# Is cross-correlation the appropriate tool to estimate delay between two measurements?

In an industrial installation, there is a pipe containing a continuous stream of product A.

The temperature of product A is measured at sampling point 1, every hour.

It is also measured a bit downstream, at sampling point 2, every eight hours.

It is comparable to a water pipe with two thermometers a few hundred meters apart.

Continuous sensors would record the following trends. I have marked with vertical bars the samples taken. The sampling times very rarely match.

Due to this, there may never be, for the same wave, repeated measurements.

   1       2
===|=======|=
-> A  A  A ->
=============


With several years of data for both measurements, I would like to determine the time product A takes to go from 1 to 2. The flow rate is assumed to be constant, but is not measured.

After reading a bit, it looks like correlation would be the right tool for the job, but it requires having datapoints for the same times.

That would require either downsampling the data from 1 a lot, or interpolate more data from 2, but both methods seem to be diminishing the quality of the data, and therefore of the end result.

Is cross-correlation the right tool for this job, or should I go an other way?

• i don't understand, you measure the temperature of product A at point 1 every hour, and you also measure the temperature of product A at point 2 every 8 hours, and these times rarely match (as in they are not in sync). Now you want to determine how long it takes for product A to go from the first point to the second? What use are any of these previous measurements then? How would you use temperature to determine speed? – user2974951 Jan 11 at 8:31
• Let's assume the plant is at a steady state, where T1 = T2 = 0°C. If there is a disturbance, and at t = 0s, T1 is measured at 10 °C, and remains at 10°C for all eternity. At t = 30 min, T2 is measured, the result is 0°C. At t = 8h30m, T2 is measured, the result is 10°C I know that the time it takes to go from A to B is somewhere between 30 minutes and 8 hours and 30 minutes. Since I have a lot of measurements, and therefore a lot of disturbances, I can narrow down the delay taking into account all occurence (or at least, this is what I would like to do!) Thank you for your help! – Maxime Jan 11 at 8:37
• I still don't understand. Are you tracking each product individually? Can you uniquely identify each product from the temperature (or with some other mean)? Does temperature change at any point in time (from beginning to end)? Does each product take an arbitrary amount of time to reach point 2? – user2974951 Jan 11 at 10:59
• My bad, it was not clear. It's a continuous process, let's say water pipe. I have edited the question to reflect that. – Maxime Jan 11 at 11:56
• So if we use your analogy: there is some water flowing through a pipe, different portions of the water have different temperatures, and temperature does not increase or decrease over time (water getting colder or warmer). You measure the first temperature in location 1, and then you also measure temperature in location 2 and both these locations are known (their position and the distance between them). You would like to determine how long it takes for water to go from lcoation 1 to 2 based on the temperature? So basically how long it takes to see a repeated measurement? – user2974951 Jan 11 at 12:09