<form id="form">
    <input type="radio" name="location" value="loc01">
    <input type="radio" name="location" value="loc02">
    <input type="radio" name="location" value="loc03">
</form>

<script>
$(function() {
    $('#form').change(function(a) {
        a.preventDefault();
        var formdata = new FormData($(this)[0]);

        $.ajax({
            method:         "POST",
            data:           formdata,
            contentType:    false,
            processData:    false,
            success:        function(a) { alert('done'); }
        });
    });
});
</script>

<?php
    // simplified
    update_user_meta( $user->ID, 'user_location', $_POST['location'] );
?>