Overview

This documentation is intended to explain how to compute spherical harmonics from model outputs for comparing with or correcting time-variable gravity measurements from the GRACE/GRACE-FO missions. This software was developed with the goal of supporting science applications for time-variable gravity. The model-harmonics projects consists of extension routines for the set of gravity-toolkit tools.

OBP

Uses outputs from the NASA-JPL Estimating the Circulation and Climate of the Ocean (ECCO) model. For ECCO near real-time Kalman-filtered (kf080i) and Rauch-Tung-Striebel (RTS) smoother (dr080i) models, reads 12-hour ocean bottom pressure data (OBP) and calculates monthly averages. For ECCO version 4 models, reads monthly ocean bottom pressure potential anomalies and converts to estimates of absolute ocean bottom pressure (OBP). Near real-time models are downloaded using the jpl_ecco_sync.py program, monthly Cube92 models are calculated using the jpl_ecco_cube92_sync.py program, interpolated Version 4 models are downloaded using the jpl_ecco_v4_sync.py program, and monthly Version 4 and 5 models in LLC tile format are downloaded using the jpl_ecco_llc_sync.py program. Because Boussinesq-type models conserve volume rather than mass, the global area average of each monthly map is removed [7]. Monthly anomalies in ocean bottom pressure are calculated by removing a multi-annual mean (typically 2003 – 2007). Ocean bottom pressure anomalies are converted to spherical harmonics following Boy and Chao [2] (Equations (1) and (2)).

(1)\[\begin{split} \left\{\begin{matrix}\tilde{C}_{lm}(t) \\ \tilde{S}_{lm}(t) \end{matrix} \right\} = \frac{3}{4\pi a\rho_{e}}\frac{1+k_l}{(2l+1)}\int \xi_l(\theta,\phi,t)~P_{lm}(\cos\theta) \left\{\begin{matrix}\cos{m\phi} \\ \sin{m\phi} \end{matrix} \right\}\,d\Omega\end{split}\]

(2)\[ \xi_l(\theta,\phi,t) = \left(\frac{a + N(\theta,\phi) - d(\theta,\phi)}{a}\right)^{l+2}\frac{p_{bot}(\theta,\phi,t)}{g(\theta,\phi)}\]

Fig. 1 ECCO Spherical Harmonics Framework

TWS

Uses GLDAS model outputs from the NASA Goddard Space Flight Center (GSFC) Hydrological Sciences Laboratory (HSL) Global Land Data Assimilation System Version 2 (GLDAS-2) [20]. GLDAS outputs are downloaded using the gesdisc_gldas_sync.py program. GLDAS version 2.1 is forced with a combination of model and observation data. Additionally, the GLDAS project produces two months of “early production stream” products that are run without the forcing data. Here, monthly terrestrial water storage (TWS) estimates are calculated by combining the GLDAS soil moisture (SM), snow water equivalent (SWE) and total canopy storage outputs. Monthly anomalies in terrestrial water storage are calculated by removing a multi-annual mean (typically 2003 – 2007). Before converting to spherical harmonics, the GLDAS terrestrial water storage estimates are masked to remove urbanized, glaciated and permafrost regions. Terrestrial water storage anomalies are converted to spherical harmonics following Wahr et al. [29] (Equation (3)).

(3)\[\begin{split} \left\{\begin{matrix}\tilde{C}_{lm}(t) \\[-4pt] \tilde{S}_{lm}(t) \end{matrix} \right\} = \frac{3}{4\pi a\rho_{e}}\frac{1+k_l}{2l+1}\int\sigma(\theta,\phi,t)~P_{lm}(\cos\theta) \left\{\begin{matrix}\cos{m\phi} \\[-4pt] \sin{m\phi} \end{matrix} \right\}~d\Omega\end{split}\]

Fig. 2 GLDAS Spherical Harmonics Framework

Reanalysis

ERA-Interim is computed by ECMWF and is available starting from 1979. ERA5 is the latest reanalysis computed by ECMWF offering much higher spatial and temporal resolution and is available starting from 1950. Differences between ERA-Interim and ERA5 are outlined here. ERA-Interim outputs are downloaded using the ecmwf_reanalysis_retrieve.py program following using the ecmwf-api-client documentation. ERA5 outputs are downloaded using the cds_reanalysis_retrieve.py program following using the cdsapi documentation. MERRA-2 is computed by the NASA Global Modeling and Assimilation Office (GMAO) and is available starting from 1980. MERRA-2 outputs are downloaded using the gesdisc_merra_download.py or gesdisc_merra_monthly.py programs. NCEP-DOE-2 is computed by the National Centers for Environmental Prediction (NCEP) and is available starting from 1979. NCEP-DOE-2 outputs are downloaded using the noaa_cdc_ncep_ftp.py program. NCEP-CFSR is computed by the National Centers for Environmental Prediction (NCEP) and is available starting from 1979 with Version 2 available from 2011 onward. NCEP-CFSR outputs are downloaded using the ucar_rda_cfsr_surface.py program. JRA-55 is computed by the Japan Meteorological Agency (JMA) and is available starting from 1958. JRA-55 outputs are downloaded using the ucar_rda_jra55_surface.py program.

Spherical harmonics from reanalysis outputs are computed here using three different schemes of complexity following Boy and Chao [2], Swenson and Wahr [24]: 1) a thin-layer 2D spherical geometry, 2) a thin-layer 2D geometry with realistic geometry incorporating model orography and estimates of geoid height (Equations (4) and (5)), and 3) a 3D atmospheric geometry integrating over the model layers (Equations (4) and (6)). Anomalies for each reanalysis are calculated relative to a multi-annual mean (such as 2003 – 2014).

(4)\[\begin{split} \left\{\begin{matrix}\tilde{C}_{lm}(t) \\ \tilde{S}_{lm}(t) \end{matrix} \right\} = \frac{3}{4\pi a\rho_{e}}\frac{1+k_l}{(2l+1)}\int \xi_l(\theta,\phi,t)~P_{lm}(\cos\theta) \left\{\begin{matrix}\cos{m\phi} \\ \sin{m\phi} \end{matrix} \right\}\,d\Omega\end{split}\]

(5)\[ \xi_l(\theta,\phi,t) = \left(\frac{a + h(\theta,\phi) + N(\theta,\phi)}{a}\right)^{l+2}\frac{p_0(\theta,\phi,t)}{g(\theta,\phi)}\]

(6)\[ \xi_l(\theta,\phi,t) = -\int_{p_0}^{0}\left(\frac{a + z(\theta,\phi) + N(\theta,\phi)}{a}\right)^{l+2}\frac{dp}{g(\theta,\phi,z)}\]

Fig. 3 Reanalysis Spherical Harmonics with Two-Dimensional Geometry Framework

Fig. 4 Reanalysis Spherical Harmonics with Three-Dimensional Geometry Framework

SMB

Uses MERRA-2 model outputs from the NASA Global Modeling and Assimilation Office (GMAO), or ERA5 model outputs computed by ECMWF. MERRA-2 Vertically Integrated Diagnostics (M2TMNXINT) and Land Ice Surface Diagnostics (M2TMNXGLC) are downloaded using the gesdisc_merra_sync.py program. ERA5 precipitation and evaporation outputs are downloaded using the cds_reanalysis_retrieve.py program following using the cdsapi documentation. For MERRA-2, monthly surface mass balance (SMB) estimates are calculated by combining the convective rain (PRECCU), large-scale rain (PRECLS), snow (PRECSN), evaporation (EVAP), and runoff over glaciated land (RUNOFF) variables. For ERA5, monthly surface mass balance (SMB) estimates are calculated by combining the total precipitation (tp) and evaporation (e) variables. ERA5 surface mass balance estimates are not including runoff as those variables are presently inaccurate over glaciated surfaces. Monthly cumulative anomalies in surface mass balance are calculated by removing a multi-annual mean (typically 1980 – 1995). Before converting to spherical harmonics, the surface mass balance estimates are masked to isolate regions of interest. Surface mass balance anomalies are converted to spherical harmonics following Wahr et al. [29] (Equation (7)).

(7)\[\begin{split} \left\{\begin{matrix}\tilde{C}_{lm}(t) \\[-4pt] \tilde{S}_{lm}(t) \end{matrix} \right\} = \frac{3}{4\pi a\rho_{e}}\frac{1+k_l}{2l+1}\int\sigma(\theta,\phi,t)~P_{lm}(\cos\theta) \left\{\begin{matrix}\cos{m\phi} \\[-4pt] \sin{m\phi} \end{matrix} \right\}~d\Omega\end{split}\]

Fig. 5 Surface Mass Balance Spherical Harmonics Framework