geophysics · impedance
Lambda-Rho / Mu-Rho (LMR)
λρ = Zp² − 2Zs², μρ = Zs² (Zp=ρVp, Zs=ρVs)
click formula to derive ↑
Inputs
g/cc
ft/s
ft/s
Description
Computes Lambda-Rho and Mu-Rho — Goodway, Chen & Downton's (1997) reparameterization of the Lame elastic constants (λ, incompressibility; μ, rigidity) multiplied through by density. Because λρ = Zp²−2Zs² and μρ = Zs², both are obtainable directly from P- and S-impedance without ever needing density as a separate seismic-inversion output, and together they split a rock's elastic response into a fluid-sensitive term (λρ) and a fluid-insensitive, lithology-only term (μρ).
Variables
| Symbol | Unit | Description |
|---|---|---|
| λρ | GPa·g/cc | Lambda-Rho |
| μρ | GPa·g/cc | Mu-Rho |
| ρ | g/cc | Bulk density of the formation, typically from the density log — same input as Acoustic Impedance. |
| Vp | ft/s | True compressional velocity (e.g. checkshot-calibrated sonic, or 1/DT already converted from slowness) — not a raw uninverted slowness curve. |
| Vs | ft/s | True shear velocity from a dipole/multipole sonic or converted-wave data — not inferred from Vp via an assumed Vp/Vs ratio, which would circularly bias the fluid/lithology split this calculator is meant to reveal. |
Assumptions
- Vp, Vs, and ρ are measured or reliably estimated at the same depth and represent the same rock volume — same requirement as Acoustic Impedance and Vp/Vs.
- The rock is treated as an isotropic elastic medium — λ and μ are the standard isotropic Lame parameters, not a full anisotropic stiffness tensor.
- Vs is a genuine shear-velocity measurement (dipole/multipole sonic or converted-wave derived), not inferred from Vp via a generic Vp/Vs assumption.
Limitations
- A single depth point's λρ/μρ pair has limited diagnostic value on its own — the method's real power is in crossplotting λρ vs μρ (or the λ/μ fluid stack) across many samples or an inverted volume to separate a fluid trend from a lithology trend, which this single-value calculator cannot show directly.
- Requires a genuine shear-velocity measurement or a simultaneous P/S seismic inversion output — not available from a compressional-only sonic log.
- Like Gassmann, this treats the rock as uniformly saturated with a single effective fluid; partial or patchy fluid saturation is not modeled.
Use Cases
- → Simultaneous inversion fluid/lithology screening: Convert Zp/Zs (or ρ/Vp/Vs) from a well or a simultaneous AVO inversion into λρ/μρ to build a rock-physics template separating fluid-charged sand from shale or tight lithology.
- → AVO / DHI QC cross-check: Cross-check an AVO-flagged bright spot against its λρ signature — a genuine fluid effect should show a λρ anomaly with comparatively little μρ change, since μ is fluid-insensitive.
- → Rock physics template calibration: Calibrate a λρ-μρ crossplot template at the well location before applying it to a 3D simultaneous-inversion volume.
Related Calculations
Region Notes
Global
λρ/μρ crossplotting benefits from local calibration even though the transform itself is lithology-independent — clean sand typically shows higher μρ relative to λρ than shale at equivalent depth, with the largest separation appearing in gas-charged sands.
Permian Basin
Wolfcamp shale-on-carbonate/anhydrite contrasts are large in μρ (rigid carbonate/anhydrite >> shale) while λρ contrasts stay comparatively muted absent a real fluid effect — useful for separating a lithology-driven amplitude anomaly from a genuine hydrocarbon indicator.
Gulf of Mexico
Classic Class III AVO gas sands (Pliocene-Pleistocene) show a strong λρ decrease with comparatively small μρ change relative to the encasing shale — the textbook LMR fluid signature this method was originally developed to isolate.
References
Primary source
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