It is my belief, at this time, that the first Biblical month of the year, Abib (Aveev) should be determined in this manner:

The first Biblical month of the year is to begin with the first new moon after the spring (vernal) equinox.

The spring equinox is that moment when the sun passes directly overhead the earth's equator.  It is called the equinox, because on that date, daylight and night are of equal length.  It is called spring equinox, because this event ends winter and brings the spring season (warmer weather) to the hemisphere that the sun is entering.

Why do some people use barley to determine the first Biblical month of the year?

Those who use barley refer to the following Scriptures as their reason for using barley to determine the first month:
Exod 9:29 (NIV) Moses replied, "When I have gone out of the city, I will spread out my hands in prayer to {Yahweh}.  The thunder will stop and there will be no more hail, so you may know that the earth is (Yahweh's).
Exod 9:31 (NIV) The flax and barley were destroyed, since the barley had headed and the flax was in bloom.
[Three chapters later]
Exod 12:1 (NIV) {Yahweh} said to Moses and Aaron in Egypt,
2 "This month is to be for you the first month, the first month of your year.

It is assumed by some that about three days passed from Exodus 9:31 to Exodus 12:1,2, but it could have been a much longer period of time.  And, nowhere in chapter 12 is barley mentioned.

It is explained that "headed" barley (Exodus 9:31 above) tells you that the next moon starts the first month. (although "headed" barley is defined differently by different people and groups).
Next, most of those who use this method require that the barley observed be "in the head" and be growing in Israel.

Which method should we use and why?

Is the barley growing in Israel the same variety of barley that was destroyed by the hail in Egypt in Exodus 9:31? (There are early maturing and late maturing varieties with up to a month difference.)  Was the barley in Exodus 9:31 early in the head or late in the head (that can be a three week difference)?

This event in Exodus 9:31 happened in Egypt.  Egyptian barley on Egyptian land with Egyptian weather. If one wants to use the event that happend in Exod. 9:31 for determining the first month of the year, shouldn't the barley be checked in Egypt each year, where the event happened?

Many variables!  Many options! when using barley as a reference.

In Exodus 12:1-2, the words abib and barley are not found:
(NIV) 1{Yahweh} said to Moses and Aaron in Egypt, 2 "This month is to be for you the first month, the first month of your year.
How can one just assume "This month" meant something about barley to Moses?

Using the first new moon after the spring equinox does not have the confusing variables of the barley method.  Perhaps that is why Yahweh gave us the sun, the moon, and the stars to determine the seasons:

Gene 1:14 (KJS) And {Yahweh} said, Let there be lights in the firmament of the heaven to divide the day from the night; and let them be for signs, and for seasons ("moed" appointed times), and for days, and years...
Where does it say that he gave us barley to determine seasons (appointed times)?

Since Genesis 1:14 tells us to use the "lights" (sun, moon, stars) for determining appointed times, let us do that!