Geology 5670/6670 Inverse Theory 20 Mar 2015 © A.R. Lowry 2015 Last time: Cooperative/Joint Inversion Cooperative or Joint Inversion uses two different types of data to invert for common parameters. For known data variance (and “perfect physics”), we seek to minimize the cost function e.g.: Generally though, data error is not the only/primary source of misfit (it will also reflect oversimplified parameterization of the model & measured but unmodeled processes). In such cases it may be necessary to weight the data empirically… Or better yet, to remove dependence of the cost function on unmodeled processes & degrees of freedom of the problem… !

Joint Inversion by Likelihood Filtering Recall from our earlier discussions on solution appraisal that the 100(1–  )% confidence interval on a model parameter space can be estimated using the Likelihood Ratio Method as: Suppose that we have two separate models (using very different data) that have one or more shared parameters. We can use the confidence intervals derived for one analysis to inform the inversion for the other via Likelihood Filtering.

First we must note that different data sets will likely have different degrees-of-freedom. We’ll denote the number of data and number of parameters for model a as N a and M a, respectively, and similarly N b and M b for model b. Following the likelihood ratio method, we can assign a confidence interval (1 –  ) to every modeled node in the parameter space for model a as: Given  2 statistics of a squared error in both, the parameter- space representation of likelihood of shared parameters m || in model b is then: using  from the confidence intervals on model a !

This provides a transformation of the confidence intervals for the degrees of freedom on the first model to the degrees of freedom on the second, and thus enables us to “condition” the modeling b of data d b by the modeling a of data d a without introducing assumptions about the nature of measurement error (or unmodeled physics) in the two models (beyond the assumption of Gaussian random variables resulting in a   distribution of norm). Example : Joint inversion of gravity and receiver function H-K stacks for crustal thickness ( H ) & v P /v S ( K ): First we note that H-K stacks are not a minimization of L 2 misfit error but rather a maximization of receiver function amplitudes sampled at times predicted for arrival of converted phases from the Moho (as a function of H and K ) and then stacked…

Receiver Function Estimates of Crustal Thickness: PPsPs Delay Time Deconvolve source-time function to get impulse response of phases converted at impedance boundaries Delay time between phase arrivals depends on crustal thickness and P- & S-velocity EARS uses iterative time-domain deconvolution [Ligorria & Ammon, BSSA, 1999]: well-suited to automation PPsPs Crust Mantle

However the H-K parameters that predict a given observed phase delay time are perfectly cross-correlated (so completely ambiguous) But not true for the primary plus reverberations… Ps P PpPs PpSs PsPs

PPsPsPpPs PpSs PsPs The reverberation phases are the bounces: PPsPsPpPsPpSsPsPsPsPs

Ps PsPs PpSs & PpPs H-K parameters that predict the observed phase delay times intersect at a point in parameter space PPsPpPs PpSs PsPs H–K Stacking: [Zhu & Kanamori, JGR, 2000]

Ps PsPs PpSs & PpPs Method stacks observed amplitudes at delay times predicted for converted Ps phase and reverberations. Max stack amplitude ideally reveals crustal thickness & V p /V s ratio. H–K Stacking: [Zhu & Kanamori, JGR, 2000] PPsPpPs PpSs PsPs (EARS H–K stack for station COR)

The Moho is not the only lithospheric impedance contrast… And crustal thickness is not constant The Problem: (EARS H–K stack for station TA.P10A)

The results is that community estimates from single-site data are prone to outliers & errors.

Example: Cascadia Top-of-Slab Serpentinized Subduction Wedge [Blakely et al., Geology, 2005] EARS Estimates of Crustal Thickness

Despite outliers, H & K have properties consistent with a fractal surface… Crustal Thickness H V p /V s Ratio K

Station TA.P10A (Central Nevada) The semivariance properties can be used to estimate a “most likely” crustal thickness and V p /V s ratio via optimal interpolation from nearby sites.

Station TA.P10A (Central Nevada) Can also model gravity predicted by estimates… And find a “most likely” model with uncertainties.

Station TA.P10A (Central Nevada) Gravity Model Likelihood Filter Optimal Interp. Likelihood Filter Combined

 Unlikely stack amplitude maxima are downweighted using likelihood statistics Station TA.P10A (Central Nevada)

Parameterizing the Gravity Model G moho  moho + G bulk + G thermal  = G obs ∂∂ ∂∂ GRAVITY

Thermal Modeling Surface Heat Flow Filtered Heat Flow Aero-Spectral Gamma Surface Temperature Surface Heat Production Depth Distribution Geotherm

Results: Gravity Residual

Results: Crustal Thickness H

Results: Bulk Crustal Vp/Vs Ratio K