The elastic bending of the lithosphere under crustal loads is a fundamental part of modern geodynamics describing a swathe of processes including glacial isostatic adjustment , foreland basin formation in compression , and the flexural response of the lithosphere to extension . The mathematical theory as applied to vertical loads deflecting the Earth's lithosphere was originally proposed in the pre-plate tectonic era by , who was interested in the wider question of compensation of loads by isostatic balance. The original theory of elastic beam bending for engineering from which it was derived is attributed to Leonhard Euler and Daniel Bernoulli in the 1750s. wrote his series of papers at the culmination of a many-decades-long debate between geodesists and geologists concerning how loads on the crust and lithosphere were compensated for by displacement of mantle material see. Geodesists had long favoured the idea that loads were all locally compensated for (Airy or Pratt isostasy). was the first person to fully realise and formulate the necessary equations describing how a load can be compensated for over a much greater distance than its own width due to the elastic strength of the lithosphere. Before him, it was clear that several authors had very similar insights but were never able to quantitatively demonstrate them (see , for a summary of the history of isostasy).

Gunn's work transformed our understanding of lithospheric mechanics, establishing how loads on the crust were balanced by elastic bending of the lithosphere as well as suggesting how this compensation would affect measured gravity anomalies. Subsequently, in the early post-plate tectonic era, published a series of manuscripts on the question of the elastic thickness of the Earth's lithosphere. Elastic thickness (h) is the key parameter in the flexural coefficient D (the rigidity), which itself is related to the bending moment of the lithosphere (Fig. 1). A high value of D means less bending of the lithosphere under loading. Following , estimating the value of elastic thickness h, especially for continental lithosphere, became one of the most strongly debated topics in geodynamics e.g.. At present, estimated values of h for the continents vary from 0 to 100 km, presumably due to a combination of factors related to thermal structure and tectonic history of a particular region of the lithosphere.

The fundamental equation describing the elastic bending of the lithosphere is our concern in this paper. Its original form presented to geologists by remains unchanged. It describes the balance of forces (bending moment, vertical shearing forces, any added loads, and restoring forces from the buoyancy of mantle below) in an elastic plate resting on a fluid (inviscid) mantle, a so-called Winkler foundation (Fig. 1). Moment and shearing forces are converted to functions of the deflection of the plate (u(x) in this paper), resulting in a fourth-order differential equation in u given as 1Du′′′′+Pu′′-ku=q, where u(x) is the vertical deflection of an originally horizontal surface of the plate due to loading, u′=dudx and so on for higher derivatives; q(x) is a term describing the applied load; P is a plate-wide stress, which is actually a uniform, compressive load in the plane of section of a plate; k=(ρM-ρF)⋅g is a constant allowing compensation of plate deflection by displacement of underlying “fluid” substrate (mantle); and D=Eh312(1-ν2) is the flexural rigidity in which E is the elastic modulus, h the elastic thickness of the lithosphere, and ν Poisson's ratio. For many solutions given in the literature D is assumed to be constant, meaning h does not vary along a plate's length. Should it be the case that h(x) is variable, however, then a modified general form of the equation above is 2(Du′′)′′+Pu′′-ku=q. Equations (1) and (2) can be reformulated in the following ways. We begin by rewriting them explicitly to show the different components of the coefficients, while also setting P=0. 3Du′′′′-(ρM-ρF)gu=ρLgq(x) We note that the load term (ρM-ρF)gu corresponds to two body forces: ρMgu is the restoring force (per unit length and width with units of Newtons) due to displacement of mantle by deflection of the plate, and ρFgu is a force exerted by material assumed to infill any surface deflections of the plate below an arbitrary reference level, but it is also clear that without modification, any solution of this problem equally assumes that infill forces are removed wherever there is a positive deflection of the plate (Fig. 2). Such infill can range from nothing (empty basins) to water (oceanic cases, for instance) to sedimentary product (foreland basins) or a mixture of any of the above in various combinations. We also note that the load term ρLgq(x) defines a separate load of potentially different density to that assumed for the infill.

Two versions of the equation can be developed from here. The first one (Fig. 3a), which has rarely been explicitly discussed or used in geodynamics, involves separating the infill load term from the mantle restoring force (see also Appendix A). 4Du′′′′-ρMgu=ρLgq(x)-ρFgu With the load term now entirely on the right-hand side of the equation, we see it consists of two parts with the infill part, ρFgu (right hand side), dependent on the deflection u(x) for which we are solving. However, a fixed load ρLgq(x) is also being applied, which is itself an arbitrary function of x. The physical interpretation of this depends on how this fixed load term is regarded. In most cases, it is probably assumed to be some kind of imposed crustal load, such as a thrust sheet or ice sheet, or, in oceanic cases, a seamount or volcanic island. Under these circumstances, it is equally clear that infill material cannot occupy space wherein the fixed load is applied unless the special circumstance applies that the top of the fixed load at some point lies below the original reference level, in which case a reduced accommodation space is available for infill material defined as the local sum u(x)+q(x). In general, implementing a solution to this form of the problem requires numerical methods, since arbitrary piecewise variations of load density may be required, and a solution will need to be iterative due to the dependence of the infill load term on u(x). It is also important to state that the fixed load term ρFgq(x) is actually an applied force scaled for a particular load thickness. Hence, arbitrary “forces” can equally well be applied to the lithosphere, whatever their origin is assumed to be.

In general, it can be seen by examination that if D=0 (no flexural bending occurs and hence no marginal basin forms beyond the end of the load, so there is no basin infill load) the solution will correspond to “Airy isostasy” (i.e. an “iceberg” model) with 5u(x)=-ρLρMq(x)q(x)≠00q(x)=0. An alternative development of the equation for solution is by division through by (ρM-ρF)g, giving 6Du′′′′(ρM-ρF)g-u=ρL(ρM-ρF)q(x). Here we see immediately that in contrast to Eq. (4), a direct, non-iterative solution is possible. This occurs because the physical meaning of the equation in this form is quite different. If we set D=0 once more, then 7u(x)=-ρL(ρM-ρF)q(x)q(x)≠00q(x)=0. Equation (7) also represents a case of Airy isostasy. However, in this form, q(x) no longer represents a load thickness, but rather a load surface topography (Fig. 3b), for which the appropriate flexural compensation function u(x) is calculated. Hence, the difference q(x)-u(x) gives the resulting finite load thickness (in Fig. 3b we explicitly show Topo(x) as the “load” term and q(x) as a term derived from Topo(x)-u(x)). As can be seen from Eq. (7), however, a problem with this formulation arises due to the different density terms employed, in particular the different values of ρL and ρF. Only when ρF=ρL is a condition of Airy isostatic balance of a load of density ρL actually calculated, and hence the appropriate deflection u(x) and resulting load thickness q(x)-u(x). In analytical solutions of this equation, the only way to avoid this problem is by assuming ρF=ρL everywhere. Otherwise, for piecewise variable density terms, a numerical solution is required.

In summary, the general equation of flexure of the lithosphere can be formulated in two different ways. In the first, a flexure-dependent load term due to infill results, regardless of density variations, and requires an iterative solution due to the fact that regions with an imposed load cannot be simultaneously occupied by infill. Moreover, this form of the equation allows imposition of arbitrary forces to an elastic plate. A second and more common formulation of the problem describes the flexural subsidence required to support a particular surface topography. Analytical solutions to any flexure problem are generally unable to account for variable density of different load components or to differentiate between fill of basins created by flexural subsidence and removal of fill in any positively deflected regions. Hence, we now consider some numerical solutions to flexure problems.

For the most part, finite-difference methods have been applied to solve the flexure equation numerically in both the one-dimensional beam-type situation and for two-dimensional, thin elastic sheets. For the simplest case of constant flexural rigidity, D, the left-hand side of Eq. (3), for instance, is discretised as (see Fig. 4) 8Ddx4ui+2-4ui+1+6-ρMgdx4Dui-4ui-1+ui-2. Over the past 40 years, a number of numerical solutions to flexure problems were proposed. A lot of this effort was aimed at solving problems for the two-dimensional extension of the flexure equation to an elastic sheet with variable elastic thickness and hence flexural rigidity . corrected what was probably the earliest attempt at a numerical solution with variable flexural rigidity . Relatively few publications have dealt with the details of the numerical, one-dimensional flexure equation discussed here for both constant and variable elastic thickness cases. It should also be noted that since the initial publication, most of the succeeding work involving numerical solutions of the flexure equation has been based on their numerical derivation .

An early paper using a numerical solution illustrates this form of the one-dimensional numerical solution to the flexure equation with variable elastic thickness. Taking Eq. (2), for instance, the product rule of differentiation is applied prior to discretisation, and hence 9Du′′′′+2D′u′′′+D′′u′′-ku=q. We note that this formulation of the problem, whilst mathematically correct, has a clear, unambiguous physical implication. Any function describing the variation of elastic thickness h(x) (and hence rigidity, D(x)) as a function of position must be continuous and at least twice differentiable (see Appendix C). It will often be possible to get solutions to this problem with other functions of D(x), but they are not consistent with the way the problem is posed in Eq. (9). The use of the product rule derivation of the problem extends to all of the aforementioned publications concerning the two-dimensional sheet-like problem as well (see passing from Eq. 3 to 7 of ). The resulting finite-difference discretisation is given in Appendix B.

Perhaps surprisingly, there is an alternative and quite different method of discretising Eq. (2) available. This has been termed the “half-station” method and avoids a product rule derivation prior to discretisation entirely. Instead, Eq. (2) is directly transformed into a finite-difference approximation, replacing the derivatives both within and outside the brackets with finite-difference approximations to second derivatives (see Appendix B). The main effect of this is that no third or fourth derivative terms in u (and hence also first and second derivative terms in D) explicitly arise. Instead, the fourth-order nature of the differential equation as well as gradients in D are implicitly contained in the numerical scheme. Somewhat remarkably, this means that there is no restriction on the nature of the function D(x). Any piecewise, arbitrary variation of D(x) (and thus h(x)) is consistent with this discretised form of the equation. The half-station method is generalisable to two dimensions, meaning an alternative discretisation arises with Eq. (3) of as a starting point.

It must be noted here that both the half-station and whole-station methods are proven to be second-order accurate, finite-difference solutions to the flexure equation. Both converge to the analytical solution to the problem, effectively by a limited case of the Lax–Milgram theorem (, as demonstrated in ). However, it is also clear that the whole-station method will never give a correct (convergent) solution for an abrupt variation (piecewise jump) in coefficient (Appendix C). Hence, for any case involving a piecewise linear or abrupt jump in the value of the coefficient, only the half-station method is convergent (see Appendix C).

We now address the question of the different behaviours of the numerical schemes with respect to variations of h(x). As we have already discussed, for the product rule version of the equation (so-called “whole-station” method; ) the formal derivation of the numerical scheme actually requires a continuous, at least twice differentiable function of h(x). There are a number of interesting and illustrative cases of elastic thickness variation, however, in which abrupt changes are required. We note that both whole-station and half-station discretisations contain, for any grid point or node i, terms involving Di+1,Di, and Di-1 (see Fig. 4 and Appendix B), meaning that a discontinuity in h(x) must be present across at least three grid nodes to take full effect. This was also commented upon by in the case of the whole-station scheme, but without detailed analysis of the results. A recognised difficulty with varying elastic thickness is how to find analytical solutions to which to compare numerical results. An indirect way of doing this, however, comes from , who showed analytically see Eqs. 13 and 16, that for the same point load, the maximum deflection of a broken plate (i.e. one loaded at its end) is 4 times that of a continuous (infinite) plate equivalently loaded in its centre. Suppose we take a continuous plate and reduce the elastic thickness to zero over three nodes, creating an elastic “break”, and place a “point” load at a single node, directly to the left or right of the elastic break. We note that the load in this case is not at a single point, but instead applied over a finite width equal to the grid spacing used in the numerical scheme, so direct comparison to an analytical solution is difficult. However, we would expect the plate, when loaded just next to the elastic break, to act like the end-loaded or broken plate, whereas the same plate loaded equally but without an elastic break should behave like the continuous one, so the relative maximum subsidence of the two numerical cases should change by a factor of 4. Results of the experiments are shown in Fig. 5. The half-station method gives exactly the result expected, showing that it corresponds to an end-loaded plate when elastic breaks or discontinuities are present within a larger plate. Due to the severe violation of the conditions of continuity and twice differentiability in the function of h(x), the whole-station method gives no meaningful result and is unable to simulate a “broken” plate.

Another interesting case to test the numerical schemes is that of what we can term an “isostatic raft” (Fig. 6). In this case, we simulate an effectively infinitely stiff plate (h≥500 km) with a central region of length ∼200 km, which is bordered at each end by an elastic break. The “raft” is loaded evenly across its centreline by a rectangular-shaped load of width ∼40 km. The relative dimensions are chosen purely to illustrate the point. If the plate segment were truly infinitely stiff, it would undergo no bending at all, and the load mass applied would be compensated for by escape of an equal mass of mantle substrate. The half-station method in this case produces an almost constant subsidence of the plate segment and causes a tiny amount of flexural bending. The resulting mass difference between displaced mantle and load is <0.2 %, showing the expected “raft-like” response. Again, for such a case, the whole-station method produces a spurious result.

A geological application of the raft analogue arises when we consider tilted crustal blocks formed in compression . This refers to the concept of short segments of crust and lithosphere bounded by basement-transecting faults. initially created a simple force balance model wherein horizontal stress across a fault-bounded block generates a moment, which turns and tilts the block against the resisting force of the mantle and is potentially augmented by the effects of erosion and sedimentation on the tilted block surface. This idea was principally used to explore the amount of compressive stress required to “break” the lithosphere in plate interiors; however, a corollary of it was to explain subsidence and basin formation as also due to the block-tilting process. The model treats the block as completely rigid by default and assumes that the horizontal compressive force is responsible for the tilting, neglecting the effects of “self-loading” due to one block overriding another and also flexural bending induced in the block.

Whilst our flexure model cannot be directly related to horizontal compressional stresses potentially involved in breaking the lithosphere, it is trivial to produce a succession of adjacent crustal segments by placing elastic breaks across a plate, creating isolated segments of the desired dimensions. By loading each segment at or near its end, thus creating a turning force, and noting that any load can be treated as an arbitrary system of forces which arise for many different reasons, we produce a result similar to that of but which also takes into account the flexural bending in the segments. Figure 7 shows two situations of identically loaded blocks with length 100 and 200 km. The longer blocks undergo substantially more bending as a result of the mantle resisting force being spread over a greater length and consequently holding the plate down more firmly, allowing it to bend elastically to a greater degree under loading. We note that a flexural model with a plate containing elastic breaks effectively parameterises lithospheric structure in terms of changes in elastic thickness. In the case of the tilted block model, this parameterisation can be thought of as the net effect of bounding faults, with the applied load characterising any combination of the possible forces acting on a block (e.g. moment on the block due to horizontal stress, friction on the fault resisting tilting, self-loading due to one block overriding another). In our illustrative models here, we have used 4 km thick, 15 km wide, distributed loads, corresponding to a net force of 1.6×1012 N. This produces relatively small uplifts at block corners, although clearly, the major component of the uplift of block corners is likely to be due to steady transport of basement faults bounding adjacent tilted blocks due to shortening. The subsidence induced in basins, by contrast, can be more directly related to the response to loads on the crust. In the case of the Laramide orogeny, for instance, maximum sedimentary thicknesses in the associated basins are ∼2–4 km , which is quite close to the ∼3 km subsidence under the load in our models.

All the preceding cases concern situations in which the lithosphere is modelled as segmented or broken. In many cases, however, we consider elastic thickness to vary more steadily e.g.. In such cases, the most straightforward spatial variation of h(x) is described by a piecewise linear function interpolated between a few points of fixed value. We find that in cases in which the gradient of the imposed linear change is not too sharp, the whole-station approximation can return reasonable results, including in cases in which h→0. Below a certain threshold, however, the error (which we take as the difference to the half-station solution) quickly reaches ≥2 %, which for studies fitting flexural curves to gravity signals, for instance e.g., will be critical. Figure 8 shows two cases with differing gradients in h(x). While the gentle gradient yields a difference of 6 m maximum subsidence (∼0.5 %) the sharper gradient reaches 340 m (∼30 %). As the gradient in h(x) increases still further, the whole-station scheme will ultimately reach a point at which it returns no solution at all, whilst the half-station method is stable for any combination of loads and variations of h(x).

A more general mathematical issue concerns the positivity of any numerical solution to a fourth-order differential equation of this kind. As can be verified (see Appendix B), all discretisations of the flexure equation using second-order finite-difference approximations will yield an identical set of linear equations when the value of h is a constant. The discretised form then becomes that shown in Eq. (8). It can also be seen that for constant values of D at least, for each line, the sum across the columns (∑jai,j) is ρMg, the mantle restoring force, since all other terms involving D sum to zero. As a result, the residual term due to the mantle restoring force is, according to the maximum principle , necessary for maintaining the positivity of the system of equations represented by Eq. (8). Hence, the composition of the main diagonal, which itself consists of a sum of two terms, 6Ddx4 and ρMg, becomes of critical importance. This is due to the issue of round-off, whereby the capacity of 64 bit representations of numbers to sum terms with large contrasts in magnitude leads to the smaller term being partly or entirely lost as the maximum number of significant figures available in arithmetic operations (approximately 15) is exceeded. In the large term, D will vary as a function of h3, where elastic thickness h may reach values of 100 km on Earth and possibly even 300 km on Mars . Grid spacing dx requires values of ∼100 m or less to ensure convergence and also to allow reasonable resolution in the representation of loads. It should be noted that currently, the highest-resolution, public, and globally available topographic databases, SRTM and ASTER GDEM , are both on 1 arcsec (∼30 m) grids, whilst TANDEM-X data are relatively freely accessible to scientists on a 0.4 arcsec (∼10–12 m) grid. Hence, 6Ddx4 will quite conceivably be of magnitude 1019 or more in “real” problems in geodynamics. The small term ρMg will always be of order 104. Under such conditions, round-off will lead to an “effectively singular” matrix and the numerical problem will fail. One work-around is relatively easily available. The use of quadruple-precision (128 bit) representation of floating-point numbers allows >30 significant figures to be taken account of in arithmetic operations. Although this costs additional memory and some speed, it ensures that any conceivable problem of flexure with “real-world” dimensions and parameters will be correctly dealt with by the numerical algorithm.

Besides the general issue described above, the additional term for the plate-wide stress P (see Eq. 2 and Appendix B) also has the potential to cause problems for the numerical solution. In particular, P, which has always been assumed to be constant throughout a plate, will interact with regions of variable elastic thickness in a potentially problematic way. For cases with elastic breaks, where D→0, for instance, a constant value of P will often lead to failure of the numerical solution. In such cases, it is probably reasonable to set P→0 on the three nodes of the discontinuity, thereby treating these as if they were an infinitely thin fault. Because the nodes either side of the break have the normal value of P applied, the continuity of P is respected to some degree.

The half-station method of discretisation we have presented here is clearly able to deal with a complete set of possible variations of elastic thickness in the flexure equation. The whole-station method (applying the product rule first), by contrast, is unable to be relied upon to do so. Perhaps the most worrying aspect of the whole-station method is that in some circumstances, it will give results that appear plausible but are in fact in error by significant amounts (Fig. 8b) and actually represent solutions among the set of transitional variations of h just before the method fails completely. The principal reason for the behaviour of the whole-station discretisation is the fact that, as posed, a condition of the equation is a twice differentiable, continuous function of D. Were we to apply such functions, the whole-station discretisation would perform safely. However, it is also known that the half-station method works with smoothly varying functions of D, and in general, the errors associated with the half-station method are always smaller than those of the whole-station method . It is also unjustifiable to impose any such constraint on the nature of variations of elastic thickness of the lithosphere. Consequently, it appears clear that the finite-difference discretisation of the flexure equation should be carried out using the half-station method.

The wider application of the half-station method to many other differential equations with variable coefficients was originally noted by . The specific application of it to the one-dimensional wave equation and the general rarity of its use have also been discussed by p. 44. It seems clear that this form of finite-difference discretisation, which allows arbitrary piecewise variations of coefficients, is a potentially significant and generally overlooked method for the wider spectrum of the physical sciences.

The use of numerical solutions for the flexure equation covers many aspects of geodynamics. On the one hand, the determination of the elastic thickness of the lithosphere can be done using a forward modelling approach with flexure models used to match gravity data and topography e.g.. In such cases it may well be necessary to look for solutions incorporating variable elastic thickness, especially around mountain fronts in foreland basins. Flexure models may also be used to study the dynamics of past flexural events for which they are often used to model subsidence patterns and explain basin formation. In many of these cases too, variable elastic thickness is likely to need taking account of. Increasingly, topics relating to global sea level rise and the melting of the polar ice caps will demand high-resolution models of flexural responses, which may require taking account of changes in elastic thickness of the lithosphere. More generally, the issue of elastic breaks within continental lithosphere has yet to be substantially explored and could have significant consequences for topics such as intraplate seismicity and seismic hazard. In short, it seems very important to make such numerical approximations in as accurate a way as possible. Current flexure models are based on the whole-station (product rule) derivation of the numerical scheme and should be revised.

Despite a long history of use in the literature and an apparent sense of being work completed, in fact a host of problems arising from a simple numerical analysis of the discretised flexure equation have remained untouched. When we examine these, we find there are significant issues with the method of discretisation used. It is not advisable under any circumstances to use a product rule derivation of an equation of this type when the coefficient varies as a function of coordinate. Realistic models of natural variations in elastic thickness (and many other coefficients in many other equations arising in natural sciences in general) will require sharp changes in those coefficients to be taken account of. A product rule scheme cannot do this successfully, especially for fourth-order differential equations.

Fundamental problems relating to the nature of the system of linear equations arising from discretisations of the flexure equation have also gone unnoticed so far. For small grid spacings, something which will inevitably become increasingly common as computer power increases, the numerical solution will rapidly become unstable and fail unless a 128 bit floating-point representation is employed. If this is used, the problem will probably be avoided at grid spacings ≥1 m, but below this threshold, instability could once more arise quite easily. Although we have presented only one-dimensional problems in this paper, it is nevertheless clear that everything shown here extends to two-dimensional, thin elastic plate formulations as well. To this end, we give the two-dimensional half-station formulation and discretisation of the problem (Appendix B). We will discuss two-dimensional solutions using this scheme in forthcoming papers.