Stress–strain characterization of seismic source ﬁelds using moment measures of mechanism complexity

39-yr focal mechanism catalogue of the San Jacinto Fault (SJF) zone and to realizations from the Graves– Pitarka stochastic rupture model. The SJF data are consistent with the SSC model, and the recovered parameters, R = 0 . 45 ± 0 . 050 and κ = 5 . 7 ± 1 . 75, indicate moderate mechanism complexity. The parameters from the Graves–Pitarka realizations, R = 0 . 49 ± 0 . 005 , κ = 9 . 5 ± 0 . 375 , imply lower mechanism complexity than the SJF catalogue, and their moment measures show inconsistencies with the SSC model that can be explained by differences in the modelling assumptions.

response to stress and, together with the stress shape parameter, determines the mechanism complexity. Inversion of the moment measures to estimate these quantities is what we mean by the stress-strain characterization (SSC) of a seismic source field.

Organization
The SSC method developed in this paper can be applied to arbitrary superposition of seismic sources, including large earthquakes as well as extended source fields comprising seismic clusters and regional seismicity. Section 2 sets up the notation of deviatoric moment-tensor fields that represent individual earthquakes and catalogueed sequences. Section 3 describes the representation of these fields by basis sets of orthogonal mechanisms and reviews the total-moment simplex defined by a basis set.
Section 4 develops a probabilistic model of mechanism complexity in which the elementary stress-aligned DCs of arbitrary orientation are exponentially distributed with the shear-traction magnitude. The strain-sensitivity factor scales the normalized traction; increasing this sensitivity concentrates the energy release into optimal mechanisms and decreases mechanism complexity.
In Section 5, a stress-oriented basis set is constructed by requiring the CMT mechanism to be its leading member. Under the SSC assumptions, this tensor defines the stress-oriented reference frame. We then select the other basis members to represent orthogonal rotations about the CMT axes. We solve the forward problem by deriving integral expressions for the expected total-moment fractions of the stressoriented basis, and we set up the inverse problem by showing how these measures of mechanism complexity depend on the differential stress ratio and the strain-sensitivity factor.
In Section 6, we construct a likelihood function for the SSC parameters that accounts for the covariances among the observations. We apply the SSC methodology to a 39-yr moment-tensor catalogue of the San Jacinto Fault (SJF) zone of Southern California. We obtain maximum-likelihood estimates of the stress parameters consistent with previous studies, as well as novel estimates of mechanism complexity and the strain sensitivity to stress implied by this complexity. We also use the moment measures to compare the SSC model with the Graves & Pitarka (2016) stochastic rupture model, highlighting the statistical differences between the two models.

M O M E N T -T E N S O R F I E L D S
A moment-tensor field is a space-time distribution of second-order, symmetric tensors used in the kinematic representation of seismic sources (Backus & Mulcahy 1976a, b). At every space-time point (r, t) in the finite space-time source volume V = R × T, we express the moment tensor field as a non-negative scalar-moment density times a source-mechanism density, m (r, t) = m(r, t)m (r, t). The scalar moment density is the Frobenius (Euclidean) norm of moment-tensor field, m = m = √ m : m = √ m i j m i j , and the local source mechanismm is a symmetric, second-order tensor with unit norm,m :m = 1.
The space of all moment tensors M can be mapped to the 6-D Euclidean vector space R 6 by a norm-preserving isomorphism, which allows us to treat mechanisms as unit vectors in R 6 (Silver & Jordan 1982). The most general source field includes an isotropic component, for example from crack opening (Matsu'ura et al. 2019), but here we assume the field is purely deviatoric, tr [m] = 0, which reduces M to the five-dimensional space of deviatoric moment tensors, denoted M D . Deviatoric mechanisms are points on the unit hypersphere in M D .
The total moment of any moment-tensor field is defined to be the integral of the scalar moment density (Jordan & Juarez 2019), For an elastic dislocation field, m(r, t) is proportional to the local strain energy drop, where the coefficient of proportionality is the ratio of the average shear driving stress to the shear modulus (Kostrov 1974;Dahlen 1977;Matsu'ura et al. 2019). It has been observed that the average apparent stress is independent of earthquake size (McGarr 1999;Ide & Beroza 2001), so M T is expected to scale with the total change in the internal elastic energy of the source region, denoted E T . We consider two types of moment-tensor fields. The first is proportional to the stress-glut density˙ (r, t) of an earthquake rupture (Backus & Mulcahy 1976a, b), We note that the √ 2 factor in eq. (2) arises from Aki's (1966) original definition of seismic moment, which pegs the scalar density of the stress glut at √ 2m(r, t). In our notation, the moment tensor defined by the zeroth moment of the stress glut-the usual seismic moment tensor (Aki & Richards 2002)-is therefore √ 2 times the net moment tensor M 0 . The centroid (r 0 , t 0 ) of a moment tensor field is the space-time point that minimizes the norm of the first polynomial moment, which is a third-order tensor (Backus 1977), The CMT is the monopole approximation of m that concentrates the energy release at the source centroid (Dziewonski et al. 1981): m(r, t) ≈ M 0 δ(r − r 0 )δ(t − t 0 ), where δ(·) is the Dirac delta function. We refer to the normalized moment tensor,M 0 = M 0 /M 0 , as the CMT source mechanism. The CMT mechanism is the moment-weighted mean of the local source mechanismsm (r, t). Accurate estimations of the CMT are now available at low magnitude thresholds both globally (Ekström et al. 2012) and regionally (e.g. Hutton et al. 2010;Godey et al. 2013;Ross et al. 2017;Aoi et al. 2020).
The second type of moment-tensor field is a discrete distribution of seismic events within a region R and time interval T, such as the seismicity of a deformation zone (e.g. Terakawa & Matsu'ura 2008;Bailey et al. 2010), an aftershock sequence (e.g. Beroza & Zoback 1993;Shearer et al. 2003;Hardebeck 2020), or a multiple CMT representation of a single earthquake (e.g. Tsai et al. 2005;Mai & Thingbaijam 2014). We assume the individual events are small enough and simple enough to be represented by deviatoric CMTs, so that the field comprises N point sources with moment tensors m n ∈ M D , seismic moments m n , and source mechanismsm n : In this case, the discretized field is represented by a CMT catalogue C = {r n , t n , m n ,m n : n = 1, . . . , N }, and the moment tensor of eq.
(3) reduces to a Kostrov summation, The Aki moment of this discrete field, M 0 = N n=1 (M 0 :m n )m n , thus scales with the net inelastic strain (Kostrov 1974), whereas its total moment, M T = N n=1 m n , scales with the total energy release E T .

M O M E N T M E A S U R E S O F M E C H A N I S M C O M P L E X I T Y
Any deviatoric moment-tensor field m(r, t) ∈ M D can be represented as a linear combination of up to five mutually orthogonal source mechanisms. The minimum number of mechanisms required to support the moment-tensor density equals the rank of the field, denoted D. It will be convenient to define the reduced dimension d = D − 1; for example a deviatoric source mechanismm is a vector on the d-sphere, which is the unit hypersphere in M D (Silver & Jordan 1982). In this section, we allow the dimensionality to be variable, but in our theoretical development of the SSC model, we will focus on the general deviatoric case, where D = 5 and d = 4. We choose the CMT mechanismM 0 to be the leading (zeroth-degree) basis member, and we complete the basis by defining d orthogonal mechanisms {M α } indexed by a degree number α = 1, . . . , d. The complete basis set, representing the D degrees of moment-tensor freedom, are normalized to unit length, Hence, the basis projections of the local source mechanismm define the direction-cosine fields, which give the coordinates ofm on the d-sphere. The moment-tensor density can be written as the sum over the basis members, The particular (stress-oriented) basis set used in this paper for the SSC of moment-tensor fields is described in Section 5. The squares of the direction cosines form a d-simplex; that is they are D = d + 1 positive numbers that sum to unity at each space-time point, Downloaded from https://academic.oup.com/gji/article/227/1/591/6293847 by guest on 16 July 2021 The substitution of (11) into (1) decomposes the total moment into D fractional moments, Mα.
The fractional moment of degree α is the integral of the scalar moment density weighted by the square of the direction cosine, The partitioning of the scalar density m(r, t) into moment fractions by eq. (13) depends on the local source orientationm but not m itself. This choice is consistent with our interpretation that M T scales with the total elastic energy change E T ; that is, all elementary sources add to E T in proportion to their scalar seismic moments, regardless of their orientation. An alternative is quadratic (2-norm) moment partitioning, in which the integral of |M α : m(r, t)| 2 defines the square of the degree-specific moment. Combining moments quadratically is inconsistent with the additive energy interpretation, however. Therefore, we adopt the simplex (1-norm) measure given by eq. (13).
Normalizing the expressions (12) and (13) by M T defines the total-moment simplex (Jordan & Juarez 2019), Fα is the total-moment fraction of degree α, which we associate with the fraction of the seismic energy released with mechanismM α . If the mechanismm is constant throughout the source volume, F0 = 1 and Fα = 0 for α ≥ 1. A value of F0 close to unity thus indicates a simple mechanism. By the same token, its complement, measures mechanism complexity. In Section 5.4, we will use the 1-simplex ( F0, F+) to set up a mechanism complexity scale.
Another gauge of mechanism simplicity is the ratio of the Aki moment to the total moment, F 0 := M 0 /M T . The moment measures F0 and F 0 are closely related quantities, but they are not equal. The Aki moment M 0 is the integral of the scalar moment density weighted by the zeroth-degree direction cosine, rather than by the square of this cosine, If the mechanism complexity is small ( F+ 1), then the expected value of cos θ 0 will be close to unity, and an application of the small-angle approximation, θ 0 1, in (15) and (16) shows that As the mechanism complexity increases incrementally from zero, the Aki moment fraction F 0 decreases linearly with F+, but at only half the rate of F0. Von Mises distributions with a single degree of rotational freedom have been used to investigate the expected values of the moment fractions over the entire range of mechanism complexity, 0 < F+ < 1 (Jordan & Juarez 2019, Section 6). Here, we extend the analysis to a statistical model of stress-aligned mechanism complexity with two degrees of rotational freedom in which the expected value is governed by the magnitude of the resolved shear traction.

S T R E S S -S T R A I N C H A R A C T E R I Z AT I O N O F M O M E N T -T E N S O R F I E L D S
We develop an SSC model based on three simplifying assumptions: 1. Shear tractions on surfaces within the source volume are specified by a constant deviatoric stress tensor . Local variations in stress are ignored.
2. The slip vectors of fault ruptures within the source volume are parallel to the shear traction exerted by across the fault. Perturbations to the stress field due to fault displacements are ignored.
3. Slip can occur on any fault plane, but more slip is likely if the shear traction across a fault plane is higher. Dependence of the slip on fault friction and other properties is ignored.
The SSC model is parametrized by the normalized (reduced) stress tensorˆ , which orients the principal-axis reference frame and specifies the differential stress ratio (shape factor) R, and by a strain-sensitivity factor κ, which governs the dependence of the seismic moment density on the shear-traction magnitude and therefore parametrizes the inelastic strain response to stress. We show that the SSC model predicts the Aki moment fraction F 0 and the total-moment simplex { Fα : α = 0, . . . , d} as functions of the stress-shape parameter R and the seismic strain-response parameter κ.

Stress-aligned double couple
Inversion of seismicity for stress is based on the widely used concept of a stress-aligned DC (Wallace 1951;Bott 1959;McKenzie 1969;Angelier et al. 1982;Gephart & Forsyth 1984;Michael 1984). The moment-tensor field m(r, t) of an extended seismic source is assumed to be a space-time distribution of elementary dislocations, each described by a DC mechanism with unit fault-normal vectorn and unit slip vectorŝ: This purely deviatoric field is said to be fully aligned with the tectonic stress if the slip vectorsŝ are everywhere parallel to the maximum shear tractions acting across the elementary fault planes, regardless of the fault orientationn. In structural geology, the alignment of fault slip with the maximum shear traction is known as the Wallace-Bott hypothesis (Dupin et al. 1993;Célérier et al. 2012;Lisle 2013;Lejri et al. 2017), named after the first authors to apply the notion to faulting in a homogeneous medium (Wallace 1951) and to reactivated fault systems (Bott 1959). We follow this usage by referring to a moment-tensor field in which slip is everywhere stress-aligned as a Wallace-Bott field. The standard derivation of a stress-aligned DC (e.g. Angelier et al. 1982) is recast here to establish our frame-invariant tensor notation. The deviatoric stress tensor is represented in its principal-axis frame = (r 1 ,r 2 ,r 3 ), The principal compressive stresses sum to zero, σ 1 + σ 2 + σ 3 = 0; they are ordered, σ 1 ≥ σ 2 ≥ σ 3 , and interrelated by the differential stress ratio, The shape parameter R is used by many authors (e.g. Gephart & Forsyth 1984;Hardebeck & Michael 2006;Célérier et al. 2012), although its alternative, φ = 1 − R, is also popular (e.g. Angelier et al. 1982;Lislie 2013). The reduced stress tensor,ˆ = Σ −1 , is normalized to unit length by the root-sum-square of the principal stresses, The vector normal to the fault element has principal-axis coordinatesn i =r i ·n,wheren 2 1 +n 2 2 +n 2 3 = 1. The traction, τ (n) := ·n = σ 1n1r1 + σ 2n2r2 + σ 3n3r3 , can be resolved into normal and shear components, The normal traction is given by the quadratic form, and the shear traction by the projection of τ orthogonal ton, The squared norm of the shear traction across the plane with normal vectorn is thus the difference of two quadratic forms (Angelier et al. 1982), The direction of the stress-aligned slip vector isŝ (n) = s −1 (n)s(n). The corresponding stress-aligned DC mechanism, expressed in the form (18), is the unit symmetric tensor, Downloaded from https://academic.oup.com/gji/article/227/1/591/6293847 by guest on 16 July 2021 The Wallace-Bott mechanism given by (26) is the DC that maximizes the shear traction s on a fault with normaln. Some algebra shows that the inner product of this mechanism with the reduced stress tensor is proportional to the shear traction resolved on the fault plane, The shear traction s(n) is plotted on the unit sphere S 1 for selected values of R in Fig. 1. Here we have scaled the stresses such that the average shear stress in the 1-3 plane, which equals the maximum of the shear traction on any plane, is unity for all 0 < R < 1, With this normalization, the principal stresses are completely determined by R, and the shear-traction magnitude and the squared stress modulus become, If 0 < R < 1, the non-negative scalar field s(n) has 18 stationary points on S 1 . It vanishes ifn is a principal axis of ; that is at the 6 points wheren = ±r i . The stress magnitude has eight saddle points atn = 1 √ 2 (±r i ±r 2 ), where i = 1 or 3, and it achieves its maximum value of unity at 4 points that lie in the 1-3 plane, which we label as ±n 0 , ±ŝ 0 . The two orthogonal directions, are the normal and slip vectors of the optimal DC of , defined to be the Wallace-Bott mechanism where the shear traction reaches its maximum value of s 0 = 1: The null axis of the optimal DC isb 0 = −n 0 ×ŝ 0 =r 2 . The end-member cases R = 0 ( σ 1 = σ 2 ) and R = 1 ( σ 2 = σ 3 ) are degenerate. At these limits, the locus where s achieves its maximum value s 0 = 1 dimensionally inflates from four discrete points to two small circles at latitudes of ±π/4 from the stress symmetry plane; s vanishes at the pole and equator of this plane (Fig. 1). This degeneracy leads to a singular behavior at R = 0, 1 for some of the moment measures (see Section 5.3), but these singularities can be handled by defining the stress states at R = 0, 1 to be the limiting states as R → 0, 1.

SSC model
Ifˆ is constant throughout the source volume, the Wallace-Bott mechanism (26) is a function of only its fault-orientation vector; therefore, any Wallace-Bott field is completely specified by a scalar moment density m(r, t) and a vector field of fault orientationsn(r, t), To formulate the SSC model, we represent the spatiotemporal source process and its response to the stress field in terms of probability distributions. The scalar moment density normalized by the total moment is a non-negative function that integrates to unity over the source space-time volume: We can think of this distribution as a probability density function (p.d.f.) on V that defines the moment-weighted expectation of a general source function g(r, t): In particular, the moment-weighted expectation of the source mechanism under f is the product of its Aki moment fraction and its CMT mechanism, Projecting (37) ontoM 0 shows that the Aki moment fraction is the expectation of the zeroth-degree direction cosine, Similarly, the total-moment fractions corresponding to the basis expansion of eq. (10) are the expected values of the squared direction cosines, The fault orientation fieldn(r, t) maps the source volume onto the fault-orientation sphere, V → S 1 . The expectation of any orientation field g(n) under f can thus be expressed as an integral over S 1 , The p.d.f. on S 1 is the moment-weighted density, p s (n) = E s [δ(n)], which is the expectation under f that the fault orientationn points in a particular directionn : This fault-orientation density integrates to unity becausen is somewhere on S 1 : If m is a Wallace-Bott field, then the mapping of the moment density f from V to S 1 via eq. (40) is well-defined and produces a unique density p s . Our SSC model assumes that faults are more likely to slip if the shear traction is high. In particular, we take the fault-orientation density to vary exponentially with s(n): The parameter κ ≥ 0 is the strain-sensitivity factor, which describes the seismic strain response to the tectonic stress. Larger values of κ concentrate p s near the maxima of s and decrease the mechanism complexity factor F+, which, for fixed R, approaches a minimum as κ → ∞. The dependence on s weakens as κ decreases, and the limit κ → 0 gives a uniform distribution of fault-plane orientations, which maximizes F+.
We define an SSC field to be a stochastic Wallace-Bott field with a probability density of fault orientations in the form of eq. (42). Each realization of an SSC field is a deterministic Wallace-Bott field that samples the fault-orientation sphere according to the p.d.f. p s (n). In the limit of dense sampling, the sample average of an orientation field g(n) converges to the expectation integral, where the normalization integral, is a function of R and κ. The scaling with κ is set by the normalization of the maximum shear traction to unity (eq. 28). We plot p s (n) for selected values of R and κ in Fig. 2.

S T R E S S -O R I E N T E D R E P R E S E N TAT I O N
Various types of source-mechanism bases are useful in representing source complexity. Principal component analysis (PCA) identifies the most efficient basis by sequentially maximizing the projection of the residual moment-tensor field onto each successive basis member ( PCA basis is not the CMT mechanism, however, and the norm of its projection is not the Aki moment, though it can be very close for simple sources. In our previous papers (Jordan & Juarez 2019, 2020, we instead focused on moment-oriented representations (MORs) involving the sequential addition of basis members that best account for the low-order polynomial moments of the residual field. The mechanism that maximizes the zeroth-order moment is the CMT monopole (a second-order tensor), and the higher-degree terms are a source dipole (a third-order tensor), a quadrupole (a fourth-order tensor), etc. Extending the basis by higher-moment optimization is an ill-posed orthogonalization process, requiring heavy regularization or strong Bayesian priors ; consequently, the MOR results can be hard to interpret.
Here we introduce an orthonormal basis set of source mechanisms that facilitates the interpretation of the moment measures of Section 3 in terms of tectonic stress and seismic strain response. We fix the zeroth-degree basis member to be the CMT mechanismM 0 . We then add basis members derived from rotations ofM 0 along itsn 0 ,ŝ 0 andb 0 axes, and we complete the basis by including the unique mechanism orthogonal to the other four. The projection of the moment-tensor field onto the completed orthonormal basis set constitutes what we call the stress-oriented representation (SOR) of a moment-tensor field. The SOR recipe can be applied to any moment-tensor field, but the name applies to the special case of an SSC field, where the expected value of the CMT mechanism always equals the reduced stress tensor.

Stress-oriented reference frame
The leading (zeroth-degree) member of the stress-oriented basis is the CMT mechanism. If the source is a realization of an SSC field, then the expected value of its CMT mechanism is proportional to the expectation ofm : Direct evaluation of this integral yieldsM 0 =ˆ , independent of κ. This equality also follows from the fact thatM 0 is the unique source mechanism that maximizes the Aki moment fraction: IfM 0 =ˆ , then eq. (27) implies that Because p s (n) ∝ e κs(n) is a strictly monotonic function of the shear traction, E s [s] maximizes E s [s ] over all linear forms s ∝M :m ; any rotation ofM away fromˆ decreases the integral in (46). The CMT mechanism of an SSC field thus provides estimates of the ordered sequence of eigenvectors = (r 1 ,r 2 ,r 3 ) and the value of R. In this stress-oriented reference frame, it becomeŝ This equality provides a theoretical basis for inferring tectonic stress from observed CMT mechanisms. We note that (49) holds only for Wallace-Bott fields with images p s (n) on S 1 that have the same symmetries as the reduced stressˆ . The expectation of an SSC field always has these symmetries (Fig. 2). A deviatoric mechanism can always be written as a linear combination of a DC and a compensated linear vector dipole (CLVD) with the same set of principal axes (Knopoff & Randall 1970;Riedesel & Jordan 1989;Vavryčuk 2015): M DC 0 is the optimal DC given by eq. (33).M CLVD 0 has anr 3 symmetry axis if R < 1 /2 and anr 1 symmetry axis if R > 1 /2. The CLVD symmetry changes discontinuously at R = 1 /2, where b = 0 (Riedesel & Jordan 1989): The two components are not orthogonal;M DC 0 :M CLVD 0 = √ 3 2 , and the normalization of (50) requires a 2 + √ 3ab + b 2 = 1. Equating (50) with (49) yields The coefficients (a, b) and eigenvalues (λ 3 ) are plotted as functions of R in Fig. 3. The DC component goes to zero as R approaches zero ( σ 1 = σ 2 ) or unity ( σ 2 = σ 3 ) (Fig. 3), whereas the CLVD component vanishes at R = 1 /2 ( σ 2 = 0).

Stress-oriented basis set
Mechanism complexity is a geometrical concept describing how the elementary sources of a moment-tensor field are mis-oriented relative to the mean mechanismM 0 . We seek moment-valued observables that measure mechanism complexity in terms of the seismic strain response Downloaded from https://academic.oup.com/gji/article/227/1/591/6293847 by guest on 16 July 2021 to tectonic stress. To solve this forward problem, we construct a basis set that separates mechanism complexity into moment measures that have clear geometrical interpretations. A physically meaningful decomposition can be achieved from the three orthogonal rotations ofM DC 0 about its DC axes,n 0 ,ŝ 0 andb 0 . Each rotational degree of freedom generates a moment-tensor field of rank D = 2, and each of these three fields can be represented as a linear combination ofM 0 and one other orthogonal mechanismM α (Jordan & Juarez 2019), which we add to our basis set.
In the special case of zero intermediate stress, R = 1 /2, the CMT mechanism is the DC, The matrix components here and elsewhere are in the principal-axis frame . For visualization purposes, we take the null axisb 0 =r 2 to be vertically oriented, so thatM DC 0 represents strike-slip motion on a vertical fault plane; then, rotations about this null axis correspond to variations in fault strike. The rotated mechanism can be expressed as a combination ofM DC 0 and the orthogonal mechanism tensor (Jordan & Juarez 2019), Similarly, the rotations ofM DC 0 about the fault normaln 0 represent rake variations and rotations about the slip vectorŝ 0 represent dip variations. The orthogonal mechanisms that account for these two types of rotation are, respectively, The four basis members (54)-(57) uniquely specify the fifth. For R = 1 /2, the basis is completed by a vertically oriented CLVD, The focal mechanisms of the R = 1 /2 basis set are plotted in the middle column of Fig. 4. In , we applied the MOR and PCA orthogonalization procedures to a single realization of the Graves & Pitarka (2016) stochastic rupture generator and recovered basis sets that approximate the canonical basis for R = 1 /2 shown in Fig. 4. This correspondence reflects the fact that the Graves-Pitarka model for a vertical strike-slip fault is specified in terms of strike, rake and dip rotations. The statistical inconsistencies of the Graves-Pitarka rupture generator with the SSC model are discussed in Section 6.3.
In the general case, 0 < R < 1, the basis members can be divided into two orthogonal subspaces, the 2-D subspace of deviatoric tensors diagonalized in the stress-oriented frame and the 3-D subspace of deviatoric tensors with all zeroes on the diagonal in this frame.

Diagonal moment tensors
and its orthogonal complement, The coefficients a and b are given by (52) and (53). The DC component ofM 4 , like the CLVD component ofM 0 , is discontinuous at R = 1 /2: The covariation ofM 0 andM 4 with R is shown in Figs 3 and 4.

Off-diagonal moment tensors
For 0 < R < 1, this subspace is spanned by the basis members {M 1 ,M 2 ,M 3 }, representing rotations ofM DC 0 about itsb 0 ,n 0 andŝ 0 axes, respectively. BecauseM DC 0 is fixed by the ordered reference frame , our choice of the off-diagonal basis set is independent of R. In the degenerate cases, where two of the principal stresses are equal,M 0 is a CLVD with either a tensional (R = 0) or compressional (R = 1) axis of symmetry (Fig. 4), and any DC that shares this tensional or compressional axis is optimal. Hence, the off-diagonal basis set is only defined within a rotation about the CLVD symmetry axis. This ambiguity is reflected by singularities in the high-degree (α ≥ 2) moment measures at these limits, which are discussed in Section 5.3.
Except in the degenerate cases, the complete stress-oriented basis {M α : α = 0, . . . , 4} is fully determined by the CMT. This feature is important in applications of the model because the CMTs of observed moment-tensor fields are usually well constrained by seismic data.
To represent sources with isotropic components, such as crack opening, the deviatoric basis set can be augmented with a sixth isotropic member,M 5 = 1 √ 3 I. The SSC model assumes that any isotropic moment fraction is negligibly small.
These basis projections are plotted on the fault-orientation sphere in Fig. 5. They display a variety of rotational and reflection symmetries that govern the behavior of the moment integrals. Three have even parity (α = 0, 1, 4), and two have odd parity (α = 2, 3).

Moment measures of zeroth degree
The projections integrate to zero over the fault-orientation sphere, except the zeroth-degree term, cos θ 0 (n), which is proportional to s(n) and everywhere non-negative. The expectation of this field is the Aki moment fraction, F 0 (R, κ) = E s [cos θ 0 ], given by eq. (48). The expectation of its square is the zeroth-degree fraction of the total moment, S 1 e κs(n;R) s 2 (n; R) dn.
The R-κ maps of these moment ratios, plotted in Fig. 6, are bilaterally symmetric about the R = 1 /2 midline. This bilateral symmetry, which characterizes all of the moment measures (e.g. Fig. 7), follows from the rotational and reflection symmetries of the integrands dictated by the SSC model. For example, s(n) for R = 1 /2 − x equals s(n ) for R = 1 /2 + x becausen is a rotation ofn about ther 2 principal axis by an angle π/2 (Fig. 5). The integral is independent of this rigid-body rotation.
When κ = 0, the density p s is a constant on S 1 , the normalization reduces to N = 4π , and F 0 and F0 are proportional to spherical averages of s(n) and s 2 (n), respectively. The square of the shear traction can be integrated analytically, Downloaded from https://academic.oup.com/gji/article/227/1/591/6293847 by guest on 16 July 2021  Substitution into (62) shows that F0(R, 0) equals 2/5 independent of R, consistent with the numerical calculations in Fig. 6(e). Therefore, the maximum mechanism complexity allowed by the SSC model is F+ = 3/5. In this limit of zero strain sensitivity, the Aki moment fraction is a weak function of R. It reaches a minimum value of F 0 = 1/ √ 3 ≈ 0.577 at R = 0,1 and rises by less than 3 per cent to a maximum value of 0.593 at R = 1 /2. Aki moment fractions calculated for finite-fault models F 0 and F0 increase monotonically with κ, so that higher sensitivity to stress magnitude yields simpler mechanism fields. In the special case R = 1 /2, both moment measures converge asymptotically to unity (Fig. 6e), and the mechanism complexity factor F+ goes to zero. At all other values of R, the high-κ limit of F+ is less than unity. Increasing the strain-sensitivity factor concentrates the density function around the maximum values of the shear traction. For 0 < R < 1, these maxima occur at the four points ±n 0 , ±ŝ 0 , and the limiting form of p s is the sum of delta functions at these points. The cases R = 0, 1 are degenerate, where the p.d.f. is delta-distributed on the two small circles at latitudes of ±π/4 from the symmetry plane.
Owing to the exponential sharpening of the p s distribution, care must be taken in computing the expectation integral (43) for large values of κ. Fortunately, in the asymptotic limit κ → ∞, the integrals (48) and (62) can be evaluated analytically. Owing to our stress normalization, s 0 = 1, all maxima have unit values; hence, for any 0 ≤ R ≤ 1 and any q ≥ 0, S 1 e κs(n;R) s q (n; R) dn = 1.
In approaching these limits, the optimal DCs given by the shear-traction maxima share a principal axis with the principal direction of the CLVD. All optimal DCs sharing this axis have the same (optimal) projection onto this CLVD, and their zeroth-degree direction cosines equal the Aki moment fraction of √ 3/4. In this case, the minimum complexity predicted by the SSC model is F+(0, ∞) = 1/4 in the high-κ limit, which compares with a maximum complexity of F+ (0, 0) = 3/5 in the low-κ limit. The non-zero value of the high-κ limit is dictated by the SSC assumption that a CLVD CMT can only arise from a superposition of Wallace-Bott DCs and not from other source types, such as crack-opening dipoles.

Higher-degree moment fractions
The expectations of the squared direction cosines give the total-moment fractions of an SSC field as functions of R and κ, Fα (R, κ) = E s cos 2 θ α = 1 N S 1 e κs(n;R) cos 2 θ α (n; R) dn, α = 0, . . . , 4. This equality holds because cos θ 2 (n) and cos θ 3 (n) are related by a reflection symmetry across ther 1 -r 2 plane. The Wallace-Bott mechanism m (n) is invariant with respect to this reflection, which only interchangesn 0 andŝ 0 . These symmetries are dictated by the physics of the SSC model.  The R-κ maps of the moment fractions display two regimes, low-κ, and high-κ, separated by κ ≈ 3 (Fig. 7). At this value of strain sensitivity, F 0 and F0 are approximately constant in R (Fig. 9b). At κ = 3, the complexity F+ is almost equipartitioned across the four higher-degree moment fractions, which all lie within the narrow range 0.095-0.126 (Fig. 9g).

High-sensitivity regime
In the high-κ regime, the complexity factor F+ decays monotonically as the strain sensitivity increases. Along the R = 1 /2 midline, all of the higher-degree moment fractions Fα≥1 decrease monotonically as a function of κ. Their values go to zero asymptotically, as required by the limit F0 → 1, and their decay rates increase with degree index α. Away from the midline, the fractions F1-F3 decay to zero. In the high-κ limit, the optimal DCs are diagonal in the stress-oriented reference frame and thus completely determined by theirM 0 andM 4 components.
The only nonzero moment fractions are F0 and F4, which are related by The domain of eq. (67) can be extended to R = 0 (and likewise to R = 1) by taking the limit F4 (0, ∞) = lim R→0 F4(R, ∞). We note that the asymptotic value of this two-variable function depends on the order in which the limits are taken, reflecting the fact that the CMT does not define an optimal DC at R = 0, 1 for any finite κ. From eq. (67), we find F4(0, ∞) := lim R→0 lim κ→∞ F4(R, κ) = 1/4, whereas direct computation yields lim κ→∞ F4(0, κ) = 1/8. This discontinuous decrease of F4 at R = 0 is compensated by discontinuous increases in F2 and F3 from 0 to 1/16, which appear as the discontinuous changes in these moment measures between R = 0 and the first positive value at 0.01 in Fig. 9j. The influence of the singularities on the three R-κ manifolds is most clearly seen in Fig. 8f, which compares the R = 0 limits (dotted lines) with the numerically integrated values at R = 0.01 (solid lines). The two sets of curves separate at κ > ∼ 10 2 , where there is an upward inflection in F4 (orange curve in Fig. 8f)  Fα(R, κ). The latter, denoted by Fα(0, ∞), is the R → 0 limit of (67).

Mechanism complexity scale
The mechanism complexity of a moment-tensor field is measured by the complexity factor F+ = 4 α=1 Fα = 1 − F0. Contours of F+ for the SSC model are plotted as a function of R and κ in Fig. 10. The complexity is maximized at κ = 0, where F0 equals 0.4 and Downloaded from https://academic.oup.com/gji/article/227/1/591/6293847 by guest on 16 July 2021 F+ equals 0.6 independently of R. It decreases smoothly to an asymptotic value as κ → ∞, which ranges from zero at R = 1 /2 to 0.25 at R = 0, 1 (eq. 67).
We divide the mechanism complexity scale into three equal intervals, designated as low complexity (0 ≤ F+ < 0.2), moderate complexity (0.2 ≤ F+ < 0.4), and high complexity (0.4 ≤ F+ < 0.6). In the special case R = 1 /2, the category boundaries correspond to κ values of about 3.75 and 9.75, respectively. These categorical choices work well for a couple of reasons: • The low-moderate boundary is a weak function of R and lies close to the transition from the low-sensitivity regime to the high-sensitivity regime. We thus equate the high-complexity region with the low-sensitivity regime in which F0 and F2 = F3 are nearly constants (Figs 7a, c and d).
• To a good approximation, the Aki moment fractions are 90-100 per cent for low-complexity fields, 75-90 per cent for moderatecomplexity fields, and 60-75 per cent for high-complexity fields. These round numbers nicely quantify the qualitative mechanism-complexity scale for the SSC model. Exact values differ from these rules-of-thumb by about 1 per cent (e.g. Table 1).
As R → 0, 1, the minimum value of F+ increases to 0.25. According to our qualitative scale, any SSC field that approaches these limits has at least moderate complexity. This complexity is the minimal required to represent a CLVD CMT as a superposition of Wallace-Bott DCs.

A P P L I C AT I O N T O E A RT H Q UA K E S A N D S E I S M I C I T Y C ATA L O G U E S
Predictions of the SSC model can be tested against moment measures retrieved from observed source fields, including earthquake stress gluts, fault-zone seismicity and seismic sequences. When the fit to the data is acceptable, we seek to interpret the estimates of the model parameters in terms of the stress and deformation fields that produced the source field. When the fit is unacceptable, we seek to understand how the deviations from SSC predictions are related to unmodelled complexities in those fields, as well as observational errors.
A complete discussion of the inference problem is beyond the scope of this theoretical paper and will be deferred to subsequent publications. Here we briefly describe two examples that illustrate how the SSC model can be applied to observed and theoretical momenttensor fields. We begin by constructing a likelihood function that captures the uncertainties and co-dependencies among the observations.

Likelihood function
The SSC model of the moment measures has five degrees of freedom: the three Euler angles that specify the ordered set of principle stress directions = (r 1 ,r 2 ,r 3 ), the differential stress ratio R, and the strain-sensitivity factor κ. The observables are the five parameters of the CMT moment tensor M 0 and the five fractional moments { Mα : α = 0, . . . , 4}. The latter sum to the total moment M T , which measures the size of the moment-tensor field but does not constrain the SSC model. We thus normalize all moments by M T and consider a set of observables comprising the CMT mechanismM 0 , the Aki moment fraction F 0 , and the total-moment simplex F = ( F0, . . . , Fd ). We parametrizeM 0 by the three Euler angles that specify its ordered eigenvector set, E 0 := (ê (0) 2 ), and the differential eigenvalue ratio, E 0 := (λ (0) 3 ). This notation distinguishes the model parameters and R from the observables E 0 and E 0 . The former are numbers, whereas the latter are random variables that account for the finite sampling, as well as the modelling and observational errors. The CMT reference frame E 0 is an estimator of the stress frame . Reference-frame uncertainties can be represented by directional distributions (Kent 1982;Silver & Jordan 1982;Mardia & Jupp 2000), although they are often small enough to be approximated by a multivariate normal distribution (Riedesel & Jordan 1989).
Our primary objective is to test the SSC model against the diagnostic observables, E 0 , F 0 , and F. As indicated by eqs (17) and (65), the zeroth-degree moment measures F 0 and F0 are strongly correlated and effectively redundant; hence, we include only the latter in the likelihood function. We further assume that, once the reference frame is fixed by the CMT, the conditional uncertainties in E 0 are uncorrelated with the simplex parameters; that is E 0 and F are conditionally independent: To express reference-frame uncertainties in the likelihood function, we marginalize the conditional distributions by integrating over E 0 , We express the likelihood function for the model parameters R and κ as the product of these unconditional distributions: E 0 is a function of the eigenvalues ofM 0 , which are stationary with respect to variations of E 0 and thus first-order insensitive to any reference-frame errors (Riedesel & Jordan 1989). Owing to this stationarity, the marginalization (69) is not necessary. The marginalization (70) is an important step, however, because F is first-order sensitive to uncertainties in E 0 , and the uncertainties induced by this dependency can be highly correlated within the simplex.
To account for these strong correlations, we approximate the d-variate joint probability density function for F by a Logistic-Normal distribution (Aitchison & Shen 1982;Frederic & Lad 2008). This choice, which is commonly applied in compositional analysis (Aitchison 1984), properly handles the unit-sum constraint and allows for arbitrary correlations within the simplex. The Logistic-Normal distribution is obtained by transforming the simplex F = ( F0, . . . , Fd ) into a vector φ ∈ R d comprising four log-ratios, In the cases examined here, d = 4. The inverse (logistic) transformation is If we assume φ is normally distributed with a mean vector μ and covariance matrix V, then F is distributed with the Logistic-Normal p.d.f. (Aitchison & Shen 1982), We have adopted F0 as the normalizing member of the simplex, but it can be shown that the statistical inferences are invariant with respect to a different choice (Aitchison 1984).

Seismicity of the SJF zone
Our first example applies the SSC model to the seismicity in the SJF zone of Southern California. The data set comprises the focal mechanisms of 1330 earthquakes in the magnitude range 2.5-4.5 that occurred during the 39-yr period 1981-2020 with epicentres in a 147 km × 22 km box centred on the SJF (Fig. 11). We downloaded focal mechanisms and moment magnitudes from the Southern California Earthquake Data Center (SCEDC; last accessed on Nov. 20, 2020), restricting the data set to mechanisms that SCEDC had refined by the methods of Hauksson et al. (2012) and Yang et al. (2012) and assigned qualities of A, B, or C , Table 1). The resulting catalogue is denoted C SJF = {r n , t n , m n ,m n : n = 1, . . . , N } , where N = 1330 events. We summed the individual moment tensors m n = m nmn to estimate the Aki moment M 0 and the CMT mechanismM 0 of C SJF , and summed the individual scalar moments to estimate the total moment M T . Previous studies of the seismicity and mechanism complexity in southern California, and in particular in SJF (e.g. Bailey et al. 2009Bailey et al. , 2010, found a right-lateral strike-slip CMT oriented parallel to the SJF. The CMT obtained by Bailey et al. (2010) has a larger CLVD component (E 0 ≈ 0.3) than our estimate (E 0 ≈ 0.46). The principal difference is that we averaged moment tensors while Bailey et al. (2010) averaged normalized potency tensors, which contain information about the orientation of the coseismic strain drop but not about its magnitude. They analysed earthquakes with magnitudes between 0.0 and 5.0 in the period between January 1984-December 2003. Using their windows did not significantly change our results.
FixingM 0 at its empirical mean, we constructed the four other members of the SOR basis set {M α : α = 1, . . . , 4} according to eqs (55)-(57) and (59), and we computed the fractional moments as the sums, The SOR mechanisms are depicted as beach balls in Fig. 12(a).
Empirical statistical distributions of the moment tensor M 0 and the total-moment simplex F were generated by a bootstrapping procedure that resampled (with replacement) the moment-tensor catalogue 1000 times, each random sample comprising 133 of the SJF events (∼10 per cent). The values calculated from each sample were aggregated into a joint empirical distribution of the diagnostic observables (E 0 , F), thus marginalizing over the samples of E 0 . The correlation coefficient of the two zeroth-degree moment fractions, F 0 and F0, was 0.97, which justifies including only the latter in the likelihood function (71). Fig. 12. displays the histograms of the total-moment fractions and the two-parameter Beta distributions that fit these empirical marginal distributions of E 0 (panel b) and F (panels c-e). The Pearson correlation matrix obtained from the resampling (panel f) shows that F0 is negatively correlated with Fα≥1, consistent with the simplex constraint that an increase in F+ must match a decrease in F0.
We fit the multivariate empirical distributions with the Logistic-Normal model of eq. (74) to obtain P( F) and the Beta model of Fig. 12(b) to obtain P(E 0 ). The likelihood function L(R, κ | E 0 , F) is contoured in Fig. 12(g), and the MLE is plotted as the black dot. We calculated the 1σ uncertainty of the estimates from the inverse of the Fisher information matrix, obtaining Our results for the reduced stress are consistent with previous studies of the SJF region, which estimated R in the range of 0.3-0.5 and principal stress directions (σ 1 ) similar to our value of N10 o E (e.g. Martínez-Grazón et al. 2016;Abolfathian et al. 2020).
The new model parameter recovered in this study is the strain sensitivity. The estimate κ SJF has a relative uncertainty of about 30 per cent, which implies a complexity factor of F+ = 0.31 ± 0.072. The SJF catalogue thus falls in the region of 'moderate complexity' on the R-κ map (Fig. 12g). The SJF zone is well known for the geometrical complexities of its fault traces, seismicity alignments, and focal-mechanism distributions (Petersen et al. 1991;Ross et al. 2017;Abolfathian et al. 2020). These fault-zone complexities have been attributed to the immaturity of the SJF system and, in particular, its low cumulative slip (Wesnousky 1990;Stirling et al. 1996).
It will be interesting to see whether the values of F+ derived from SSC modelling are consistent with more traditional measures of fault-zone complexity. Such calibration will require the systematic application of the SSC model to fault-zone seismicity catalogues in California and elsewhere, including an explicit accounting for the observational bias introduced by mechanism uncertainties, which is one focus of our current research (Juarez & Jordan 2021). An obvious goal is to understand to what degree mechanism complexity is controlled by cumulative slip and other evolutionary variables.

Graves-Pitarka rupture simulator
In our second example, we ask whether the moment-tensor fields produced by the stochastic rupture simulator of Graves & Pitarka (2016;abbreviated GP16) are consistent with an SSC model. Graves & Pitarka designed the GP16 simulator to obey earthquake scaling relations and mimic the statistics of dynamic rupture simulations, such as those of Shi & Day (2013) and Trugman & Dunham (2014), but they did not impose the Wallace-Bott condition. Their model has been incorporated into two computational systems widely used for earthquake hazard simulations, the Broadband Platform (Maechling et al. 2015) and the CyberShake Platform (Graves et al. 2011;Jordan et al. 2018). In , we applied the MOR and PCA decompositions to a single GP16 realization of a strike-slip rupture on a vertical fault and recovered basis sets similar to the SOR for R = 1 /2 (Fig. 4). We attributed this alignment of the basis sets to the statistical structure of GP16, which generates mechanism complexity by perturbating the strike, rake, and dip of small subfaults. The variations of strike and dip are computed from the gradient of a perturbed fault surface with a roughness amplitude linearly scaled to the fault length. The rake directions are computed as samples of a Gaussian distribution with a specified mean direction and a standard deviation of 15 • , uncorrelated with the fault roughness.
To quantify the mechanism variability, we computed 1000 realizations of a GP16 rupture by randomizing the initial seeds of the GP16 stochastic model. We specified the rupture by the same parameters as : a magnitude-6.8 rupture on a vertical rightlateral fault of dimension 15 km × 80 km, discretized into 0.1 km × 0.1 km subfaults. The slip amplitudes and the rupture isochrons are displayed for a typical realization in Fig. 13, which also shows the histograms of strike, rake, dip and total slip compiled from the 120 000 subfaults in this example. The angular distributions have means close to the target values for the strike (0 • ), rake (180 • ) and dip (90 • ) and standard deviations of 7 • , 15 • and 8 • , respectively.
We performed the same analysis with this GP16 1000-rupture set as we did with the SJF catalogue, fitting a Logistic-Normal model to the joint empirical distribution of subfault moment fractions, contouring the likelihood function L(R, κ | E 0 , F), and locating the MLE (Fig. 14): The small 1σ uncertainties reflect the large sample, which comprises a total of 1.2 × 10 8 subfault mechanisms. The basis set recovered from the GP16 ensemble (beachballs in Fig. 14) accurately reproduces the SOR basis for R = 1 /2 (Fig. 4).
The best fit of the SSC model to the GP16 rupture set yields a complexity factor of F+ = 0.20 ± 0.021, which places the MLE on the boundary between low and medium complexity (Fig. 14g). The corresponding estimate of the Aki moment fraction is F 0 = 0.89 ± 0.013. However, Fig. 14(c) shows that the SSC model overestimates the complexity of the GP16 ruptures. The mean of the empirical distribution for F0 is about 0.85, implying a raw complexity factor of 0.15, which moves the estimate into the low-complexity region. Therefore, the mechanism complexity of the GP16 rupture realizations is significantly lower than that of the SJF seismicity catalogue.
The other GP16 moment measures show sharp empirical distributions (Fig. 14) that are also statistically inconsistent with the predictions of the SSC MLE (coloured dots in the figure). The MLE value of F1 lies near the centre of its GP16 marginal distributions, but F2 underestimates its marginal, whereas both F3 and F4 overestimate their marginals. The p-value of the MLE computed from the parametrized joint distribution is only 2.3 × 10 −4 , which tells us that the MLE does not resemble any likely sample of the GP16 stochastic process. We can reject with even higher confidence the more stringent hypothesis that the mean moment measures of the GP16 simulations are consistent with the expected values of the SSC model. The reasons for the inconsistency between GP16 and the stress-aligned model are twofold. In a stress-aligned model with a vertical strike-slip CMT, the variations in rake and dip are equal, whereas in GP16, the former is twice the latter (Fig. 13), resulting in an empirical F2 mean (0.058 ± 0.0024) that is significantly larger than the F3 mean (0.019 ± 0.0049). The second reason is that GP16 total-moment simplex has only three degrees of freedom (rotations aboutb 0 ,n 0 ,ŝ 0 ), whereas that of the stress-aligned model has four. For R = 1 /2, the fourth-degree mechanismM 4 is a vertical CLVD (eq. 58). The stress-aligned mechanisms with the largest projections onto a vertical CLVD are dip-slip DCs with vertical principal axes. The shear-traction magnitudes across such dipping planes are suboptimal (s < 1; see Fig. 1), which is why F4 decreases rapidly with κ (Fig. 9). In GP16, however, the tight monomodal distribution of rake (Fig. 13d) effectively precludes any dip-slip mechanisms. Consequently, the mean of F4 (0.004 ± 0.0010) is significantly smaller than predicted by a stress-aligned model of comparable complexity.

D I S C U S S I O N A N D C O N C L U S I O N S
The SSC model developed in Section 4 assumes that the moment-tensor field comprises only DCs satisfying the  and that its image on the fault-orientation sphere, given by the mapping (40), is exponentially dependent on the shear-traction magnitude (eq. 42). Exponential dependence is a plausible modelling assumption that appears to work well in describing the moment measures derived from the observed seismicity of fault zones, such as the SJF catalogue analysed in Section 6.2, but the SSC methodology requires further experimental development and physical justification. One practical issue not discussed in this theoretical paper is the observational bias in the moment measures due to mechanism uncertainty. The CMT mechanismM 0 of an SSC field equals the reduced stress tensorˆ and thus determines the ordered set of principal stress directions = (r 1 ,r 2 ,r 3 ) and the differential stress ratio R. The moment measures of an SSC field obtained from the stress-oriented basis defined byM 0 (Fig. 4) are parametrized by the total moment M T , the stress-shape parameter R and the strain-sensitivity parameter κ.
The physical assumptions underlying the SSC model are more restrictive than the CMT stress-inversion method of Terakawa & Matsu'ura (2008, 2010, which neither imposes the Wallace-Bott slip condition nor requires the stress to be homogeneous. They instead equate the moment tensor to the volume integral of stress drop over a sufficiently large elastic region surrounding the source and assume that the tectonic stress has the same orientation and shape as this stress drop. Each CMT is represented as the spatial integral of the reduced tectonic stress with a Gaussian kernel scaled by the Aki moment. In regions of constant stress, the Terakawa-Matsu'ura model also implies that the reduced stress tensor equals the CMT mechanism, so the two methods of stress characterization are compatible. As Terakawa & Matsu'ura (2008) have pointed out (see also Angelier 2002), direct inversion of seismic moment tensors for reduced stress tensors avoids the need to resolve the fault-plane ambiguity, which has long been a troublesome requirement for the methods that invert observed slip vectors (e.g. Gephart & Forsyth 1984;Michael 1984Michael , 1987Lund & Slunga 1999;Hardebeck & Michael 2006;Vavryčuk 2014;Martínez-Garzón et al. 2016).
The CMT does not, by itself, measure source-mechanism complexity. We need to know how much smaller the zeroth-degree moment is than the total moment. This reduction is quantified by F + , which we determine from the higher-degree moment fractions { Fα : α = 1, . . . , 4} introduced in Section 2. In Section 5, we cast these measures in terms of projections onto a stress-oriented basis set of mechanisms uniquely defined by the CMT mechanism and its orthogonal rotations (Fig. 4), and we explored the dependence of the SSC moment measures on R and κ, giving special attention to their limiting values, which are summarized in Table 1.
We tested the SSC model against a catalogue for the SJF zone and found good agreement between the maximum-likelihood estimates and the empirical distributions (Fig. 12). The SSC constraint F2 = F3 is honored by the data, and the overall distribution of SJF mechanisms on the fault-orientation sphere is consistent with the SSC model. Our results for the stress orientation and shape agree with previous studies.
The new parameter estimated from the data is the strain sensitivity, κ = 5.7 ± 1.5, which corresponds to the mechanism complexity factor, F+ = 0.31 ± 0.072. Hence, about 70 per cent of the SJF seismic energy release maps onto the CMT and 30 per cent onto mechanisms orthogonal to the CMT. Using the qualitative scale defined in Fig. 10, the SJF estimate lies in the 'moderate complexity' region of the R-κ diagram, as defined in Fig. 10. By applying these methods to other fault zones in California and elsewhere and properly accounting for mechanism-uncertainty bias, we should be able to calibrate the F+ values against other indicators of fault-zone complexity. We hope to shed light on how fault-zone complexity is mechanically controlled by cumulative slip and other evolutionary variables.
Although the SSC of seismic catalogues can be obtained by straightforward projections and summations, the mechanism complexity of individual large earthquakes is much more difficult to resolve, because their stress gluts evolve on the same spatiotemporal scales as the seismic waves they excite. We have investigated methods for inverting seismic wavefields to constrain the mechanism complexity based on an expansion of the stress glut into polynomial moments (Jordan & Juarez 2019, 2020 and have applied them to successfully recover the dipole components of the 2016 Kaikōura earthquake . The stress-oriented basis of Section 5.2 can be used to simplify the inverse problem (Juarez & Jordan 2021).
In Section 6, we applied the SSC method to stochastic rupture simulations developed by Graves & Pitarka (2016). We were able to ascribe the inconsistencies of the GP16 mechanism distributions with the SSC model to differences in the modelling assumptions. The complexity factor obtained from the GP16 empirical distribution is F+ ≈ 0.15, which is significantly lower than from the SJF seismicity catalogue. One question to be addressed in future research is whether the mechanism complexity of large earthquakes is significantly less than the complexity derived from catalogues of small events in the same fault zone.
This question is related to the interpretation of κ, the parameter that scales the logarithm of mechanism orientation density to the normalized shear traction (eq. 42). Higher values of κ concentrate the mechanism distribution closer to the traction maxima on the faultorientation sphere, increasing the seismic strain sensitivity to stress variations and thereby reducing mechanism complexity. If the mechanism density is exponentially sensitive to variations in the normalized shear-traction magnitude over the fault-orientation sphere, as assumed in the SSC model, then we can plausibly conjecture that this density is also exponentially sensitive to the absolute traction. In this sense, κ functions like a stress magnitude.
This suggests that κ may serve as a proxy for the average absolute (unscaled) differential stress 1 /2(σ 1 − σ 3 ) operating during the source process. If true, a seismicity catalogue for a fault zone with high ambient differential stress would be expected to show less mechanism complexity than one with lower ambient stress, provided the state of fault-zone development (e.g. cumulative slip) is the same. The latter caveat allows κ to depend on mechanical factors such as cumulative fault slip in addition to stress magnitude. Our preliminary studies of California seismicity show that κ is correlated with cumulative slip (Juarez & Jordan 2021). In the case of large earthquakes, low complexity relative to the ambient seismicity may signify high dynamic stresses on dominant fault planes that focus the energy release onto the zeroth-degree (CMT) mechanism.
How good are the assumptions that underlie the SSC model? Many stress inversions and palaeostress assessments have characterized the faulting process as Wallace-Bott field driven by constant tectonic stress (e.g. Angelier et al. 1982;Gephart & Forsyth 1984;Michael 1984). In the course of these seismological and geological studies, the constant-stress assumption and the Wallace-Bott condition (26) have been repeatedly scrutinized. One issue is whether the stress at seismogenic depths is smoothly varying, as described in assumption (1) of Section 1, or locally rough with high-amplitude, small-scale heterogeneities (Smith & Dieterich 2010;Smith & Heaton 2011). The rough-stress hypothesis has been discounted by comparisons of mechanisms before and after larger earthquakes, which indicate little change in stress alignment (Hardebeck 2010), but the matter remains a subject of controversy (e.g. Schoenball & Davatzes 2017;Simón 2019).
The validity of the Wallace-Bott hypothesis (assumption 2) can be compromised by anisotropies in fault strength and local stress perturbations due to material heterogeneities and earthquake-mediated interactions with nearby faults. Various researchers have used theoretical analysis and numerical simulation to investigate these limitations and identify situations (e.g. at fault intersections) where slip will be measurably misaligned with the regional stress (Reches 1987;Marrett & Allmendinger 1990;Pollard et al. 1993;Cashman & Ellis 1994;Twiss & Unruh 1998;Lisle 2013;Simón 2019). These studies indicate that the local perturbations of the slip vectors are not large and systematic on a regional scale (Dupin et al. 1993;Célérier et al. 2012;Lejri et al. 2017).
If these analyses are correct, the small-scale stress heterogeneities are unlikely to substantially bias the moment measures used here, which are integrals over the entire source volume. Of course, local stress variations can be expected to increase the variances of those measures. Much of the scatter in the empirical distributions of Fig. 12, as well as the small offsets of the means from the SSC estimates, may reflect small-scale stress and strength variations in addition to observational errors.
Although the applicability of the SSC method is subject to the same caveats as the stress inversions, it has the advantage of providing its own set of internal consistency checks. The SSC model reduces the number of degrees of freedom in the total-moment simplex from four to two. In particular, an SSC field requires the moment fractions F2 and F3 to be equal. As shown by the SJF results (Fig. 12), the empirical distributions of these two moment measures are almost identical, validating the SSC constraint.
By the same token, observational evidence that F2 = F3 is cause for questioning the model. This type of inconsistency is illustrated by the GP16 realizations, which generate empirical distributions for F2 and F3 that are widely separated from each other (Fig. 14d). A discrepancy is to be expected, however, because the GP16 model assumes that the variance of the rake distribution is about twice that of the dip distribution (Fig. 13), which violates the Wallace-Bott condition required by the SSC model.
In real-world situations, the F2-F3 symmetry can be broken if there is a preferred orientation of fault planes, say due to an inherited tectonic fabric misaligned with the current stress field. Such anisotropy in a strike-slip environment could also bias F1 to higher values. The SSC model can be extended to accommodate this type of quenched anisotropy but at the cost of introducing more parameters.
One advantage of the SSC formulation is that expectation integrals over the source volume, such as the net moment tensor M 0 (eq. 37) and the total-moment fractions Fα (eq. 38), can be recast as expectation integrals on the fault-orientation sphere (eqs 45 and 66), where the form of the density function is prescribed by the SSC model. These expected values are, by assumption, independent of the support of m(r, t) in V, such as the spatial distribution of faults. From a practical point of view, this usefully separates the measurement of mechanism complexity from the full stress-glut inverse problem. For example, the mechanism summations needed to compute the total-moment simplex of an event set (eq. 76), like M 0 itself (eq. 7), do not depend on the event locations in space or time.

A C K N O W L E D G E M E N T S
We thank Victor Tsai and Mitsuhiro Matsu'ura for detailed reviews that pointed out weaknesses in our presentation and provided useful suggestions for improvements. This research was supported by the Southern California Earthquake Center (Contribution No. 10945) under National Science Foundation Cooperative Agreement EAR-1600087, U.S. Geological Survey Cooperative Agreement G17AC00047 and by a grant from the W. M. Keck Foundation.

DATA AVA I L A B I L I T Y
The earthquake data used in this paper are available from the Southern California Earthquake Data Center at https://scedc.caltech.edu/rese arch-tools/altcatalogs.html, last accessed on 20 November 2020. The data were restricted to mechanisms that SCEDC had refined by the methods of Hauksson et al. (2012) and Yang et al. ( 2012) and assigned qualities of A, B, or C, according to Yang et al. 2012, Table 1.