A REE Inverse Model From Bulk Distribution Coefficients and Boundary Conditions: Results for Shield and Rejuvenated Stage Hawaiian Volcanoes

A major challenge in mantle geochemistry is determining the source composition and melt fraction involved in melting. We provide a new Rare-Earth Element (REE) inverse model that provides source concentration, source and melt mineral modes, and melt fraction based on the difference between separate determinations of bulk distribution coefficients and constrained by boundary conditions. An analytical inverse of the batch melting equation provides expressions for source, $C_o^i$, and bulk distribution coefficient of the mantle, $D_o^i$, with two unknowns, the initial concentration of La in the mantle, $C_o^{La}$, and Pⁱ, the bulk distribution coefficient of the melt. We traverse through a range of $C_o^{La}$ steps and examine thousands of melt modes, Pⁱ, at each step. Thousands of trial melt modes fail by generating $D_o^i$ that are inconsistent with partition coefficients. Many surviving trials cannot be inverted to estimate a mantle mode. Other boundary conditions eliminate even more trials. Surviving trials are ordered by the difference between $D_o^i$ calculated from the REE data of a lava suite and $D_c^i$ calculated from partition coefficients and mantle mode. We select the solution with the closest fit that passes all the boundary conditions. We tested our new model with lava suites from Hawaii where different lines of evidence suggest that they melted from different mantle sources, Mauna Kea representing shield-stage lava and submarine Kiekie representing rejuvenated stage lava. Our inverse determination of mantle composition and melting parameters was consistent with earlier models based on assumptions of HREE composition.