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If all the points in your 'point stack' have a common ID (i.e., "STACK_ID"), try using the Summary Statistics tool. You'll find the MAX of the DEPTH field and your Case Field is your STACK_ID. The result will be a table that will have a record for each unique STACK_ID with the maximum depth. What it won't have is the alkalinity field value for that point. But you do have enough information (the STACK_ID and the DEPTH) to find that unique point using an attribute query. The way I'd do this is to create a new field on the point feature class that concatenates STACK_ID and DEPTH. This would be a text field. See this blog post on how to concatenate field values. If you can create this field on your point feature class, you can also create it on the output of Summary Statistics. Then you have a common key field to use in joins and relates. You could use Join Field to join alkalinity from the point features to the statistics table for example.
If you have an advanced license, you could use the Sort tool to sort your points on STACK_ID and DEPTH (descending) , then use Summary Statistics with the FIRST statistic to find the alkalinity value. STACK_ID would be your Case field. This'll return the first value of alkalinity found for each unique STACK_ID. Since the table is sorted, the first occurrence is the deepest point.
There's probably a better way to do this using an SQL query, but I can't think of it right now.
If all the points in your 'point stack' have a common ID (i.e., "STACK_ID"), try using the Summary Statistics tool. You'll find the MAX of the DEPTH field and your Case Field is your STACK_ID. The result will be a table that will have a record for each unique STACK_ID with the maximum depth. What it won't have is the alkalinity field value for that point. But you do have enough information (the STACK_ID and the DEPTH) to find that unique point using an attribute query. The way I'd do this is to create a new field on the point feature class that concatenates STACK_ID and DEPTH. This would be a text field. See this blog post on how to concatenate field values. If you can create this field on your point feature class, you can also create it on the output of Summary Statistics. Then you have a common key field to use in joins and relates. You could use Join Field to join alkalinity from the point features to the statistics table for example.
If you have an advanced license, you could use the Sort tool to sort your points on STACK_ID and DEPTH (descending) , then use Summary Statistics with the FIRST statistic to find the alkalinity value. STACK_ID would be your Case field. This'll return the first value of alkalinity found for each unique STACK_ID. Since the table is sorted, the first occurrence is the deepest point.
There's probably a better way to do this using an SQL query, but I can't think of it right now.