Discretization of Multipole Sources in a Finite Difference. Setting for Wave Propagation Problems

Transcription

1 Discretization of Multipole Sources in a Finite Difference Setting for Wave Propagation Problems Mario J. Bencomo 1 and William Symes 2 1 Institute for Computational and Experimental Researc in Matematics, Brown University, Providence, RI USA 2 Te Rice Inversion Project, Rice University, Houston, TX USA (June 20, 2018) Running ead: Multipole Source Discretizations ABSTRACT Seismic sources are commonly idealized as point-sources due to teir small spatial extent relative to seismic wavelengts. Te acoustic isotropic point-radiator is inadequate as a model of seismic wave generation for seismic sources tat are known to exibit directivity. Terefore, accurate modeling of seismic wavefields must include source representations generating anisotropic radiation patterns. Suc seismic sources can be modeled as linear combinations of multipole point-sources. In tis paper we present a metod for discretizing multipole sources in a finite difference setting, an extension of te moment matcing conditions developed for te Dirac delta function in oter applications. We also provide te necessary analysis and numerical evidence to demonstrate te accuracy of our singular source ap- 1

2 proximations. In particular, we develop a weak convergence teory for te discretization of a family of symmetric yperbolic systems of first-order partial differential equations, wit singular source terms, solved via staggered-grid finite difference metods. Numerical experiments demonstrate a stronger result tan wat is presented in our convergence teory, namely, optimal convergence rates of numerical solutions are acieved point-wise in space away from te source if an appropriate source discretization is used. 2

3 INTRODUCTION Seismic sources are commonly idealized as concentrated at a source point due to teir small spatial extent relative to seismic wavelengts. A simple and familiar example of a wave propagation model wit spatially concentrated source is te isotropic point-radiator problem for te acoustic wave equation in Euclidean 3-space (Courant and Hilbert, 1962): 2 t 2 p(x, t) c2 2 p(x, t) = f(x, t) w(t)δ(x), p(x, t) = 0, t << 0, (1) δ(x) being te Dirac delta function. Te solution p is sperically symmetric: p(x, t) = w ( t r ) c 4πc 2, r = x r T x. (2) Bot active and eartquake seismic sources, owever, generate spatially asymmetric wavefields (see for example Searer (2009), Yilmaz (2001)). Te acoustic isotropic point-radiator is terefore inadequate as a model of seismic wave generation and propagation because of its prediction of spatial symmetry. Te symmetry of te solution arises in part from te sperical symmetry of te rigt-and side in equation 1. Terefore, accurate modeling of seismic wavefields must include energy source (rigt-and side) representations generating anisotropic radiation patterns. A multipole, or multipole source in te context of source modeling, is a finite linear combination of partial derivatives of te spatial Dirac delta function. Suc sources combine localization of energy and anisotropic radiation pattern. In fact, Peetre s Teorem (Hörmander, 1969) implies tat any function of space and time f(x, t) concentrated en- 3

4 tirely at a point in space (of point support) is a multipole of finite order N 0, N f(x, t) = w s (t) D s δ(x x ), (3) s =0 in wic we ave introduced multi-index notation: for spatial dimension d = {1, 2, 3} and multi-index (integer d-tuple) s = (s 1,..., s d ), te s-mixed partial derivative operator, denoted D s, and its (total) order s, are defined as D s = d ( ) si, s = i=1 x i d s i. (4) i=1 Te coefficient time functions w s (t) may be scalar-, vector-, or tensor-valued, according to te nature of te quantity (pressure, velocity, or stress) in te equation in wic f(x, t) appears as rigt-and side. Multipoles may approximate arbitrary sources igly localized on te wavelengt scale, in te sense of generating approximately te same field away from te source location, and for tis reason ave enjoyed widespread use in te representation of seismic sources (Searer, 2009). Wile finite element metods are also used in eartquake seismic modeling and inversion (Komatisc et al., 2000; Coen, 2002; Epanomeritakis et al., 2009), tis paper focuses on regular (rectangular) grid finite difference metods, in particular te staggered-grid variety, wic are widely used for basin- and exploration-scale modeling; see Moczo et al. (2006) for an excellent overview and many older references. Suc metods pose an immediate problem for singular source models suc as multipoles: finite difference algoritms know only gridded fields, so a source located at an arbitrary point x in space must be represented someow by virtual sources at nearby grid points. Tis task is complicated by te nature 4

5 of te field, as is evident for instance from inspection of te 3-D analytical function 2: solutions of te acoustic equations wit singular rigt-and sides are generally temselves singular at te source point, so te Taylor-series based analysis of finite difference accuracy does not apply. Te focus of tis paper is to study singular source discretizations, particularly in te context of staggered-grid finite difference solvers for a family of wave propagation differential equations (e.g., linear acoustics and elasticity in first-order form) wit multipole sources. Specifically, we are interested in te following class of symmetric yperbolic systems of first-order partial differential equations: find vector-valued fields (u, v) suc tat A t u + P T v = f B t v P u = g (5) for given source terms (f, g). Operators A and B are bounded symmetric matrix-valued and represent medium parameters. P is a first-order spatial differential operator wit constant coefficients, e.g., P is te gradient in te acoustic equations 23 wit P T being te negative of te divergence. We directly discretize multipole sources using discrete moment conditions as developed by Waldén (1999), and elaborated by Tornberg and Engquist (2004) for multi-dimensional approximations. Let (u, v ) denote te finite difference solution to 5 via staggered-grid finite differences, wit discretized source terms (f, g ), were > 0 denotes te grid size. Te class of staggered-grid finite difference metods we deal wit are second-order in time and 2p-order in space accurate, and we refer to suc a sceme as being of order 2-2p. Our main contribution is a weak convergence teory, a novel approac to te singular 5

6 source approximation problem, tat is applicable to 5 in tat (u, v ) converge to (u, v) given tat source approximations (f, g ) converge to (f, g) as 0, bot in a weak sense wic we make clear in te teory section. In particular, we sow (u, v), ( f, g) (u, v ), ( f, g) = O( q + t 2 + ( 2p + t) N d/2 ) (6) given smoot test functions ( f, g). Inner products, are interpreted as standard L 2 space-time inner products in a continuum or discrete sense depending on teir arguments. Estimate 6 tus depends on singular source approximation order q, spatial alf-order p for a staggered-grid finite difference sceme of order 2-2p, te maximum multipole order between f and g denoted by N, and spatial dimension d. Numerical results presented ere, consistent wit oter similar works (e.g., Petersson and Sjögreen (2010)), appear to indicate, owever, stronger convergence results: optimal convergence point-wise away from source. In particular, we report second and fourt order rates wen studying te spatial convergence of 2-2 and 2-4 finite difference metods respectively, were te source approximation order matced te spatial order of te numerical sceme. Te analysis and numerical examples presented by Waldén (1999), toug limited to te 1-D Helmoltz equation, demonstrated point-wise convergence of numerical solutions wit optimal convergence rates (as suggested by te numerical sceme) away from te source location wen appropriately discretizing te singular source term. Te teory of singular source approximations as been furter extended to a range of applications, most notably for te Dirac delta function in te context of te immersed boundary metod (Peskin, 2002). Several autors ave addressed questions regarding te convergence of source approximations and subsequently teir effect on solutions to more complicated differential equations. 6

7 Consider te following abstract problem: find u suc tat Lu = f, (7) for some differential operator L and singular source term f. Define te regularization of problem 7 by replacing f wit some regular (at least piecewise continuous) function f ɛ parameterized by regularization parameter ɛ > 0, tat is, find u ɛ suc tat Lu ɛ = f ɛ. (8) Regularized source term f ɛ is said to approximate f in tat f ɛ f as ɛ 0 in some sense. Te end goal is of course to ave u ɛ approximate true solution u, i.e., lim ɛ 0 uɛ u X = 0 under some suitable norm X. Tornberg and Engquist (2003) ave studied te regularization error u ɛ u X, pointwise away from te source location. Teir analysis is based on a simple ODE case were tey prove convergence of regularized solutions u ɛ if f ɛ satisfy wat we call te continuum moment conditions. We use te qualifier continuum to differentiate at times between te discrete moment conditions. Recent work by Hosseini et al. (2016) addresses te mode of convergence of f ɛ f subject to regularized source terms satisfying te continuum moment conditions, mainly convergence in a weak- topology (distribution sense) and in a weigted Sobelev norm. Bot Tornberg and Engquist (2003) and Hosseini et al. (2016) argue tat f ɛ f implies u ɛ u as ɛ 0, point-wise away from te source location, 7

8 in particular for elliptic operators L. Tis argument inges on te integral representation of elliptic operators and te smootness of teir kernel (i.e., Green s functions) away from source location. Suppose tat te regularized problem 8 is discretized wit mes or cell size > 0; L u ɛ = f ɛ. (9) In te context of finite difference metods L is te finite difference approximation of differential operator L. Te discrete source term f ɛ can be interpreted as te discretization (e.g., sampling over grid points) of te regularized source term f ɛ generated by te continuum moment conditions, or as te direct discretization of f troug te discrete moment conditions. In practice te regularization parameter ɛ is related to te discretization parameter, tat is, ɛ = ɛ() suc tat lim ɛ() = 0. 0 Tornberg and Engquist (2004) provided insigt into te convergence of f ɛ as 0 for direct discretizations of f via te discrete moment conditions, in particular for discretizations wose support is proportional to ɛ and ɛ() = O(). Consistent wit results by Waldén (1999), Tornberg and Engquist (2004) demonstrated te convergence of numerical solutions u ɛ for problem 9, in particular u ɛ u X = O( p ) were p is te convergence rate of te numerical sceme and f ɛ satisfies a sufficient number of moment conditions. Te norm X in tis case coincides wit te sup-norm wit a deleted neigborood containing te source location. Teory presented by Tornberg and 8

9 Engquist (2004) is based on analysis of te 1-D Poisson equation discretized by second- and fourt-order finite difference approximations. Green s functions for yperbolic problems are singular at te wavefront tey propagate as well as at te source location, tus te analysis of Tornberg and Engquist (2004) does not apply. Recent work by Petersson et al. (2016) presented some of te most relevant analysis based on centered difference approximations to te 1-D advection equation wit singular source term. Tey sow tat te discrete moment conditions are necessary but not sufficient for te convergence of numerical solutions at optimal rates away from te source location. Tey demonstrated tat smootness conditions are also required to acieve convergence; tese conditions are based on Fourier analysis and te ability of te finite difference operators at play to resolve spurious modes injected by te singular source approximation. Te remainder of te paper is organized as follows: In te teory section we begin by presenting an overview of te singular source approximation metod via continuum and discrete moment conditions. We focus primarily on source approximations of narrow support, as discussed in Tornberg and Engquist (2004), toug we provide explicit formulas for approximations of arbitrary order. Moreover, we sow tat te discrete moment conditions in fact define a sequence of continuum functions tat converge to target distributions in a weak sense, a new result. Te teory section also covers known convergence results of staggered-grid finite difference metods via energy estimates, applied to our set of differential equations under smoot coefficients and smoot source terms. At tis point we present our weak convergence teory. Te last section covers numerical results, mainly for staggered-grid finite difference solutions to te 2-D acoustic equations in first-order form 9

10 wit multipole sources. Consistent wit numerical results presented in te literature (see for example Petersson and Sjögreen (2010)), we observe optimal convergence rates of our numerical solutions wen te source discretization satisfied te proper number of moment conditions. 10

11 THEORY Singular Source Approximation (Continuum) Moment Conditions We begin by noting tat te delta function and its derivatives (and tus multipoles) are not actually functions but rater so-called distributions, functionals tat return a real number wen applied to a test function. Let D denote te space of test functions over R d, tat is, te space of C0 (R d ) endowed wit te standard topology of test functions. Te set of distributions is given by te dual of te space of test functions, denoted by D. It is conventional to represent te application of a distribution on a function by te integral of te product, even wen te distribution is not actually a function tat can be integrated in te usual sense. For example, given multi-index s = (s 1,..., s d ), te s-mixed partial derivative of te Dirac delta function, sifted by x R d, is defined by R d D s δ(x x ) ψ(x) dx = ( 1) s D s ψ(x ), ψ D. Te key idea for constructing approximations to D s δ(x x ) is based on mimicking te beavior of te target distribution on polynomials, reminiscent of finite difference approximations for differential operators. Consider ψ(x) = (x x ) α, wit multi-index α = (α 1,..., α d ), were multi-indexed monomials are interpreted as te product of monomials in eac dimension, x α = d k=1 x α k k. 11

12 It can be sown tat R d D s δ(x x ) ψ(x) dx = s!( 1) s δ sα were δ sα is te Kronecker delta, defined as follows for multi-indexes, d δ sα := δ sk α k. k=1 Given η L 1 0(R d ) (i.e., integrable function of compact support) and multi-index α, te α-moment of η centered at x R d, denoted by M α (, x ), is defined as M α (η, x ) := η(x) (x x ) α dx. R d (10) For given nonnegative integer q and multi-index s, te function η is said to satisfy te continuum (q, s)-moment conditions at x R d if M α (η, x ) = s!( 1) s δ sα, α = 0,..., q + s 1. (11) If η satisfies te (q, s)-moment conditions at x, ten its associated distribution, tat is R d η(x) ψ(x) dx, ψ D, is an approximation to D s δ(x x ) in tat it is exact on polynomials of order q + s 1. Te following teorem states tat a sequence of (regular) distributions of compact support, satisfying te (q, s)-moment conditions, will converge in te weak- topology at a rate q to te target distribution as te widt of te supports approac zero. Let B(x, ɛ) denote te 12

13 d-dimensional ball of radius ɛ centered at x. Teorem 1. Let nonnegative integer q, multi-index s, and x R d be given. Suppose {η ɛ } L 1 0(R d ) is a sequence of functions as ɛ 0, were supp(η ɛ ) B(x, ɛ). Furtermore, suppose tat tere exists a constant K > 0 independent of ɛ suc tat R d η ɛ (x) (x x ) α dx K, α = s. (12) If {η ɛ } satisfy te (q, s)-moment conditions at x, equation 11, ten te sequence of distribution tey generate converges to D s δ(x x ) in te weak- topology as ɛ 0. In particular, if ψ is of class C q+ s over B(x, ɛ), ten E := D s δ(x x ) ψ(x) dx η ɛ (x) ψ(x) dx = O(ɛq ). (13) R d R d Proof. We first apply multi-variate Taylor s teorem to ψ, centered at x and truncated after N = q + s 1 terms (Königsberger, 2013), assuming ψ is C q+ s over B(x, ɛ), η ɛ (x) ψ(x) dx = η ɛ (x) R d R d = N α =0 N α =0 D α ψ(x ) α! D α ψ(x ) α! (x x ) α + β =N+1 ( ) η ɛ (x) (x x ) α dx + R d R β (x) (x x ) β dx β =N+1 R d η ɛ (x)r β (x) (x x ) β dx, were R β is te remainder term, R β (x) = β β! 1 0 (1 t) β 1 D β ψ(x + t(x x )) dt. Note tat te term in te parentesis in te bottom equation corresponds to te α-moment 13

14 centered at x wit α q + s 1, ence te (q, s)-moment conditions apply; R d η ɛ (x) ψ(x) dx = N α =0 1 ) α! Dα ψ(x ) (s!( 1) s δ sα + = ( 1) s D s ψ(x ) + β =N+1 β =N+1 R d η ɛ (x)r β (x) (x x ) β dx. R d η ɛ (x)r β (x) (x x ) β dx Te remainder term is bounded uniformly over B(x, ɛ), sup x B(x,ɛ) R β (x) C(β, ψ) := 1 β! max α = β max y B(x,ɛ) Dα ψ(y), using te fact tat ψ C N+1 in B(x, ɛ). Tis gives te following error estimate, E β =N+1 C(β, ψ) η ɛ (x) (x x ) β dx. B(x,ɛ) Let γ be a multi-index suc tat γ = q, tus β γ = s. Tis yields, E β =N+1 β =N+1 = O(ɛ q ) C(β, ψ) C(β, ψ) ( ) sup (x x ) γ x B(x,ɛ) ( ) sup (x x ) γ x B(x,ɛ) K η ɛ (x) (x x ) β γ dx B(x,ɛ) Teorem 1 and moment conditions given in equation 11 are extensions of wat is presented in Hosseini et al. (2016) for s 0. Given equation 13, we refer to q as te singular source approximation order and η ɛ as being a q-order approximation of D s δ(x x ). 14

15 Discrete Moment Conditions We define te regular grid centered at x 0 R d wit cell size = ( 1,..., d ) > 0, as te collection of points denoted by G(x 0, ), G(x 0, ) = {x n = (x 1,n1,..., x d,nd ), n Z d }, were x k,nk = x 0,k + k n k, k = 1,..., d. Note tat tere is no reason to assume tat te grid cell is cubical, it may ave different lengts along different axes. However, we assume tat te grid is refined by scaling a caracteristic grid cell size (e.g., = max k ) and wic we will use to denote grid k functions and oter grid-dependent quantities wit subscript even in dimensions iger tan one. Te obvious definition of te discrete α-moment, centered at x R d (note tat x need not coincide wit a grid point in G(x 0, )), of a grid function η : G(x 0, ) R is given as follows: ( d ) M α (η, x ) := k η (x) (x x ) α. k=1 x G(x 0,) It is wort pointing out tat te discrete moment defined above is dependent on coice of grid, in particular dependent on te source location x relative to te grid. Similar to te continuum moment conditions (equation 11), grid function η is said to satisfy te discrete 15

16 (q, s)-moment conditions at x R d if M α (η, x ) = s!( 1) s δ sα, α = 0,..., q + s 1. (14) Te following teorem is a discrete analogue of teorem 1. Teorem 2. Let nonnegative integer q, multi-index s, and x R d be given. Suppose {η ɛ } is a sequence of grid functions η ɛ : G(x 0, ) R as ɛ 0. Furtermore, assume tat te support of η ɛ is contained in B(x, ɛ) wit ɛ = O(), and tat tere exists constant K > 0 independent of ɛ suc tat ( d ) k η ɛ (x) (x x ) α K, α = s. k=1 x G(x 0,) If {η ɛ } satisfy te discrete (q, s)-moment conditions at x (equation 14) and ψ is of class C q+ s over B(x, ɛ), ten ( d ) D s δ(x x ) ψ(x) dx k R d k=1 x G(x 0,) η ɛ (x) ψ(x) = O(q ). Proof. Te proof of tis teorem is omitted since it is nearly identical to tat of te continuum case (teorem 1), replacing integrals wit summations over grid points. Te jump from O(ɛ q ) to O( q ) follows from ɛ = O(). Teorem 2 and discrete moment conditions 14 are generalizations of work by Tornberg and Engquist (2004) for s = 0. Te discrete moment conditions are also an extension of Waldén (1999) for dimensions iger tan one. We now discuss wit more detail ow to construct said sequences of gridded and continuum functions starting in 1-D. 16

17 1-D Constructions We make te coice of aving 1-D gridded approximations be centered at source location x R d, and define tem to be zero outside te interval [ ɛ + x, ɛ + x ), wit 2ɛ = N for some positive integer N. In oter words, tere exists N grid points, denoted by { x l } N l=1, suc tat tey are contained in te interval [ ɛ+x, ɛ+x ) for a given grid G(x 0, ). Tese grid points { x l } N l=1 are referred to as te stencil points of te approximation. We assume tat te stencil points are ordered, i.e., x 1 < x 2 < < x N. Te discrete (q, s)-moment conditions tus results in a N (q+s) system of equations for te grid function η ɛ evaluated at stencil points, Ad = b wit {A} kl = ( x l x ) k 1, {d} l = η ɛ ( x l), {b} k = s!( 1)s δ s,k 1, for l = 1,..., N and k = 1,..., q + s. Note tat A is a Vandermonde matrix of full rank and is guaranteed a solution if N q + s and no solution for N < q + s under general x R. We coose te case were N = q + s, wic we refer to as grid functions of narrow support. Te system above will result in a unique solution for a given x R. In fact te inverse matrix for A can be written explicitly using te following Vandermonde matrix 17

18 inverse formula: {A 1 } lk = ( 1) N k 1 m 1 < <m N k N m 1,...,m N k l 1 m N m l ( x m1 x ) ( x mn k x ) 1 m N m l ( x l x m ), for 1 k N 1, for k = N. ( x l x m ) Given te particular form of rigt-and side vector b, it follows tat d is simply te scaled (s + 1)-column of A 1, wence η ɛ ( x l) = s!( 1) N 1 1 m 1 < <m q 1 N m 1,...,m q 1 l N ( x m1 x ) ( x mq 1 x ) 1 m N m l (l m), for q > 1 (15) N s!( 1) s, for q = 1. (l m) 1 m N m l Note tat te equation above above is dependent on x. To be more precise, approximation η ɛ is actually dependent on te relative position of x wit respect to te stencil points { x l } N l=1, or equivalently dependent on x 0 and. For example, sifting te source location by would result in te same grid function η ɛ toug sifted by a grid point. However, arbitrary sifts in x yield approximations tat may vary by more tan a simple translation. Given tat equation 15 is unique for a particular source location, we sow tat imposing te discrete moment conditions over all x R indeed defines a function over te reals. Moreover, we sow tat tese continuum functions satisfy te continuum moment 18

19 conditions and tus define a sequence of distributions tat converge to D s δ in te weak- topology, a result previously not known. Connection between Continuum and Discrete Moment Conditions We first focus on te x = 0 case and define our continuum approximation η ɛ to be zero outside [ ɛ, ɛ), wit 2ɛ = N for N = q + s. Furtermore, we define η ɛ to be piecewise polynomial over N invervals: P l (x), x [a l, a l+1 ), for l = 1,..., N η ɛ (x) = 0, oterwise (16) were P l is some polynomial over te considered interval, and a l = ɛ + (l 1) for l = 1,..., N + 1. Given te support of our approximation, and a regular grid G(x 0, ), it follows tat tere are N grid points contained in te interval [ ɛ, ɛ), again denoted by { x l } N l=1. In fact, x l [a l, a l+1 ), l = 1,..., N. Let l be te index suc tat 0 [a l, a l +1) and define ζ (0, ] by ζ = a l +1 x l. Tus, if we vary x witin te interval ( ζ, ζ] if follows tat x l x [a l, a l+1 ), l = 1,..., N. Let η ɛ ( ; x ) denote te grid function tat satisfies te discrete (q, s)-moment conditions for a given x ( ζ, ζ]. Ten η ɛ ( x l x ) := η ɛ ( x l; x ) defines η ɛ over [a l, a l+1 ) by allowing x to vary over te prescribed interval. Moreover, sligtly modifying equation 15 19

20 as a function of x = x l x defines te P l (x) polynomials of η ɛ, P l (x) = s!( 1) N 1 1 m 1 < <m q 1 N m 1,...,m q 1 l N ((m 1 l) + x) ((m q 1 l) + x) N 1 m N m l (l m), for q > 1 s!( 1) s, for q = 1. (l m) 1 m N m l (17) Inspection of equation 17 reveals tat P l is a polynomial of degree q 1. Teorem 3. Let nonnegative integer q, positive integer s, and x R be given. Suppose η ɛ is constructed according to equations 16 and 17 for a given > 0. Ten it follows tat η ɛ (x) is a q-order approximation of D s δ(x). Proof. In order to apply teorem 1 we need to verify tat η ɛ indeed satisfies te (continuum) (q, s)-moment conditions at 0 as well as te estimate given by equation 12. We first evaluate te α-moment of η ɛ for α = 0,..., q + s 1; M α (η ɛ, 0) = = R N η ɛ (x) x α dx al+1 l=1 a l P l (x) x α dx. Applying te following cange of variables, x = a l + ξ wit ξ [0, ), over eac interval 20

21 yields M α (η ɛ, 0) = = N l= [ P l (a l + ξ)(a l + ξ) α dξ ] N P l (a l + ξ)(a l + ξ) α l=1 dξ. Note tat te term in te bracket coincides wit te discrete α-moment of η ɛ wit respect to a uniform grid G(a 1, ) (containing stencil points {a l } N+1 l=1 ) for a source located at x = ξ. In oter words, N P l (a l + ξ)(a l + ξ) α = M α (ηɛ, ξ) l=1 were η ɛ : G(a 1, ) R defined by η ɛ (x) = ηɛ (x + ξ) satisfies te discrete (q, s)-moment conditions at ξ by construction. We can conclude M α (η ɛ, 0) = 0 1 [s!( 1)s δ sα ] dξ = s!( 1) s δ sα. Lastly, since η ɛ consist of piecewise polynomials of order q 1 divided by a factor of N, were N = q + s, and ɛ = O(), we ave tat sup η ɛ (x) = O(ɛ s 1 ). x B(0,ɛ) Tus B(0,ɛ) dx η ɛ (x) x s sup η ɛ (x) B(0,ɛ) B(0,ɛ) dx x s = O(1), as required for estimate 12 21

22 Constructions in Higer Dimensions via Tensor Products We construct approximations in general d-dimension by taking tensor products of 1-D approximations, similar to work by Tornberg and Engquist (2004) for s = 0. Namely, given approximation order q and multi-index s, multivariate continuum approximation η : R d R is given by d η(x) = η k (x k ) (20) k=1 were η k : R R is a continuum function over te k-t axis as given by equations 16 and 17, satisfying te (q, s k )-moment conditions for eac k = 1,..., d. It follows tat tese tensor approximations are indeed approximations to multipoles in te iger spatial dimension. Teorem 4. Let nonnegative integer q, multi-index s, and x R d be given. Suppose η : R d R is a multi-variate grid function given by te tensor product of 1-D approximations η k : R R. If η k satisfy te discrete (q, s k )-moment conditions at x k for eac k = 1,..., d, ten it follows tat η satisfies te discrete (q, s)-moment conditions at x. Proof. Suppose for eac k = 1,..., d tat η k satisfies te (q, s k )-moment conditions at x k. Let α be some multi-index wit α q + s 1. Note tat, d M α (η, x ) = M α k (η k, x k ). k=1 Clearly, if α k q + s k 1 for all k = 1,..., d, ten te result follows from te supposition. Same applies for α = s. Suppose ten, tat tere exists index l suc tat α l > q + s l 1, 22

23 tat is a l = q + s l 1 + i for some i N. Tus, α = k l α k + α l = k l α k + q + s l 1 + i q + s 1 = k l α k + i k l s k. wic implies tat α k < s k for at least one k l; for tis particular k, it follows tat M α k (η k, x k ) = s k!( 1) s k δ sk α k = 0 since it as been establised tat α k s k, i.e., te product over k is zero if s α. Examples Numerical convergence rate tests in te following section will employ te singular source approximation discussed ere, replacing multipole terms wit grid functions to be used in finite difference scemes. Equation 15 gives an explicit formula for suc grid functions, depending on te source location x, multi-index s, approximation order q, and of course te underlying finite difference grid. Alternatively, equations 16 and 17 define te continuum form of te approximations, were grid functions are obtained by sifting and sampling over te grid. Te following figures plot 1-D and 2-D continuum approximations η ɛ, in particular, we plot second- and fourt-order approximations of D s δ(x) for s = 0, 1, 2 in 1-D and s = (0, 0), (0, 1), (0, 2) in 2-D, wit = 1. Two dimensional approximations are constructed via tensor product of 1-D approximations as discussed. In particular, te q = 2 approximation for te 1-D Dirac delta function (s = 0) is none oter tan te well known at/triangular function of unit mass. 23

24 Convergence Teory of Wave Equation Solutions wit Multipole Sources We prove weak convergence of finite difference solutions wit discrete multipole sources satisfying te discrete moment conditions discussed above. Convergence teory for smoot solutions is in some sense standard. Te easily available results (tose based on Lax s Equivalence Teorem) pertain to L 2 convergence for smoot solutions and smoot source terms, wic we begin wit. Teoretical results presented ere pertain to te family of symmetric yperbolic systems of first-order partial differential equations for vector-valued fields (u, v), defined over a domain Ω R d in d = {1, 2, 3} dimension space and some interval in time, given by 5 wic we restate wit greater detail. Let k 1 N and k 2 N denote te dimension of vector-fields u and v respectively. Te systems we are interested in are of te following form: A(x) t u(x, t) + P T v(x, t) = f(x, t) B(x) t v(x, t) P u(x, t) = g(x, t) (21) were u L 2 loc (R, H 1) and v L 2 loc (R, H 2) wit Hilbert spaces H 1, H 2 ; coefficient operators A and B are k 1 k 1 and k 2 k 2 matrix-valued functions respectively; we assume tey are symmetric for all x Ω and uniformly-positive 24

25 definite, i.e., tere exists constants 0 < A A and 0 < B B suc tat A I A(x) A I, x Ω, B I B(x) B I, x Ω; (22) P : H 1 H 2 is a constant coefficient, first-order differential operator of te form P u = d j=1 P j u x j, P j R k 2 k 1 ; source terms (f, g) are for te time being assumed to be smoot, i.e., f L 2 loc (R, H 1) and g L 2 loc (R, H 2) ; we assume solution (u, v) and source terms (f, g) are causal, i.e., for t < 0 u(x, t) = f(x, t) = 0, v(x, t) = g(x, t) = 0. See Blazek et al. (2013) for proof of existence of solutions, stability, and oter matematical properties for a larger class of partial (integro-)differential systems for wave modeling, applicable to system 21. As an example, consider te acoustic equations in first-order (pressure-velocity) form: 1 p(x, t) + v(x, t) = f(x, t) κ(x) t (23) ρ(x) t v(x, t) + p(x, t) = g(x, t) were u = p is te scalar pressure field and v = v R d is te vector particle velocity field; we take H 1 = L 2 (Ω) and H 2 = L 2 (Ω) d. Coefficient operator A(x) = 1/κ(x) and 25

26 B(x) = ρ(x)i, wit κ denoting bulk-modulus and ρ density of te medium; ere I R d d is te identity matrix. Lastly, te differential operator P coincides wit te gradient and its adjoint wit te negative of te divergence, P = x 1. [, P T = x 1,..., x d ]. x d Te discretization of continuum problem 21 is given by te following abstract staggered grid dynamical system: A (u n+1 B (v n+1/2 u n ) + P (r) T v n+1/2 = r f n+1/2 v n 1/2 ) P (r)u n = r gn (24) were u n H 1, and v n+1/2 H 2, for a family of Hilbert spaces H 1,, H 2, ( space of spatial-grid functions ), wit > 0 ( cell size ). Superscript indexes in u n and v n+1/2 refer to te discretized time axis at times n t and (n + 1/2) t respectively, wit time step size t = r; r will play te role of te CFL constant, crucial for stability of te discretization. Note tat te fields u and v are staggered in time; bounded self-adjoint positive-definite operators (discretizations of A and B) A : H 1, H 1,, B : H 2, H 2, wit upper and lower bounds uniform in : tere exists constants 0 < A A and 26

27 0 < B B suc tat A u 2 A u, u A u 2, u H 1,, B v 2 B v, v B v 2, v H 2, ; bounded operators P (r), Q : H 1, H 2, suc tat tere is a constant Q > 0 were Q Q for all > 0, P (r) = rq ; discrete source terms f n+1/2 ten H 1, and g n H 2,; if (f, g) are smoot source terms, f n+1/2 = S 1, f(, (n + 1/2) t), g n = S 2,g(, n t), were we define bounded operator S 1, : H 1 H 1,, and S 2, : H 2 H 2, for sampling continuum fields onto te spatial grids. Consider again te acoustic case (related to problem 23). For simplicity we assume we are dealing wit rectangular grids, in 2-D for tis example, of te form G(0, ) = {x i,j = (i, j) : i, j Z}. Te simplest staggered-grid finite difference sceme, second-order in time and space, is 27

28 given by 1 1 κ(i, j) t [ ] (p ) n+1 i,j (p ) n i,j + ρ((i + 1/2), j) 1 t ρ(i, (j + 1/2)) 1 t 1 [ ] (v 1, ) n+1/2 i+1/2,j (v 1,) n+1/2 i 1/2,j + (v 2,) n+1/2 i,j+1/2 (v 2,) n+1/2 i,j 1/2 = (f ) n+1/2 i,j ] [ (v 1, ) n+1/2 i+1/2,j (v 1,) n 1/2 i+1/2,j [ ] (v 2, ) n+1/2 i,j+1/2 (v 2,) n 1/2 i,j+1/2 ] [ (p ) n i+1,j (p ) n i,j = (g 1, ) n i+1/2,j [ ] (p ) n i,j+1 (p ) n i,j = (g 2, ) n i,j+1/2 (25) were finite difference solution (p, v 1,, v 2, ) approximates te continuum fields (p, v 1, v 2 ), (p ) n i,j p(i, j, n t), (v 1, ) n+1/2 i+1/2,j v 1((i + 1/2), j, (n + 1/2) t), (v 2, ) n+1/2 i,j+1/2 v 2(i, (j + 1/2), (n + 1/2) t). We empasize tat te velocity fields for finite difference sceme 25 are staggered wit respect to spatial grids, more specifically eac component is sifted by alf a cell size in its respective axis. Relating system 25 wit 24, we see (u ) i,j = (p ) i,j, (v ) i,j = (v 1, ) i+1/2,j (v 2, ) i,j+1/2. Te space H 1, corresponds to te set of square summable scalar-valued functions on rectangular grids, and H 2, is te set of square summable R 2 -valued functions on alf-cell sifted 28

29 grids, bot equipped wit a discrete L 2 inner-products, u, ũ = 2 i,j (u ) i,j (ũ ) i,j for u, ũ H 1, v, ṽ = 2 i,j (v ) i,j (ṽ ) i,j for v, ṽ H 2,. Inner-products and norms of te spaces H 1, H 2, H 1,, H 2, are all denoted by, and and interpreted given te context, unless oterwise specified. Sampling operators S 1, and S 2, coincide wit evaluating continuum functions on grid or sifted grid points accordingly, (S 1, u) i,j = u(i, j), (S 2, v) i,j = v 1 ((i + 1/2), j) v 2 (i, (j + 1/2)). Discretized coefficient operators A and B are given by, (A u ) i,j = 1 κ(i, j) (u ) ij, (B v ) i,j = ρ((i + 1/2), j)(v 1, ) i+1/2,j ρ(i, (j + 1/2))(v 2, ) i,j+1/2. Lastly, P and P T correspond to second-order approximations of te gradient and te negative of te divergence respectively; (u ) i+1,j (u ) i,j (P (r)u ) i,j = r (u ) i,j+1 (u ) i,j ] (P (r) T v ) i,j = r [(v 1, ) i+1/2,j (v 1, ) i 1/2,j + (v 2, ) i,j+1/2 (v 2, ) i,j 1/2, again, wit r = t/. In general, te differential operator P can be approximated by a 29

30 family of central difference operators P of even order 2p for p N, resulting in te 2-2p staggered-grid finite difference scemes. Energy Estimates Define te following energy form: E n := E(u n, vn 1/2 ) := 1 ( A u n 2, un + B v n 1/2, v n 1/2 + P T vn 1/2, u n ). Teorem 5. For a solution (u, v ) of system 24 wit (f, g ) 0, it follows tat E n is independent of time index n. Moreover, for sufficiently small r, tere exist 0 < C C suc tat C ( u 2 + v 2) E(u, v ) C ( u 2 + v 2). (26) Proof. First we sow te energy conservation property: E n+1 E n = A (u n+1 u n ), (un+1 + u n ) + B (v n+1/2 v n 1/2 ), (v n+1/2 + v n 1/2 ) + P T vn+1/2, u n+1 ) P T vn 1/2, u n ) = P T vn+1/2, (u n+1 + u n ) + P u n, (vn+1/2 + v n 1/2 ) + P T vn+1/2, u n+1 ) P T vn 1/2, u n ) = 0. Te upper bound in 26 is clear from -uniform bounds on A, B, P. Te lower bound follows from E(u, v ) A u 2 + B v 2 rq u v wic is a positive definite form in u, v wen rq < A B. In oter words, we 30

31 ave establised positive definiteness of E(, ), and more importantly, tat te energy form is equivalent to a norm. L 2 Convergence of Smoot Solutions Given teorem 5, tat is, equivalency of between te L 2 and energy norms, we focus on proving stability estimates wit respect to te energy norm in order to imply L 2 convergence. Teorem 6. For r sufficiently small tat E is positive definite, stability follows, i.e., tere exists K 0, λ > 1 so tat E n λ n E 0 + K n 1 m=0 λ n m+1 ( f m 1/2 2 + g m 2 ) were (u, v ) satisfy te inomogeneous system 24 wit source terms (f, g ). Proof. Te same aritmetic as used in te omogeneous case leads to E n+1 E n = r f n+1/2, u n+1 + u n + r gn, vn+1/2 + v n 1/2 r 2α n+1/2 ( f g n 2 ) + α2 r (E n+1 + E n ) C for any α (0, 1). Coose α so tat α 2 r < C (tat is, can make fixed coice of α for r small enoug), and set λ = ( ) ( ) α2 r 1 α2 r and K = C C ( r 2α α2 r C ). (27) 31

32 Ten λ 1 E n+1 E n = K( f n+1/2 2 + g n 2 ) = λ (n+1) E n+1 λ n E n = λ n K( f n+1/2 2 + g n 2 ) wence λ n E n = E 0 + K n 1 m=0 λ m+1 ( f m 1/2 2 + g m 2 ). Teorem 7. Suppose tat (u n, v n 1/2 ) H 1 H 2, and (u n, vn 1/2 ) solves te discretized system 24 wit initial data u 0 = S 1,u 0, v 1/2 = S 2, v 1/2. Set δf n+1/2 = 1 r δg n = 1 r ( A (S 1, u n+1 S 1, u n ) + P T S 2,v n+1/2) + f n+1/2 ( B (S 2, v n+1/2 S 2, v n 1/2 ) P S 1, u n) + g n. (28) Moreover, let T > 0 be given, independent of. If (δf, δg ), as defined above, satisfy te estimate δf m 1/2 2 + δg m 2 L 2 2p (29) for 0 m N, N t = T, ten E(u n S 1,u n, v n 1/2 S 2, v n 1/2 ) M 2 L 2 2p. (30) 32

33 Proof. Define K, λ as in 27, ten it follows from previous teorem tat E(u n S 1,u n, v n 1/2 S 2, v n 1/2 ) K n 1 m=0 λ n m+1 ( δf m 1/2 2 + δg m 2 ). wic implies E(u n S 1,u n, v n 1/2 S 2, v n 1/2 ) KL 2 2p+1 λn 1 λ 1 using estimate 29. Note tat if is sufficiently small, ten λ < α2 r C so λ N exp(2 α2 C T ), were T = Nr. If is peraps smaller yet, ten λ > 1 + α2 r 2C. Putting tis all togeter, tere is M 2 depending on T and all of te oter constants in te setup, so tat if nr < T so tat 30 olds. If (u, v) is a smoot solution of te acoustic (pressure-velocity) system 23 wit smoot rigt-and sides (f, g), and (u, v ) is te staggered-grid finite difference approximation of order 2-2p, ten, from standard truncation error calculations, for eac time slice of compact support we ave, δf n+1/2 = O( t 2 + 2p ) δg n = O( t2 + 2p ) point-wise and uniformly in i, j. Since te support is uniformly bounded if 0 n t T, te O statements apply to te L 2 norms as well. Tus, by equivalency of te energy form wit te L 2 -norm, and te corollary above, we can conclude u S 1, u + v S 2, v = O( t 2 + 2p ), 33

34 tat is, 2-2p order convergence in te L 2 sense. Weak Convergence of Singular Solutions Let n min, n max be integers. Suppose tat te grid-function sequence (u n, vn+1/2 ), satisfies 24 wit source term (f n+1/2, g n ) and vanis identically for n < n min. Let (ũ n+1/2, ṽ n ) be anoter sequence of grid functions tat vanis for n > n max. Ten, t n ( ũ n+1/2, f n+1/2 + ṽ n, gn ) = n ( ũ n+1/2, A (u n+1 ) u n ) + P T vn+1/2 + ṽ n, B (v n+1/2 v n 1/2 ) P u n = n ( A (ũ n 1/2 ũ n+1/2 ), u n P ũ n+1/2, v n+1/2 + B (ṽ n ṽn+1 ), v n+1/2 + P T ṽn, un ) = n ( A (ũ n 1/2 ũ n+1/2 ) P T ṽn, un + B (ṽ n ṽn+1 ) + P ũ n+1/2, v n+1/2 (31) Define ( f n, gn+1/2 ) H 1, H 2, by A (ũ n 1/2 ũ n+1/2 ) P T ṽn = r f n B (ṽ n ṽ n+1 ) + P ũ n+1/2 = r g n+1/2. (32) Ten, identity 31 can be re-written as t n ( ũ n+1/2 ), f n+1/2 + ṽ n, gn = t n ( f n, un + gn+1/2 ), v n+1/2. (33) Now specialize to te acoustic case - te elastic case is similar. In tis case g (and 34

35 g) is vector-valued, and f (and f) is scalar-valued. Once again, P is a discrete gradient; assume tat it is accurate of order 2p, as before. P T is a discrete (negative) divergence, also accurate of order 2p. Te adjoint system 32 is recognized as anoter discretization of te acoustic system 23, wit two differences: te time index is dual to te one used in 24, tat is, sifted by t/2, and te inomogenous terms get a negative sign. Evidently, te truncation error analysis is exactly te same. Suppose tat ( f, g) are smoot in bot temporal and spatial variables, and tat f n = S 1, f n, g n+1/2 = S 2, g n+1/2. (34) Denote by (ũ, ṽ) te (weak) solution of te acoustic system 23 wit rigt-and sides ( f, g) tat vanises for t > T. Te standard teory for yperbolic systems sows tat (ũ, ṽ) is also smoot. Let (ũ n+1/2, ṽ n ) be te solution of te discrete adjoint system 32 wit rigt-and sides ( f n, gn+1/2 ). Ten (as before assuming tat P is accurate of order 2p) δũ n+1/2 = ũ n+1/2 S 1, ũ n+1/2, δṽ n = ṽn S 2,ṽ n solve 32 wit rigt-and side tat is O( t 2 + 2p ), so according to te error estimate 30 (wic applies ipso facto to te system 32), δũ n+1/2 2 + δṽ n 2 = O( t 2 + 2p ) (35) over any finite range of n t. We now ave all of te ingredients to prove weak convergence, wic we now state: 35

36 Teorem 8. Let (u, v) be a solution to continuum problem 21 wit singular source terms (f, g) of te form f(x, t) = w(t)d s 1 δ(x x ), g(x, t) = z(t)d s 2 δ(x x ), were w(t) R k 1 and z(t) R k 2 are smoot vector-valued functions in time. Also, let (u, v ) denote te corresponding finite difference solution wit singular source approximates (f, g ) of order q. Ten, for any smoot test functions ( f, g), we ave te following error estimate E := T 0 { u, f } + v, g dt t N { u n, f } n + vn+1/2, g n+1/2 n=0 (36) = O( t 2 + q + ( t 2 + 2p ) N d/2 ), wit N = max{ s 1, s 2 }. Again, ( f, g ) as given by 34. Proof. Using 33, and its continuum version, error E is rewritten in terms of inner products wit te singular source terms, bot in te continuum and discrete sense, i.e., = T 0 { } f, ũ + g, ṽ dt t N { n=0 f n+1/2 }, ũ n+1/2 + g n, ṽn. Note ( p, ṽ) are solutions to problem 21 wit smoot source terms ( f, g), and (ũ, ṽ ) is te respective staggered grid finite difference solutions wit sources ( f, g ). Furter- 36

37 more, E = T 0 E 1 + E 2. { } f, ũ + g, ṽ dt t N { n=0 f n+1/2, ũ n+1/2 ± S 1, ũ n+1/2 + g n, ṽn ± S 2,ṽ n } wit E 1 := T 0 { } f, ũ + g, ṽ dt t N E 2 := t { n=0 N n=0 f n+1/2 { f n+1/2, S 1, ũ n+1/2 + g n, S 2,ṽ n } }, δũ n+1/2 + g n, δṽn. T 0 For E 1, we first focus on te terms involving f and ũ; f, ũ dt t N f n+1/2, ũ n+1/2 + N t n=0 n=0 T 0 f, ũ dt t N n=0, S 1, ũ n+1/2 f n+1/2 { f n+1/2, S 1, ũ n+1/2 f n+1/2, ũ n+1/2 }, wic is noting more tan quadrature error for te time integration and te singular source discretization error. In particular, T 0 f, ũ dt t N f n+1/2, ũ n+1/2 = O( t2 ) n=0 from standard error estimates of te midpoint rule, and N t { } f n+1/2, S 1, ũ n+1/2 f n+1/2, ũ n+1/2 = O( q ) n=0 from te singular source approximation teory (teorem 2). Similar error estimates can be 37

38 derived for te terms involving g and ṽ. 1 We bound E 2 using L 2 -error estimates of finite difference solutions for smoot problems, and using L 2 -bounds of te discrete singular source approximations; one can sow f 2 = O( s 1 d/2 ) and g 2 = O( s 2 d/2 ). Wence, E 2 t N { n=0 f n+1/2 } 2 δũ n+1/2 2 + g n 2 δṽ n 2 = O(( t 2 + 2p ) N d/2 ). Estimate 36 tus follows. Consider te simpler case were only multipoles of order zero are present, tat is N = 0. According to estimate 36, we ave weak convergence at rate 2 d/2 if we coose q = 2p and t = O(). In oter words, convergence (in te weak sense) is guaranteed toug at a suboptimal rate. For multipoles of order N > 0, to retrieve te smoot solution beavior, we would require 2p > N + d/2 and t 0 like a positive power of. Te (weak) error estimate we present ere can be improved by making te following observation: te d/2 factor originates from L 2 -bounds of multipole source approximations. In particular, we can remove tis factor by using L 1 -bounds instead, mainly f 1 = O( s 1 ) and g 1 = O( s 2 ). 1 O( t 2 ) quadrature error estimates follow from noting tat te summation term coincides wit te trapezoidal rule, assuming tat g(t) and ṽ(t) satisfy omogenous initial and final time conditions respectively. 38

39 Applying Hölder s inequality to E 2 yields E 2 t N { n=0 f n+1/2 } 1 δũ n+1/2 + g n 1 δṽ n = O(( t 2 + 2p ) N ). (37) We conjecture point-wise bounds old for smoot δũ and δṽ, wence E = O( t 2 + q + ( t 2 + 2p ) N ), (38) yielding optimal rates for te N = 0 case, again if q 2p. Note tat 37 follows troug if L error terms δũ and δṽ ave optimal rates. Te autors are not, owever, aware of any suc L error estimates for te types of yperbolic systems considered ere. One could potentially attempt to prove stability in L in order to apply te Lax equivalence teorem and imply L estimates (see Brenner et al. (2006)) or alternatively proof L estimates requiring only L 2 stability (see Layton (1982)). 39

40 NUMERICAL TESTS AND RESULTS We ave implemented te singular source approximation as discussed in te teory section using te C++ packages IWave and Rice Vector Library (RVL); (Symes et al., 2011; Padula et al., 2009). Te IWave package is a framework for finite difference solvers over uniform grids wile te RVL package provides a system of classes for expression of gradient-based optimization algoritms over Hilbert spaces. IWave and RVL come togeter to form a modeling engine for seismic inversion and migration. Implementation of multipole sources as RVL objects enables straigtforward composition wit IWave solvers and inclusion in inversion algoritms powered by RVL optimization code. Any oter wave equation solver wrapped in te appropriate RVL interfaces could be coupled to te multipole source objects in te same way. A convergence rate study is performed to corroborate teoretical results pertaining to te accuracy of moment-consistent approximations to multipole sources. In particular, our numerical experiments explore te semi-discrete error of staggered-grid finite difference solutions (time discretization errors are minimized by taking sufficiently small time steps). We used te IWave implementation of staggered-grid finite difference scemes for te acoustic system 23 (Virieux, 1984), of order 2 in time and orders 2 and 4 in space - we refer to tese as te 2-2 and 2-4 order scemes respectively. In tese experiments we use only scalar (pressure) sources and pressure trace data. Similar results are obtained wit oter coices. Boundary conditions are of PML type, as described by Hu et al. (2007), Te numerical experiments carried out concern multipoles in 2-D of te form f(x, t) = w(t)d s δ(x x ), 40

41 for s = (0, 0), (0, 1), (0, 2). Te discretizations of D s δ are cosen as to acieve a target order of convergence for te difference scemes, in most cases te nominal spatial order (q = 2 for te 2-2 sceme, q = 4 for te 2-4 sceme). Again, te time step is fixed small enoug tat te time discretization error plays essentially no role in te global error - it reflects truncation error of te spatial derivative and te source approximation only. Timedependent function w(t) is cosen to be a Ricker wavelet wit peak frequency of 5Hz, see figure 3a. We approximate te convergence rate R(x) at a given location x via Ricardson extrapolation, ( ) p (x, ) p /2 (x, ) R(x) = log 2 p /2 (x, ) p /4 (x, ) were p denotes te computed pressure field via finite difference using a grid size respectively. Te norm is cosen to be p(x, ) := t k p(x, t k ) 2, Coordinates are aligned suc tat x = (z, x), were z and x refer to dept and orizontal distance respectively. Te computational domain consists of a 4km by 4km square medium wit a constant bulk modulus of 9GP a and buoyancy of 1cm 3 /kg, tus a constant velocity of 3km/s. Te source is placed sligtly off of te center, (z, x ) = ( 2003m, 2003m), as to not coincide wit a grid point for any of te computational uniform grids. Te following are some oter specifications tat apply to all tests carried out ere: total recording time is 1.5s; 41

42 spatial grid sizes = 40m, /2 = 20m, /4 = 10m; time step t = 0.5ms; Note tat te coarsest grid cell size is 40m wic implies tat at least 15 grid points per wavelengt (gpw) at peak frequency of 5Hz or 5 gpw at 15Hz (see figure 3b), in all numerical experiments, tus minimizing te effects of grid dispersion on approximated convergence rates. Results Figures 4a 4c plot a snapsot of te computed pressure field for different multipole sources. Consistent wit analytical formulas for te omogeneous unbounded medium case, te observed pressure field exibited a polarity reversal (or lack of), a symmetry about te x-axis (i.e., z = 2003), and an overall decrease in amplitude dependent on te multipole. Convergence rates are plotted for te s = (0, 0) case in figure 5, over te entire domain and at a particular dept of z = 2000 in te left and rigt column plots respectively. Te first row of graps sows results wen using te 2-2 finite difference sceme and te secondorder source approximation. We observe tat second order convergence is indeed acieved away from te source. Similarly, fourt order convergence results are observed wen using te 2-4 finite difference sceme and te fourt-order source approximation, i.e., second row in figure 5. Lastly, te tird row of plots demonstrates te negative effects of using a lower order source approximation relative to te spatial finite difference order, namely a secondorder source approximation wit a fourt-order metod in space. Clearly, fourt order rates are not acieved and moreover, tere seems to be regions were rates dip below or above 42

43 te expected second order. Convergence results for s = (0, 1) and s = (0, 2) are given in figures 6 and 7 respectively wit similar results. DISCUSSION Numerical results presented ere validate te accuracy of moment-consistent singular source discretizations for controlling te propagation of finite difference truncation error for multipole sources. In particular, optimal spatial convergence rates for te 2-2 and 2-4 staggered grid finite difference metods are acieved, point-wise in space away from source location, wen te source approximation order is equal to tat of te spatial finite difference approximation order. Te onset distance of optimal order convergence is consistent wit te support of te source approximation for te coarsest grid used in te convergence study, tat is, (q + s ) wit = 40m. Te erratic beavior of numerical convergence rates witin te support of source approximations can be explained in part by ow convergence rates were approximated. Namely, numerical solutions over consecutively refined grids are essentially compared, toug over te coarsest grid. It follows tat te support of f, te source approximation wit respect to grid wit cell size, will contain te support of f /2 and a region in te -grid outside te direct influence of f /2, resulting in irregular convergence rates. Work by Petersson et al. (2016) demonstrated tat optimal convergence rates can be acieved if te source discretization satisfied moment as well as smootness conditions. Again, teir work was centered around central difference approximations of te 1-D ad- 43

44 vection equation. As our numerical tests demonstrate, smootness conditions were not necessary ere, wic is attributed to te fact tat we use staggered-grid finite difference scemes. In particular, te central finite difference operator used in Petersson et al. (2016) ave an associated grid spacing of 2 wit respect to an -sized grid, coupled wit a source approximation of narrow support, i.e., 2ɛ = (q + s ), results in te triggering of spurious modes unresolved by te numerical solution. Te finite difference operators used ere owever ave a grid spacing of given tat tey approximate derivatives over staggered grids. Tus, in essence our narrow source approximations seem twice as wide, and terefore smooter, from te point of view of te difference operator P. For tis reason, smootness conditions are not necessary for staggered-grid finite difference scemes. CONCLUSION In tis paper we ave covered te singular source approximation teory based on moment conditions, wic essentially as approximations mimic te beavior of te target distribution D s δ(x x ) on polynomials. Moreover, we give explicit forms of source approximations wit narrow support, based on te discrete moment conditions in te context of finite difference solvers: diameter 2ɛ = (q + s ) for a multipole of order s over a grid of size. As a new result, we connect te discrete and continuum singular source approximation teory by proving tat continuum functions generated from te discrete moment conditions indeed satisfy te continuum moment conditions. Our main contribution was te development of a weak convergence teory tat is applicable to a large set of wave propagation problems (including acoustics and elasticity in first-order form) solved via a family of staggered-grid finite difference scemes, larger tan 44

45 wat is reported in te current literature. Posing te convergence mode of numerical solutions in terms of weak convergence was indeed a natural coice given tat source terms are derivatives of te Dirac delta function, tat is, distributions. Numerical results, owever, give evidence of stronger convergence, namely optimal convergence rates given by numerical sceme under smoot conditions, point-wise away from source location for appropriate source discretizations and in particular for multipoles of order s = 0, 1 and 2. ACKNOWLEDGEMENTS We are grateful to te sponsors of Te Rice Inversion Project for teir long-term support, and to te Rice Graduate Education for Minorities (RGEM) and XSEDE scolarsip programs for teir support of M. Bencomo s P.D. researc. Tis material is also based upon work supported by te National Science Foundation under Grant No. DMS wile te autor was in residence at te Institute for Computational and Experimental Researc in Matematics in Providence, RI, during te Fall 2017 semester. 45

46 REFERENCES Blazek, K., C. C. Stolk, and W. Symes, 2013, A matematical framework for inverse wave problems in eterogeneous media: Inverse Problems, 29, :1 34. Brenner, P., V. Tomée, and L. B. Walbin, 2006, Besov spaces and applications to difference metods for initial value problems: Springer, 434. Coen, G. C., 2002, Higer order numerical metods for transient wave equations: Springer. Courant, R., and D. Hilbert, 1962, Metods of matematical pysics: Wiley-Interscience, II. Epanomeritakis, I., V. Akcelik, O. Gattas, and J. Bielak, 2009, A Newton-cg metod for large-scale tree-dimensional elastic full-waveform seismic inversion: Inverse Problems, 24, 24: (26pp). Hörmander, L., 1969, Linear partial differential operators, 3rd ed.: Springer Verlag. Hosseini, B., N. Nigam, and J. M. Stockie, 2016, On regularizations of te dirac delta distribution: Journal of Computational Pysics, 305, Hu, W., A. Abubakar, and T. Habasy, 2007, Application of te nearly perfectly matced layer in acoustic wave modeling: Geopysics, 72, SM169 SM176. Komatisc, D., C. Barnes, and J. Tromp, 2000, Simulation of anisotropic wave propagation based upon a spectral element metod: Geopysics, 65, Königsberger, K., 2013, Analysis 2: Springer-Verlag. Layton, W. J., 1982, Simplified l estimates for difference scemes for partial differential equations: Proceedings of te American Matematical Society, 86, Moczo, P., J. O. A. Robertsson, and L. Eisner, 2006, Te finite-difference time-domain metod for modeling of seismic wave propagation: Advances in Geopysics, 48, Padula, A. D., W. Symes, and S. D. Scott, 2009, A software framework for te abstract 46

47 expression of coordinate-free linear algebra and optimization algoritms: ACM Transactions on Matematical Software, 36, 8:1 8:36. Peskin, C. S., 2002, Te immersed boundary metod: Acta numerica, 11, Petersson, N. A., O. O Reilly, B. Sjögreen, and S. Bydlon, 2016, Discretizing singular point sources in yperbolic wave propagation problems: Journal of Computational Pysics, 321, Petersson, N. A., and B. Sjögreen, 2010, Stable grid refinement and singular source discretization for seismic wave simulations: Comm. Comput. Pys., 8, Searer, P. M., 2009, Introduction to seismology: Cambridge University Press. Symes, W., D. Sun, and M. Enriquez, 2011, From modelling to inversion: designing a well-adapted simulator: Geopysical Prospecting, 59, (DOI: /j x). Tornberg, A., and B. Engquist, 2003, Regularization tecniques for numerical approximation of pdes wit singularities: J. Sci. Comput., 19, , 2004, Numerical approximation of singular source terms in differential equations: J. Comp. Pys., 200, Virieux, J., 1984, SH-wave propagation in eterogeneous media: Velocity stress finitedifference metod: Geopysics, 49, Waldén, J., 1999, On te approximation of singular source terms in differential equations: Numerical Metods for Partial Differential Equations, 15, Yilmaz, O., 2001, Seismic data processing, in Investigations in Geopysics No. 10: Society of Exploration Geopysicists. 47

48 LIST OF FIGURES 1 Plots of η ɛ (x), 1-D approximations to D s δ(x) wit = 1. (a) s = 0 and q = 2, (b) s = 0 and q = 4, (c) s = 1 and q = 2, (d) s = 1 and q = 4, (e) s = 2 and q = 2, (f) s = 2 and q = 4. 2 Plots of η ɛ (x), 2-D approximations to D s δ(x) wit = (1, 1). (a) s = (0, 0) and q = 2, (b) s = (0, 0) and q = 4, (c) s = (0, 1) and q = 2, (d) s = (0, 1) and q = 4, (e) s = (0, 2) and q = 2, (f) s = (0, 2) and q = 4. 3 Ricker wavelet wit peak frequency 5Hz: (a) time plot, (b) power spectrum plot. 4 Snapsot of pressure field (t = sec) computed using 2-2 finite difference sceme ( = 10m, t = 0.5 ms) and second-order approximation of multipole source D s δ(x x ): (a) s = (0, 0), (b) s = (0, 1), and (c) s = (0, 2). 5 Convergence rate results for s = (0, 0). Using 2-2 FD sceme and second-order source approximation (a), 2-4 FD sceme and fourt-order source approximation (c), 2-4 FD sceme and second-order source approximation (e). Plots on rigt column correspond to convergence rates at z = 2000, related to te left column. 6 Convergence rate results for s = (0, 1). Using 2-2 FD sceme and second-order source approximation (a), 2-4 FD sceme and fourt-order source approximation (c), 2-4 FD sceme and second-order source approximation (e). Plots on rigt column correspond to convergence rates at z = 2000, related to te left column. 7 Convergence rate results for s = (0, 2). Using 2-2 FD sceme and second-order source approximation (a), 2-4 FD sceme and fourt-order source approximation (c), 2-4 FD sceme and second-order source approximation (e). Plots on rigt column correspond to convergence rates at z = 2000, related to te left column. 48

49 49

50 (a) (b) (c) (d) (e) (f) Figure 1: Plots of η ɛ (x), 1-D approximations to D s δ(x) wit = 1. (a) s = 0 and q = 2, (b) s = 0 and q = 4, (c) s = 1 and q = 2, (d) s = 1 and q = 4, (e) s = 2 and q = 2, (f) s = 2 and q = 4. 50

51 (a) (b) (c) (d) (e) (f) Figure 2: Plots of η ɛ (x), 2-D approximations to D s δ(x) wit = (1, 1). (a) s = (0, 0) and q = 2, (b) s = (0, 0) and q = 4, (c) s = (0, 1) and q = 2, (d) s = (0, 1) and q = 4, (e) s = (0, 2) and q = 2, (f) s = (0, 2) and q = 4. 51

52 (a) (b) Figure 3: Ricker wavelet wit peak frequency 5Hz: (a) time plot, (b) power spectrum plot. 52

53 (a) (b) (c) Figure 4: Snapsot of pressure field (t = sec) computed using 2-2 finite difference sceme ( = 10m, t = 0.5 ms) and second-order approximation of multipole source D s δ(x x ): (a) s = (0, 0), (b) s = (0, 1), and (c) s = (0, 2). 53

54 (a) (b) (c) (d) (e) (f) Figure 5: Convergence rate results for s = (0, 0). Using 2-2 FD sceme and second-order source approximation (a), 2-4 FD sceme and fourt-order source approximation (c), 2-4 FD sceme and second-order source approximation (e). Plots on rigt column correspond to convergence rates at z = 2000, related to te left column. 54

55 (a) (b) (c) (d) (e) (f) Figure 6: Convergence rate results for s = (0, 1). Using 2-2 FD sceme and second-order source approximation (a), 2-4 FD sceme and fourt-order source approximation (c), 2-4 FD sceme and second-order source approximation (e). Plots on rigt column correspond to convergence rates at z = 2000, related to te left column. 55

56 (a) (b) (c) (d) (e) (f) Figure 7: Convergence rate results for s = (0, 2). Using 2-2 FD sceme and second-order source approximation (a), 2-4 FD sceme and fourt-order source approximation (c), 2-4 FD sceme and second-order source approximation (e). Plots on rigt column correspond to convergence rates at z = 2000, related to te left column. 56