Fix Missing Aerosol Scattering Jacobians and Restore Backscatter Logic

[BenjaminTJohnson]

Description

This PR addresses critical gaps in the CRTMv3 Aerosol Scattering module by restoring missing Jacobian calculations for key state variables and re-enabling backscatter coefficient logic that had been inadvertently disabled. These changes ensure that sensitivities for aerosol concentration, relative humidity, and effective radius are correctly propagated through the Tangent Linear (TL), Adjoint (AD), and K-Matrix interfaces.

Key Changes

1. Restored Relative Humidity Jacobians

• Re-implemented Relative_Humidity sensitivity logic in CRTM_AerosolScatter.f90 (Forward, TL, and AD).
• This is critical for hygroscopic aerosol schemes such as GOCART-GEOS5 and NAAPS, where extinction, single-scattering albedo, and asymmetry factor are strongly humidity-dependent.
• Prior to this fix, the Jacobian for relative humidity was identically zero for these schemes.

2. Restored Backscatter & Asymmetry Logic

• Re-integrated Backscat_Coefficient (kb) and Asymmetry_Factor (g) interpolation and accumulation into the ASvar_type structure and aerosol scattering routines.
• These quantities are required for active sensor simulations (lidar/radar) and accurate phase function calculations.

3. Comprehensive Unit Testing

• Tangent Linear (TL): Added test_AerosolScatter_TL.f90 to verify non-zero sensitivities for relative humidity and aerosol concentration using MODIS (visible/IR) channels instead of ATMS (microwave), which bypasses aerosol scattering.
• Adjoint (AD): Added test_AerosolScatter_AD.f90 to validate TL/AD consistency via the dot-product test (passes to machine precision).
• K-Matrix: Added test_AerosolScatter_K.f90 to confirm correct propagation of sensitivities to the top-level Atmosphere_K structure.

Validation

• Verified that d(OpticalDepth)/d(Relative_Humidity) is non-zero for hygroscopic aerosols (e.g., sea salt).
• Confirmed TL/AD consistency: ⟨TL(x), y⟩ = ⟨x, AD(y)⟩.
• Ensured K-Matrix output exhibits physically consistent behavior.

Impact

This fix is mandatory for assimilation and retrieval applications involving aerosols where humidity dependence or backscatter (active sensors) is relevant. Without these corrections, the model assumes zero sensitivity to key aerosol properties, leading to incorrect gradients and potential convergence failures in variational (e.g., 4D-Var) systems.

[chengdang]

All ctests pass, and all changes look good.
I think we should merge #288 first because it has a lot less but related code changes
Then I'll redo the test for this PR merge.

[BenjaminTJohnson]

I removed the IF aerosol scheme GEOS5 type lines, it's unnecessarily restrictive.