Custom post archive with search, is_search() is false?

I have a custom post type, and have created an archive template, archive-custom_post_type.php, which includes a search form.

I am then using pre_get_posts to add parameters to the query to filter my results.

However, to make sure this only happens on this archive page, I want to check a few things. First I am checking if the post type matches.

But then I wanted to check the is_search() parameter, only to see that it is false.

How and when is this defined? Can I do anything to let WP know that a search is happening?

pre_get_posts callback

$post_type = get_query_var( 'post_type' );

if ( $post_type === 'document' ) {
    $params = $_POST;

    if ( $params ) {
        $query->set( 's', $params['keyword'] );
        $query->set( 'order', $params['order'] );
        $query->set( 'orderby', $params['order_by'] );
    }
}

archive-document.php

<?php get_header(); ?>

<?php get_template_part( 'my-slug', 'document-filter' ); ?>

<div id="search-results">

<?php if ( have_posts() ) : while ( have_posts() ) : the_post(); ?>

    <div>
        <?php the_title(); ?>
    </div>

<?php endwhile; else : ?>
    <p>Sorry, no posts matched your criteria</p>
<?php endif; ?>

</div>

<?php get_footer(); ?>

document-filter.php

<form id="document-filter" method="post">
    <select name="order_by">
        <option value="date">Date</option>
        <option value="title">Name</option>
    </select>
    <select name="order">
        <option value="desc">DESC</option>
        <option value="asc">ASC</option>
    </select>
    <input name="keyword" type="text" placeholder="Filter by keyword" value=""/>
    <input type="submit" />
</form>