Well,

The formation of water networks originally occurred in earlier geological eras, when the terrain was formed. From the beginning of the geological ages until the last era of the fourth geological era, the lifting and lowering of the surface of the crust played a key role in the formation of most of the existing water networks and later overlapped and updated some changes due to the interference of dry land and water. The erosion operations carried out by streams over a long period of time played an important role in the water systems and formed large alluvial fans that changed its course and then formed new branches within the water network and the water networks differ in importance, depending on the length of the main valleys or short as well as the tributaries of the drainage network are given ranks according to their importance.

River ranks are divided into the following:

1. The smallest stream order is given the number (1), which are small valleys that are not connected by smaller branches of which less than 5 km in length.

(2) The second order is given the number (2), which is the valleys formed as a result of the union of two or more tributaries of the first order.

3. The third order is given the number (3), and consists of the union of two or more tributaries of the previous order (2) Thus, the higher-order in the water network, the more important the water network (as in the attached Figure ).

You might be interested in my researchgate publication on stream types and management implications. In that analysis, I used 1:24K and 1:100K topographic maps, where two or more contour crenellations (indentions) indicated a drainage depression to estimate first order stream. In some earlier work with GIS specialist, I used 5 acres or about 2 hectares to help estimate stream network, and automate stream ordering in the areas of Blue Ridge and Piedmont we managed. It did reasonably well, but not as good as manually analyzing the contour indicators from DEMs.

In 2015, in a different analysis, I was able to compare the 1:24K detail based DEM with a LiDAR based DEM in a portion of the Blue Ridge Mtns. The results showed in most instances there were two added stream orders identified with more detailed DEM. In field surveys on the analysis area, it showed the LiDAR based channels existed with signs mostly as past ephemeral flow.

You may note note in a separate analysis (Sediment from a Small Ephemeral Gully), that severe erosion and sediment delivery can occur from extremely small areas of the eroded and gullied piedmont such as 0.1 hectare (1/4 acre) in this instance.

In another analysis (2015, in my researchgate), we looked at LiDAR improvements in estimating coastal plain hydrologic boundaries and stream network. Information is presented on how much LiDAR based DEM improved hydrologic information.

Based on on my experience, it is very difficult to set a standard size for order 1 hydrologic units. If you look through some of the first paper citations, I think you will find instances as the Bad Lands in South Dakota, where drainage density is very extreme.

This is a very interesting question. This may or may not be helpful: My initial reaction is to think of this as a regulatory question. That is, in at least some places in the U.S., often we use "blue line" streams for regulatory purposes. That is, if it's officially identified as blue line stream, it's a stream. If not, then not. ... But I haven't been able to dig up any clear information for New Jersey or for the U.S. (Army Corps of Engineers) about how they determine blue line streams. (I'm sure it's somewhere.). I wonder if in humid climates, we would call only perennial streams, blue line streams. ... I don't know, tho, if stream order is used in any meaningful way, in say, New Jersey...

"Does anybody really know what time it is[?]"

--songwriter Robert Lamm, Chicago

Does anyone really know what stream order the creek they're standing in is?

--me!

My read of the literature--stream ordering is an intellectual exercise for classification/counting purposes, the results of which change depending on the quality of your mapping data. And you can see that from the answers given so far. Hansen suggests exactly that--what a stream is numbered changed when the resolution of the LiDAR data changed. Mangiafico mentions the use of "blue line" streams (which come from USGS paper mapping products). But I guarantee you that a USGS "blue line" counted as a first order stream would be a 2 or 3 order stream if you were to repeat a LiDAR-type evaluation in the same area.

The basic question is: Why do you care what order the stream is? What's your purpose in assigning an order to the tributaries in a drainage network? I think a good understanding of that purpose (you may have this figured out and just didn't share it yet) will help you choose thresholds or mapping products that give you the order attribution you need to accomplish your task. Good luck.

Since most of the topographic maps and DEMs in US have used 1:24,000 scale, and not LiDAR, the 1:24K has been a helpful tool in describing stream order densities in sloping terrain. However in the well vegetated coastal plain, both hydrologic boundaries, stream density and georeferencing is substantially improved with LiDAR. The famous geomorphologist Luna Leopold had a leadership position in the USGS, and he indicated that the topographic lines on contour maps were put in by cartographers using aerial photogrammetry methods. He suggested the blue lines were put in for looks, with no significant intent to be coincident with perennial streams. For me in several areas I worked in, I expanded the stream network using the contour crenulations, and used my knowledge of the stream ordering to estimate likelihood of perennial, intermittent or ephemeral channels, and estimated which streams needed stream buffers based on the estimate of their flow perminence. In the piedmont and mountains, at 1:24k scale, order 3 and larger were perennial, order 2 were intermittent, order 1 ephemeral.