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"Typhoon Abby was the second typhoon to hit Japan within a span of a few days." - I would wikilink Japan here. Also, maybe you should substitute "hit" with like "strike" and then wikilink that to Landfall (meteorology).
"After peaking in intensity early on August 9, with winds of 140 mph (230 km/h), Abby slowly weakened, though the storm briefly re-intensified on August 11." ---> "After reaching peak intensity with winds of 140 mph (230 km/h) early on August 9, Abby slowly weakened, though the storm briefly re-intensified on August 11."
"However, meteorologists continued monitoring the storm for six more days. After causing five casualties in the Philippines, Typhoon Abby brought several consecutive days of rain to southern and central Japan, resulting in widespread damage." - Between these two sentences, I would split the lede into two paragraphs.
"After causing five casualties in the Philippines, Typhoon Abby brought several consecutive days of rain to southern and central Japan, resulting in widespread damage." - First, this is sorta of a run-on mentioning 5 deaths in Philippines, several consecutive days of rain in southern and central Japan, and then widespread damage. Second, elaborate more on the deaths in the Philippines. So I want you separate the Philippines and Japan and state how the fatalities in the Philippines occurred.
"About a dozen people were injured during a flash flood that destroyed three houses while three more dwellings were washed away when a river overflowed its banks." - Ever heard of a comma? :P
"it failed to organized significantly until August 3, when thunderstorm activity" - What "failed to organized" (grammar error: "organized" --> "organize")? Guam? :P
"Although forecast by the JTWC to move west," - This is a bit odd. Since "forecasted" isn't really a word (I think), replace it was something like "predicted".
"Even though a Hurricane Hunter aircraft measured a pressure of 888 mbar (26.2 inHg)," - I'm not sure if a layman would know what you are referring to when you say "pressure".
"but due to the massive size of Abby, this had little effect on Abby." - Avoid using a name twice in the same sentence. Replacing the second use of "Abby" with a term such as "the storm" or "the typhoon" would be preferred.
"Continuing to trek north-northwest, at 0000 UTC on August 17, the JTWC downgraded into a tropical storm." ---> "Continuing to trek north-northwest, the JTWC downgraded Abby to a tropical storm at 0000 UTC on August 17."
"Typhoon Abby brought widespread damage to Japan;[19] Abby was the second storm to strike Japan in a week." - Avoid using "Japan" twice. On the second use, substitute it with "the country" or "the nation".
"Moreover, Amagisan also saw peak 24-hours and hourly totals of 627 millimetres (24.7 in) and 73 mm (2.9 in) respectively." - "saw" --> "observed". Also, put a comma before the word "respectively".