Combining the results of two surveys Someone has surveyed a number of people and put the results in a database (Survey 1). Each observation has additional information that, for any subpopulation (men only, young only, etc.), gives a national-level estimate of the number of people in that subpopulation, as well as a confidence interval for that estimate. As expected, the sum of the estimates for mutually exclusive subgroups (number of men plus number of women) gives the estimate for the total number of people in the population. 
I don't know how the survey was conducted, the sampling method, etc. All I have is the database. All estimated counts coming from the database are assumed to be log-normally distributed. 
Someone else has done another survey (Survey 2). A lot more people were interviewed. This survey was not meant to estimate anything -- it was just meant to give information on those people who were interviewed.
For the population as a whole, and for any subpopulation, Survey 2 gives an undercount, since not everyone in the population was interviewed. Often, the estimate based on Survey 1 is greater than the count from Survey 2, but that's not always the case.
Question: What is the best way to combine the information from the two surveys? I am fine with an approximate solution.
If I only had Survey 1, my point estimate for the number of people in subpopulation A would be E(A). However, from Survey 2, I know that A > $min(A)$. So should I be computing E(A|A > $min(A)$)?
Doing so leads to a contradiction. Namely, the sum of estimated counts in mutually exclusive subpopulations comes out to be greater than the estimated count for the entire population.
Thank you for your help. I hope this is clear. If not, please ask, I will try to explain. :-)
 A: To be able to estimate to a population from a survey, the sampling method needs to be understood. The sampling method is used to create the sampling weights, which are then used to basically multiply up the survey estimates to be population estimates. There are all sorts of ways of creating weights, but they need to be based on the survey design.
If you don't have the survey design information and/or you don't know how the weights were constructed, the key pieces of information you need to create the population-level estimates are missing. In particular for the two surveys, you need to be confident that the sampling method was appropriate, e.g. no quotas were used to stop sampling people of a particular age/sex combination, a convenience sample wasn't used. If either survey had these particular characteristics in the design, any population estimates (and even subpopulation estimates) are going to be wrong. 
There are some aspects of your question that I don't understand. For example, why do you wish to combine the two surveys - did they ask different questions? And surveys don't routinely sample the entire population - when that happens, we call it a census, so I don't understand your comment about the second survey.
Can you give any more information about the survey design, and also if there are weights in the datasets and what these weights look like? 
Update for clarity: I am not sure that survey 2 will add anything other than bias to survey 1. In your question you state that survey 2 was not meant to estimate anything - that makes it sound like survey 2 has a convenience sample design. When dealing with convenience samples, it is not possible to weight up to the population because the sampling method used is biased rather than random. For example, a survey of supermarket shoppers at 10am on a weekday is biased (it will undercount full-time workers and overcount adult females, for example). With a biased sample it is not possible to weight the data to take account of the bias because the probability of being sampled is unknown for some groups, and may even be zero for others, but you don't know what these probabilities are. Therefore it is impossible to construct weights to account for the sampling when a biased sample has been used.
Because it sounds like survey 1 has the better design for population estimation, I recommend that you use survey 1 for your estimates.  
A: It looks like your survey 2 is a convenience sample. I don't know what it can be useful for. Without a clear sampling strategy, you cannot generalize to the population in any meaningful way. At best, you might be able to utilize Survey 2 to build a model of how variables are interrelated, and then try to improve your estimates from Survey 1 using some sort of generalized regression estimation but to get there, you need to make sure your sample is not biased.
For instance, suppose you want to predict how much income taxes can be collected in the economy. Suppose you use something like the US Current Population Survey as your survey 1. This is a very well designed survey, with weights, poststratification, bells and whistles, whatever have you. Then you also have a survey that you hand out in the local unemployment offices, and only hope that most people will write something in. This is your survey 2. You don't know how well it reaches to your population -- in all likelihood, you are more likely to reach those who are seeking jobs more actively, and show up in the local offices more often. You won't reach the frustrated workers who quit looking for a job, or those who are not eligible for the unemployment benefits but would be looking for a job otherwise, some seasonal workers, and a number of other people. You don't know any of that though: somebody just handed you Survey 2 and said, "This is our rich data base, make sense out of it". Well, this is a biased sample to begin with. If you fit a regression model of individual's earnings using this data set, you will likely get wrong estimates: the sample censors out those with higher earnings in full time, permanent jobs, and probably has way more people with low education than there are in general population. So what's the use of Survey 2 for you? As I said, I doubt it has much value in this purpose.
It is not helping at all that you give zero background information about what the survey is about, what sampling units are, etc. I understand that you are probably bound by your employer, or your client, or whatever form of supervisor you have. But without more detail, all we can give you is some sort of handwaving advice. I can point you to technical literature about combining information from several surveys (using Bayesian or empirical likelihood methods), but I am not sure it will help much at this point.
