Skill assessment of different quadruplet wave-wave interaction formulations in the WAVEWATCH-III model with application to the Gulf of Mexico

•The GQM was implemented in WWIII model and skill assessed along with DIA, GMD and WRT methods in duration- and fetch-limited test cases.•A combined error index was defined based on both Hs and Tm to assess the model performance at buoys as well as satellite tracks.•Calibrated ST3 package slightly outperformed calibrated ST6 package in GOM during the fair weather as well as extreme conditions of Hurricanes Harvey and Irma.•The DIA method with calibrated ST3 package was reliable and successful method in reproducing in-situ data during study period in GOM.•More accurate Snl formulations than DIA, such as GMD and GQM, did not effectively improve the model results; most likely due to design and calibration of Sin and Sds terms.