= Solution
The <asthenosphere> exists because the <mantle>'s ability to deform is strongly dependent on <temperature>, <pressure>, composition, <water> and deformation mechanism. Cooling from the surface creates a strong <lithosphere>. Beneath it, hotter <mantle> deforms by <material creep> at much lower <stress> on geological timescales. \b[The <asthenosphere> is predominantly solid weak <rock>, not a global ocean of <magma>.] On the much shorter timescale of a <seismic wave>, it still supports <S waves>.
The hot shallow <mantle> can be close to its <solidus>. <Water weakening of rocks> and, in some settings, a small connected <melt fraction> can further lower effective <viscosity>. At greater depths, changes in <pressure>, <homologous temperature>, mineral structure and <water> storage modify the deformation laws; increasing depth does not imply indefinitely decreasing <viscosity>. Explaining a weak upper-mantle interval does not require every location to contain the same amount of <melt>.
There are several physically distinct boundaries to test. The thermal <lithosphere-asthenosphere boundary> describes a thermal transition, the mechanical boundary a contrast in long-term strength or <viscosity>, and the <seismic lithosphere-asthenosphere boundary> a change in <seismic velocity> or other wave properties. They need not have identical depths. A smooth <geothermal gradient> can explain broad thermal weakening, but a very sharp decrease of <seismic velocity> may require changes in <melt fraction>, hydration, composition or <seismic anisotropy>. A sharp shallow reflector within a <craton> might instead be an internal lithospheric boundary.<A global view of the lithosphere-asthenosphere boundary by Catherine A. Rychert and Peter M. Shearer>[Abstract and pp. 495–498: converted-wave sharpness and the possibility that the imaged cratonic discontinuity lies within thicker lithosphere.]{ref}
Begin with a thermal null model. A <half-space cooling model> gives
$$
T(z,t)=T_s+(T_m-T_s)\operatorname{erf}\!\left(\frac{z}{2\sqrt{\kappa t}}\right).
$$
If a thermal boundary corresponds to a fixed <temperature> $T_*$, its depth is
$$
z_*(t)=2\sqrt{\kappa t}\,\operatorname{erf}^{-1}\!\left(\frac{T_*-T_s}{T_m-T_s}\right),
$$
so it deepens as the square root of plate age. Test that prediction against independently determined seafloor ages, <bathymetry> and <heat flux>. An <oceanic plate cooling model> adds finite thickness and basal heat supply, allowing the old-age depth to approach a finite value. For continents, include <radiogenic heating>, crustal thickness, the history of <continental rifting> or <tectonic thickening>, and advective heating by <magma> or <mantle flow>. A cold, dry <craton> and a recently extended volcanic region should not share the same thermal profile.
Then calculate the observable consequences of competing mechanisms rather than merely assigning a boundary to a map:
* Use <receiver functions> from both <P-to-S conversion> and <S-to-P conversion> to constrain the depth and sharpness of a negative <seismic velocity> contrast. Check for crustal reverberations and whether the result changes with incident direction or frequency. Combine these with <Rayleigh waves>, <Love waves> and <seismic tomography> to constrain the broader <shear-wave velocity> profile.
* Measure <seismic attenuation> and <seismic anisotropy>. Elevated <temperature>, small amounts of <melt> and aligned <minerals> can all affect apparent <seismic velocity>, but predict different attenuation, polarization and directional patterns. Convert laboratory elastic and anelastic measurements into synthetic seismic observations at the relevant frequencies.
* Obtain <magnetotelluric sounding> constraints on <electrical conductivity>, together with surface <heat flux>. Connected <melt> or hydrated material can enhance <electrical conductivity>, but the interpretation depends on composition and connectivity. Joint thermal, seismic and electrical modelling is more discriminating than any single proxy.
* Use <mantle xenoliths> and erupted <magmas> to constrain <pressure>, <temperature>, <water> content, <melt depletion> and composition, recognizing that transport can alter a sample. Relate these observations to <geodynamic modelling> of <mantle convection>, <slab> flow, <mantle plumes> and <lithospheric rejuvenation>.
Compare models against the observed boundary depth \i[and its sharpness], including uncertainty and resolution. An age–depth trend with matching <heat flux> favours thermal control; a sharp discontinuity that fails the thermal prediction but coincides with an electrical anomaly can support a melt- or hydration-related transition. Neither observation alone proves a unique cause. The spatially varying boundary is best understood as the result of thermal evolution interacting with composition, fluids and deformation history.
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