The Pacific Tsunami Warning Center in Hawaii says it would have expected its initial tsunami warning on Thursday to have been made public in New Zealand, despite it later being downgraded.
Several hundred people left their homes in Gisborne after international media reported the tsunami warning.
New Zealand Civil Defence did not issue a notice until the warning was lifted, at about 6am.
Gisborne Civil Defence controller Richard Steele says he expected any public tsunami warning to have been dealt with by the national headquarters in Wellington.
Steele says public announcements about tsunami threats are supposed to be released by the Civil Defence headquarters but this did not happen in the way the regional offices had expected.
Gisborne Mayor Meng Foon says keeping residents informed about what they should do and how the council can help is important and will be a council priority.
Canterbury Civil Defence Management Group chair, Sue Wells, says the delay in contacting regional civil defence staff was totally unacceptable.
Wells says that if there had been a real threat, the civil defence staff members would have been an hour behind in their preparations.
She says the staff members expect to be contacted in such a case, and no one would have minded being stood down an hour later if the warning was withdrawn.
A geophysicist with the Pacific Tsunami Warning Center, Stuart Coyanagi, says the first warnings come at an early stage of the threat assessment and are based on seismological information, not on actual water level.
However he says the public should at least be aware that the country has been placed under a warning status.
Coyanagi says tsunami warnings and cancellations do not happen very often.
The international media is now being blamed for its coverage of Thursday's tsunami scare and Civil Defence Minister Rick Barker has singled out BBC World Service television for incorrectly reporting a wave was heading towards Gisborne.
But the BBC has defended reporting the warning.
Head of news Richard Porter says it did not mean to cause unnecessary alarm and was reporting credible sources. He says those included the Pacific Tsunami Warning Center and a correspondent in New Zealand who had spoken to Auckland police.
An international aid agency says it alerted its network in New Zealand minutes after an earthquake near Tonga triggered a possible tsunami warning in the Pacific.
New Zealand Red Cross operations manager Andrew McKie says he received a text message from their Geneva headquarters about 20 minutes after the quake.
He says he was then able to alert authorities in the Pacific and place the two people he has on standby in New Zealand to be deployed overseas.
McKie says the Red Cross warning system is independent of Civil Defence's.
Civil Defence officials met in Wellington on Friday to de-brief Thursday's response to the tsunami warning.
The meeting discussed how protocol was followed on a minute-by-minute basis, and is standard procedure after every Civil Defence activation.
The criticism Civil Defence received for not officially informing the public until three hours after the initial tsunami warning will be discussed in a wider review ordered by Barker.
Meanwhile, an earthquake measuring 6.0 has hit the islands of Tonga late on Thursday, according to the US Geological Survey website which is monitored in London.
The quake came a day after the larger quake of 7.9 was recorded near Tonga on Thursday, generating a small tsunami and sparking fears across the south Pacific of a major disaster.
A 10cm wave was registered in the French territories in response to the quake.
The website said the aftershock was recorded at 12.25am local time, (11.25pm NZT) on Thursday, 135km north east of Nuku'alofa, Tonga.
It was not clear if the latest quake caused any damage.
The last major event in the region occurred in October 1997 when a magnitude 7.7 earthquake was recorded.
