a natural and important class of problems in computer science deals with random generation of objects satisfying certain properties .more precisely , one is interested in an efficient algorithm that , given a _compact _ description of a set of objects , outputs an element in the set uniformly at random , where the exact meaning of `` compact '' depends on the specific problem in question .uniform sampling typically arises for problems in .namely , given an instance belonging to a language in , one aims to produce a witness uniformly at random . here , the requirement is stronger than that of decision and search problems . in a seminal paper , jerrum ,valiant and vazirani gave a unified framework for this problem and showed that , for polynomial - time verifiable relations , uniform sampling of a witness for a given instance is reducible to approximate counting of the witnesses , and hence , can be efficiently accomplished using a oracle .it is natural to ask whether the requirement for an oracle can be lifted .in fact , this is the case ; a result of bellare , goldreich , and petrank shows that an oracle is sufficient and also necessary for uniform sampling of witnesses .the sampling problem can be equivalently stated as follows : given a boolean circuit of polynomially bounded size , sample an input that produces the output ( if possible ) , uniformly at random among all possibilities .this problem can be naturally generalized to models of computation other than small boolean circuits , and an interesting question to ask is the following : for what restricted models , the uniform ( or almost - uniform ) sampling problem is efficiently solvable ( e.g. , by polynomial - time algorithms or polynomial - sized circuits ) without the need for an additional oracle ?of course if the role of the oracle in can be replaced by a weaker oracle that can be efficiently implemented , that would immediately imply an efficient uniform sampler .while for general relations the full power of an oracle is necessary , this might not be the case for more restricted models .  in this work , we study the sampling problem for the restricted model of _ polynomial functions_. a polynomial function of degree over a field ( that we assume to be finite ) is a mapping such that every coordinate of the output can be computed by an -variate polynomial of total degree at most over . the corresponding sampling problem ( that we call _ variety sampling _ )is defined as follows : given a polynomial function , find a pre - image of a given output ( that can be considered the zero vector without loss of generality ) uniformly at random .hence , in this problem one seeks to sample a uniformly random point on a given algebraic variety .it is not difficult to show that this problem is , in general , -hard .hence , it is inevitable to relax the generality of the problem if one hopes to obtain an efficient solution without the need for an oracle .accordingly , we restrict ourselves to the case where  1 .the co - dimension of the variety ( or , the number of the polynomials that define the variety ) is _ small _ , 2 .the underlying field is _ sufficiently large _ , 3 .the output distribution is only required to be _ statistically close _ to the uniform distribution on the variety .it is shown in that almost uniform generation of witnesses ( with respect to the statistical distance ) is possible without using an oracle for self - reducible relations for which the size of the solution space can be efficiently approximated .the relation underlying the variety sampling problem consists of a set of -variate polynomials over and a point , and it holds if and only if is a common zero of the polynomials . obviously , assuming that field operations can be implemented in polynomial time , this is a polynomial - time verifiable relation .moreover , the relation is self reducible , as any fixing of one of the coordinates of the witness leads to a smaller instance of the problem itself , defined over variables .approximate counting of the witnesses amounts to giving a sharp estimate on the number of common zeros of the set of polynomials .several such estimates are available .in particular , a result of lang and weil ( theorem [ thm : langweil ] ) that we will later use in the paper gives general lower and upper bounds on the number of rational points on varieties .moreover , there are algorithmic results ( see and the references therein ) that consider the problem of counting rational points on a given variety that belongs to a certain restricted class of varieties over finite fields .thus , it appears that the result of already covers the variety sampling problem .however , this is not the case because of the following subtleties :  1 .our relation is not necessarily self - reducible in the strong sense required by the construction of .what required by this result is that partial fixings of the witness can be done in steps of at most logarithmic length ( to allow for an efficient enumeration of all possible fixings ) .namely , in our case , a partial fixing of amounts to choosing a particular value for one of the variables .the portion of corresponding to the variable being fixed would have length , and in general , this can be much larger than .the general lang - weil estimate gives interesting bounds only when the underlying field is fairly large .the algorithmic results mentioned above , being mostly motivated by cryptographic or number - theoretic applications such as primality testing , focus on very restricted classes of varieties , for instance , elliptic or hyperelliptic curves ( or general plane curves that are only defined over a constant number of variables ) , or low - dimensional abelian varieties .moreover , they are efficient in terms of the running time with respect to the logarithm of the field size and the dependence on the number of variables or the degree( whenever they are not restricted to constants ) can be exponential .hence , over large fields , fine granularity of the self - reduction can not be fulfilled and over small fields , no reliable and efficient implementation of a counting oracle is available for our problem , and we can not directly apply the general sampler of . in this work , we construct an efficient sampler that directly utilizes the algebraic structure of the problem . the main theorem that we proveis the following :  * ( main theorem ) * [ thm : main ] let the integer be any absolute constant , and be positive integers , be an arbitrarily small parameter , and be a large enough prime power .suppose that  ] .when the statistical distance of and is at most , we say that and are -close .we will also use the notion of a convex combination of distributions , defined as follows :  let be probability distributions on a finite set and be nonnegative real values that sum up to .then the _ convex combination _ is a distribution on given by the probability measure for .there is a simple connection between convex combinations and distance of distributions :  [ prop : convexclose ] let and be probability distributions on a finite set such that for some , .then is -close to .in this section we introduce a sampling problem on graphs , and develop an algorithm to solve it .we will later use this algorithm as a basic component in our construction of samplers for varieties .the problem is as follows :  [ prob : bisamp ] let be a bipartite graph defined on a set of left vertices and of right vertices .suppose that the degree of every vertex on the right is between and , for some , and the degree of every vertex on the left differs from an integer by at most .we are given an oracle that returns an element of chosen uniformly at random ( and independently at each call ) , and an oracle that returns the neighbor list of a given vertex .construct an algorithm that outputs a random vertex in almost uniformly .  intuitively , for a bipartite graph which is regular from left and right , sampling a vertex on the left amounts to picking a random edge in the graph , which is in turn possible by choosing a random edge connected to a random vertex on the right side .here of course , the graph is not regular , however the concentration of the left degrees around allows us to treat the graph as if it were regular and get an almost uniform distribution on by picking a random edge .we will compensate the irregularity from right by using a `` trial and error '' strategy .the pseudocode given in algorithm [ alg : bisamp ] implements this idea .the algorithm in fact handles a more general situation , in which a call to can fail ( and return a special failure symbol ) with some probability upper bounded by a given parameter . given as in problem [ prob : bisamp ] , and denoting the failure probability of .let be as in problem [ prob : bisamp ] .   with probability , output an element of uniformly at random and return .output an arbitrary element of .[ alg : bisamp ]  [ lem : vertexsamp ] the output distribution of algorithm [ alg : bisamp ] is supported on and is -close to the uniform distribution on .first we focus on one iteration of the * while * loop in which the call to has not failed , and analyze the output distribution of the algorithm conditioned on the event that line  returns a left vertex . in this case, one can see the algorithm as follows : add a special vertex to the set of left vertices .bring the degree of each right vertex up to by connecting it to as many times as necessary .hence , the graph now becomes -regular from right .now the algorithm picks a random element and a random neighbor of and independently repeats the process if is picked as a neighbor .  let be a non - empty subset of the left vertices ( excluding ) in the graph .we want to estimate the probability of the event .we can write this probability as follows : = \sum_{r \in { \mathcal{r } } } \pr[t \mid r = r ] \pr[r = r ]    = \frac{1}{|{\mathcal{r}}| } \sum_{r \in { \mathcal{r } } } \pr[t \midr = r ]   = \frac{1}{d |{\mathcal{r}}| } \sum_{r \in { \mathcal{r } } } |t \cap \gamma(r)| ,  \end{aligned}\ ] ] where in the last equation is the set of neighbors of in the graph .hence the summation can be simplified as the number of edges connected to .this quantity is in the range , because the left degrees are all concentrated around , ignoring which is by assumption not in .that is , = \pr[t , \lnot v_0 ] = \frac{|t|\ell}{d |{\mathcal{r}}|}(1 \pm \delta ) , \label{eqn : prt}\ ] ] where we use the shorthand to denote a quantity in the range  ] , each of degree at most .suppose that is a fixed point of .then the fraction of -dimensional affine spaces passing through that properly intersect is at least , where is independent of and polynomially large in .  without loss of generality , assume that is the origin , and that .denote by the set of -dimensional linear subspaces that can be parametrized as where is a set of indeterminates in .note that , and define the polynomial ring  ] , where for each  ] the ideal generated by . for every  ] is generated by a polynomial .each coefficient of can be written as a polynomial in with total degree at most , where for a fixed , is polynomially large in .this can be shown using an elimination method , e.g. , generalized resultants or grbner bases ( cf .take any coefficient of which is a nonzero polynomial in . the number of the choices of which makes this coefficient zero is , by lemma [ lem : schzip ] , at most .this also upper bounds the number of the choices of that make identically zero .a union bound shows that for all but at most choices of none of the polynomial is identically zero , and hence the solution space of is zero dimensional ( and obviously non - empty , as we already know that it contains ) .this gives an upper bound of on the fraction of bad subspaces in .  by proposition [ prop : subspace ] ,the set contains at least a fraction of all -dimensional subspaces of .hence , the fraction of -dimensional subspaces of that properly intersect is at least the claim follows by taking .[ prop : properaffine ] let be positive integers , and be an affine variety of co - dimension defined by the zero - set of polynomials  ] ( as the degree of is at most ) .  by proposition [ prop : properlinear ] , for every , at least affine subspace pass through and properly intersect .hence in total affine spaces properly intersect , where we have counted every such affine space at most times ( this is because the intersection of and an affine space that properly intersects it is of size at most , and is counted once for each point at the intersection ) .thus , the fraction of distinct affine subspaces that properly intersect is at least   by the fact that and the lower bound on , we conclude that this fraction is at least as , this proves the claim .  now having the above tools available, we are ready to give the reduction from variety sampling to the vertex sampling problem introduced in the preceding section and prove our main theorem : * proof of theorem [ thm : main ] .* let be the affine incidence graph of .we will use algorithm [ alg : bisamp ] on . to show that the algorithm works ,first we need to implement the oracles and that are needed by the algorithm .the function simply samples a -dimensional affine space of uniformly at random , and checks whether the outcome properly intersects . to do so , one can parametrize the affine subspace as in the proof of proposition [ prop : properlinear ] and substitute the parametrization in to obtain a system of polynomial equations in unknowns , each of degree at most which is polynomially large in .as is an absolute constant , it is possible to solve this system in polynomial time using multipolynomial resultants or the grbner bases method combined with backward substitutions .if at any point , the elimination of all but any of the variables gives the zero polynomial , it turns out that the system does not define a zero - dimensional variety and hence , does not properly intersect .also , if the elimination results in a univariate polynomial that does not have a solution in , the intersection becomes empty , again implying that does not properly intersect . in both cases fails , and otherwise , it outputs .furthermore , if the intersection is proper , the elimination method gives the list of up to points at the intersection , which one can use to construct the oracle .now we need to show that the graph satisfies the conditions required by lemma [ lem : vertexsamp ] . by the argument above, the degree of every right vertex in is at least and at most , which is polynomially large in .let denote the failure probability of .then proposition [ prop : properaffine ] implies that when .  to bound the left degrees of the graph , note that each left node , which is a point on , is connected to all -dimensional affine subspaces that properly intersect and pass through the point .the number of such spaces is , by proposition [ prop : subspace ] , at most ( assuming ) , and by combination of proposition [ prop : subspace ] and proposition [ prop : properlinear ] , at least now if we choose , the left degrees become concentrated in the range .  putting everything together , now we can apply lemma [ lem : vertexsamp ] to conclude that the output distribution of the algorithm is -close to the uniform distribution on .  to show the efficiency of the algorithm , first note that algorithm [ alg : bisamp ] calls each of the oracles and at most times , which is upper bounded by a polynomial in .hence it remains to show that the implementation of the two oracles are efficient .the main computational cost of these functions is related to the problem of deciding whether a system of polynomial equations of bounded degree in unknowns has a zero dimensional solution space , and if so , computing the list of at most solutions of the system .as in our case is a fixed constant , elimination methods can be efficiently applied to reduce the problem to that of finding the zero - set of a single uni - variate polynomial of bounded degree .a randomized algorithm is given in for this problem that runs in expected polynomial time .thus , we can use this algorithm as a sub - routine in and to get a sampling algorithm that runs in expected polynomial time .then it is possible to get a worst - case polynomial time algorithm by using a _ time - out _ trick , i.e. , if the running time of the sampler exceeds a ( polynomially large ) threshold , it is forced to terminate and output an arbitrary point in .the error caused by this can increase the distance between the output distribution of the sampler and the uniform distribution on by a negligible amount that can be made arbitrarily small ( and in particular , smaller than ) , and hence , is of little importance .  finally , we need an efficient implementation of the field operations over .this is again possible using the algorithm given in .moreover , when the characteristic of the field is small , deterministic polynomial time algorithms are known for this problem .we showed the correctness and the efficiency of our sampling algorithm for varieties of constant co - dimension over large fields .though our result covers important special cases such as sampling random roots of multivariate polynomials , relaxing either of these requirements is an interesting problem . in particular , it remains an interesting problem to design samplers that work for super - constant ( and even more ambitiously , linear in ) co - dimensions ( in our result , the dependence of the running time on the co - dimension is exponential and thus , we require constant co - dimensions ) .moreover , in this work we did not attempt to optimize or obtain concrete bounds on the required field size , which is another interesting problem .finally , the error of our sampler ( i.e. , the distance of the output distribution from the uniform distribution on the variety ) depends on the field size , and it would be interesting to bring down the error to an arbitrary parameter that is given to the algorithm .we would like to thank frdric didier for fruitful discussions .

Abstract: we consider the problem of uniform sampling of points on an algebraic variety . specifically , we develop a randomized algorithm that , given a small set of multivariate polynomials over a sufficiently large finite field , produces a common zero of the polynomials almost uniformly at random . the statistical distance between the output distribution of the algorithm and the uniform distribution on the set of common zeros is polynomially small in the field size , and the running time of the algorithm is polynomial in the description of the polynomials and their degrees provided that the number of the polynomials is a constant .   mahdi cheraghchi  amin shokrollahi


the quality of golf data offers a unique opportunity to investigate the excitation mechanism of low degree p modes . in the region of 4 - 6 minutes, several modes can be extracted and analysed one by one with a short enough time resolution ( days ) over one year , allowing an accurate statistical treatment of the data obtained .+ here we deal with the `` superficial '' energy per unit of mass , associated with the mode at the surface of the sun .we do not address the issue of the `` global '' energy of the mode , which is related to the superficial energy through the shape of the eigenfunction inside the sun .+ woodard ( 1984 ) first revealed the exponential nature of the distribution of spectral power in each frequency bin , using acrim data .+ goldreich & keeley ( 1977 ) first considered the excitation of p modes by turbulent motions near the surface of the convection zone . usingthe analogy of a damped oscillator excited stochastically , kumar , franklin & goldreich ( 1988 ) described analytically how the theoretical distribution of energy , averaged over a given time - window , should depend on the damping time of the mode. comparisons with real data were performed by toutain & fr " ohlich ( 1992 ) with 160 days of iphir data .they found in particular that the damping times deduced from the linewidths were compatible with those expected in the model of stochastic excitation .+ chang ( 1996 ) pointed out that strong localized peaks in the energy variations of a damped oscillator excited stochastically do not necessarily correspond to a strong excitation , but rather to an exceptional coherent addition of random phases .+ this is true provided that the number of independent excitations per damping time of the mode is large enough , like in the case of excitation by solar granules . in this context , two different modes excited by exactly the same sources would have uncorrelated energies .+ different modes , however , would be correlated , if the timescale between two excitations from a common source were longer than their damping time .+ the correlation between the modes energies is therefore directly related to the characteristics of their source of excitation .+ using 160 days of iphir data at the end of 1988 ( just before the solar maximum , ) , baudin ( 1996 ) concluded that the p modes were likely to be correlated .an anticorrelation between the mean solar magnetic field and the p - mode power was found by gavryusev & gavryuseva ( 1997 ) in iphir data , while no clear correlation has been detected yet in golf data ( baudin 1997 ) .+ after checking in sect .[ sectexpo ] the exponential nature of the distribution of energy of p modes in golf data , we address the issue of their relative independence in sect .[ sectindep ] , using statistical tests based on montecarlo simulations .+ these same tests are used to re - analyse iphir data in sect .[ sectiphir ] .the energy integrated over a time interval , the power of the mode , was computed by chaplin ( 1995 ) using a fourier transform over short subseries .more sophisticated methods based on the wavelet analysis were developped by baudin , gabriel & gibert ( 1994 ) in order to analyse the variations of power both with time and frequency .+ frequency resolution is not required for our study . since the distribution of energy is likely to be mathematically simpler than the distribution of power ( kumar , franklin & goldreich 1988 ) , we have prefered to extract the energy directly .+ let be the oscillatory velocity ( integrated over the surface of the sun ) , filtered in the fourier domain through two windows of width centred on the eigenfrequencies .its fourier transform is therefore equal to zero out of these windows . the time evolution of the energy of this isolated mode can be obtained by a bivariate spectral analysis , as in toutain & fr " ohlich ( 1992 ) . herewe favour a simpler method based on the inverse fourier transform of the line , translated around .it is shown in appendix a that the energy of this mode can be written as follows : this approach is equivalent to the one used by chang & gough ( 1995 ) , by means of the hilbert transform of the velocity , since .+ if the distribution of velocities is gaussian , then the real and imaginary parts of the function are two independent gaussian distributions with identical amplitudes and variances . thus eq .( [ energ ] ) directly implies that the distribution of the energy is exponential , as expected .( [ defg ] ) shows clearly that the time resolution of the energy , reconstructed by eq .( [ energ ] ) , is related to the size of the filtering window : denoting by the total length of the observation , the frequency resolution of the fourier transform is , and the filtering window contains points .the inverse fft algorithm is used to compute eq .( [ defg ] ) and define the energy at successive instants .( [ resol ] ) then guarantees that the resulting energy is not oversampled .we have considered the set of p modes corresponding to , and , between 11th april 1996 and 14th february 1997 ( a publication concerning the calibration procedure is in preparation ) .the fourier transform of the resulting velocity over these 310 days allows a filtering window size of ( days ) for this set of modes .the window is symmetric with respect to the centroid of the line , , which is determined according to lazrek ( 1997 ) .the two -components of the mode , however , are not separated .in contrast with iphir , the width of the window is determined by the proximity of another mode ( ) , rather than by the level of noise which is here very low .[ etl01 ] shows the time evolution of the energy of the 18 selected modes and , normalized to their mean energy .the golf instrument was stopped for one day on 8th september 1996 .four days of signal were removed from our statistical study ( around the 156th day on fig .[ etl01 ] ) in order to account for the stabilisation of the instrument .the resulting sample is made up of 210 points .following the picture of a thermodynamic equilibrium between the random motions of the convective cells and the oscillating cavity ( goldreich & keeley 1977 ) , we wish to compare the observed sample of energies , with an exponential distribution .any exponential distribution is defined by a single parameter , its mean value .[ de22 ] shows a typical histogram and cumulative distribution for the modes extracted from the golf data ( the cumulative distribution is defined as the primitive of the density of probability , it increases monotonously from 0 to 1 ) .they are compared to an exponential distribution whose mean value is estimated from the sample of points . using the maximum likelihood approach ,the best unbiased estimator of for an exponential distribution is the following :   \(i ) the variance test + a simple test consists in checking that the first moments of the distribution ( mean value and variance ) are compatible with those of a theoretical exponential distribution .+ the variance of an exponential distribution coincides with the square of its mean value .we check this property by computing , for each mode of the golf data , the ratio of the estimated variance ( denoted by ) to the estimated mean value squared : each value is interpreted owing to the cumulative distribution of , obtained if were built from a true exponential distribution . is computed numerically using a montecarlo method , with exponential samples of points . for each of the modes selected , is the fraction of these trials leading to a value of _ larger _ than the one observed .since we are interested only in knowing whether the observed is typical of an exponential distribution or not , we shall give equal importance to the lowest and highest values of the variance by measuring the quantity .\(ii ) the kolmogorov - smirnov test + while the variance test depends only on one particular moment of the observed distribution , a more global comparison is achieved with the kolmogorov - smirnov ( ks ) test on the cumulative distribution .this test measures the maximum distance between and a theoretical exponential cumulative distribution .if the mode energies were exponentially distributed , the statistics of would be described by a cumulative distribution denoted by .since the mean value of the reference ditribution is estimated from the data , we can not use the standard formulae ( numerical recipies 1992 , chapt .14.3 ) to fit .+ instead of doing this , we have used a montecarlo method of samples in order to define the cumulative distribution of the distance . therefore indicates the fraction of these trials leading to a distance _ larger _ than the value observed .+ for each of the modes selected , a value of close to would indicate that the observed distribution is too far from the theoretical one .a value of close to is just as improbable , but would indicate an exceptionnal agreement between the theoretical distribution and the observed one .\(iii ) autocorrelation of the artificial exponential distributions used in the montecarlo method . +all of the p modes selected are autocorrelated over a timescale comparable to their damping time ( 2 to 4 days ) , usually deduced from the full width at half maximum ( fwhm ) of their lorentzian fit in the fourier space .for the sake of accuracy , we have therefore used exponential distributions with comparable autocorrelation in order to compute the theoretical cumulative distributions and in our montecarlo simulations .each one is obtained by first creating a time series of a damped oscillator excited by a gaussian noise , and then extracting the energy with the method described in sect .[ extractm ] .the damping time of the oscillator is chosen such that it corresponds to a fwhm of in the fourier space . + the output of the tests , however , is only slightly modified if distributions made of _ independent _ points are used .+ for both tests , fig .[ ks9 ] shows a very good agreement for the set of modes selected . as an exception ,the energy of the mode , is not exponentially distributed ( , ) . +although the global shape of this distribution can be made compatible with an exponential distribution by adopting a mean value smaller than the estimated value ( for ) , its variance is too large to be reconciled with the variance of an exponential distribution .+ we have also analysed the distribution build with the 18 modes altogether ( each mode is normalized by its mean energy ) .even with this improved statistics of points , the variance and ks tests have not detected any significant deviation from an exponential distribution ( , ) .no striking general correlation appears when looking at the set of 18 modes displayed on fig .[ etl01 ] . nordoes it stand out from the computation of the correlations of these modes , two by two .although some large correlations are measured ( between the modes , and , ) , even larger anticorrelations are also found ( between the modes , and , ) .no general trend is visible , the mean value being ( fig .[ corgolf ] ) .+ the statistical error of the estimator of the correlation coefficient between exponential distributions , from a sample of points , scales like . for our sample of points ,no effect smaller than can therefore be detected .altogether , of the couples ( 17 out of 306 couples ) present correlations contained , in absolute value , between 2 and 3 standard deviations , which is not very significant ( would be expected for a normal distribution of the statistical error ) .+ nevertheless , a more sensitive indicator can be constructed , in order to determine the mean correlation coefficient more accurately .+   if distributions are independent ( null hypothesis ) , the variance of their sum should be equal to the sum of their variances .this test was performed by baudin ( 1996 ) with iphir data , who normalized the distribution by their level of noise , and found some discrepancy . +a fundamental feature of our method is the use of the exponential nature of each individual energy distribution , in order to compute the standard deviation of our estimate of the correlation , and therefore the confidence level of our conclusions .+ we denote by the sum of distributions of energy , made of events , where each of the distributions is normalized by its estimated mean energy . since each distribution appears to be exponential within the statistical error ( section 2 ) , ought to resemble a gamma - distribution of order ( denoted by ) if they are independent , or an exponential distribution of mean value if they are all identical .the null hypothesis can therefore be tested by comparing the observed distribution with the theoretical -distribution , using the variance and ks tests .+ denoting by the correlation coefficient between the modes and , and their mean value , the variance of is directly related to these correlations : if the modes are independent , the standard deviation of the variance estimator of is consequently , another way of testing the null hypothesis is the comparison of the variance of with var , in units of the statistical error .+ even if the distributions defining are independent and exponential , an additional error of the order of is introduced in the estimation of the mean value of each exponential distribution .we use a montecarlo method of trials made from independent exponential distributions , in order to define the cumulative distributions for the outcome of the variance test , and for the outcome of the ks test . hereagain , we have used autocorrelated exponential distributions in the montecarlo simulations .( [ varcor2 ] ) indicates that the variance of the distribution gives a direct measure of the mean correlation among the modes : this formulae , however , is not directly useful without an expression of the statistical error associated with the estimator of the variance .if the modes are correlated , computing it requires some additionnal assumptions about the properties of the correlation ( sect .[ lambhyp ] ) .nevertheless , coincides with eq .( [ sist0 ] ) to first order .together with eq .( [ delvar ] ) , the smallest correlation detectable with this method scales as follows : which is a factor smaller than the sensitivity of the correlation coefficient two by two .a better sensitivity is therefore obtained by summing a large number of modes .however , the hypothesis of a constant correlation between the modes might be questionable if the range of frequencies is large , especially since the mean energy and the lifetime of the modes vary significantly with frequency .   both tests are of course very sensitive to the presence of a gap in the data .if of the data were filled with zeros due to an interruption of the instrument , the variance of would be increased by a factor .consequently , we have carefully removed from our samples the points corresponding to these gaps .+  the sum of the energies of the 9 modes , , normalized to their mean energy , is shown in fig .[ gaml0t9 ] .we note in passing that the clear gap in the data appearing around the 156th day confirms the validity of our procedure for the extraction of the energy . as before ,four days of signal have been removed from our statistical study to account for the stabilisation of the instrument , resulting in a sample made up of 210 points .their distribution is successfully compared to a -distribution in fig .[ gaml0d9 ] ( and ) .the same test performed on the 9 modes , , obtains and .applied to these 18 modes altogether , the tests confirm again the null hypothesis ( and ) .+ the correlation being low , we may use eq .( [ delvar ] ) with the statistical error given by eq .( [ sist0 ] ) , and obtain a confirmation of the absence of correlation , with an error bar : for 9 modes , and for 18 modes .a refined estimate of the correlation can be obtained by making some assumptions about its origin .we build in appendix b a simple model where the excitation is a mixture of two types of sources , which , taken separately , would result in uncorrelated / highly correlated modes energies respectively .+ the first type represents the granules , which produce so many excitations per damping time that the correlation among the modes energies is close to zero .+ the second type is hypothetical .it could be produced by some isolated events , possibly of magnetic origin , separated by a time comparable to or larger than the damping time of the modes considered .+ we assume that the mode response to an excitation is linear , and therefore the response to a mixture of sources is a superposition of the answers to the two types separately .our model depends on a single parameter , namely the fraction of the energy of each mode due to the second type of sources .+ we define in appendix b the theoretical distribution function corresponding to such correlated modes energies .if the model is applicable , the distribution ought to converge , for , towards a well defined distribution denoted by , such that : the variance and ks test can therefore be used , for various values of , in order to test this `` -hypothesis '' .+ with the definition of appendix b , the correlation coefficient is related to the coefficient as follows :   the statistical error associated with the estimator of the variance also depends on according to eq .( [ astat ] ) .( [ delvar ] ) can then be used to determine the mean correlation , with a consistent statistical error .+ we do not expect higher order moments of the distribution to be more sensitive to a correlation between the modes , since we prove in appendix b that they also vary like at first order .+ we also demonstrate that the shape of the cumulative distribution varies like .the sensitivity limit of the ks test is therefore expected to be comparable to the sensitivity limit of the variance test .+ here again , normalizing the distributions by their estimated mean value introduces a bias , which we take into account using a montecarlo method . for each value of , we compute from trials the theoretical distribution , and use other trials to define the cumulative distributions and which are used for our variance and ks tests . for the sake of simplicity , the effect of the autocorrelation of each mode is neglected here .indeed , we know from sect .[ statest ] and [ method ] that it introduces very small corrections on the cumulative distributions and .+ we define the error bar of the correlation coefficient as the range of values of within which the test ( or ) remains inside the upper region , by analogy with the statistics of normal distributions .[ figogolf ] shows that the variance and ks tests , applied to 18 modes for various values of , stays within the upper region for , which coincides with the statistical limitation expressed by eq .( [ lamin ] ) .+ nevertheless , while the measurements made by golf are compatible with a total lack of correlation of the modes , our tests can not exclude that up to of the energy is common to the modes .since the conclusions of baudin ( 1996 ) about 160 days of iphir data were obtained from time variations of the power instead of the energy , using a different normalization and a different time resolution , we have first re - analysed these data with the method described above , using the same 11 modes ( , , and , ) .the central frequency is taken from toutain & fr " ohlich ( 1992 ). the higher level of noise limits the size of the filtering window to , leading to a time resolution of days , and a statistical study on points .we have removed the data surrounding two gaps in the series , around the 5th and the 61st day .the resulting sample is shortened to points only . + according to fig .[ gamiph ] , the distribution of energy of each of the 11 modes of iphir is compatible with an exponential distribution ( apart from the mode , ) .+  the correlation coefficient of the modes energy , two by two , is shown in fig . [ coriphir ] .the mean value is , the statistical error being .altogether , of the couples ( 2 out of 45 couples ) present correlations contained , in absolute value , between 2 and 3 standard deviations , which is comparable to the expected for a normal distribution of the statistical error .we have also extracted the same modes , with the same filtering window , from the first 153 days of golf data to obtain a comparable sample of 78 points .the difference between the two series appears on the distribution shown in fig .[ gamgo ] , where both the variance and the ks tests indicate that the modes are likely to be less independent in iphir data than in golf data .+ the tests applied to golf data are compatible with the null hypothesis ( , ) , which is consistent with the results of sect .[ sectindep ] .+ by contrast , the same tests applied to iphir data _ reject the null hypothesis _ with a confidence level with the variance test , and a confidence level for the ks test ( , ) .while the cumulative distribution of golf does not show any systematical trend when compared to , the cumulative distribution of iphir shows a clear trend .this trend is successfully suppressed when compared to the distribution ( fig .[ gamgo ] ) .[ figophir ] shows that within our simple model , the signal of iphir would be absolutely normal as regards our tests ( , ) if a fraction of each mode energy were common to all the modes , correponding to a mean correlation . + error bars are obtained by varying the parameter : the variance test leads to , while the ks test obtains .+ moreover , the correlation computed from eq .( [ delvar ] ) with the statistical error given by eq .( [ astat ] ) is .although this analytical estimate is less reliable than the tests based on montecarlo simulations ( eq . ( [ astat ] ) neglects the error in estimating the mean energy of each mode ) , it is useful as a quick check of the results . + it is therefore comforting to notice , as can be seen in fig .[ figophir ] , that the range of correlations defined by these three methods overlap in the range , correponding to .of course , this overlapping region can not be directly interpreted in terms of a standard deviation .we shall adopt the conservative range obtained with the ks test , which takes the full distribution into account : , corresponding to a fraction .   in order to check the possibility that the correlation might come from a multiplicative noise ( such as due to a pointing noise ) ,we have computed the correlation between windows of noise centered ( resp . ) to the right of each mode .this test indicates that the noise itself is not correlated , with and ( resp . and ) .+ we have checked the effect of changing the size of the filtering window to 4 ( no noise , but low statistics of 52 points ) and 8 ( good statistics of 106 points , but iphir is influenced by the noise ) .while a smaller filtering window still favours , a larger window takes into account a significant fraction of uncorrelated noise , as expected , resulting in a slightly lower value of .+ one might also suspect that the discrepency between the iphir distribution and a distribution is due to the mode , which is not well fitted by an exponential distribution ( see fig .[ gamiph ] ) .nevertheless , performing the same analysis without this particular mode leads to the same conclusion : and if , while and if .the exponential nature of the energy distribution of each mode has been used to compute their mean correlation coefficient with a consistent error bar . + two tests based on montecarlo simulations , and one analytical formulae , applied to 310 days of golf data , support the null hypothesis of no mean correlation among the modes , , with an accuracy of . + our analysis of the modes correlation in iphir data , using these statistical tools , gives an accurate statistical support to the tentative conclusions of baudin ( 1996 ) .the variance test rejects the null hypothesis with a confidence level .+ the presence of a clear correlation among p - mode energies in iphir data strongly constrains the standard picture of stochastic excitation .if really of solar origin , it suggests the existence of an additional source of excitation , other than the granules .we have built a one parameter model of random excitations separated by a time comparable to the damping time of the modes , added to the usual granule excitations .iphir data are fully compatible with this `` -hypothesis '' , if a fraction of each mode energy is due to this additionnal source of excitation , resulting in a mean correlation among the modes .+ on the other hand , the absence of correlation in golf data support the standard picture of stochastic excitation by the granules only .+ this difference between iphir and golf data can be interpreted as a change from in iphir data to less than in golf data .+ this evolution could be related to the change in magnetic activity , since the golf data correspond to a period close to the solar minimum while the iphir data correspond to a period closer to the solar maximum .if this is true , a confirmation will be obtained by performing this same analysis on golf data when we approach the solar maximum , in a couple of years .virgo data will also be useful in order to identify the possible role of the measurement techniques ( velocity / intensity ) in the determination of the correlation .+ however , the mechanism by which the magnetic field influences the excitation of the modes , the nature of these hypothetical exciting events remains to be explored in more detail .we thank claus fr " ohlich and thierry toutain for providing access to iphir data cleaned of satellite pointing noise .we are grateful to michel tagger and romain teyssier for many useful discussions , and thierry appourchaux for constructive comments about the manuscript .soho is a mission of international cooperation between esa and nasa .let us consider the real displacement filtered through a double window with a width , centred on the frequencies : where is the complex conjugate of the first term .by analogy with an oscillator of eigenfrequency , the energy is defined as the sum of a kinetic and a potential part : from eq .( [ defg ] ) and ( [ yt ] ) , the filtered displacement and velocity can be written as follows : we first deduce from the relation that and are related as follows : from eq .( [ defg ] ) , the non - zero fourier components of ( and ) correspond to frequencies between and .if , as is usually the case , , the high frequency oscillations of are well separated from the slower variations of and , and eq . ( [ em ] ) is approximated by eq .( [ energ ] ) .the amount of energy which is coherent among the modes can be estimated by constructing a simple one - parameter model as follows . indexing the modes by , we assume that the velocity residual of each mode is made of a superposition of two independent signals , where is common to all the modes , and all the , , are independent .+ using the filtering method described in section 1 , we introduce the parameter as : where , , are independent normalized normal distributions .the quantity can be interpreted as the ratio of the energy in the common signal to the total energy of the mode .we denote by the energy of the signal filtered in the fourier space , normalized to its mean value : the correlation between two modes is then : for the sake of simplicity , is assumed to be independent of the mode .this is equivalent to assuming that the correlation is uniform among the modes .+ the sum of the normalized energies is defined as : the variance of the estimator of the variance depends on the fourth moment of the distribution , and is equal to : .\label{astat}\end{aligned}\ ] ] let us show that the higher order moments of the distribution vary like , to first order .+ where denotes the expectation value of the distribution . as the transformation does not change the distribution defined by eq .( [ ej ] ) , only the even powers of can contribute to .it can therefore be expanded into powers of .let us develop the product in eq .( [ mup ] ) and prove that the term of order is zero .we use below the special relation between the centred moments of a -distribution of order : let us now show that the probability density of also varies like , to first order .+ given the eqs .( [ ej])-([defs ] ) defining , we can expand in powers of . moment of order is defined as : the fact that every moment varies at least like ( eq . [ egalzero ] ) implies : the only function satisfying eq .( [ intenul ] ) for any value of is . therefore both and its primitive ( the cumulative distribution of involved in the ks test ) vary like to first order . + we conclude that the sensitivity of the ks test is comparable to the sensitivity of the variance test .    baudin f. , gabriel a.h . , garci r. , foglizzo t. , gavryusev v. , gavryuseva e. , gough d. , ulrich r.k . and the golf team , 1997 , proceedings of the iau symposium 181 , f - x schmider & j. provost eds , observatoire de nice , france , in press        lazrek m. , baudin f. , bertello l. , boumier p. , charra j. , fierry - fraillon d. , fossat e. , gabriel a.h . ,garca r.a ., gelly b. , gouiffes c. , grec g. , pall p.l ., prez hernandez f. , rgulo c. , renaud c. , robillot j .- m . , roca corts t. , turck - chize s. , ulrich r.k .1997 , sol .special soho issue , in press

Abstract: we have studied the statistical properties of the energy of individual acoustic modes , extracted from 310 days of golf data near the solar minimum . the exponential distribution of the energy of each mode is clearly seen . the modes are found to be uncorrelated with a accuracy , thus supporting the hypothesis of stochastic excitation by the solar convection . + nevertheless , the same analysis performed on the same modes just before the solar maximum , using iphir data , rejects the hypothesis of no correlation at a confidence level . + a simple model suggests that of the energy of each mode is coherent among the modes studied in iphir data , correponding to a mean correlation of .


global tropospheric chemistry transport models ( ctm ) are used to address important issues ranging from air quality to climate change . in order to continuously improve their performance ,it is of crucial importance to understand and quantify the diverse sources of uncertainties and errors present in them .we group these in three different categories , ( _ i _ ) errors and uncertainties coming from observations and data used in our models ( such as emission inventories , wind fields , reaction rates ) ; ( _ ii _ ) errors coming from our choice of governing equations ( or mathematical model ) , parametrizations , and the level of complexity of the physical modules included in our formulation ; and ( _ iii _ ) numerical errors coming from the choice of algorithms we use to solve the governing equations using computers .+ in this study , we focus our attention on estimating the magnitude of numerical errors ( _ iii _ ) , in particular , those arising from the choice of operator splitting technique utilized to integrate in time the transport and chemistry operators in real - life global ctms . in order to achieve this, we numerically extend the results introduced for the linear diffusion - reaction case in , to a non - linear 1-d chemistry - transport numerical model .the latter numerical results provide us with a framework to estimate upper bounds for operator splitting errors in the fully non - linear 3-d state - of - the - art global ctm : geos - chem . to the best of our knowledge ,our contribution is the first in estimating operator splitting errors in the context of real - life global atmospheric chemistry simulations .+ ctms simulate the dynamics of chemical species in the atmosphere by numerically integrating a set of coupled nonlinear partial differential equations of the type : for ; where represents the spatio - temporal evolution of the concentration of species ( typically over a hundred species are considered ) , is the wind velocity , is the air density , the eddy diffusivity matrix , are the nonlinear production terms , are the destruction terms , are the volume emission sources , and are the sinks ( ex . precipitation or in - cloud removal ) .see for a detailed description of these equations .+ due to the dimensions of grid boxes in global ctms , like geos - chem ( with hundreds of kilometers in the horizontal versus tens to hundreds of meters in the vertical ) , intertial vertical transport processes in this global models are simulated ( a ) using vertical mass fluxes schemes that ensure that the horizontal air flow is divergent - free ( ) , ( b ) using convection parametrizations , and ( c ) using a boundary layer mixing algorithm .in addition , horizontal diffusion due to numerical errors in transport schemes are typically higher than their eddy difusivity counterpart , as measured by aircraft missions . as a consequence , the first term of the right - hand side of equation ( [ eq : advec - reac ] ) , which models the dynamics of intertial vertical transport as an eddy diffusion process , is not explicitly integrated in global ctms ; and the governing equations ( [ eq : advec - reac ] ) are sometimes written in a simplified way as   the chemistry operator on the right - hand - side of equations ( [ eq : advec - reac2 ] ) models the chemical interaction of atmospheric species whose lifetimes range from milliseconds to many years .the chemistry operator is very stiff as a consequence of this large range of time - scales and thus , implicit - in - time methods are an appropriate choice to integrate equations ( [ eq : advec - reac ] ) .traditional methods , such as the method of lines , aimed at achieving this task in realistic 3d simulations , involve solving for an enormous number of degrees of freedom at each time step in a coupled fashion ( 100 chemical species in grid cells , for a spatial resolution ) .this is due to the inter - species coupling in the chemistry operator and the spatial coupling in the transport operator . in practical situations , however , efficient computational algorithms to integrate equations ( [ eq : advec - reac ] ) use operator splitting strategies that allow the explicit time integration of the transport and implicit time integration of the chemistry operators separately and sequentially , thus , reducing significantly the degrees of freedom solved in a coupled fashion at a given time step .this is done at the expense of a loss of accuracy in the approximate solution . + estimating the magnitude of the numerical errors introduced by the time integration of equations ( [ eq : advec - reac ] ) in realistic 3-d computer simulations is a hard task since no relevant analytic solution can be used as a reference to estimate them . in theory , estimates of these errors depend directly on the regularity properties of the analytic solution of equations ( [ eq : advec - reac ] ) , the set of initial and boundary conditions , and the chosen numerical scheme . in this study, we assume that the analytic solution of equations ( [ eq : advec - reac ] ) is unique and regular enough so that numerical error estimates can be expressed as inequalities of the form ( [ eq : error_est ] ) .operator splitting errors , as well as numerical errors arising from the time integration of the chemistry operator depend explicitly on the magnitude of the chosen time steps , while numerical errors coming from the time integration of the transport operator depend both on the time step and on the grid size .this fact , in combination with an expression of the analytic solution of equations ( [ eq : advec - reac ] ) , is exploited to obtain the exact magnitude of operator splitting errors in our one - dimensional proto - type transport - chemistry numerical model .+ our one - dimensional numerical experiments show three main results : ( a ) operator splitting sequences where the stiff non linear chemistry operator is evaluated at the end of the time step are more accurate than those where the transport is evaluated lastly , independently of the operator splitting time - step , as in the linear case introduced in ; ( b ) the results of numerical simulations that use different operator splitting strategies differ by at most 10% ; and ( c ) numerical errors coming from the integration of the transport operator are much bigger than those coming form the operator splitting technique for spatial and temporal scales comparable to those used in global ctm .we use this fact , and evidence from papers such as , to suggest that in realistic 3d simulations , errors due to operator splitting are much smaller than those introduced by transport schemes .upper bounds of the numerical errors introduced by solving partial differential equations with regular boundary and initial conditions , using a given numerical scheme , can be expressed by inequalities represented as where is the true solution of the partial differential equation , the numerical approximation , and are the time step and grid size respectively , and are exponents ( typically larger than one ) that determine the order of convergence of the method in time and space respectively , and are constants that depend on the regularity of the true solution and parameters in the equation , and is the norm in the appropriate banach space . for a convergent method , as and , the numerical error vanishes , ( _ i.e. _ ) andthe numerical approximation converges to the true solution , in the normed space .more details about the integral representation ( equation [ eq : error_est ] ) of numerical errors due to discretization of partial differential equations can be found in : + for the specific set of partial differential equations ( [ eq : advec - reac ] ) , operator splitting errors and errors coming from the numerical integration of the chemistry operator ( where no coupling in space exists ) contribute to the first term on the right - hand - side of inequality ( [ eq : error_est ] ) , whereas , numerical errors from the integration of the transport operator contribute to the first and second terms of the right - hand - side of inequality ( [ eq : error_est ] ) . quantifying the independent contribution of each processes to each term of inequality ( [ eq : error_est ] ) is not simple in practical applications . in the following section ,we show how to estimate the magnitude of operator splitting errors in the absence of other numerical error coming from the time integration of the transport and chemistry operators .+   classical approaches to estimate the numerical errors introduced by operator splitting approaches are based on asymptotic expansions of exponential operators ( linear case ) and lie operator formalism ( nonlinear case ) . for completeness ,we briefly describe important results of the linear analysis of operator splitting methods in this section .we refer the reader to and the references therein for more details . in this section, it is assumed that the time integration of each operator separately can be found exactly giving rise to no numerical error , _i.e. _ the numerical errors discussed below come only from the choice of the operator splitting technique .+ we use as an example the linear evolution equation , where and are linear operators .one of these operators could represent the linear spatial differential operator ( transport ) in equations ( [ eq : advec - reac ] ) . the analytic solution for this problemis given by : the simplest operator splitting method , called godunov and denoted by , can be obtained for  ] for km , so that the plume would stay within the domain for the whole simulation time , $ ] for hours , and in order to not introduce any errors due to boundary conditions in the numerical advection operator .the values of and were chosen for the stiff chemistry operator .the effective stiffness of the chemistry with respect to the transport is since .the initial conditions are given by , where \\     0 & \text{elsewhere}.    \end{array }   \right.\ ] ]  in a 10-hour simulation time period , the initial release is advected exactly km to the right , and the concentrations of all species have reached chemical equilibrium . according to expression ( [ eq : steady_1d_2 ] ) , , and . the exact solution at time hours is explicitly given by and .this is our reference solution .+  km and different time steps , when compared to the analytic solution.,title="fig : " ] km and different time steps , when compared to the analytic solution.,title="fig : " ] km and different time steps , when compared to the analytic solution.,title="fig : " ]  for the numerical simulations , we implemented an explicit , second order accurate ( in space ) , one - dimensional advection - scheme based on the lax - wendroff method with superbee slope limiters ( see , pp 112 for details ) , and used for the chemistry , the built - in implicit stiff - ode integrator ode23 from matlab .in order to minimize contributions to the numerical error , to the first term in inequality ( [ eq : error_est ] ) , from both the advection scheme and chemistry integrator , we utilized a very small internal advection time step , seconds , and set the convergence relative - tolerance parameter to in the routine ode23 ( it adaptively chooses a small internal time step in order to meet the prescribed error convergence criterion ) .+ we solved equations ( [ eq : toy_syst])-([eq : toy_syst3 ] ) using multiple first order godunov operator splitting approaches ( where transport and chemistry were evaluated in different orders ) for multiple operator - splitting time - steps , and seconds , and for multiple grid sizes and km ( the three largest grid sizes were chosen to resemble spatial resolutions of , , and , in current 3d global ctms ) .the results of these numerical simulations and the exact solution are plotted in the top plot of figure [ fig : ozone_1d ] .the numerical solutions corresponding to the multiple operator splitting approaches , for a given value of , appear as a single curve since their differences were smaller than the line - width chosen for the plot . +the quantification of numerical errors was performed using the modified relative root mean square ( rrms ) , commonly used in 3d atmospheric chemistry simulations given by where and are the concentrations of species calculated in simulations and , respectively , is the set of grid - boxes where exceeds a threshold , and is the number of such grid - boxes .we used , thus neglecting concentrations smaller than with respect to the original concentration . in our one - dimensional experiments ,simulation is the exact solution , and simulation was one of the multiple godunov operator splitting approaches .the second plot of figure [ fig : ozone_1d ] shows the quantity for species , for the multiple values of and . in this plot ,the red triangles represent simulations where transport was evaluated last , ( ) , and the green dots where chemistry was evaluated last ( ) .this plot confirms what is observed in the top plot , _i.e. _ , the fact that the differences across the multiple operator splitting approaches , for a given , are very small ( ) .+ in the bottom plot of figure [ fig : ozone_1d ] , we further show the values of the numerical error for the two sequences , and , for km , for the multiple values of the operator splitting time - steps .we found this plot to be representative of the behaviour of the numerical error for other values of . note that while the differences across the multiple approaches are very small , the interesting mathematical behaviour of the numerical error , discussed in section [ sec : op_split ] , can be observed .indeed , the numerical error of the sequences , where the chemistry ( the stiff process ) is evaluated last , produce better numerical results than their counter parts .moreover , sequences appear to be almost insensitive to the magnitude of the operator splitting time - step ( the error even seems to grow as as reported in sportisse , 2000 ) making them a preferred choice , since larger operator splitting time steps allow faster computations when exploiting the intrinsic parallelizable nature of the chemistry operator .the quality of results produced by sequences where transport is evaluated last , follows the traditional behaviour of linear analysis where the numerical error decreases as the operator splitting time decreases .since the magnitude of these first order operator splitting errors was so small , we chose to not implement higher order operator splitting approaches . + while the bottom plot of figure [ fig : ozone_1d ] shows a clear picture of the magnitude of operator splitting errors ( ) , we performed transport - only simulations in order to verify the magnitude of the numerical errors coming from the numerical advection scheme itself .the results of these simulations are shown in the top plot of figure [ fig : ozone_1d_transp ] .note that while the magnitude of the concentration of in these simulations is exactly one ( since no chemistry is present ) , the numerically simulated profiles , for the different values of , look very similar to those in the top plot of figure [ fig : ozone_1d ] . indeed , when computing the modified rrms error associated to these simulations , as shown in the bottom plot of figure [ fig : ozone_1d_transp ] , the behaviour of the relative errors resembles the one observed in the middle plot of figure [ fig : ozone_1d ] . in short , the numerical errors coming from the choice of operator splitting are eclipsed by the largest component of the numerical error coming from the spatial discretization ( second term in inequality ( [ eq : error_est ] ) ) in the numerical advection scheme .+ having chosen an initial condition in the shape of a step function in our experiments , caused our second order numerical advection scheme to behave as a first order scheme .indeed the numerical error decreases close to linearly in our numerical experiments when using the -norm instead of the modified rrms ( plot not shown ) .estimates for the numerical errors , in the form of an effective numerical diffusion , , for 1d first order numerical advection schemes place their value at , where is the mean flow velocity and the grid spacing . in our 1d experiments ,these numerical diffusion is of the order m/s .numerical diffusion in 3d global models ( ) is estimated to be around m/s .these 3d estimates place our one - dimensional experiments within a relevant range .determining the exact magnitude of numerical errors in 3d global ctm simulations in the exact same way we did for our 1d prototype is not possible .this is due to the lack of an analytic expression for the solution to equations ( [ eq : advec - reac ] ) in realistic circumstances ( time - dependent winds , time - dependent chemistry rates changing throughout the geographic domain due to photolysis , time - dependent emissions ) . in order to estimate operator splitting errors in 3d ctms, we can only compare the output of simulations where everything is kept the same except for the operator splitting sequence and the operator splitting time step .this is the strategy we present in this section , which in combination with the results from our one - dimensional simulations , allowed us to determine upper bounds of operator splitting errors in geos - chem . in order to further understand additional numerical errors , due to the time integration of relevant processes inherently solved with operator splitting approaches and not present in our 1d toy example , we performed multiple additional 3d global simulations . in these simulations, we gradually included inhomogeneous boundary conditions ( emission processes ) to the time integration ( ) , and vertical processes ( convection and dry deposition ) .+ geos - chem is a state - of - the - art 3d global eulerian model of tropospheric chemistry driven by assimilated meteorological observations from the goddard earth observing system ( geos ) of the nasa global modeling and assimilation ofice ( gmao ) .the model simulates global tropospheric ozone - nox - voc - aerosol chemistry .the full chemical mechanism for the troposphere involves over a hundred species and over three hundred reactions .the ozone- chemical mechanism of geos - chem has been described by and recently updated by .details of the chemical reactions and rate constants are reported in the chemical mechanism document ( http://acmg.seas.harvard.edu /geos / wiki_docs / chemistry /chemistry _ updates_v6.pdf ) . in figures4 the chemical species are arranged in the order of their chemical lifetimes in the atmosphere , from oh ( 1 second ) and ( hour ) , to co and ( 23 months ) .the chemical mass balance equations are integrated using a gear - type solver .stratospheric chemistry is not explicitly simulated and it instead uses the `` synoz '' cross - tropopause ozone flux boundary condition of .the model uses the flux form semi - langrangian advection scheme of .we used the geos - chem model ( v8 - 02 - 03 ) driven by the geos-5 data at the 4 x 5 horizontal resolution and 47 levels in the vertical .detailed descriptions of the model are given by and .in this study , we initiate the model simulations on january 1 , 2005 with model fields from a 6-month spin - up run , and focus on the weekly averaged model results for january 1 - 7 , 2005 .   our strategy consisted of comparing the instantaneous concentration of several chemical species , after multiple one - week long , 4 x 5 horizontal resolution , geos - chem simulations ( version v8 - 02 - 02 ) , using two versions of the ( default ) second order strang operator splitting method given by the sequences : for different values of the operator - splitting time step .these sequences are denoted as and respectively in the subsequent paragraphs .we used mins . in all these simulations , transport and chemistry were the only active mechanisms , all other mechanisms were turned off .the inactive mechanisms include : emissions , convective transport , deposition , and planetary boundary layer mixing .emissions correspond to inhomogenous boundary conditions that are treated numerically as production rates distributed in the boundary layer and solved together in the chemistry operator .+ we used the modified rrms ( [ eq : rms ] ) with a threshold molecules to quantify the numerical differences in our global simulations .figure [ fig : rms_opsplit_all ] shows the relative differences between the reference simulation with mins , and the other operator splitting approaches for multiple s .note that the maximum differences across simulations ( and species ) are of the order of .+ using our one - dimensional prototype and fixing km , we compared the results of two operator splitting strategies ( and ) for multiple values of , with the sequence and sec set as a reference .the results are displayed in figure [ fig:1d_opsplit_tch_as_ref ] .note that while the bottom plot of figure [ fig : ozone_1d ] shows that operator splitting ( relative ) errors are less than ( when comparing to the analytic solution ) , the relative differences between simulations using alternative operator splitting methods may be as large as .this is roughly the same magnitude of the differences observed between the 3d ( transport - chemistry ) simulations in the top panel of figure [ fig : rms_opsplit_all ] .+ note that we chose the sequence with mins as the reference simulation for our 3d experiments , instead of the sequence with that would have been suggested by our 1d experiments ( as in figure [ fig:1d_opsplit_tch_as_ref ] ) .the reason for this is shown in the top panel of figure [ fig : rms_op_split_dt ] , where we can see that the differences between ( transport - chemistry ) simulations with different operator splitting sequences but with the same time step , get smaller as gets smaller .this behaviour would be expected from a converging operator splitting method where none of the operators is stiff and where the order of evaluation of the operators is not relevant .an alternative explanation could be that the operator splitting errors are very small and what we are observing is the convergence of the time integration of each operator , separately , as gets small .this would suggest that the numerical errors of the time integration of the transport and the chemistry contribute significantly to the first term ( involving ) on the right hand - side of inequality ( [ eq : error_est ] ) , and should be comparable , in magnitude , to those observed between different operator splitting sequences . + in order to investigate this , we plotted the differences between simulations where the only active mechanism was either chemistry or transport , for multiple s , while keeping all other parameters exactly the same as in the previous simulations .the results are plotted in figure [ fig : rms_chem_only ] .these two plots show that indeed the numerical errors arising from the time integration of each of the operators separately lead to differences of the same magnitude as those observed in the operator splitting simulations .we also observe that the differences get smaller as decreases suggesting numerical convergence .this comparable differences make it hard to disentangle a sharp estimate of the operator splitting in 3d .+ note also that in our one dimensional prototype a cleaner analysis was achieved since we chose a smaller internal time step ( = 90 seconds ) to integrate the ( explicit - in - time ) transport operator than the operator splitting time step ( 180 seconds ) .this choice reduced the contribution to the numerical errors involving in inequality ( [ eq : error_est ] ) from the transport integration . in order to save computational time in geos - chem ( and in most ctms ) , however , the time step of the ( explicit - in - time ) transport scheme is chosen to be equal to the operator splitting time step leading to larger numerical errors . + in our one dimensional prototype , the chemistry operator was solved using an adaptive time integration routine with very tight convergence constraints , thus reducing numerical errors . the time integration of the chemistry operator in geos - chem uses an adaptive time stepping strategy ( ) in order to meet convergence requirements ( absolute and relative numerical error tolerances ) at every user - defined time step .these parameters have been internally set to keep simulation times reasonable while maintaining acceptable numerical accuracy .we kept these settings as they are typically used in global simulations for our numerical experiments .figure [ fig : rms_chem_only ] shows the differences between chemistry only simulations for different user - defined chemistry time - steps .presumably these errors could be decreased by fine tuning the error tolerances in the time integration routine appropriately , but this approach may increase processing times considerably .+ despite all of these numerical issues , we highlight the fact that we can establish an upper limit of about a for the magnitude of operator splitting errors based on the results of our multiple simulations in 3d . moreover ,we show that differences of the single chemical species with largest discrepancies across simulations , isoprene , are not significant in figures [ fig : isoprene_chem_only ] , [ fig : isoprene_transp_only ] , and [ fig : isoprene_op_split_only ] , for chemistry only simulations , transport only simulations , and different sequences of operator splitting methods , respectively . from these plots and the results of our one - dimensional prototype ,we hypothesize that the operator splitting errors may be much smaller than .+ we also highlight the fact that we did not pursue further efforts to show that the sequences evaluating the chemistry at the end of the time step in 3d compare better with observations , since our one - dimensional prototype , as well as multiple studies in global ctms , suggest that the numerical errors associated to the transport integration , at current spatial resolutions , are significantly larger than those observed in operator splitting methods .in addition , uncertainties in emission fields and deposition mechanisms may pose further difficulties in addressing this question . in our one - dimensional proto - type , subsequent reductions in the spatial resolution lead to significant improvements in the accuracy of the numerical solution globally ( for any operator splitting sequence ) .whereas a better choice of operator splitting ( where chemistry is evaluated last ) leads to a very modest improvement at a given spatial resolution .other important processes in 3d simulations are integrated in time using operator splitting strategies .as noted in and the time integration of inhomogeneous boundary conditions , such as emission processes in global simulations , using operator splitting strategies may lead to considerable numerical errors .additionally , the time integration of vertical processes such as convection and deposition using operator splitting may also lead to important numerical errors .  in order to investigate the magnitude of numerical errors due to these processes , we performed additional 3d simulations that gradually included inhomogeneous boundary conditions ( emissions ) and vertical process .in other words , aside from the 3d `` transport - chemistry '' simulations discussed in the previous sections , we performed simulations with ( i ) `` transport , chemistry , and emissions '' and simulations with ( ii ) `` transport , chemistry , emissions , convective transport and deposition '' . when emissions are included , they are integrated within the chemistry solver , using the chemistry time step .convective transport and deposition are solved using the standard setting of geos - chem , which integrate these two processes ( sequentially ) during the chemistry time step .  the differences between these two sets of simulations ,using the same methodology explained in the previous section , are plotted in the two lower panels of figures [ fig : rms_opsplit_all ] and [ fig : rms_op_split_dt ] .as these figures show , the additional numerical errors coming from the inclusion of inhomogenous boundary conditions ( emissions ) are significant .indeed , the differences between the simulations that include `` transport , chemistry , and emissions '' are roughly double the magnitude of the differences between the simulations that include only `` transport - chemistry '' for different operator splitting strategies .the incorporation of convective transport and deposition to the simulations does increase the differences between simulation , mainly when changes in time steps are large , as shown in the bottom panel of figure [ fig : rms_opsplit_all ] .when time - steps are fixed and operator splitting approaches are different , these vertical processes do not seem to lead to larger differences in the different simulations .   and ) , fixing km , for multiples time steps .the reference solution is obtained with the sequence for seconds . ]   to months .active processes in these simulations are as follows : transport and chemistry ( top panel ) ; transport , chemistry and emissions ( middle panel ) ; trasnport , chemistry , emissions , convective transport and deposition ( bottom panel ) . ]mins.active processes in these simulations are as follows : transport and chemistry ( top panel ) ; transport , chemistry and emissions ( middle panel ) ; trasnport , chemistry , emissions , convective transport and deposition ( bottom panel ) . ]we have presented a way to characterize operator splitting errors in the context of atmospheric chemistry modeling .our approach numerically extends one dimensional linear results to non - linear 1d and 3d cases .these numerical findings are relevant to global atmospheric chemistry modeling .our findings suggest that stiff operators should be evaluated lastly in operator splitting methodologies .this results is consistent with the linear results presented in , and previous studies in numerical weather prediction .differences of approximately across species are found when comparing the outputs of global simulations using different operator splitting approaches , using multiple splitting time steps .this , in combination with our one - dimensional results , suggests that operator splitting errors do not exceed relative errors in global simulations .we show also , that in current spatial resolutions , the numerical diffusion errors introduced in global atmospheric chemistry models eclipse errors emerging from operator splitting techniques .future studies should identify whether operator splitting strategies that evaluate fast dynamics operators lastly in global simulations lead to simulations that improve the match between simulations and observations .further exploration is also required regarding the effect of different operator splitting strategies in the time integration of of the governing equations of aerosol dynamics and different choices of boundary layer mixing schemes .additional `` toy - 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Abstract: we present upper bounds for the numerical errors introduced when using operator splitting methods to integrate transport and non linear chemistry processes in global chemical transport models ( ctm ) . we show that ( a ) operator splitting strategies that evaluate the stiff non linear chemistry operator at the end of the time step are more accurate , and ( b ) the results of numerical simulations that use different operator splitting strategies differ by at most 10% , in a prototype one dimensional non linear chemistry - transport model . we find similar upper bounds in operator splitting numerical errors in global ctm simulations .   atmospheric chemistry , operator splitting , model verification , numerical errors