How do I insert a variable into the resulting url parameter string after I send my GET form?

I have a form that queries a library database on another site. It works perfectly if I only include input fields for the "format" and "keyword" because the resulting parameters passed to the url come out in the correct order and the destination site recognizes it.

However, I also need my form to have an input field to select to search by "Author", "Title", "Subject" etc. This is what causes the problem. The destination site only recognizes this parameter if it is in the middle of the url (inside the keyword parameter).

My current resulting url: http://alpha2.suffolk.lib.ny.us/search~S50/X?SEARCH=harry&searchscope=50&SORT=D&m=b

What I need the url to look like: http://alpha2.suffolk.lib.ny.us/search~S50/X?SEARCH=t:(harry)&searchscope=50&SORT=D&m=b

If you compare the two you will notice a couple differences. First, ignore the parentheses around harry. That doesn't make a difference. The real issue is how do I get the "t:" to be inserted into my url? The "t:" comes from selecting "Title" as the thing to search by.

Here is my current form HTML (If you remove the "searchtype" select box at the bottom the form will execute without errors, but I need it to execute with it.)

<form class="form-inline" role="search" method="get" name="searchform" id="searchform" action="http://alpha2.suffolk.lib.ny.us/search~S50/X">
                  <div class="form-group" style="float: left; margin-top: 6px;">
                    <label class="form-title">Search Collection</label>
                  </div>
                  <div class="form-group">
                    <input type="text" value="" name="SEARCH" id="SEARCH" placeholder="Enter Search Terms..." />
                    <input type="hidden" value="50" name="searchscope" />
                    <input type="hidden" value="D" name="SORT" />
                  </div>
                  <div class="form-group" style="float: left; margin-top: 3px;">
                    <label for="searchformat">For:</label>
                    <select name="m" id="m">
                            <option value="">ANY</option>
                            <option value="a">BOOK</option>
                            <option value="e">EBOOK DOWNLOAD</option>
                            <option value="l">LARGE PRINT BOOK</option>
                            <option value="b">BLU-RAY</option>
                            <option value="g">DVD</option>
                            <option value="i">AUDIO BOOK CD</option>
                            <option value="h">AUDIO BOOK MP3CD</option>
                            <option value="x">EAUDIOBOOK DOWNLOAD</option>
                            <option value="q">PLAYAWAY</option>
                            <option value="j">MUSIC CD</option>
                            <option value="p">MAGAZINE/NEWSPAPER</option>
                            <option value="n">EMAGAZINE DOWNLOADS</option>
                            <option value="v">PLAYAWAY VIEW</option>
                            <option value="s">VIDEO GAME</option>
                            <option value="r">CD-ROM</option>
                            <option value="d">VHS</option>
                            <option value="t">GAMES/PUZZLES</option>
                            <option value="f">DIGITAL IMAGE</option>
                            <option value="z">EVIDEO DOWNLOADS</option>
                            <option value="y">EMUSIC DOWNLOADS</option>
                            <option value="c">SHEET MUSIC</option>
                            <option value="m">MAP</option>
                            <option value="w">ONLINE RESOURCE</option>
                            <option value="o">OTHER MATERIALS</option>
                    </select>
                  </div>
                  <div class="form-group" style="float: left; margin-top: 3px;">
                    <label for="searchtype">By:</label>
                    <select name="" id="searchtype">
                            <option value="a:"> Author</option>
                            <option value="t:"> Title</option>
                            <option value="d:"> Subject</option>
                            <option value="N:"> Note</option>
                            <option value="" selected="selected"> Keyword</option>
                    </select>
                  </div>
                  <button class="btn btn-primary" id="searchsubmit" type="Submit">GO</button>
                </form>

EDIT:

The attempted Javascript (as requested in the comments):

<script>
function searchSubmit() {
    m = document.getElementById("m").value;
    t = document.getElementById("searchtype").value;
    a = document.getElementById("searcharg").value;
    var newurl = "alpha2.suffolk.lib.ny.us/search/X~S22?SEARCH="; + t + a + "&searchscope=50&SORT=D&m=" + m;
    var searchform = document.getElementById("searchform");
    searchform.action = newurl; searchform.submit();
}
</script>


What I would do is get the searchtype-value and add it to the input-search before submitting. Here is a JS-fiddle example:

$(document).ready(function(){
    $("#searchform").on("submit", function(e){
        var keyWord = $("#SEARCH").val();
        var searchtype = $("#searchtype").val();
        $("#SEARCH").val(searchtype + keyWord);
    });
});