Recordings of surface waves and overtones provide excellent
constraints on the structure of the Earth's crust, upper mantle, and
transition zone. This is because they provide good global coverage, and are
sensitive to elastic and anelastic structure in both the crust and the
mantle. Yet, in order to determine the seismic velocities and
anisotropy in the mantle, we must separate the effects of the crust
from those due to the sought-after mantle structure. Despite recent
improvements in the global mapping of crustal structure (e.g.
CRUST2.0: *Bassin et al.*, 2000), the difficulties associated with
accurately modeling the effects of the crust seismic on waves mean that these
improvements do not automatically translate into better corrections
for crustal effects.

Crustal corrections in long period waveform modeling were first
applied in a linear fashion, by calculating the effects of
perturbations in Mohorovicic depth and surface topography on the
(eigen)frequencies of Earth's free oscillations (*Woodhouse and
Dziewonski*, 1984). However, variations in crustal thickness are
often large enough to produce non-linear effects on the
eigenfrequencies. In order to account for this non-linearity,
Montagner and Jobert (1988) proposed a two-step approach in which the
eigenfunctions and eigenfrequencies are calculated exactly for a set
of tectonic settings (thereby capturing the non-linear effects), while
perturbations away from these canonical 1D profiles are handled using
linear corrections. This approach has recently been implemented in
full-waveform analyses (*Marone and Romanowicz*, 2007). However,
when applied to higher modes and to high frequencies, these non-linear
crustal corrections can be computationally very expensive.

Here, we present an alternative method for performing crustal corrections. Like the aforedescribed methods, we calculate exactly the eigenfunctions and eigenfrequencies for a set of tectonic settings, but instead of using these directly, we solve for scaling coefficients, which, when applied to standard linear crustal corrections, mimic the non-linear effect. The main advantage of this approach is that, once the correction factors have been calculated, it requires no additional computational costs aside from those associated with linear corrections. This allows it to be more easily applied to overtones and to higher frequencies than the standard quasi-non-linear approach.

The effects of lateral heterogeneity, such as variations in crustal
velocities and thicknesses, can be approximated by considering how
this heterogeneity shifts the frequencies of Earth's free
oscillations, compared to their frequencies in a spherically symmetric
reference model such as PREM (*Dziewonski and Anderson*, 1981).
For the case relevant to this study, in which only the radii of
discontinuities in the Earth are perturbed by , for an
isolated mode branch, local frequency shifts can be calculated in a
linear fashion through the use of sensitivity kernels , the
expressions for which can be found in Woodhouse and
Dahlen (1978). Note that these kernels are calculated for the
spherically symmetric reference model:

In order to quantify the inadequacy of linear crustal corrections, we
divide the Earth's surface into 7 regions with similar crustal
thicknesses and ocean depths. We base this regionalization on
Mohorovicic depth from CRUST2.0, with the first 6 regions
characterized by Moho depth range of 10-25km, 25-40km, 40-50km,
50-60km, and 60km. The 7th region is introduced to capture the
strong effect of a shallow ( 2km) ocean layer that characterizes
the continental shelves. Figure 2.29 shows the
geographical extent of the 7 regions. For each region, we calculate
an average radial profile of density () and shear () and
compressional () wave velocity. Armed with a set of radial
models that define 7 canonical crustal types, we proceed to calculate
the frequencies of the fundamental modes for each model.
These frequencies are then compared with those of PREM, and frequency
shifts between PREM and each of the 7 regional models are calculated
as

We can use the kernels that we obtained for the reference
spherically symmetric model, in this case PREM, to predict the
linearized effects of the canonical crustal structures on the normal
mode frequencies. To do this, we only consider the differences in the
radii of the discontinuities between each canonical crustal model and
PREM, neglecting the differences in crustal velocities and
density. This is an often used approximation of the true linear
crustal effect, and is appropriate because crustal velocities have
been shown to have minimal effect on long period waves (e.g. *Stutzman and Montagner*, 1994). Henceforth, which shall refer to the
frequency shifts calculated in this standard, linear fashion as
. The dotted lines in Figure 2.30 show the
for each of the canonical crustal structures.

A comparison of the approximate terms with the calculated before (and displayed as solid lines) confirms that linear crustal corrections are inadequate, even at long periods. Therefore, we are interested in ways of correcting the so that they better track . In order to accomplish this task, we are confronted with a crucial choice.

We must decide which term or terms in equation 2.4 to correct. Since needs to be calculated for each mode, correcting this term can be computationally expensive. This is what is done in the aforedescribed non-linear corrections. Correcting , on the other hand, does not increase computational costs. The gradual change with frequency of the differences between and change gives us hope that modifying might significantly improve the accuracy of .

We start the procedure by rewriting equation 2.4 in matrix
notation, where we only consider N fundamental modes and identify
perturbations relating to the Mohorovicic with a subscript and
those pertaining to the surface with :

where the vector contains the non-linear frequency shifts . The least-squares solution to this minimisation problem is given by:

where the apostrophe indicates the transpose.

We could have introduced a multiplicative correction term, instead of the additive one described above. However, solving for such a term becomes unstable when the 's are small. Given that discontinuity topography is likely to vary both above and below its depth in the reference model, the accompanying zero-crossings of might have adverse effects.

Because the non-linearity of crustal effects depends strongly on both crustal and mode type, we perform the minimisation in equation 2.8 for spheroidal and toroidal modes separately for each crustal type. Once the set of factors appropriate for a given mode type are obtained, we modify the surface and Mohorovicic topography of CRUST2.0 at each point on the surface by the correction factor appropriate for the relevant crustal type (obtained from Figure 2.29). Therefore, the crustal type and correction factor information is fused into a single file that specifies a modified discontinuity topography for each mode type.

The dashed lines in Figure 2.30 show the frequency shifts predicted by our modified discontinuity radii. Henceforth, we label them . The improvement in fit to is significant, and good for a large frequency range. When only long period waves () are considered, excellent agreement between and can be achieved by only correcting the Mohorovicic topography.

We propose and validate a new method for improving linear crustal corrections. By considering a set of 7 crustal types, we quantify the inadequacy of standard linear corrections at accounting for the effects of the crust on the fundamental mode surface waves. Then, we improve the accuracy of linear corrections by introducing additive factors to the discontinuity topographies. Incorporating an additive correction factor to the discontinuity topography as opposed to the kernels, results in no additional computation costs, compared to standard linear corrections. The correction factors depend on the local crustal type, on the discontinuity considered, and on the reference model used for calculating the sensitivity kernels, as well as mode type.

This research was supported by National Science Foundation grant NSF/EAR-0308750 and an National Science Foundation Graduate Fellowship held by VL.

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Marone, F. and B. Romanowicz, Non-linear crustal corrections in high-resolution regional waveform seismic tomography. *Geophys. J. Int., 101* 22245-22272, 2007.

Montagner, J.-P. and N. Jobert, Vectorial tomography, II, Application to the Indian Ocean. *Geophys. J. R. Astron. Soc. 94,* 309-344, 1988.

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