized to the albedo of low clouds for which we have a reference spectrum from the gome data .[ cols="<,<,^,^,^ " , ]   the reconstructed scenes of the earth , from which the results below are extracted , have been made using a sun declination of , corresponding to the earth - sun configuration for may .the earth phase viewed by the observer is reconstructed for several inclination .the angle is the orbital phase of the earth viewed from the location of the observer : is for western maximum elongation , for superior conjunction , for eastern elongation and for inferior conjunction .  in table[ vre_albedo ] , the vre and albedo values obtained from the model are given for the different periods .for the modern earth , the albedo values obtained with the isccp data are higher than with biome3.5 , due to a lower cloud cover with biome3.5 , leading to a lower vre .this suggests that the biome3.5 model probably overestimates the vre. nevertheless , relative comparisons between biome3.5 results remain consistent although the absolute values may be overestimated .the main result from this study is that the climate differences between the quaternary extrema and the modern earth s climate have little impacts on the vegetation signal , since the spectral signature of vegetation is not washed out during the lgm and does not increase very much during the holocene optimum .there are nevertheless interesting differences in vegetation visibility between these two extrema and the situation today , depending of the observer s position : when the northern hemisphere is mainly in view , vegetation is indeed more visible during the holocene optimum as shown in fig .[ vre-24h - variation_nord ] and table [ vre_albedo ] : the vre was typically 6% , thus higher than today ( 5% ) or during the lgm ( 4% ) .this is mainly due to a greener sahara as shown in fig .[ vre-24h - variation_nord](a ) .this is true for an observer with mostly the northern hemisphere in view , i.e. inclination . in inferior conjunction configuration, the crescent - shaped earth mainly shows its northern part .there are no significant differences between modern and holocene earths ( vre ) and vegetation becomes almost undetectable for the earth with the lgm large polar ice cap ( ) .if the observer is above the northern pole ( observing from ecliptic north pole ) with in view a half - illuminated earth , there are again no significant differences between modern and holocene earths ( vre ) . however , as the north pole ice is much more extended during the lgm , the signal of the vegetation is significantly weaker ( only 2% ) , due to the higher reflectance of a larger polar zone and the consequently smaller vegetated areas is view .  the three bottom lines from table [ vre_albedo ] give results for earth views in which southern hemisphere contribution is high , with inclination ranging from 90 to .as pointed out in section [ sect_biome_spectra ] , our model probably overestimates the vre values of scenes with the southern hemisphere mostly in view as our calibration was made during fall time ( may ) in that hemisphere and deciduous species should have a lower vre than with the spectra used in the simulation that have been collected in the northern hemisphere .nevertheless , there are two trends to be mentioned : the first one is that the difference between holocene and modern period becomes negligible , because the sahara contribution decreases or even vanishes from these southern view points .the second remark is that , paradoxically , the southern hemisphere vre during the lgm is higher : this is induced by a greener australia during the lgm , as we obtained a simulated wider distribution of temperate grassland and less desert in australia using the biome3.5 model ( fig .[ biomes_maps ] ) at the holocene optimum . asdiscussed above , this does not agree with observations of past vegetation and may arise from an incorrectly simulated climate over this region .figures [ earth_scenes_spectra_45 ] and [ earth_scenes_spectra_45_biome35 ] finally show modern earth s scenes with corresponding spectrum obtained following the method described in sect .[ method ] . in fig .[ earth_scenes_spectra_45 ] , we show the spectrum for central longitudes of ( africa and europe ) and ( northern pacific ocean and alaska ) at which the model gives the maximum and minimum vre , respectively of 8.1 and 1.5% . fig .[ earth_scenes_spectra_45 ] show the earth spectra based on biome3.5 data for central longitudes of ( africa and europe ) at which the model gives the maximum vre , i.e. 8.6 , 11.0 and 6.3% at 0 , 6 and 21 kyrbp , respectively .will next space observatories be able to detect vre - like features in an earth - like planet spectrum ?the terrestrial planet finder coronagraph mission ( tpf - c ) is aimed to make the direct detection and the characterization ( multi - spectral imaging at resolution up to 70 ) of nearby terrestrial extrasolar planets in the visible range .the telescope would have an elliptical aperture of  .an earth - like planet in the habitable zone is fainter than its parent star , making such planets very faint objects in the v=28 to 31 range . according to , exposure time of the order of one daywill be needed to detect them and spectral characterization will add weeks of additional integration time per target. more quantitatively , in a comparison of different types of coronagraphs , the authors conclude that an earth - like planet at about 10 pc could be detected in 2.3 h with a signal - to - noise ratio of snr=7 with a 100 nm bandwidth from 0.5 to with a 4-m telescope , and even in 14 minutes with a 6-m telescope , both equipped with one of best performers of the comparison , a phase - induced amplitude apodisation coronagraph . now if we want to be able to detect a 4% vre - like feature in a spectrum at a resolution of 70 around 700 nm , the snr should be about 80 for a 3-sigma detection , or 125 for a 5-sigma detection . with the 4-m coronagraphic telescope metioned above , the total exposure time becomes vey long , 18 and 44 weeks for 3 and 5-sigma detection , respectively .this in agreement with the order of magnitude given by . with a 6-m telscope, the total exposure times decrease to 13 days and one month , respectively .if a larger telescope than the tpf - c - like telescopes mentioned above is considered , the exposure time will decrease again . according to , a planetcould be detected in 2 minutes with a 12-m telescope .the total exposure times to detect a 4% feature in a spectrum would then decrease to 2 and 5 days , for 3 and 5-sigma detection , respectively .these very long exposure times suggest that features of a few percents will be extremely difficult to detect , maybe even not within reach of a first - generation 4-m tpf .a 6-m telescope seems to be the minimum to envisage the detection of seasonal variations . with exposure times of 2 to 5 days, a 12-m telescope would probably not freeze the rotation of the planet and rather give an integrated spectrum of the planet over several rotations , but atmospheric changes at times scales of the order of a week or longer would be possible to detect and follow up , as well as longer seasonal variations .  unfortunately tpf - c might not fly before 2025 , and in the meantime , a smaller instrument could look for bigger and closer terrestrial planets : a 2-m class space coronagraph like see - coast could indeed detect a vre on super - earths closer than 3 to 6 pc , depending on their radius whose maximum value is presently unclear .these super - earths are as well habitable than earth - sized planets .more ambitious instruments that might fly later this century will allow us to make a step furthermore in the characterization of these planets and the observation of exo - vegetation. hypertelescopes in space - interferometric sparse arrays of small telescopes - will indeed allow us to see earth - like planets as _small resolved disks _ several resels across and this clearly will help us to detect photosynthetic life on these planets !for example , a 150-km hypertelescope would provide 40 resolution elements ( resels ) across an earth at 3 pc in yellow light . anda formation of 150 3-m mirrors would collect enough photons in 30-min to freeze the rotation of the planet and produce an image with at least resels , and up to thousands depending on array geometry ( fig .[ labeyrie99 ] ) .at this level of spatial resolution , it will be possible to identify clouds , oceans and continents , either barren or perhaps ( hopefully ) conquered by vegetation .   higher .central longitude is decreasing , according to the eastward earth rotation ., width=491 ]  .central longitude is decreasing , according to the eastward earth rotation . , width=491 ]  , at of central longitude .scenes are for , a wavelength at which vegetation is brighter than in the visible . for a better readability ,the radiance is shown here at power 0.7 to enhance the darker regions of the scene and underline the shape of the sunlit earth seen by the observer.,height=680 ]   ( table [ vre-24h - variation_nord ] and fig .[ vre-24h - variation_nord ] ) , for central longitudes of ( top , africa and europe ) and ( bottom , northern pacific ocean and alaska ) at which the model gives the maximum and minimum vre , respectively .the earth spectrum is in white , and a vegetationless model in yellow ( moreover with and bands corrected ) allows to visualize and calculate the vre . for the record ,cloud and ocean spectra are shown in blue .scenes are for , a wavelength at which vegetation is brighter than in the visible . for a better readability ,the radiance is shown here at power 0.7 to enhance the darker regions of the scene and underline the shape of the sunlit earth seen by the observer.,width=566 ]   ( table [ vre-24h - variation_nord ] and fig .[ vre-24h - variation_nord ] ) , for central longitudes of ( africa and europe ) at which the model gives the maximum vre .the earth spectrum is in white , and a vegetationless model in yellow ( moreover with and bands corrected ) allows to visualize and calculate the vre . for the record ,cloud and ocean spectra are shown in blue .scenes are for , a wavelength at which vegetation is brighter than in the visible . for a better readability ,the radiance is shown here at power 0.7 to enhance the darker regions of the scene and underline the shape of the sunlit earth seen by the observer.,width=566 ]in order to mimic an earth - like extrasolar planet observed without spatial resolution , i.e. as a single dot , in a different climate state than our contemporary earth , we considered the earth during the last climatic extrema , 6 and 21 kyr ago , and studied the vegetation spectral signature integrated over the full illuminated phase seen by the observer .the results presented in this paper suggest that vegetation disk - averaged signature remained visible in earth spectra during these climatic extrema , with vre values close to those for the modern earth .our discussion emphasized some differences induced for example by the greener sahara during the holocene optimum , or by the large northern ice cap during the lgm .results would probably benefit in accuracy from radiative transfer calculations , an implementation of the sun glint effect and the brdfs of all scattering surfaces included in the model .the spectra would thence also display a more accurate rayleigh scattering slope and absorption bands depth .further , improved gcm estimates for these past periods are now available , including coupled ocean - atmosphere simulations , as well as dynamic vegetation models , both of which may help improve the past vegetation distributions obtained here .all these improvements could be considered for future developments .  of course a next step in this workcould also be the consideration of the earth at much older epochs , when continents had a very different shape than today and climates maybe much more extreme than those considered in this work , although paleoclimatic models are probably less confident for these epochs .it would although probably provide us with relevant illustrations of earth spectrum variability over one full day , and related vre variations .current climate change , thence biome maps for the end of this century , might also tell us if vegetation will remain a detectable biomarker for observers out there .* acknowledgements * this work was supported in part by the french cnrs _gdr exobiologie_. the authors thank jean - claude lebrun ( service daronomie , cnrs , verrires - le - buisson , france ) who provided the modtran spectra of and , and andreas richter ( institute of environmental physics , university of bremen , germany ) for sciatran data and discussions on rayleigh scattering .arnold l. , bron f .-, brewer s. , guiot j. , jacquemoud s. , schneider j. 2003 , `` the green vegetation as a biosignature on earth and earth - like planets : polder data and earthshine observations '' in sf2a-2003 : scientific highlights 2003 , bordeaux , france , 16 - 20 june 2003 , eds .f. combes , d. barret , t. contini and l. pagani , edp - sciences , les ulis , 133 - 136  arnold l. 2008 , space science reviews 135 , issue 1 - 4 , 323 - 333 ( doi : 10.1007/s11214 - 007 - 9281 - 4 ) also in `` strategies of life detection '' , space sciences series issi 2008 , ed . o. botta _ et al . _ ,springer , ( doi:10.1007/978 - 0 - 387 - 77516 - 6_1 )   braconnot p. , b. otto - bliesner , s. harrison , s. joussaume , j .- y .peterchmitt , a. abe - ouchi , m. crucifix , e. driesschaert , th .fichefet , c. d. hewitt , m. kageyama , a. kitoh , a. lan , m .- f .loutre , o. marti , u. merkel , g. ramstein , p. valdes , s. l. weber , y. yu , and y. zhao 2007 , clim .past 3 , 261 - 277   clark , r. n. 1999 , `` chapter 1 : spectroscopy of rocks and minerals , and principles of spectroscopy '' , in manual of remote sensing , vol .3 , remote sensing for the earth sciences , a.n .rencz , ed . , john wiley and sons , new york , 3 - 58           leemans r. , cramer w. 1991 , `` the iiasa climate database for mean monthly values of temperature , precipitation & cloudiness on a terrestrial grid '' , international institute for applied systems analysis , laxenburg , germany       schneider j. , boccaletti a. , mawet d. , baudoz p. , beuzit j .-l . , doyon r. , marley m. , stam d. , tinetti g. , traub w. , trauger j. , aylward a. , cho j. y. k. , keller c .- u ., udry s. & the see - coast team 2008 , experimental astrononomy , in press ( doi:10.1007/s10686 - 008 - 9129 - 2 )  shaklan s. b. , balasubramanian k. , ceperly d. , green j. j. , hoppe d. j. , lay o. p. , lisman p. d. & mouroulis , p. z. 2006 , `` terrestrial planet finder coronagraph instrument design '' , in `` direct imaging of exoplanets : science & techniques '' , proceedings of the iau colloquium # 200 , edited by c. aime and f. vakili , cambridge university press , cambridge ( uk ) , pp.577 - 580  sitch s. , smith b. , prentice i. c. , arneth a. , bondeau a. , cramer w. , kaplan j. , levis s. , lucht w. , sykes m. , thonicke k. & venevski s. 2003 , `` evaluation of ecosystem dynamics , plant geography and terrestrial carbon cycling in the lpj dynamic vegetation model '' , global change biology 9 , 161 - 185

Abstract: the so - called vegetation red - edge ( vre ) , a sharp increase in the reflectance around , is a characteristic of vegetation spectra , and can therefore be used as a biomarker if it can be detected in an unresolved extrasolar earth - like planet integrated reflectance spectrum . here we investigate the potential for detection of vegetation spectra during the last quaternary climatic extrema , the last glacial maximum ( lgm ) and the holocene optimum , for which past climatic simulations have been made . by testing the vre detectability during these extrema when earth s climate and biomes maps were different from today , we are able to test the vegetation detectability on a terrestrial planet different from our modern earth . data from the biome3.5 model have been associated to visible gome spectra for each biome and cloud cover to derive earth s integrated spectra for given earth phases and observer positions . the vre is then measured . results show that the vegetation remains detectable during the last climatic extrema . compared to current earth , the holocene optimum with a greener sahara slightly increases the mean vre on one hand , while on the other hand , the large ice cap over the northern hemisphere during the lgm decreases vegetation detectability . we finally discuss the detectability of the vre in the context of recently proposed space missions .   _ keywords : _ extrasolar planets , earth - like planets , vegetation , biomarker , biosignature , earth spectrum , modern earth , holocene optimum , last glacial maximum .