Stable recovery of coefficients in an inverse fault friction problem

We consider the inverse fault friction problem of determining the friction coefficient in the Tresca friction model, which can be formulated as an inverse problem for differential inequalities. We show that the measurements of elastic waves during a rupture uniquely determine the friction coefficient at the rupture surface with explicit stability estimates.


Introduction
The study of earthquake physics remains highly challenging through its complex dynamics and multifaceted nature.Nearly all aspects of earthquake ruptures are controlled by the friction along a fault, where these commonly occur, that progressively increases with tectonic forcing.Indeed, in a recent Annual Review of Earth and Planetary Sciences, it was stated that "determining the friction during an earthquake is required to understand when and where earthquakes occur" (Brodsky et al. [2]).Some common approach has been developed retrieving the stress evolution at each point of the fault as dictated by the slip history obtained from the kinematic inverse rupture problem; we mention work by Ide and Takeo [13], who determined the spatiotemporal slip distribution on an assumed fault plane of the 1995 Kobe earthquake by "waveform inversion" and then numerically solved the elastodynamic equations to determine the stress distribution and constitutive relations on the fault plane.However, seismologists studying earthquake dynamics have reported that both stress and friction on a fault are still poorly known and difficult to constrain with observations (Causse, Dalguer and Mai [4]).Here, we address the question whether this is possible, in principle.
We study the recovery of a time-and space-dependent friction coefficient via the slip rate and normal and tangential stresses, using the Tresca model (see e.g. the book of Sofonea and Matei [20]), at a pre-existing fault from "near-surface" elastic-wave, that is, seismic displacement data.This dynamic inverse friction problem can be regarded as an inverse problem for differential inequalities, as the Tresca friction model can be formulated through variational inequalities as seen in many contact mechanics problems (e.g.[6,20]).While inverse problems for differential equations have been widely studied, inverse problems for differential inequalities have not yet received much attention.Our approach is based on the quantitative unique continuation for the elastic wave equation established in our recent work [5], where we studied the kinematic inverse rupture problem of determining the friction force at the rupture surface from seismic displacement data.Itou and Kashiwabara [14] recently analyzed the Tresca model on a fault coupled to the elastic wave equation; we exploit their results in our study of the inverse problem.We also mention recent work by Hirano and Itou [11] on deriving an analytical solution to the slip rate distribution of self-similar rupture growth under a distance-weakening friction model.As a disclaimer, while we address the most fundamental question, we do ignore more complex physics such as thermo-mechanical effects.
We remark that in the past two decades, quite many studies have been devoted to the practical determination of fault frictional properties by analyzing slowly-evolving afterslip following large earthquakes.For very recent results, see Zhao and Yue [23].Afterslip is the fault slip process in response to an instantaneous coseismic stress change, in which the slip velocity decrease corresponds to the stress releasing by itself.Its "self-driven" nature provides a framework to model the slip process with the fault frictional properties alone.For a review, see Yue et al. [22].Afterslip is analyzed with quasi-static deformation, that is, with the elastostatic system of equations, typically using geodetic data.
Let M ⊂ R 3 be a bounded connected open set with smooth boundary, modelling the solid Earth.Let Σ f ⊂ M be a connected orientable embedded smooth surface with nonempty smooth boundary satisfying Σ f ∩ ∂M = ∅, modelling the rupture surface.Consider the elastic wave equation ( 1) with the Tresca friction condition (e.g.[6,14,15]) on the rupture surface Σ f : (2) Here σ n = (σ(u)n) • n and σ τ = σ(u)n − σ n n are the normal and tangential components of the stress tensor σ(u), where n is the unit normal vector of the rupture surface Σ f .The stress tensor σ(u) is defined as The notation [ • ] + − stands for the jump across the rupture surface, more precisely, where u τ is the tangential component of u, with respect to the unit normal vector field n smoothly extended to a neighborhood of Σ f .The friction force g at Σ f is of the form where F > 0 is the friction coefficient.Note that the friction force g and the friction coefficient F may depend on time.
Regarding the direct problem for the Tresca friction model above, the weak formulation is understood in the variational sense (see e.g.[14,6,20]).Let Recall that with the Dirichlet condition on the boundary ∂M , the problem of finding u satisfying the Tresca friction model (1-2) is formulated as finding u such that for all v ∈ H 1 (V 0 ) and all t ∈ (0, T ), the following variational inequality holds: where dV, dA stand for the volume and area element of M and Σ f , respectively, and a is a bilinear symmetric form defined by a(v, w) := M \Σ f tr σ(v) T ε(w) dV .If one assumes and the compatibility conditions at t = 0: then the friction problem above has a unique solution [14], In this paper, we consider the inverse problem of determining the friction coefficient F in the Tresca friction model above.For the sake of presentation, we consider the elastic wave equation (1) on the time interval (−T, T ).Let M ⊂ R 3 be a bounded connected open set with smooth boundary, modelling the solid Earth.Let Σ f ⊂ M be a connected orientable embedded smooth surface with nonempty smooth boundary, modelling the rupture surface.Denote the a priori bound for the norm of the elastic wave u in the space (7) by (8) ∥u∥ W 1,∞ (−T,T ;H We impose the following assumption on the regularity of the normal stress F n and the friction coefficient In addition, assume that the parameters ρ, λ, µ in the elastic wave equation (1) to be smooth and time-independent on M \ Σ f .
We prove the following result on the inverse fault friction problem during a rupture.
for some constant c 1 > 0 and a priori norm (8) for the elastic wave u.Suppose that we are given the elastic wave u on an interior open set U satisfying U ⊂ M \ Σ f up to sufficiently large time T .Then we have the following conclusions.
(1) The friction coefficient (2) Suppose that we have two systems with friction coefficients F 1 , F 2 , and we are given the corresponding elastic waves u 1 , u 2 on U × [−T, T ].Then there exist constants ε 0 , C, c > 0 such that for any 0 < ε 0 < ε 0 , if then the friction coefficients satisfy , where the constants ε 0 , C depend on c 0 , C 0 , Λ 0 , c 1 , T , parameters of the elastic wave equation and M, U, Σ f , and c is an absolute constant.
We remark that one could also formulate the result assuming a lower bound for [ ∂ t u τ ] + − on a subset of Σ f and then recovers the friction coefficient F in the same subset.Theoretically speaking, our method can also work with measurements of the elastic waves on an open subset of the boundary ∂M if additional regularity of the elastic waves on ∂M can be assumed, see [5,Remark 1] or [3,Theorem 3]; however, this additional boundary regularity required is not provided by the regularity class (7).
The proof of Theorem 1 is divided into two parts: the measurements of u on U determine u near Σ f , and the latter determines F under the Tresca friction condition.The first part, also known as the kinematic inverse rupture problem [10], has mostly been done in our recent work [5].Our method was based on the quantitative unique continuation for the elasticity system, motivated by [21,7].However, regularity issues remain: the actual regularity of waves is not enough for the quantitative unique continuation arguments to work on the whole domain, which is addressed in Section 2. The second part is discussed in Section 3.
Acknowledgement.The authors thank Erik Burman for pointing out the reference on the regularity of the direct problem, and thank the anonymous referee for pointing out a mistake in an earlier version of the paper.MVdH was supported by the Simons Foundation under the MATH + X program, the National Science Foundation under grant DMS-2108175, and the corporate members of the Geo-Mathematical Imaging Group at Rice University.M.L. and J.L. were supported by the PDE-Inverse project of the European Research Council of the European Union and the Research Council of Finland, grants 273979 and 284715.L.O. was supported by the European Research Council of the European Union, grant 101086697 (LoCal), and the Research Council of Finland, grants 347715 and 353096.Views and opinions expressed are those of the authors only and do not necessarily reflect those of the European Union or the other funding organizations.

Interior regularity
Let N ⊂ R 3 be a bounded connected open set with smooth boundary.Let δ > 0 be small, and denote the interior by (10) N δ := {x ∈ N : d(x, ∂N ) ≥ δ}.
Let u be an elastic wave satisfying the elastic wave equation (1) on the whole set N (without the presence of a rupture surface).Recall [7, Lemma 5.1] that the elastic wave equation can be decomposed into a system of hyperbolic equations for (u, div u, curl u).Let U ⊂ N be a connected open subset, and we choose δ sufficiently small such that U ⊂ N δ .We apply Proposition 3.2 in [5] to (u, div u, curl u) in N δ : for sufficiently small h and sufficiently large T (specified in [5 where n = dim(N ) = 3, and Observe that Λ δ essentially asks for H 2 regularity, while the solution of the original direct problem (7) is only in H 1 in space.One way to resolve the issue is to use interior regularity estimate, and then use Sobolev embedding to get an immediate estimate for the boundary layer.
We assume that the elastic wave u in N is in the following energy class, The following H 2 interior regularity estimate holds for the elastic wave equation which gives a bound where C T depends only on T , and C N depends only on geometric parameters of N .
Proof.Suppose that φ is a weak solution of the elliptic equation on N , Let χ be a cut-off function satisfying χ| N δ = 1, χ| ∂N = 0 and ∥χ∥ C 2 (N ) ≤ Cδ −2 .Then Hence φ 0 ∈ H 2 (N ) by boundary regularity for elliptic equations (e.g.[9, Theorem 4 in Chapter 6.3]).Then it follows that φ| N δ ∈ H 2 (N δ ), and ( 15) The constant C N depends on the first Dirichlet eigenvalue on N which is uniformly bounded below by diameter and curvature bounds (e.g.[16,Theorem 8]).The same argument is valid Now we switch to the notations in our first paper [5], Recall that the elastic wave equation ( 1) can be decomposed into the following system ([7, Lemma 5.1] or [5, Lemma A.1]): where A i are first order and has no time derivative.Then consider the second equation.Since On the other hand, since u ∈ W 2,∞ (−T, T ; L 2 (N )) from ( 13), then Thus from the interior regularity argument (15) (for L 2 -H −1 ), we have with constant C N δ −2 in the regularity estimate.
Next consider the first equation on N δ .Using the improved interior regularity for v| N δ , we see that Thus by shrinking the domain by another δ and using (15), we obtain with constant C N δ −4 in the regularity estimate, i.e., (18) Thus from ( 18) and ( 13), we obtain the estimate , where the constants ε 0 , C depend on Λ 0 , T , parameters of the elastic wave equation and geometric parameters of N , and c is an absolute constant.
Proof.We have already estimated the L 2 -norm of u on N δ × [−T /2, T /2] from (11) and Lemma 2.1.The L 2 -norm on the remaining part follows from the Sobolev embedding theorem.Namely, consider Apply the Sobolev embedding theorem (e.g.Theorem 4.12 in [1]) to the space N × [−T, T ] of dimension n + 1 = 4, which satisfies the uniform cone condition (Definition 4.8 in [1]), Then, .
Hence, we have From here, we need to balance the parameters δ, h, ε 0 .Choose δ = h such that the three terms on the right-hand side of ( 19) are equal, and we get (20) δ for some constant c depending only on n = 3, and for some constant C independent of h.The condition h < h 0 gives the choice for ε 0 : Inserting the choice of δ, h back into (19) gives the desired form of the estimate.□

Inverse friction problem
Using Proposition 2.2, we consider the inverse problem of determining the friction coefficient F in the Tresca friction model.As Proposition 2.2 is formulated without the presence of the rupture surface Σ f , we apply the result to a smooth manifold whose boundary extends Σ f .Of course the choice of such a manifold can be arbitrary and in this paper we use a type as illustrated in Figure 2.
where ε 1 := C(log | log ε 0 |) −c .The same estimate also holds for the components σ n , σ τ .The constant C depends on Λ 0 , T , parameters of the elastic wave equation and M, U, Σ f , and c is an absolute constant.
Proof.First, we construct a connected open set N ⊂ M with smooth boundary such that U ⊂ N and Σ f ⊂ ∂N .Let n be a smooth unit normal vector field on Σ f , and consider the ray z + rn, r ≥ 0, from a point z ∈ Σ f .Since Σ f is smoothly embedded in R 3 , there exists sufficiently small r 0 > 0 such that the nearest point of z + rn in Σ f is z for all r ∈ [0, r 0 ] and all z ∈ Σ f .This gives a smooth embedding f : Note that r 0 is chosen such that the closure of the image of f does not intersect with ∂M or U .The boundary of the image f (Σ f × [0, r 0 ]) extends Σ f , and can be smoothened over a neighborhood of ∂Σ f without intersecting ∂M or U .This construct a compact smooth manifolds D ⊂ M with smooth boundary satisfying Σ f ⊂ ∂D, see Figure 2. Note that the complement M \D is connected due to ∂Σ f ̸ = ∅.Moreover, the manifold D can be constructed such that the curvature tensor of ∂D and its covariant derivatives are bounded depending on r 0 , d(Σ f , ∂M ), d(Σ f , U ) and geometric parameters of Σ f .Hence Proposition 2.2 is applicable to the connected open set N := M \ D with smooth boundary ∂N = ∂M ∪ ∂D.(8).By interpolation, we have (22) ∥u∥ Recall the system ( 17) for (u, v, w) := (u, div u, curl u).Then ( 23) To proceed further, we recall the H (k,s) -norm in R n+1 (see [12,Definition B.1.10])defined as with respect to the coordinates y = (x ′ , x n ) ∈ R n × R. Note that when s = 0, the H (k,0)norm above is equivalent to the usual H k -norm.The idea is using partial hypoellipticity (e.g.[12,Appendix B] or [8,Chapter 26.1]) to trade regularity between normal and tangential components.
Since the H (k,s) -norm was defined above for functions on R n+1 , we apply the technique to local coordinates and patch up using partition of unity in the standard way.Let {B k } N k=1 be a finite open cover of a small neighborhood of Σ f in N , and let {χ k } N k=1 be a partition of unity subordinate to the open cover.Setting the diameter of each open set B k smaller than the injectivity radius of N , one can work in the boundary normal coordinate (y 1 , y 2 , y 3 ) ∈ R 3 of N , where y 1 = d(y, ∂N ) is the coordinate normal to ∂N .Let Φ k : B k → R 3 y 1 ≥0 be the smooth (boundary normal) coordinate function on each B k such that Φ k maps Σ f ∩ B k to an open set of {(y 1 , y 2 , y 3 ) ∈ R 3 : y 1 = 0}.Using the notation (24), we write v, w ∈ H (−α,0) from ( 23) and thus , where P 2 = ρ 2µ+λ ∂ 2 t − ∆ by the second equation in (17).We denote by P 2,k the push-forward of the operator In the coordinate y = (y 1 , y 2 , y 3 ) of Φ k (B k ) ⊂ R 3 y 1 ≥0 , the operator P 2,k has the standard form where the constants C depend on the C 2 -norms of Φ k , Φ −1 k , χ k , which, in turn, depend on the geometry of Σ f .Note that the commutator [P 2 , χ k ]v in the above is first order in v.
Let 17), and denote by P 1,k the push-forward of the operator just obtained above, we have where the constants C depend on the C 1 -norms of Φ k , Φ −1 k , χ k .Note that we have used the fact that A 1 is a first-order linear operator in u, v. Using [12, Theorem B.2.9] again to and Changing back to the original Euclidean coordinate, we have and thus with the same norm estimate as above.Hence, Then the regularity for the stress tensor σ(u) on Σ f immediately follows from the definition (3).Contracting the stress tensor σ(u) gives the same estimate for the components σ n , σ τ .□ To prove our main result we need the following lemma.
Denote by T ρ f = ρ • f the composition operator.Then T ρ is a continuous operator from H 1 (Ω; R m ) to H 1 (Ω; R m ), and for all j, where the function ρ j : R m → R is the component of ρ, namely, According to [18,Theorem 1], this follows after showing that ρ j is locally Lipschitz and that there is a uniform constant C > 0 such that almost everywhere in R m , for any j = 1, • • • , m.
For |ξ| ̸ = c, the function ρ j is smooth and it is straightforward to check that Hence it suffices to verify that ρ j is Lipschitz across |ξ| = c.Let |ξ| ≤ c and |η| ≥ c.Then and This shows that ρ j is Lipschitz on R m with uniform Lipschitz constant 2/c, for all j = 1, • • • , m.Moreover, [18,Theorem 1] gives an estimate for the norm For the continuity of T ρ , it is necessary to use the specific form of the function ρ in (26), as the continuity does not hold for all composition operators in general, see e.g.[19,Section 1].In our case, for ξ It is clear that h is uniformly Lipschitz on R m as ρ is uniformly Lipschitz on R m .We apply [19,Example 3.1] to the Lipschitz function h with S = {ξ ∈ R m : |ξ| ≤ c}, the closed ball of radius c.Since h is identically 0 in S and h is smooth in the complement of S, it is enough to verify that, for every ξ 0 ∈ ∂S, (30) lim ξ→ξ 0 ξ / ∈∂S ∇h(ξ) • τ = 0, for all τ ∈ T ∂S (ξ 0 ), where T ∂S (ξ 0 ) denotes the tangent space of ∂S = {ξ ∈ R m : |ξ| = c} at ξ 0 ∈ ∂S.For ξ ∈ S \ ∂S, we see ∇h(ξ) = 0 since h is identically 0 in S. On the complement R m \ S, since h| R m \S can be smoothly extended to a function h(ξ) := ξ/|ξ| − ξ/c on a neighborhood of ∂S, we have lim In the above, the fact that ∇ h(ξ 0 ) • τ = 0 is due to the definition of tangent vector, namely, by differentiating ( h j • γ)(s) at s = 0, where h j , j = 1, • • • , m, are the components of h and γ(s) is a smooth curve in ∂S with the initial vector γ ′ (0) = τ ∈ T ∂S (ξ 0 ).Then the claim immediately follows since h j • γ = 0 due to h = 0 on ∂S.This verifies (30), and then [19,Example 3.1] gives the continuity of T h and thus the continuity of T ρ .□ Now we prove Theorem 1.
Proof of Theorem 1.Since [ i.e., the fault is slipping everywhere, it follows from ( 9) that (31) On the other hand, the friction condition (2) implies that (32) where N, D are the manifolds with smooth boundary constructed in the proof of Lemma 3.1, see Figure 2. Let us denote As u + ∈ E(N ) by ( 7), we can extend u + to a small neighborhood N of N in M so that the extension of u + is in the same regularity class E( N ).Similarly, we can extend u − to a small neighborhood D of D in M so that the extension of u − is in the same regularity class E( D).
Let V be a small open neighborhood of Σ f in M .As Σ f ⊂ ∂D, in this way, the functions u + , u − are both extended to V and the extensions are in E(V ).Then and thus the tangential component of the difference Recall the definition of tangential component near Σ f in (4) where the unit normal vector field n is smoothly extended to a neighborhood of Σ f .By Lemma 3.2, the function Denote the composition operator T ρ f := ρ • f , where ρ : R 3 → R 3 is the Lipschitz function with uniform Lipschitz constant 4/c 1 defined in (26), taking c = c 1 .In the above we have shown that tr(T ρ f ) ∈ H 1/2 (Σ f × [−T, T ]), where tr stands for the trace operator onto Σ f .Now we show that tr(T and by the continuity of the composition operator T ρ from Lemma 3.2, Since T ρ (trf k ) = tr(T ρ f k ) for smooth functions, we see that T with an estimate of the norm depending on Λ 0 , C 0 , c 1 and Σ f .
a bounded connected open set with smooth boundary and U ⊂ N be a connected open set.Let u be a solution of the elastic wave equation (1) in N × [−T, T ] with a priori norm(13).Then there exist constants ε 0 , C, c > 0 such that for any 0

Figure 2 .
Figure 2. Applying the quantitative unique continuation to the connected open set N = M \ D with smooth boundary, where D is a compact smooth manifold whose boundary extends Σ f .

Lemma 3 . 2 .
Let c > 0 be a constant and let Ω ⊂ R n , n > 2, be a bounded open set.Suppose f ∈ H 1 (Ω; R m ) with value in R m , m ≥ 1.Let ρ : R m → R m be defined by